Nonlinear mechanisms in passive microwave devices

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NONLINEAR MECHANISMS IN PASSIVE MICROWAVE DEVICES PhD Thesis Dissertation by EDUARD ROCAS Submitted to the Universitat Politècnica de Catalunya (UPC) in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY June 2011 Supervised by Dr. Carlos Collado PhD program on Signal Theory and Communications Nonlinear mechanisms in passive microwave devices Copyright 2011 Eduard Rocas 2 ABSTRACT This work comprises the study, analysis and modeling of the nonlinear mechanisms in a variety of passive microwave devices. Bulk acoustic wave resonators and transmission lines made of high temperature superconductors, ferroelectrics or regular metals and dielectrics are the subject of this work. Both phenomenological and physical approaches are considered and circuit models are proposed and compared with measurements. The nonlinear observables, being harmonics, intermodulation distortion, and saturation or detuning, are properly related to the material properties that originate them. The obtained models can be used in circuit simulators to predict the performance of these microwave devices under complex modulated signals, or even be used to predict their performance when integrated into more complex systems. Index Terms: Nonlinearities, microwaves, transmission lines, bulk acoustic waves, thermodynamics, self-heating, harmonics, intermodulation distortion, materials. 3 Aquesta tesi està dedicada als meus pares, Joana i Guillem. 4 PREFACE The current tendency in the telecommunication market is a natural push for higher frequency, higher power and smaller devices. In conjunction with that, advanced materials are gradually being incorporated into electronics to achieve better capabilities like fast frequency tuning, ultra-low losses or superior power handling. However, inherent nonlinearities associated with materials’ field-dependent properties give rise to undesired effects like harmonics, intermodulation distortion, detuning or saturation, which can potentially degrade the system’s performance. In this scenario, with rising stringent requirements for electronics and increasing importance of signal integrity issues, the prediction of devices performance at the design stage is of crucial importance. I initially started this work with the goal of studying the nonlinear phenomena in bulk acoustic wave devices, ignoring the complexity of the topic and the lack of success of previous researchers on this area. I soon started to also get motivated for other topics like ferroelectrics and high temperature superconductors, and developed my own interests, motivated by the research group I have been working with, which has a wide experience on nonlinear characterization and modeling. The years at the National Institute of Standards and Technology (NIST) have been of crucial importance and that stage allowed me to deepen my interests for materials science and physics. The success of this thesis is the study of the thermal phenomena in microwave devices and how this impacts the nonlinear behavior. As recognition, two articles were selected as finalists at the Best Paper Competition in the 5 International Ultrasonics Symposium, Rome, 2009 and the International Microwave Symposium, Anaheim, 2010. This work wouldn’t have been possible without the support of my thesis supervisor at UPC, Carlos Collado, and my advisor at NIST, James C. Booth, who have kept me motivated all the time. Special thanks to Jordi Mateu, Alberto Padilla and Joan O’Callaghan, from UPC, and Nathan D. Orloff, from NIST, for their invaluable help. I also want to thank Robert Aigner, from TriQuint Semiconductor, for his collaborative attitude. 6 TABLE OF CONTENTS ABSTRACT....................................................................................................................... 3 PREFACE .......................................................................................................................... 5 TABLE OF CONTENTS................................................................................................... 7 INTRODUCTION ............................................................................................................. 9 CHAPTER I - NONLINEAR TRANSMISSION LINES ............................................... 15 On the Relation between the Nonlinear Surface Impedance and the Superfluid Current Density in High-Temperature Superconductors ............................................. 17 Superconducting Multiplexer Filter Bank for a Frequency-Selective Power Limiter . 21 Third order Intermodulation Distortion and Harmonic Generation in Mismatched Weakly Nonlinear Transmission Lines........................................................................ 27 CHAPTER II - BULK ACOUSTIC WAVE RESONATORS ........................................ 37 Third order Intermodulation Distortion in Film Bulk Acoustic Resonators at Resonance and Antiresonance ..................................................................................... 39 Nonlinear distributed model for IMD prediction in BAW resonators ......................... 43 Nonlinear Distributed Model for Bulk Acoustic Wave Resonators ............................ 47 First-Order Elastic Nonlinearities of Bulk Acoustic Wave Resonators ...................... 59 CHAPTER III - ELECTRO-THERMO-MECHANICAL NONLINEARITIES............. 67 Third-Order Intermodulation Distortion due to Self-heating in Gold Coplanar Waveguides .................................................................................................................. 69 Passive Intermodulation Due to Self-Heating in Printed Transmission Lines ............ 73 Modeling of Self-Heating Mechanism in the Design of Superconducting Limiters ... 85 Unified Model for Bulk Acoustic Wave Resonators’ Nonlinear Effects .................... 89 APPENDIX A TO CHAPTER III - A Large-Signal Model of Ferroelectric ThinFilm Transmission Lines ............................................................................................. 95 APPENDIX B TO CHAPTER IIIElectro-thermo-mechanical Model for Bulk Acoustic Wave Resonators ........................................................................................ 103 CONCLUSION .............................................................................................................. 115 IMPACT FACTORS OF THE PUBLICATIONS ........................................................ 117 VITA .............................................................................................................................. 118 7 8 CHAPTER I - NONLINEAR TRANSMISSION LINES The use of HTS in microwave applications raised a lot of expectations when the technology was discovered. In spite of that, the HTS microwave systems have not succeeded in entering the general market, but remain as an irreplaceable technology for some specific applications. However, the HTS technology has intrinsic nonlinearities that have been a matter of research since its discovery and can potentially degrade the performance of the devices. This chapter analyzes the intrinsic nonlinear phenomena in HTS transmission lines at microwave frequencies. The first article is a theoretical study of the origins of nonlinearity in HTS, and results into a distributed nonlinear circuit model that captures the HTS nonlinear behavior. The study is based on the two-fluid model for HTS and relates the nonlinear surface impedance to the superfluid current density. The second article illustrates how the inherent nonlinear behavior of HTS is used to implement a multiplexer filter that can limit the input power in nanoseconds time scale. The presented study allows one to properly design limiting resonators for specific maximum acceptable power levels. Finally, the third article presents the analysis and study of impedance mismatched nonlinear transmission lines and how this impacts the nonlinear observables, which can be applied to the study of ferroelectric and HTS mismatched transmission lines. The articles in Chapter 1 are: 15  E. Rocas, C. Collado, A. Padilla, J. C. Booth, “On the Relation Between the Nonlinear Surface Impedance and the Superfluid Current Density in HighTemperature Superconductors”, IEEE Transactions on Applied Superconductivity, Accepted for publication and available online at IEEExplore.org as Early Access.  E. Rocas, A. Padilla, J. Mateu, N. Orloff, J. M. O'Callaghan, C. Collado, J. C. Booth, “Superconducting Multiplexer Filter Bank for a FrequencySelective Power Limiter”, IEEE Transactions on Applied Superconductivity, Accepted for publication and available online at IEEExplore.org as Early Access.  J. Mateu, C. Collado, N. Orloff, J. C. Booth, E. Rocas, A. Padilla, J. M. O'Callaghan, “Third-Order Intermodulation Distortion and Harmonic Generation in Mismatched Weakly Nonlinear Transmission Lines”, IEEE Transactions on Microwave Theory and Techniques, vol. 57, no. 1, pp. 10-18, Jan. 2009 16 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY On the Relation Between the Nonlinear Surface Impedance and the Superfluid Current Density in High-Temperature Superconductors Eduard Rocas, Student Member, IEEE, Carlos Collado, Senior Member, IEEE, Alberto Padilla, and James C. Booth Abstract—The nonlinear surface impedance is related to a local description of the nonlinear superfluid current density that depends on the condensate velocity, by use of the definition of surface impedance as the ratio between the electric field and magnetic field at the conductor surface. To obtain this relation, we follow a rigorous approach based on the time-domain nonlinear London equations and the two-fluid model. The formulation is compared with harmonic balance simulations of an equivalent transmission line circuit that models a plane wave propagating across a vacuum-superconductor boundary, and shows very good agreement for the resulting surface impedance as a function of current density at the surface. Index Terms—HTS, nonlinear surface impedance, superconductor, superfluid current density, surface reactance, surface resistance. I. INTRODUCTION THE surface impedance is one of the most useful and most often applied material parameters used to characterize the properties of a superconductor. It directly describes the electromagnetic properties of the superconductor material, such as the Meissner effect and the square-law scaling of the losses with frequency. Furthermore, the surface impedance can be easily related to measurable figure-of-merit of resonators, such as quality factor and resonance frequency [1]. For these reasons, the surface impedance is the most common parameter used to characterize the linear electromagnetic properties of superconductors. The surface impedance has also been extensively applied to characterize superconductor nonlinear properties. In a linear regime, a formula for is found by use of the two-fluid model in phasors, by assuming sinusoidally varying Manuscript received August 03, 2010; accepted September 30, 2010. This work was partially supported by the Spanish Ministry of Science and Innovation through Grant TEC-2009-13897-C03-01/TCM, by the AGAUR, Gen. de Catalunya (2008-BE2-00196), and by the Spanish Ministry of Education through Ph.D. fellowships for E. Rocas (BES-2007-16775) and A. Padilla (AP200802235). This work is partially supported by the U.S. Government, which is not subject to U.S. copyright. E. Rocas is with the Universitat Politècnica de Catalunya (UPC), Barcelona 08034, Spain, and also with the National Institute of Standards and Technology, Boulder, CO 80305 USA. C. Collado and A. Padilla are with Universitat Politècnica de Catalunya (UPC), Barcelona 08034, Spain (e-mail: [email protected]). J. C. Booth is with the National Institute of Standards and Technology, Boulder, CO 80305 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TASC.2010.2085030 quantities in the steady state. In the nonlinear case experimental characterization is usually directly related to measurements in the linear regime. For example, the resonance frequency and quality factor of a resonator are measured to obtain , but in the nonlinear case, the measurements are done as a function of the input power. They show measurable detuning and saturation for sufficiently high input power. These measurements give , as a function of the peak magnetic field; for example, . Subsequently, can be related to physically-relevant nonlinear parameters, such as the local pair-breaking current density , as described in [2]. Obtaining is extremely useful for different materials comparison and it can contribute to better understand the nature of superconductivity phenomena in high temperature superconductors [3]. Unfortunately it is not straightforward to connect with , because of the nonlinear aspect of the problem. This implies, for example, that the expressions for the complex linear can no longer be simply generalized to the nonlinear regime by simply replacing the London penetration depth with a nonlinear version. This is because the linear expressions were derived in the frequency domain, and nonlinear analysis is better handled in the time domain. In this work, we obtain new formulas for the nonlinear surface impedance, which is defined as the ratio between tangential electric and magnetic fields at the surface of a conductor sheet [4], as derived from the time-domain nonlinear London equations and the two-fluid model. These formulas relate the experimental observables to the nonlinear device-independent parameters, such as the current density. They differ from the widely used phenomenological formulas, and will be compared with circuit-based simulations of a plane wave propagating through a nonlinear HTS material. II. MAGNETIC FIELD INSIDE THE HTS NONLINEAR PENETRATION DEPTH In a superconducting material, the nonlinear problem can be formulated by starting with the definition of current including condensate density nonlinearity [5] (1) where is the superfluid current density in the two-fluid model, which depends on the condensate velocity .For intrinsic nonlinearities and -wave superconductors, can be written as [5], [6]: (2) 17 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY where is a temperature-dependent coefficient that sets the strength of the nonlinearity and is the superfluid current density for . With (2), (1) can be written as [5] (3) This model of nonlinearity is one of the simplest expressions that we can find in the literature. In spite of its simplicity, the model can explain most of the experimental data for intermodulation distortion (IMD) in resonators fabricated from high temperature superconductor (HTS) thin films, as it implies a 3:1 slope for the logarithmic plot of IMD power vs. the input power [7]. If we assume an incident plane wave on a superconductor surface, we can calculate the current and field distributions within the superconductor. In a linear problem, this is usually done by solving the wave equation for the electric and magnetic fields, which leads to an exponential decrease of both quantities within the superconductor. For the nonlinear problem, the wave equation needs to be re-analysed with the appropriate generalization of the relevant quantities. The condensate velocity is related to the volume current density by use of [5] (4) Reference [5] proposes to solve a nonlinear version of the London equation that comes from substituting (3) into (4) and assuming that : (5) where is the London penetration depth: . Now the superconductor is considered to occupy the halfspace and with a uniform incident plane wave propagating towards positive z. The magnetic field is denoted as and the electric field as , so the screening current flows parallel to ; and . Equation (5) is then simplified to (6) Due to the continuity of tangential fields at , the boundary condition is (7) This equation is solved in [5], and the solution assuming is (8) The magnetic field inside the superconductor is then (9) with . Equation (9) follows a more complicated relation than the low-signal exponential behavior of the linear case , although a plot of both distributions as a function of shows that they are very similar. However, by use of (9), the initial decay rate is different from . The decay rate at the surface depends on the magnitude of and its inverse is defined as an effective penetration depth , resulting in (10) if is assumed. Note that is also the ratio between the current at the surface and the magnetic field . In terms of , we can write (11) with . In a similar way, we could rewrite of (2) at the surface as (12) and note that is different from the nonlinear London penetration depth obtained by replacing for in , as is usually done: (13) As we will see in the next section, the fact that is no longer equal than has a significant effect on the calculation of the nonlinear surface reactance. III. SURFACE REACTANCE We now apply this local model to calculate the surface impedance, which will be obviously nonlinear and dependent on the strength of the magnetic field at the surface. The surface impedance is defined as the ratio between tangential electric and magnetic fields at the surface of a conductor sheet [4] (14) We can calculate the surface impedance by deriving the tangential electric and magnetic fields at the surface and applying (12). By use of the two-fluid model, the electric field inside the superconductor is related to the superfluid current density by and with (1), we can write (15) Therefore, with (12), the current density and the electric field at are related by (16) 18 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ROCAS et al.: RELATION BETWEEN THE IMPEDANCE AND THE CURRENT DENSITY IN SUPERCONDUCTORS Eqn. (16) can be written in powers of if : (17) Note that and are time-dependent quantities. If we assume that the incident wave is sinusoidal we can approximate as and obtain the fundamental frequency component of ,,as (18) The nonlinear ratio between electric field and current density can now be calculated by use of equation (18). In a linear problem, and we can think about extending this equality to the nonlinear problem, and use (18) to calculate the surface impedance. Nevertheless, this would be erroneous because the ratio between and is given by the effective penetration depth derived in the previous section. Instead, we maintain the assumption , and by use of (10), we can evaluate (17) at the surface : (19) and by use of (11) and (13), for we obtain, (20) Hence, as previously done, the fundamental harmonic component is (21) Then we can write the surface reactance directly as (22) Note that the coefficient , which quantifies the strength of the nonlinearities, is one fourth as large as the one obtained if the quotient between and is used to obtain the surface impedance. IV. SURFACE RESISTANCE The formulation involving the nonlinear surface resistance is more complicated than for the nonlinear surface reactance. We begin by rewriting the ratio between and at the surface, but now consider that instead of . The two-fluid model gives a conductivity that is related with normal condensate fraction by (23) Therefore, its corresponding nonlinear dependence on the current density can be written as (24) with (25) At the surface , with (15) and , we obtain (26) which can be rewritten as (27) if , as is usually the case. Now, we replace (13) in (27) to obtain the ratio between the -frequency components and : (28) As is shown in the following section, this equation has been verified with nonlinear simulations using a circuit model approach, and show agreement within 0.5% level. The surface resistance involves the magnetic field and not just the current density, and we should extend the concept of effective penetration depth of (11) to relate and , but consider instead of . We should repeat the procedure of Section III but start with (29) instead of (6). Then, we need to solve: (30) We have found no solution for this nonlinear equation, but by doing simulations with the equivalent circuit explained in the following section, we have found that, at the surface, the ratio between magnetic field and current density phasors follows the relation (31) The surface impedance can then be written multiplying (28) and (31) as (32) This expression agrees well with the simulations described in the next section. Note that the nonlinear term of the surface re19 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY Fig. 1. Two-fluid equivalent circuit for a plane wave that propagates in a superconductor. Fig. 2. (Circles, red line) Magnetic and (squares, blue line) electric fields in a superconductor, obtained via a circuit simulation approach. Fig. 3. (Solid line) Surface impedance obtained with (32) and (circles) harmonic balance simulations. Dashed line corresponds to a direct non-linearization of the usual surface impedance. actance is half that compared to the value obtained directly by use of (13) instead of in the linear expression of the reactance. In the case of the resistance, the nonlinear terms of (32) are around to be one-fifth as large. V. EQUIVALENT CIRCUIT SIMULATIONS OF THE TWO-FLUID MODEL From Maxwell’s equations, the electric and magnetic fields inside the semi-infinite superconductor slab are related by (33) where by use of the two-fluid model, and (34) Fig. 1 shows an equivalent transmission line circuit that corresponds to (33) and (34) in the TEM approximation. The current density of and are calculated by use of at each segment of length dz. The fields inside the superconductor are calculated by means of numerical techniques to analyse the cascading of several nonlinear circuits such as that described in Fig. 1. We have done this by use of an electromagnetic circuit simulator. Fig. 2 shows the electric and magnetic fields as a function of . Fig. 3 shows the surface impedance obtained from simulations and by use of (32). The agreement between the simulated and closed-form expressions is very good. Fig. 3 also shows the surface impedance that would be obtained by use of a direct generalization of the usual linear surface impedance expressions. VI. CONCLUSION We have developed a phenomenological expression for the nonlinear surface impedance consistent with the nonlinear simulations of an incident plane wave on a superconductor surface. We have also shown that the present formulation yields significantly different results than those obtained from a direct extrapolation of the linear formula of the surface impedance derived in the frequency domain. REFERENCES [1] D. Pozar, Microwave Engineering. New York: Wiley, 1998, pp. 183–187. [2] P. P. Nguyen, D. E. Oates, G. Dresselhaus, and M. S. Dresselhaus, “Nonlinear surface impedance for YBa2Cu3O7-x thin films: Measurements and a coupled-grain model,” Physical Review B, vol. 48, no. 9, pp. 6400–6412, Sept. 1993. [3] J. C. Booth, B. M. Andersen, and P. J. Hirschfeld, “Disorder effects on the intrinsic nonlinear current density in YBa2Cu3O7–  ,” IEEE Trans. Appl. Supercond., vol. 17, no. 2, pp. 906–909, June 2007. [4] S. Ramo, J. Whinnery, and T. V. Duzer, Fields and Waves in Communication Electronics. New York: Wiley, 1994. [5] D. Xu, S. K. Yip, and J. A. Sauls, “Nonlinear Meissner effect in unconventional superconductors,” Physical Review B, vol. 51, no. 22, p. 16233–16233, June 1995. [6] T. Dahm, D. Scalapino, and B. Willemsen, “Phenomenological theory of intermodulation in HTS resonators and filters,” Journal of Superconductivity, vol. 12, no. 2, p. 339–339, Apr. 1999. [7] C. Collado, J. Mateu, and J. M. OCallaghan, “Analysis and simulation of the effects of distributed nonlinearities in microwave superconducting devices,” IEEE Trans. Appl. Supercond., vol. 15, no. 1, pp. 26–39, Mar. 2005. 20 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY Superconducting Multiplexer Filter Bank for a Frequency-Selective Power Limiter Eduard Rocas, Student Member, IEEE, Alberto Padilla, Jordi Mateu, Senior Member, IEEE, Nathan Orloff, Juan M. O’Callaghan, Senior Member, IEEE, Carlos Collado, Senior Member, IEEE, and James C. Booth Abstract—This work proposes a superconducting multiplexer filter bank configuration to be used as a frequency-selective power limiter. The proposed configuration limits narrowband high-power signals within a single frequency band without degrading the signal performance in the rest of the frequency bands. To accomplish this, we need to calculate the limiting power of superconducting filters implemented by means of half-wavelength transmission line resonators. The limiting power is obtained as a function of the resonator geometry, filter bandwidth, and filter order. Practical issues occurring in superconducting filters operating at high power, such as the reduction of the quality factors and de-tuning, are also analyzed and shown to not adversely affect the overall multiplexer power limiter performance. Index Terms—Critical current, filter, filter bank, limiter, multiplexer. I. INTRODUCTION ONE of the fundamental properties of superconducting materials is their ability to switch from a high-loss normal state to a low-loss superconducting state, when cooled below the critical temperature [1]. But these materials, while in the superconducting state below , can also switch to the high-loss state when a high power signal is injected, because of their nonlinear surface impedance [1]. This property can be used to develop microwave power limiter devices, in which the current flows with high losses when the current density reaches the material-dependent critical current density and consequently the signals are attenuated at the output of the limiter. The advantages of superconducting materials, for the design of microwave power limiters, over other technologies are mainly their short transition period from the superconducting to the normal state, their very low-loss behavior in the superconducting state, and their ability to reversibly switch Manuscript received August 03, 2010; accepted November 08, 2010. This work was supported in part by the Spanish Ministry of Science and Innovation under Grant TEC-2009-13897-C03-01/TCM, by the AGAUR, Generalitat de Catalunya (2008-BE2-00196) and MAT-2008-06761-C03-02; and by Spanish Ministry of Education through PhD fellowships for E. Rocas (BES-2007-16775) and A. Padilla (AP200802235). E. Rocas, A. Padilla, J. M. O’Callaghan, and C. Collado are with Universitat Politècnica de Catalunya (UPC), Barcelona 08034, Spain. J. Mateu is with Universitat Politècnica de Catalunya (UPC), Barcelona 08034, Spain. He is also with the Centre Tecnològic de Telecomunicacions de Catalunya (CTTC). N. Orloff and J. C. Booth are with the National Institute of Standards and Technology, Boulder, CO 80305 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TASC.2010.2093554 between low-loss and high-loss states. Previous work has shown that superconducting transmission line limiters can be used to protect downstream electronics from high-power transients, with nanosecond or less turn-on periods [2], [3]. These result in broadband limiters that provide attenuation over the entire frequency band of interest whenever a high-power signal is present. A more desirable approach is to restrict the limiter response to a narrow frequency band around the high-power signal, without degrading the device performance in the remainder of the frequency band. To accomplish this, we implemented a multiplexed superconducting limiter filter bank. The limiting power in superconducting transmission line limiters was set by the operating temperature and by the cross-section of the implemented transmission line [2], [3], usually resulting in very narrow line-widths (10 to 20 ). It is expected the use of resonant structures as limiting devices could provide a better control of the limiting power, because additional factors may be used for control, such as quality factor (Q) and coupling between resonators and input and output ports [4]. This approach could also lower the threshold powers significantly compared to those of transmission line limiters. In this work, we use the measured response of transmission line limiters as the basis for the design of narrowband signallimiting filters at microwave frequencies. We calculate the current density distribution of the individual resonators that constitute the filter from the incident power, as a function of the filter topology, bandwidth, quality factor, and dimensions, and subsequently determine the filter response as a function of incident power. We also studied the detuning of the filter produced by a high power signal and its effect on the overall multiplexer response. II. MULTIPLEXING ARCHITECTURE Fig. 1 outlines a multiplexing architecture in which the incident signal is split into several frequency bands, and subsequently re-combined. Each frequency band, set by a microwave filter, takes only a portion of the incident signal. When the signal power at one or several frequency bands exceeds the limiting power value, set by the critical current density , the superconducting transmission line resonators switch from the low-loss superconducting state to the high-loss normal state. Therefore, the signal in the high power channel is attenuated and thus the downstream circuitry is protected from the effects of large transient input signals. Because the transition from the superconducting state to the normal state is reversible and occurs on nanosecond timescales, the architecture of Fig. 1 can also be 21 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY Fig. 1. Outline of the multiplexing architecture, used to protect an analog to digital converter (ADC) from high-power transients. understood as a high-performance, nanoseconds tunable notch filter. III. LIMITING POWER AND GEOMETRY IN SUPERCONDUCTOR TRANSMISSION LINES The superconducting transmission line limiters reported in [2], [3] essentially consist of typical coplanar waveguides (CPW) transmission line, structured to exceed the critical current density for a given input power level. Such devices were fabricated from a 180 nm YBCO thin film deposited on an -plane sapphire substrate by pulsed-laser deposition, with a thin (20 nm to 50 nm) CeO buffer layer. Sapphire substrates were used for their high thermal conductivity and low microwave loss. The measured surface resistance at 3 GHz and 70 K was approximately in . The CPW transmission lines had three different center-conductor line widths (55 ,22 , and 11 ) and gap spacings (105 ,40 , and 20 , respectively) between the center conductor and ground planes on either side, to achieve a 50 characteristic impedance. Measurements of the limiting behavior revealed an average critical current density at 70 K of . In this section the material parameters extracted from previous work [2], [3] are used to obtain the saturation power in a different set of CPW transmission lines. Each CPW transmission line has a center conductor that ranges from 10 to 450 wide, and the gap is set to obtain 50 transmission lines. In a 50 superconducting transmission line, matched both with the source and the load, the total peak current flowing through a cross-section of the transmission line can be related to the input available power by (1) Note, therefore, that for a given input available power, a whole range of transmission lines of different dimensions would yield the same total currents, as long as the same characteristic impedance is preserved. The total current is distributed along the cross-section of the transmission line, note that the current density peaks are at the edges of the center conductor and at the inner edges of the ground planes. The current density profile can be calculated by means of numerical techniques such as the Weeks-Sheen method [5], [6]. Fig. 2 displays the maximum current density obtained in the center conductor (in solid line) for an input power Fig. 2. Dashed line: saturation power as a function of the line width. Solid line: maximum current density as a function of the center conductor width (W), for an input power P=5  W . of 5 , as a function of the center conductor width. As expected, the current density increases for narrower transmission lines. These values are then used, along with (1), to obtain, as a function of center conductor width, the incident power that gives a current density that exceeds the critical current density of . The results in Fig. 2 (dashed line) show an almost linear dependence of saturation power on the conductor width. This result shows explicitly that the limiting power in a superconducting transmission line can be set by use of the transmission line width. In the following section, we make use of these material properties to obtain the limiting power for frequency-selective resonators. IV. SUPERCONDUCTING RESONATOR LIMITERS Here we evaluate the saturation power in CPW and microstrip superconducting half-wavelength resonators. Note that, in contrast to the traveling wave transmission line, a resonator need not have a characteristic impedance of 50 . This allows us to use microstrip configurations of several center conductor widths, in addition to CPW configurations. In a half-wavelength transmission line resonator the current distribution along the line follows approximately a sinusoidal profile , where and is the maximum peak current achieved in the middle of the line. We also know that the value depends on the factor of the resonator. Hence, highresonators reach high-current values, which in turn results in lower saturation powers. To achieve high s in planar technology, designs typically employ relatively wide transmission lines, and concurrently reach high current peaks . We propose the CPW and microstrip resonator configurations outlined in Figs. 3(a) and 3(b), respectively, which consist of non-uniform transmission lines, where the width of the central region (in grey) is made narrower than the width of the rest of the resonator. Additionally, since the nonlinear behavior in superconductors is a distributed effect [12], this configuration would reduce the nonlinear effects produced in the resonant transmission line. In non-uniform transmission line resonators the factor is (2) 22 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ROCAS et al.: SUPERCONDUCTING MULTIPLEXER FILTER BANK A FREQUENCY-SELECTIVE POWER LIMITER Fig. 3. Outline of the resonators: a) CPW, b) microstrip. Fig. 4. Q factors as a function of the transmission line width, for microstrip and coplanar resonator configurations and for values of l 0 l = = 5 , = 10 and = 20 . where and are the inductance and resistance per unit length, corresponding to the wider region, and and are the inductance and resistance per unit length corresponding to the narrower region. The distributed inductance and resistance values can be obtained by means of the Weeks-Sheen method [5], [6], for the widths and thicknesses of each region. The and terms are geometrical factors, given as and , respectively, where is the total length of the wider region of the transmission line resonator of Fig. 3, and is the length of the narrower region. Note that when the current distribution differs from a sinusoidal pattern the values of and have to be recalculated accordingly. The relation between the driven available power and the maximum current in the resonator is (3) where is the loaded and the coupling coefficient between the input (output) port and the resonator. By use of (2), we extract the factor for a set of CPW and microstrip resonators as a function of the line width on the center region of the conductor and for several values of ( , and ). Note that the wider parts of the transmission lines are set to a fixed value of 450 , which results in 50 regions. The data of Fig. 4 are used, along with (3), to obtain the maximum current and maximum current density for the set of CPW and microstrip resonators. For a critical coupling value and for a fixed drive power , the results for CPW and microstrip resonators are summarized in Figs. 5(a) and 5(b), respectively. The maximum current increases with the width of the conductor due to the dependence, whereas the maximum current density decreases when the center conductor width increases. Additionally, by use of Fig. 5. Maximum current I (left ordinate) and maximum current density J (right ordinate) as a function of the conductor width and for several values of l 0 l = = 5 , = 10 and = 20 . (a) CPW; (b) microstrip. Fig. 6. Saturation power as a function of the conductor width and for l 0 l = = 5 , = 10 and = 20 , for CPW and microstrip. (3), we also obtain the saturation power for the resonant structures presented above. For the CPW and microstrip resonators, the results are shown in Fig. 6. Note that the saturation power is reduced considerably compared to the case for the transmission line limiters. V. POWER LIMITING SUPERCONDUCTOR FILTER BANK Next, we need to find the total current flowing through the filter resonators, as was previously done to obtain the saturation power, so that the total current can then be linked to the current density distribution. The resonant structures assumed for this investigation are those presented in the previous section, and their factors are summarized in Fig. 4. 23 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY Fig. 7. Outline of the filter circuit model made of in-line coupled resonators. We start by considering a circuit model of a filter made of coupled resonators (see Fig. 7), which can be formulated with its general admittance matrix [7]. Note that for clarity Fig. 7 corresponds to an in-line filter configuration. Nevertheless, the formulation below is general for any filter topology. The admittance lumped elements correspond to the RLC parallel resonator model [8]. The admittance matrix is then used to obtain the voltage at each node of the equivalent-circuit model as a function of the current source , which is related to the driven power . The resulting dissipated power is (4) where is the lumped resistance that accounts for the resonator losses and can be related to the quality factor as , where is the inductance of the RCL resonator model [8]. The filter configurations assumed in this section use the last resonator in the filter network as the limiting resonator. Note however that the formulation and results could be extended for any other location of the limiting resonator or even when more than a limiting resonator is used. The final position of the limiting resonator would depend on the multiplexing architecture used in the prototype, not yet properly addressed in this work. The dissipated power at each resonator can also be obtained from the current distribution at each resonator, for half-wavelength resonators, the current distribution at the ith resonator is . Then, the resulting dissipated power can be obtained as: (5) By equating (4) and (5), we can extract the maximum current at each resonator, which is used to obtain the maximum current density and therefore the saturation power. By doing that, we can find the saturation power for several filters, of different order and bandwidth (BW). Fig. 8 shows the saturation power in a fourth order Chebyshev filter with return losses of 20 dB, centered at 3 GHz and for 20 MHz and 200 MHz bandwidths, as a function of the center conductor width. The results reveal that the required power to saturate the filter decreases when the filter bandwidth decreases and approaches that required in a single resonator for very narrow band filters. This conclusion might be very significant from a practical point of view, since we can set the limiting power by designing filters of different bandwidths. In addition, Fig. 9 shows the saturation power for a Chebyshev filter (with 20 dB return losses) for several orders, , 4, 6 and 8. In this case, we can be observed that the saturation power is largely independent of the order of the filter. Fig. 8. Saturation power for 4th order Chebyshev filters of 20 MHz and 200 MHz bandwidth. The saturation power for a single microstrip resonator is also depicted for comparison. Fig. 9. Saturation power for third-, fourth-, sixthand eighth-order Chebyshev filters with 20 MHz bandwith. Dotted: eighth, dashed: sixth, dash-dotted: fourth, and solid: third. VI. PRACTICAL CONSIDERATIONS ON THE IMPLEMENTATION OF A SUPERCONDUCTING POWER LIMITER MULTIPLEXER The formulation and procedure presented here allow us to extract the limiting power of a superconducting filter by obtaining the total current that flows through the resonators when the maximum current density exceeds the critical current density. However, the inherent nonlinear nature of superconductors [9], which gives rise to the limiting behavior, can also produce effects such as detuning and saturation at high current densities. Therefore, we may expect a variation of the resonant frequency and reduction in for the resonators in the filter as the current density increases. The quantification of the detuning and reduction can be obtained by means of modeling the nonlinear distributed effects of the superconducting resonators [9]–[13]. Assessment of the detuning effects and reduction in a single filter is necessary when the filter is used in a multiplexing architecture, because an out-of band variation of the input reflective coefficient may adversely affect the entire system performance [7]. Fig. 10 shows the above-mentioned effects in a sixth-order Chebyshev filter centered at 3 GHz with a 200 MHz bandwidth, and whose initial Q values are set to . Figs. 10(a) and 10(b) show the frequency response of the filter when the of the last resonator is reduced from to and , respectively. The results show that the input reflection coefficient remains fully reflective when out-of-passband, 24 14 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 1, JANUARY 2009 Fig. 4. Current at the end of the line for all spurious frequency components when the fundamental tones f and f are set to 3 and 3.5 GHz, respectively. Solid lines correspond to 2 ! 0 ! , 2 ! + ! , and 3 ! , and dashed lines correspond to 2 ! 0 ! , 2 ! + ! , and 3 ! . correspond to the values evaluated by the use of the closed-form expressions, all agreeing very well with the simulated results. The expected linear transmission line length dependence of the spurious signals can be used to confirm the distributed origin of the nonlinear effects and, therefore, rule out contributions from possible external nonlinear sources. Although the results in Fig. 3 show an increment of the spurious signals as a function of the length, they do not follow a linear dependence, as it occurs in a perfectly matched transmission line. The fluctuating length dependence, due to the mismatched effects, may give higher nonlinear effects in shorter lines. This makes it difficult to predict, from the raw data, the distributed origin of the nonlinearities without using simulations or the closed-form expressions developed here. Since the fluctuating behavior comes from the mismatched effects, it depends on the operating frequency and is, therefore, different for each spurious signal. This phenomenon, if not taken into account, may lead to misleading conclusions, such as asymmetries on the IMD (i.e., differences between the signal power at and ) due to memory effects [17]. Although not reported in this section, the closed-form expressions have also been verified when the nonlinear effects come only from the dielectric part and from both conductor and dielectric parts. We have also verified the obtained expressions for a several values of characteristic impedance ranging from 10 to 100 and different frequencies. As an example, to emphasize the effects of mismatch, this section also evaluates the length dependence of the spurious signals for lines longer than those of Fig. 3 (up to 0.2 m) fed with two input tones whose frequencies ( and ) are 3 and 3.5 GHz, respectively, much farther apart than those of Fig. 3. Fig. 4 shows how the asymmetries between the spurious signals are more pronounced. Moreover, unlike in Fig. 3, the IMD-LF traces in Fig. 4 have fluctuations for only limited ranges of length. From Figs. 3 and 4, we also see that small deviation in the length determination of the line may incur in a few decibel difference of the predicted spurious signal. Fig. 5. Measurements and analytical results of the spurious signals occurring in a set of ferroelectric nonlinear transmission lines. V. VERIFICATION:MEASUREMENTS Simulations using the circuit model of Fig. 2 have been extensively used to analyze the distributed nonlinear effects in superconducting transmission line, confirming the expected length and frequency dependence [18]. Since in superconducting transmission lines the dielectric constants of the substrates are usually well known, one can build transmission lines with characteristic impedances matched to 50 . For this case, we do not expect to observe any fluctuating effect due to the length of the line. The closed-form expressions developed here have also been used to explain such results. However, this is not the case for substrates with large or nonlinear dielectric constants, such as ferroelectrics [2] or magnetoelectrics [3]. Fabrication of transmission lines incorporating these materials then results in a nonlinear mismatched transmission line. This section applies the developed closed-form expressions to explain the nonlinear behavior occurring in CPW transmission lines incorporating an SrTiO (STO) thin film of 400-nm thickness grown on a LaAlO substrate with conductors defined by a 0.3m Au layer on top. Here we report the intercept point at 0-dBm input power of the spurious signals occurring in four CPW transmission lines, all with a 50m width of center conductor and 20m gap between the center conductor and ground planes. The four lines have different length: mm, mm, mm, and mm. Fig. 5 shows the measured intercept point at 0-dBm input power and outlines the length dependence predicted from the closed-form expression developed here. Circles and squares indicate the measured intercept points of the spurious signal at and , respectively. The input tones and were set to 6 and 6.1 GHz, respectively. Experiments performed to obtain measurements of the spurious signals are detailed in [11]. We then used the current at the end of the line, obtained from (22)–(28), to extract the power at the output port [19]. The output power generated at spurious frequencies depends on the propagation constant and characteristic impedance of the line—obtained from a multiline thru-reflect line (TRL) calibration [4] as at each frequency point—and the nonlinear terms and , which are unknown. By equating the resulting output 31 MATEU et al.: THIRD-ORDER INTERMODULATION DISTORTION AND HARMONIC GENERATION Fig. 6. Outline of a mismatched transmission line. power expressions with the measured spurious signals, one can extract the nonlinear terms and . Note that, in this example, where all nonlinear effects are due to the dielectric part, the nonlinear effects from the conductor part are . This same procedure has been used in previous studies [9], [11], but the extraction of the nonlinear terms was carried out by iterating circuit simulations to match measured and simulated spurious signals. Fig. 5 reproduces the fitting of the experimental results reported in [11]. Dashed–dotted and dashed lines show the length dependence obtained from the closed-form expressions. Fig. 5 shows very good agreement between measurements and the length dependence obtained from the closed-form expressions. VI. CONCLUSION This study has developed general closed-form expressions to obtain the spurious signals resulting from a weakly nonlinear mismatched transmission line with quadratic nonlinear effects in both the conductor and dielectric parts of the circuit. The resulting closed-form expressions have been verified with circuit simulations. Assessment of the transmission line length dependence of the spurious signals reveals how the mismatch effect may mask the expected length dependence caused by distributed nonlinear effects. Without an accurate circuit model and extensive simulations, or without the use of the closed-form expressions developed here, nonlinear measurements may give misleading results due to the fluctuating length dependence and the asymmetry between spurious signals. Moreover, since the closed-form expressions show the interaction of the different sources of nonlinear effects, they offer the possibility of evaluating the nonlinear response and identifying the origin of the nonlinear effects in order to discern between different sources. APPENDIX CURRENT AND VOLTAGE DISTRIBUTION AT FUNDAMENTAL FREQUENCIES In a transmission line for which the characteristic impedance is neither matched to the impedance of the source, nor the load (see Fig. 6), the current and voltage distribution along the line of the fundamental signals is obtained from basic theory of circuit analysis [19]. We start by obtaining the forward current component at the input of the line ,,at ( for the fundamental components might by 1 or 2) as (28) indicates the current driven by the source and is the reflection coefficient to the source and load both at . The input impedance may be found as (29) The propagation constant and the characteristic impedance of the transmission line , respectively, can be obtained from the linear distributed circuit parameters , , , and of the line, as (30) (31) From the expressions above, i.e., (28)–(31), the current and voltage distribution along the line at fundamental frequency , being or , are (32) (33) NONLINEAR VOLTAGE AND NONLINEAR CURRENT GENERATORS Assuming a quadratic nonlinear dependence of the distributed circuit parameters , , , and , as outlined in (13) and (14), we found the resulting nonlinear sources at the mixing products and harmonics of the fundamental components at and by applying (13) and (14) into (11) and (12), respectively. Although this Appendix only details the nonlinear voltage and current generators at , , and , their counterparts at , , and , respectively, could be directly derived. The nonlinear voltage and the nonlinear current generators, respectively, at are (34) (35) 32 16 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 1, JANUARY 2009 where and . Substituting (34) and (35) into (19), we may isolate the forward and backward components (36) (37) where (38) with and . By the used of analogous expressions to (34)–(38), the intermodulation products at may be written as (39) (40) where now and . The forward and backward components are (41) (42) where (43) with and . For the third harmonic at , the nonlinear voltage and current generators are (44) (45) where and . The forward and backward components then result as follows: (46) (47) where (48) with and . REFERENCES [1] D. E. Oates, P. P. Nguyen, G. Dresselhaus, M. S. Dresselhaus, G. Koren, and E. Polturak, “Nonlinear surface impedance of YBCO thin films: Measurements, modelling and effects in devices,” J. Superconduct. Novel Magn., vol. 8, no. 6, pp. 725–733, 1995. [2] M. J. Lancaster, J. Powell, and A. Porch, “Thin-film ferroelectric microwave devices,” Superconduct. Sci. Technol., vol. 11, pp. 1323–1334, 1998. [3] N. Orloff, J. Mateu, M. Murakami, I. Takeuchi, and J. C. Booth, “Broadband characterization of multilayer dielectric thin-films,” in IEEE MTT-S Int. Microw. Symp. Dig., 2007, pp. 1177–1180. 33 MATEU et al.: THIRD-ORDER INTERMODULATION DISTORTION AND HARMONIC GENERATION [4] R. B. Marks, “A multiline method for network analyzer calibration,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 7, pp. 1205–1215, Jul. 1991. [5] D. F. Williams and R. B. Marks, “Transmission line capacitance measurements,” IEEE Microw. Guided Wave Lett., vol. 1, no. 9, pp. 243–245, Sep. 1991. [6] D. F. Williams and R. B. Marks, “Characteristic impedance determination using propagation constant measurements,” IEEE Microw. Guided Wave Lett., vol. 1, no. 6, pp. 141–143, Jun. 1991. [7] J. C. Booth, J. Mateu, M. Janezic, J. Baker-Jarvis, and J. A. Beall, “Broadband permittivity measurement of liquid and biological samples using microfluidic channels,” in IEEE MTT-S Int. Microw. Symp. Dig., 2006 , pp. 1750–1753. [8] J. Mateu, N. Orloff, M. Renihart, and J. C. Booth, “Broadband permittivity of liquids extracted from transmission line measurements of microfluidic channels,” in IEEE MTT-S Int. Microw. Symp. Dig., 2007, pp. 523–526. [9] C. Collado, J. Mateu, and J. M. O’Callaghan, “Analysis and simulation of the effects of distributed nonlinearities in microwave superconducting devices,” IEEE Trans. Appl. Superconduct., vol. 15, no. 1, pp. 26–39, Mar. 2005. [10] J. R. Ott, P. Lahl, and R. Wördenweber, “Nonlinear microwave properties of ferroelectric thin films,” Appl. Phys. Lett., vol. 84, no. 21, pp. 4147–4149, 2004. [11] J. Mateu, J. C. Booth, and S. A. Schima, “Frequency tuning and spurious signal generation at microwave frequencies in ferroelectric SrTiO thin-film transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 2, pp. 391–396, Feb. 2007. [12] R. Hammond, E. Soares, B. Willemsen, T. Dahm, D. Scalapino, and J. Schrieffer, “Intrinsic limits on the Q and intermodulation of low power high temperature superconducting microstrip resonators,” J. Appl. Phys., vol. 84, no. 10, pp. 5662–5667, 1998. [13] D. E. Oates, “Microwave superconductivity,” in Nonlinear Behaviour of Superconducting Devices, ser. NATO Sci. E: Appl. Sci.. Brussels, Belgium: NATO, vol. 375, ch. 5. [14] D. Seron, C. Collado, J. Mateu, and J. M. O’Callaghan, “Analysis and simlation of distributed nonlinearities in ferroelectrics and superconductors for microwave applications,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1154–1160, Mar. 2006. [15] K. S. Champlin and D. R. Singh, “Small-signal second-harmonic generation by a nonlinear transmission line,” IEEE Trans. Microw. Theory Tech., vol. MTT-34, no. 3, pp. 351–353, Mar. 1986. [16] A. B. Kozyrev and D. W. van der Weide, “Nonlinear wave propagation phenomena in left-handed transmission-line media,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 238–245, Jan. 2005. [17] J. C. Pedro and N. B. Carvalho, Intermodulation Distortion in Microwave and Wireless Circuits. Norwood, MA: Artech House, 2003. [18] J. C. Booth, L. R. Vale, and R. H. Ono, “On-wafer measurements of nonlinear effects in high-temperature superconductors,” IEEE Trans. Appl. Superconduct., vol. 11, no. 1, pp. 1387–1391, Mar. 2001. [19] D. Pozar, Microwave Engineering. New York: Wiley, 1998. Jordi Mateu (M’03) received the Telecommunication Engineering and Ph.D. degrees from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 1999 and 2003, respectively. Since October 2006, he has been with the Department of Signal Theory and Communications, UPC, and with the Centre Tecnològic de Telecomunicacions de Catalunya (CTTC). From May to August 2001, he was a Visiting Researcher with Superconductor Technologies Inc., Santa Barbara CA. From October 2002 to August 2005, he was Research Associate with CTTC. Since September 2004, he has held several guest researcher appointments with the National Institute of Standards and Technology (NIST), Boulder, CO, where he was a Fulbright Research Fellow from September 2005 to October 2006. In July 2006, he was Visiting Researcher with the Lincoln Laboratory, Massachusetts Institute of Technology (MIT). From September 2003 to August 2005, he was a Part-Time Assistant Professor with the Universitat Autònoma de Barcelona. His primary interest includes microwave devices and system and characterization and modeling of new electronic materials including ferroelectrics, magnetoelectric and superconductors. Dr. Mateu was the recipient of the 2004 Prize for the best doctoral thesis in fundamental and basic technologies for information and communications presented by the Colegio Oficial de Ingenieros de Telecomunicación (COIT) and the Asociación Española de Ingenieros de Telecomunicación (AEIT). He was also the recipient of a Fulbright Research Fellowship, an Occasional Lecturer Award for visiting MIT and a Ramón y Cajal Contract. Carlos Collado (A’02–M’03) was born in Barcelona, Spain, in 1969. He received the Telecommunication Engineering degree and Ph.D. degree from the Technical University of Catalonia (UPC), Barcelona, Spain, in 1995 and 2001, respectively. In 1998, he joined the faculty at UPC, where he has been teaching courses on theory of electromagnetism, microwave laboratory, and high-frequency devices and systems. In 2004, he was a Visiting Researcher with the University of California Irvine. Since April 2005, he has been an Associate Professor with UPC. His primary research interests include microwave devices and systems, electrooptics applications, and superconducting devices. Nathan Orloff was born in Columbia, SC, on August 10, 1981. He received the B.S. degree in physics (with high honors) from the University of Maryland at College Park, in 2004, and is currently working toward the Ph.D. degree in physics at the University of Maryland at College Park. His doctoral thesis concerns the study and extraction of microwave properties of materials including ferroelectrics, magnetoelectrics, superconducting materials, and fluids. Mr. Orloff was the recipient of the 2004 Martin–Monroe Undergraduate Research Award and the 2006 CMPS Dean’s Award for teaching assistants. James C. Booth received the B.A. degree in physics from the University of Virginia, Blacksburg, in 1989, and the Ph.D. degree in physics from the University of Maryland at College Park, in 1996. His doctoral dissertation concerned novel measurements of the frequency-dependent microwave surface impedance of cuprate thin-film superconductors. Since 1996, he has been a Physicist with the National Institute of Standards and Technology (NIST), Boulder, CO, originally as a National Research Council (NRC) Post-Doctoral Research Associate (1996–1998) and currently as a Staff Scientist. His research with NIST is focused on exploring the microwave properties of new electronic materials and devices including ferroelectric, magneto-electric, and superconducting thin films, as well as developing experimental platforms integrating microfluidic and microelectronic components for RF and microwave frequency characterization of liquid and biological samples. Eduard Rocas (S’06) was born in Palafrugell, Catalonia, Spain, in 1982. He received the Telecommunication Engineering degree from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 2005, and is currently working toward the Ph.D. degree at the UPC. While working toward the Telecommunication Engineering degree, his final project was associated with the creation of the Intelligent Communications and Avionics for Robust Unmanned Aerial Systems (ICARUS) Research Group. From September 2005 to July 2006, he was involved with the simulation and modeling of advanced SONARs with the Computer Vision and Robotics Group (VICOROB), University of Girona. His research concerns new materials and structures for novel RF/MW devices. Mr. Rocas was the recipient of an FPU grant and and FPI grant. 34 18 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 1, JANUARY 2009 Alberto Padilla was born in Barcelona, Spain, in 1984. He received the Telecommunication Engineering degree from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 2008, and is currently working toward the Ph.D. degree at the UPC. While working toward the Telecommunication Engineering degree, his final project was associated with the mitigation of nonlinear behavior of high-temperature superconducting planar devices. Since March 2008, he has been with a Microwave Engineer the Department of Signal Theory and Communications, UPC. His research concerns a new class of synthesis for microwave filters for satellite communications. Juan M. O’Callaghan (SM’01) received the Telecommunication Engineering degree from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 1987, and the M.S. and Ph.D. degrees from the University of Wisconsin–Madison, in 1989 and 1992, respectively. He is currently a Full Professor with the UPC. He was an Intern with the Systems Research Center, Honeywell, Bloomington, MN, where he was involved with noise measurement methods for field-effect transistors (FETs) at Ka -band. From 2003 to 2006, he was Manager for MERIT, a consortium of European universities delivering a joint Master’s program in information technologies within the Erasmus Mundus Program. He is currently Vice-Dean of academic affaires with Telecom BCN, the telecommunication engineering school of the UPC. He has authored or coauthored over 45 papers in peer-reviewed international magazines. He holds three patents. His research interests include microwave devices and materials and microwave photonics. He has been involved with noise characterization, large-signal properties of GaAs FETs, and advanced microwave materials such as superconductors and ferroelectrics. 35 36 CHAPTER II - BULK ACOUSTIC WAVE RESONATORS Bulk acoustic wave technology is having a great success, as a filtering solution, in the market of mobile applications. Its small footprint, high power handling and high quality factor are key requirements for today’s mobile communications, and BAW is substantially better than previous technologies like surface acoustic wave or ceramic filters. However, BAW filters are known to exhibit a high second harmonic and intermodulation distortion, which can potentially limit the potential of this technology by causing undesired effects like receiver desensitization or interference. This chapter focuses on the intrinsic nonlinearities in BAW resonators from a phenomenological modeling perspective. The first article describes a lumped model approach of the nonlinear behavior of BAW resonators. The model is based on the modified Butterworth Van Dyke (MBVD) and provides with a model that is simple to set up and can reproduce the intermodulation distortion from a phenomenological perspective. The second and third articles provide a distributed nonlinear model approach, based on the Krimtholz, Leedom and Matthaei (KLM) model, to reproduce the distributed nature of acoustic propagation through the materials stack. The fourth article focuses on the second harmonic generation and proposes a methodology to accurately extract the effects of the measurement setup from the results. The articles in Chapter 2 are: 37  E. Rocas, C. Collado, J. Mateu, H. Campanella, J. M. O'Callaghan, “Third order intermodulation distortion in Film Bulk Acoustic Resonators at resonance and antiresonance”, 2008 IEEE MTT-S International Microwave Symposium Digest, pp. 1259-1262, 15-20 June 2008  E. Rocas, C. Collado, A. Padilla, J. Mateu, J. M. O'Callaghan, “Nonlinear distributed model for IMD prediction in BAW resonators”, 2008 IEEE International Ultrasonics Symposium, pp. 1557-1560, 2-5 Nov. 2008  C. Collado, E. Rocas, J. Mateu, A. Padilla, J. M. O'Callaghan, “Nonlinear Distributed Model for Bulk Acoustic Wave Resonators”, IEEE Transactions on Microwave Theory and Techniques, vol. 57, no. 12, pp. 3019-3029, Dec. 2009  C. Collado, E. Rocas, A. Padilla, J. Mateu, J. M. O'Callaghan, N. D. Orloff, J. C. Booth, E. Iborra, R. Aigner, “First-Order Elastic Nonlinearities of Bulk Acoustic Wave Resonators”, IEEE Transactions on Microwave Theory and Techniques, Accepted for publication and available online at IEEExplore.org as Early Access. 38 Third order Intermodulation Distortion in Film Bulk Acoustic Resonators at Resonance and Antiresonance Eduard Rocas1, Carlos Collado1, Jordi Mateu1,2, Humberto Campanella3, Juan M. O’Callaghan1 1 Universitat Politècnica de Catalunya (UPC), Campus Nord, Barcelona, Spain 2 Centre Tecnològic de Telecomunicacions de Catalunya (CTTC), PMT, Castelldefels, Spain 3 Centro Nacional de Microelectrónica (CNM-CSIC), Campus UAB, Bellaterra, Spain Abstract — This paper presents recent measurements and modeling of the third order intermodulation products of a Film Bulk Acoustic Resonator (FBAR), for a various values of frequency spacing between driving tones. The frequency dependence of voltage and current in the acoustic branch rules out a voltage-dependent nonlinearity. The results show different slopes at resonance and antiresonance, which are correctly adjusted by the model with a current dependent inductor and/or capacitor. The intermodulation distortion is found to be dependent on the frequency spacing between driving tones, indicating memory effects. Index Terms — Thin-film bulk acoustic wave resonators, intermodulation, nonlinear characterization, resonance, antiresonance. I. INTRODUCTION Film bulk acoustic wave resonators (FBARs) allow to achieve very compact high-performance RF and microwave passive components [1]. The ever growing wireless market and a constantly increasing demand for spectrum are steadily pushing for more stringent requirements of such components. This is also the case of FBAR components and their nonlinear performance. In spite of the significance of the nonlinear effects in wireless communication systems [2], and the wellknown existence of nonlinearities in piezoelectric based devices such as FBARs [3],[4], there are not many published works focusing on the nonlinear characterization and modeling of the FBAR components and their impact in communication systems. Several physical phenomena occurring in a FBAR (piezoelectricity, elasticity, thermal effects, etc.) have a potential nonlinearity and may give rise to many nonlinear effects such as resonator detuning, saturation, and generation of harmonics and intermodulation distortion (IMD). The ability to predict all these effects with realistic and easy-to-use models would be of great help to designers and would facilitate the use of FBAR devices in future systems. In contexts other than FBAR devices (superconducting and ferroelectric devices, for example), IMD has shown to be of great value to relate nonlinear phenomena with their measurable effects [5],[6],[7]. Following that guide, in this work we measure the IMD of an FBAR at resonance and antiresonance for a range of input power and for various values of frequency separation between the two tones feeding the resonator. From the measurements we infer the basic features of a nonlinear version of the Butterworth Van Dyke model which we adjust to fit the measured data. The resulting model shows to be useful to predict the IMD at resonance and antiresonance. To our knowledge, there are no previous publications on nonlinear FBAR models able to account for IMD at these two resonant frequencies. II. TEST DEVICE AND LINEAR MODELING The FBAR tested in this work was implemented following the fabrication process described in [8]. This consists in a piezoelectric aluminium nitride (AIN) membrane (50x70 µm and 1 µm thick) sandwiched between two titanium/platinium layers and deposited on a silicon substrate. The geometry and dimensions of the FBAR tested were set to have resonant frequencies around 2.3 GHz. Embedded input and output coplanar waveguide (CPW) feeding ports were introduced on wafer in the test structure. A picture of the FBAR can be seen in Fig. 1. A. Frequency Response and Linear Circuit Elements Since the FBAR nonlinear performance strongly depends on its linear parameters, we first extracted these (or, equivalently, the FBAR equivalent circuit linear elements) from the smallsignal frequency response following the procedure described below. We used an Agilent 8510 network analyzer to obtain the scattering parameters defining the FBAR frequency response from 1 GHz to 2 GHz. An on-wafer thru-reflect-reflect-match (LRRM) calibration set was previously used to calibrate the Fig. 1. FBAR tested in this work with coplanar transmission feed lines. The Area of the 1 µm thick AlN membrane is 50x70 µm. 39 measurement set-up [9]. These measurements were performed for a power delivered by the network analyzer of -10 dBm, which ensured the linear regime of the measured FBAR response. Details of the frequency response, S21 and S11, are outlined in Fig. 2, in solid lines. Note that the frequency response exhibits a resonance frequency at 2.312 GHz and anti-resonance frequency at 2.337 GHz B. Circuit Model The frequency response has been fitted using a modified Butterworth Van Dyke (MBVD) model. As in [8] additional capacitances (Cox) and resistances (Rs and Rsub) have been introduced to consider the effects of the CPW feeding ports. The circuit model is outlined in Fig.3. A commercial software, [10], has been used to de-embed the circuit elements. Table I details the resulting parameters defining the circuit model, where the capacitances are in fF, the inductance in nH and the resistances in ohms. Figure 2 also depicts, in dashed lines, S21 and S11 resulting from the circuit model response, showing very good agreement over the whole frequency range. III. IMD MEASUREMENTS IN FBAR We based the nonlinear characterization of the FBAR on performing intermodulation measurements. Due to the strong IMD produced by the FBAR, instead of using sophisticated setups that we used previously [5], we could use a simple one in which the two sources synthesized at f1 and f2 are then combined to feed the nonlinear device, whose output is then driven to a spectrum analyzer to measure the magnitude of the intermodulation products at 2f1-f2 and 2f2-f1. We systematically performed a set of IMD measurements at both resonance and antiresonance frequencies. In all these measurements we kept the two input tones balanced, that is P1=P2, where P1 and P2 are the powers of the tones at f1 and f2, respectively. The power driving the device was swept from - 10 dBm to 2 dBm and measurements were carried out for ∆ f (being ∆f = f2-f1) of 100 kHz, 50 kHz, 10 kHz, 5 kHz 1 kHz and 100 Hz. Figure 4 shows the measured fundamental and IMD at resonance and antiresonance, respectively, for a tone spacing of 1 kHz. Circles and triangles represent the power at fundamental frequencies f1 and f2, respectively, and diamonds and squares represent the IMD at 2f1-f2 and 2f2-f1, respectively, all as a function of the power driving the FBAR, P1. No IMD C ox C m R sub C ox R sub R s /2 R s /2 C 0 R p R m L m Fig. 3. Equivalent circuit model. TABLE I CIRCUIT PARAMETERS Cm R m L m C o R p R s C ox R sub 10.2 14 464.5 551.1 11 60 438.3 220 Fig. 2. In solid, measured S21 and S11 . In dashed, simulated S21 and S11 modeling the FBAR with the circuit parameter of Fig.3 and Table I. Fig. 4. Measured fundamental and IMD at resonance (top) and antiresonance (bottom) 40 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 12, DECEMBER 2009 Nonlinear Distributed Model for Bulk Acoustic Wave Resonators Carlos Collado, Member, IEEE, Eduard Rocas, Student Member, IEEE, Jordi Mateu, Member, IEEE, Alberto Padilla, Student Member, IEEE, and Juan M. O’Callaghan, Senior Member, IEEE Abstract—This work expands the model proposed by Krimtholz, Leedom, and Matthaei (KLM) model to account for the nonlinear effects occurring in acoustic devices due to the nonlinear stiffened elasticity. We show that a nonlinear distributed capacitance in the acoustic transmission line of the KLM model can account for the distributed nature of the nonlinear effects. Specifically, we use the nonlinear telegrapher’s equation to find closed-form equations for intermodulation distortion and harmonic generation. We confirm the validity of these equations by comparing their results with those provided by a KLM equivalent circuit in which the nonlinear transmission line is implemented by cascading many cells having a voltage-dependent capacitance. To further confirm the model, we show measured nonlinear effects in a thin film bulk acoustic resonator in close agreement with the equivalent circuit simulations. Index Terms—Bulk acoustic wave (BAW), film bulk acoustic resonator, harmonic generation, intermodulation distortion, nonlinear Krimtholz, Leedom, and Matthaei (KLM), nonlinear stiffened elasticity, nonlinearities. I. INTRODUCTION BULK ACOUSTIC WAVE (BAW) technology is capable of producing miniature high resonators, which are essential elements in compact filters having low-insertion loss and high-frequency selectivity. A widespread use of this technology is expected in the ever-growing wireless market, where handheld devices need to accommodate for requirements such as spectrum crowding, high bandwidth demand, miniaturization, and low cost [1]. However, there are still limitations that may exclude the use of BAW resonators in some microwave applications. In particular, their inherent nonlinear behavior [2] may cause intermodulation distortion (IMD), harmonic generation, and detuning and/or saturation of the filter frequency response. Manuscript received December 29, 2008; revised August 26, 2009. First published November 13, 2009; current version published December 09, 2009. This work was supported in part by the Spanish Government (CICYT) under Grant TEC-2006-13248-C04-02/TCM. C. Collado, E. Rocas, A. Padilla, and J. M. O’Callaghan are with the Department of Signal Theory and Communications, Universitat Politècnica de Catalunya (UPC), Barcelona 08034, Spain (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). J. Mateu is with the Department of Signal Theory and Communications, Universitat Politècnica de Catalunya (UPC), Barcelona 08034, Spain and also with the Centre Tecnològic de Telecomunicacions de Catalunya (CTTC), Castelledefels, Barcelona 08860, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2009.2034211 Although the nonlinear effects may be due to several causes, most previous publications point to thermal effects and the nonlinearity in stiffness, piezoelectric coefficient, and permittivity as the dominant ones [3]. Quantifying such nonlinear effects with material parameters such as the nonlinear stiffened elasticity is crucial to fully understand the nonlinear behavior of BAW resonators. On the other hand, equivalent circuits of BAW resonators are needed to predict the nonlinear effects occurring in more complex devices, such as filters with several resonators. A definition of the equivalent circuit elements (such as a voltage-dependent capacity) consistent with the material parameters (such as the nonlinear stiffened elasticity) would be very useful to relate material properties with the final system performance. Tiersten [4] reported on a nonlinear lumped equivalent circuit for rotated Y-cut quartz resonators and its corresponding closed-form expressions to predict the third-order IMD. Thereafter, other works [5]–[9] used lumped approaches based in Tiersten’s equations or in nonlinear versions of the Butterworthvan-Dike equivalent circuit (BVD) [10] in which one or several lumped elements of the acoustic branch (series circuit) are nonlinear. Some of our previous work [11] also uses a phenomenological approach based on the BVD circuit to model the IMD occurring in a film bulk acoustic resonator (FBAR). These different approaches based on lumped element equivalent circuits are simple and useful for modeling some limited manifestations of the nonlinear effects. However, the values of the lumped elements in these equivalent circuits are jointly affected by material parameters and device-specific parameters, such as resonator size and geometry. In these conditions, it is hard to consistently relate the values of equivalent circuit elements to material parameters. A distributed model can be found in [12], in which the authors extend Mason’s linear circuit to the nonlinear region. In this paper, we address this problem by extending our preliminary work in [13], which proposes a distributed nonlinear equivalent circuit based on the lineal model proposed by Krimtholz, Leedom, and Matthaei (KLM) [14]. As stated in [14], the roles of the mechanical and electrical parts of the circuit are more clearly distinguished in the KLM equivalent circuit than in the Mason’s one, which simplifies the nonlinear extension. Our version of equivalent circuit is made by replacing the acoustic line in the KLM model with many cascaded elemental cells. These cells have nonlinear elements that are consistent with the stress–strain curve of the piezoelectric layer in the BAW resonator. With this model we are able to relate the nonlinearities in the equivalent circuit (specifically a voltage-depen47 3020 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 12, DECEMBER 2009 dent distributed capacitance in a transmission line) with nonlinearities in the material (stiffened elasticity). An additional advantage of the model we are proposing is its validity over wide frequency ranges, which makes the model valid for predicting the generation of second harmonics and out-of-band intermodulation products. This is unlike the simpler version of the BVD model, which is limited to a narrow frequency range if no additional acoustic branches are considered. As an example, we use the equivalent circuit to fit measured linear and nonlinear data in a thin film bulk acoustic resonator. Furthermore, we use the equivalent circuit to derive closed-form equations for some of the most relevant nonlinear effects: intermodulation distortion and second harmonic generation. II. NONLINEAR ACOUSTIC TRANSMISSION LINE MODEL As mentioned above, the proposed equivalent circuit is based on an extension of the linear KLM model to account for the distributed nonlinear effects in the acoustic wave. This section reviews the basic concepts of the KLM model and details the additional considerations used to include the nonlinear effects in the model. A. Linear KLM Model The conventional KLM model [14] (see Fig. 1) includes a transmission line that accounts for the usual equivalences in acoustic wave devices (voltage is equivalent to force and current to velocity). The characteristic impedance and phase velocity in this transmission line are given by (1) (2) where and are, respectively, the area of the electrodes and the density of the piezoelectric material. The term represents the stiffened elasticity and is obtained from the constitutive equations describing the stress–strain curve in a piezoelectric material [15]. Its value is equal to the elasticity under constant electric field plus an additional term that depends on the piezoelectric constant and the dielectric constant as (3) The transformation ratio , electrical capacity , and reactance shown in Fig. 1 are given by (4) (5) (6) Fig. 1. Equivalent KLM circuit of a BAW resonator [14]. The acoustic transmission line of length l extends along the x -axis and is connected to an electrical network at x =0 . Ports 1 and 2 are the input and output ports. where is a piezoelectric constant of the crystal and is its thickness. The telegrapher’s equations of the equivalent transmission line [15] for an acoustic wave propagating along the -axis can be written as (7) (8) being the distributed capacitance and the distributed inductance. B. Nonlinear KLM Model Although it is supposed that the elasticity, which relates the stress with the strain at constant electric field, is the main contribution to the nonlinear response [16], there are also other sources that may be considered, such as the piezoelectric constant or the permittivity [3]. These sources may be drawn together in a unique term in (3) which depends on the stress as (9) This function models the piezoelectric stress–strain curve, which for weak nonlinearities can be expanded in a Taylor’s series as (10) where account for the strength of the nonlinear effects. In the KLM model, the voltage is equivalent to the force, which is uniform in a cross section of the acoustic transmission line for the propagating mode of interest. Therefore, we can scale the independent variable in with the area using [15]. We can then denote (11) 48 COLLADO et al.: NONLINEAR DISTRIBUTED MODEL FOR BULK ACOUSTIC WAVE RESONATORS Since, in the equivalent circuit, the distributed capacitance in the transmission line depends on through can be modeled by a nonlinear capacitance (12) where represents the linear term of the distributed capacitance. The additional nonlinear term can also be expanded in a Taylor’s series as , where, for example, the first two terms and can be written as (13) (14) Note that (13) and (14) relate the device-independent nonlinear terms of the piezoelectric constants and with the nonlinear terms of the equivalent circuit and . Using the nonlinear distributed capacitance (12) into the telegrapher’s (7) and (8), we obtain (15) (16) with (17) Equations (15)–(17) describe the nonlinear behavior of an acoustic transmission line and will be used in Section III to derive closed-form expressions that relate nonlinear measured IMD and harmonics with the stiffened elasticity of the piezoelectric layer. Before going into details on the formulation, we summarize the assumptions considered throughout the paper. 1) The equivalent circuit of an elemental section of an acoustic transmission line used in this work is outlined in Fig. 2. For simplicity, we do not consider the series resistance and shunted conductance that would model the losses in an acoustic transmission line [15], however note that those elements could be easily included in the electrical model of Fig. 2. 2) Other sources of nonlinearities beyond those included in the definition of in (3), such as self-heating effects, changes in the density, or nonlinear friction effects are not considered. If necessary, the model could be easily expanded with a nonlinear distributed inductance, series resistance, or shunted conductance. 3) We are not taking into account the nonlinearities arising from nor those arising from and (4)–(6). We assume that the nonlinearity in due to and the changes in and with the stress are negligible compared with those arising inside the resonator due to the nonlinear stiffened elasticity. Fig. 2. Lossless transmission line cell modeling and infinitesimal length section of the acoustic transmission line. III. CLOSED-FORM EXPRESSIONS FOR IMD AND SECOND HARMONIC In this section, we derive the equations for power of the intermodulation distortion and second harmonic signals generated in the nonlinear acoustic line having a voltage-dependent distributed capacitance (12). We assume that the line is driven by two tones at fundamental frequencies being . If both driving tones are at resonance, their spatial distribution will be that of a standing wave pattern (18) (19) where the reference is taken at the center of the resonant line of length (see Fig. 1). For simplicity, only the first two nonlinear terms of , i.e., first-order nonlinearities and second-order nonlinearities will be considered. Note however that the procedure could be extended to consider different nonlinear dependences if necessary [17]. The term causes third-order intermodulation products, occurring at and , referred here as IMD3, whereas the nonlinear term produces second-order intermodulation products, at and , referred here as IMD2, second harmonic at and , and also additional contribution to IMD3 due to the mixing of the second harmonic with the fundamental signals. A. Second-Order Nonlinearities Assuming a quadratic nonlinear dependence of the nonlinear distributed capacitance, (12) becomes (20) We obtain the IMD3 occurring in a nonlinear acoustic transmission line following a similar procedure to that in [17]. Using (20) into (17), we may write the nonlinear differential current as (21) which can be Fourier transformed to obtain the frequency component at . This results in where indicates the Fourier transform at . For the third-order intermodulation (denoted as ), we can write (22) 49 3022 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 12, DECEMBER 2009 Note that this term might be seen as a current generator distributed along the acoustic line and following a spatial distribution given by the standing-wave pattern at resonance affected by the order of the nonlinearities resulting in a term . This nonlinear source (22) gives rise to a voltage , whose spatial distribution should match, except for a multiplicative constant, the one at resonance as in (18). Note that this is true as long as the driving frequencies are both at resonance, i.e., . The voltage can be analytically obtained by equating the power generated at to the sum of the power dissipated in the resonator and loads, and times the net reactive stored energy: . Since at resonance , we can write [17] (23) where is the loaded quality factor. The voltage at can be written as where describes its spatial dependence normalized to a maximum value, and is the magnitude to obtain. Define the normalized stored energy in the resonator as a function of the maximum voltage as (24) and the spatial coefficient as (25) We may substitute (24) and (25) into (23) to obtain (26) Using the assumption , i.e., , the normalized stored energy (24) and the spatial coefficient (25) can be analytically obtained as and , respectively. Then, (26) becomes (27) Note that this formulation requires that and be at resonance. In fact, if we consider the resonance frequency as the frequency where the current (or voltage) is maximum, this frequency is slightly shifted to higher frequencies than the maximum (usually called series resonance) [18]. This is because the acoustic transmission line is loaded by the external components. However, we can assume that both frequencies are close enough to consider and that follows a standing wave pattern as done in (18). By using conventional network analysis (Appendix I), we obtain the dissipated power to the load from the maximum voltage into the acoustic transmission line (27). From this analysis, we have derived the power dissipated at the load for Fig. 3. Left axis: Simulated S (dashed line) and S (dotted line) of an AlNFBAR. Right axis: maximum current reached into the resonator (solid line). TABLE I COMPARISON OF (27) AND (28) WITH SIMULATIONS the case of a two-port FBAR with 50 at the source and load ports (28) where and are the input power at and , respectively, and is the external coupling coefficient defined as the ratio between the dissipated power at one of the external loads and the dissipated power inside the resonator. This coupling coefficient can be calculated from the scattering parameters as , being the maximum in the frequency response of . Equations (27) and (28) have been validated by performing harmonic balance simulations with commercial software [19]. The simulated resonator has 2 m of AlN thickness and an area of 3500 m . The maximum of occurs at 2.7145 GHz (see Fig. 3), which is slightly shifted down from the frequency that gives maximum current ( 2.719 GHz) at the ends of the acoustic transmission line ( and in Fig. 1). The unloaded quality factor is set to 970 by loading the ends of the transmission lines with 0.1-m lumped resistances. The nonlinearities are set to 10 F/(V m) which corresponds in order of magnitude to the nonlinear reported in [22]. The 2m length of the acoustic transmission line is modeled by 160 nonlinear cells like the one in Fig. 2 (minimum 100 cells per half-wavelength are required to obtained accurate results [17]). The resonator is then fed with two 10-dBm tones spaced 1 kHz and centered at 2.7145 GHz. Table I compares the results of the simulations with those from (27) and (28). From Table I, we see that the agreement is very good although the predicted phase of the voltage differs slightly from the simulated one. If simulations are done at the frequency that gives 50 COLLADO et al.: NONLINEAR DISTRIBUTED MODEL FOR BULK ACOUSTIC WAVE RESONATORS Fig. 4. Voltage (left axis) and current (right axis) distribution in a 1-  m length acoustic transmission line from x =0 to l= 2 for the fundamental signals (squares) and for the 2H (circles). Dotted line is the distribution corresponding to cos( x=l ) and sin( x=l ) . maximum voltage ( 2.719 GHz) the phase error is lower than 0.5 . This small error is due to the assumption made in our analytical calculations, which does not fully hold in the simulations. B. First-Order Nonlinearities For the analysis of the 2H and IMD2 generation, we only consider the first term of Taylor’s series expansion of (29) The spurious signals occurring at and by (29) are due to the mixing of the fundamentals at and . On the other hand, the spurious signal at is due to the mixing of the second harmonic at with the fundamental at . 1) Second Harmonic and Generation: To obtain the second harmonic, we need to consider the nonlinear current generators at as (30) Unlike in the case described in Section III-A, the nonlinear sources at (30) are weakly coupled to the second resonant mode. This is because the impedance seen at from the center of the acoustic transmission line ( in Fig. 1) to the electrical part of the circuit is very high so the current flowing through the transformer is very small. This high impedance and the symmetrical distribution of the nonlinear sources along the line, with a maximum at the center of the acoustic line, forces a zero current at as occurring at the fundamental frequency in the middle of a line with shorts at both ends. Fig. 4 shows the simulated distribution (from to ) of the voltage and current at and in the FBAR assessed in Section III-A with and . The spatial pattern of results from the combination between the conditions set by the signal generators distributed along the transmission line (30) and the standing wave pattern TABLE II VALIDATION OF (31), (33), AND (34) WITH SIMULATIONS corresponding to . Therefore, since the current at is almost zero at the center of the line, the generated in a at flows directly through the distributed capacitance (see Fig. 2) generating a voltage drop given by (31) This is the voltage that directly couples to the electrical circuit part producing measurable second harmonic at the load. The same procedure is used for the IMD2 at but replacing the of (30) by (32) which implies that (33) that is, the dissipated power at the load at should be 6 dB greater than that at if the source signals and are kept balanced . The power coupled to the load is obtained by analyzing the linear circuit at (or adding 6 dB) as described in Appendix I, resulting in (34) where is the source and load impedance, is (4) evaluated at , and is given by (48). The normalized energy will be . Expressions (31), (33), and (34) have been validated by performing harmonic balance simulations [19], using the FBAR described in Section III-A, where the nonlinear terms are now and 10 F/(V m). Comparison between simulated results and closed-form expressions are summarized in Table II, showing very good agreement. 2) Third-Order Intermodulation Product :The results of mixing the spurious signal at (and ) with the fundamental signal at (and ) give rise to IMD3 at (and ), both within the resonance band of the FBAR structure. We have checked that no other components like contribute significantly to the IMD3 since the voltages reached inside the resonator at these low frequencies are very low. 51 3024 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 12, DECEMBER 2009 The equation for the third-order intermodulation distortion produced by a first-order nonlinear effect can be derived following steps similar to those used in Section III-A. In this case, the nonlinear sources are (35) where is a standing wave pattern described by a function which resembles a cosine function as shown in Fig. 4. Its maximum value is given by (31). As done in Section III-A, we equate the power generated at with the power dissipated using (23). This results in (36) where and . We can also write this equation as (37) where the factor is defined by (38) The term quantifies the error assuming that the second harmonic follows the standing-wave pattern at , i.e., (see Fig. 4). We have numerically evaluated (38) for several examples being always less than 1.15. Therefore, the error will be within 15% if the approximation is used to avoid numerical procedures. Again we are interested in obtaining the dissipated power to the load. In this case, we can write (see Appendix I) (39) It is remarkable to note that the power of the IMD3 generated by a nonlinear model of first-order (39) can be larger than the measured power 2H or IMD2 (34). For example, the ratio between maximum voltages at IMD3 and 2H results in (40) From (40), we see that the ratio between IMD3 and 2H is proportional to the fundamental signal , the nonlinear coefficient , and the loaded quality factor. This is because, unlike 2H signals, the IMD3 spurious signals are at resonance, so their amplitude is affected by the quality factor of the resonator. Again we use circuit simulations [19] to validate (37) and (39). Table III summarizes the comparison between simulated TABLE III VALIDATION OF (37) AND (39) WITH SIMULATIONS values and the values obtained analytically, showing good agreement. In this case, (38) has also been numerically solved resulting in , therefore . Note than the IMD3 is 10 dB higher than the IMD2 (see Table II) despite the fact that we are only considering nonlinear effects due to first-order nonlinearities. IV. EXPERIMENTAL RESULTS The following section uses the circuit model and the closedform expressions presented in Sections II and III to extract the nonlinear material parameters of a BAW resonator from measurements of the spurious signals at IMD3, IMD2, and 2H. A. Test Device The FBAR resonator tested has a 1m-thick aluminum nitride (AIN) membrane having an area of 4000 m . The electrodes are made of titanium (30 nm thick) and platinum (150 nm thick). All materials are deposited onto a silicon substrate. Fabrication details are given in [20]. The resulting resonant frequency is around 2.3 GHz. B. Scattering Parameters and Linear Circuit Model A preliminary fitting of the linear response (scattering parameters) is required for a proper modeling and characterization of the nonlinear effects occurring in the FBAR presented above. To do that, we measured the scattering parameters from 2 to 6 GHz to include the fundamental resonance and second harmonic frequencies in the analysis. The input power was fixed to 10 dBm to ensure that the FBAR is operating in linear regime, and we performed on-wafer line-reflect-reflect-match calibrations [21]. Fig. 5 depicts the frequency response of the transmission and reflection coefficient , for a narrow frequency range (from 2.2 to 2.4 GHz). The effective electromechanical coupling coefficient is 2.2% and the loaded quality factor is at the resonance (maximum ), and at the antiresonance (minimum ). This low causes high insertion losses which are incremented by the parasitic effects of the pads [20]. Fig. 6 shows the proposed linear model containing parasitic elements to account for the substrate and pads [20]. The FBAR is modeled using the KLM model (Fig. 1) and the acoustic transmission line is divided into 160 cells containing the distributed linear and nonlinear parameters, like that in Fig. 2. The acoustic line is loaded with two transmission line sections at each side modeling the electrodes. These two transmission lines have a significant impact on the FBARs resonance frequency but we will consider that these electrodes do not contribute to the nonlinearities since the titanium and the platinum have a much more linear strain–stress curve than the AlN [22]. For simplicity, each 52 COLLADO et al.: NONLINEAR DISTRIBUTED MODEL FOR BULK ACOUSTIC WAVE RESONATORS Fig. 5. Simulated (dotted) and measured (solid line) scattering parameters. Fig. 6. KLM equivalent circuit including electrodes and parasitic elements ( R , C , and R ). 180-nm Ti/Pt electrode is modeled as a unique linear transmission line with 3440 m/s and 0.207 , which is additionally loaded by a lumped resistor whose value is higher than that of the air impedance, and accounts for acoustic losses. The values of the circuital components are adjusted by using optimization routines to fit the measured scattering parameters especially at frequencies close to resonance. Fig. 5 shows the frequency response of the simulated circuit model of Fig. 6 and the measurements. The acoustic losses are modeled with 4 10 at the ends of the electrodes and the values of the parasitic elements are 9.8 , 282 and 9 pF. These parasitic elements are consistent with those extracted from other resonators on the same wafer [11]. C. Nonlinear Characterization To characterize the nonlinear response of our device under test (Fig. 6), we performed intermodulation and second harmonic measurements. For these experiments, we kept both sources balanced in power (i.e., ) and swept their powers from 10 to 10 dBm. Both tones and are set at resonance with a frequency spacing Fig. 7. Measured output power at resonance frequency versus input power for: fundamental power (circles), IMD3 (squares), IMD2 (diamonds), and 2H (triangles). The solid lines represent the simulated results of Section IV-D. Fig. 8. Normalized current (dotted) and voltage (solid) distribution along the acoustic transmission line: piezoelectric layer and electrode (see Fig. 6). ranging from 100 Hz to 1 MHz. From these experiments, we obtain the output power of IMD2 , IMD3 ( and ), and 2H ( and )as a function of the input power of the fundamental signal. The dependence of the spurious signals on the fundamental tone frequency spacing is also evaluated. Fig. 7 outlines the measured spurious and fundamental signals, when the two fundamental tones are centered at 2.292 GHz (resonance frequency) and 100 Hz apart. The slope of the fundamental tones is 1 : 1 in log–log scale up to 8 dBm, indicating that saturation effects do not occur. As one may expect from (28) or (39), and (34), the measured power dependence of IMD3 is 3 : 1 (in log–log scale), and 2 : 1 (in log–log scale) for 2H and IMD2. Next, we extract the nonlinear capacitance that fits the measurements. We first extract from the measured 2H and IMD2 experiments using (34); we evaluate if the IMD3 is consistent with this using (39), and if not, we obtain using (28). The parasitic elements will be considered as part of the resonator in order to apply the formulation based on the analysis of 53 3026 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 12, DECEMBER 2009 resonators for the third-order intermodulation. See Appendix II for further details. 1) Second Harmonic 2H and IMD2: We use (34) to obtain the value of that better fits the 2H dissipated power at the external load 50 . To do that, we first obtain: and using (4) and (51), respectively, from the fitted parameters, 6.44 10 F/m from the properties of the material, using [24], and using [25], where is the phase of . To calculate the normalized energy , we need to know the standing wave pattern in the acoustic transmission line for the fundamental frequencies. Fig. 8 shows this distribution for the voltage and current along the acoustic line and the electrodes. Since the distributions of Fig. 8 do not follow a cosine function, we need evaluate (24) numerically to obtain (see Appendix II for more details). This results in , that is, a value 54% larger than the value we would obtain if we would consider an acoustic transmission line of total length , where is the thickness of piezoelectric plus electrodes. Using (34), we find a value of that fits a measured point of Fig. 7. We obtain 1.1 10 F/(Vm). This value also fits the measured IMD2 since the measured value is 6 dB larger than . 2) IMD3 From First Order Nonlinearities :Applying the value of 1.1 10 F/(Vm) obtained above into (39) and using , we obtain an IMD3 output power of 120 dBm, which is much smaller than the measured value. This indicates that the IMD3 cannot be only due to and there must be an additional contribution due to . 3) IMD3 From Second-Order Nonlinearities :Now we use (28) to extract from the measured IMD3. In this case, the geometrical factor is calculated using (25) from the voltage shown in Fig. 8. Only nonlinear contributions from the piezoelectric layer are considered, since we assume that the electrodes do not contribute to the nonlinearities. This results in , where is the piezoelectric thickness. Doing so we obtain 2.5 10 F/(V m), and the resulting nonlinear capacitance is . D. CAD-Based Nonlinear Characterization To validate the data obtained from closed-form expressions, we have found by fitting the measured data to computer-aided design (CAD) simulations of the equivalent circuit [19]. These simulations are not subject to the simplifying assumptions made in the derivation of (28), (34), and (39). Furthermore, they can easily account for changes in the equivalent circuit due to electrodes and other extraneous elements. For example, the equivalent circuit may be easily modified from that in Fig. 1 to the one in Fig. 6 to include the effect of the electrodes. Using the same input power than in Section IV-C as a fitting point, we obtain , which is within a 8% error in both and . This agreement indicates that (28) and (34) are quite general as long as the parasitic effects and electrodes are considered when evaluating some terms used in these equation (see details in Appendix II). Fig. 7 Fig. 9. IMD3 and 2H output power as a function of the tone spacing 1 f for an input power of + 5.5 dBm. Triangles correspond to 2 ! 0 ! , circles to 2 ! 0 ! , squares to 2 ! , and stars to 2 ! . shows the measured and simulated results with these extracted nonlinear parameters versus input power. V. DISCUSSION AND FURTHER EXPERIMENTS The nonlinear coefficients extracted from the measurements and can then be related with the material nonlinear parameter through (13) and (14). From (13), we obtain 1.1 10 F/(Vm) and the value of the first-order nonlinear elastic stiffness . This value is six times greater than that reported in [22], which was obtained from the simulations of mechanical displacements. Using (14) with 2.3 10 F/(V m), we obtain the second-order nonlinear elastic stiffness 2.1 10 N m . This value is three orders of magnitude greater than the one reported in [22]. In fact the value reported in [22] would not be measurable doing IMD experiments since the IMD3 generated by would be greater than that generated by such small . The difference between measured and reported results suggests the existence of other causes for the second-order nonlinear effects. In order to further investigate the nonlinear effects, we performed additional IMD and 2H measurements versus the frequency spacing between the fundamental tones . The dependence of IMD power level on the frequency spacing between tones is an indication of dependence with the period of the envelope, and may give additional information to discern between different sources of nonlinear effects. Fig. 9 depicts the IMD3, IMD2, and 2H as a function of the frequency spacing for an input power of 5.5 dBm. While 2H and IMD2 show to be independent of the frequency spacing, that is a constant value of , IMD3 decreases when the frequency spacing increases, which would result in depending on the frequency spacing between the input tones. This later effect is not considered in the circuit model presented in this paper. The observed IMD3 dependence on the frequency spacing between the fundamental tones and the difference in the extracted quadratic nonlinear stiffness is an added indication that 54 COLLADO et al.: NONLINEAR DISTRIBUTED MODEL FOR BULK ACOUSTIC WAVE RESONATORS additional sources may contribute to the second-order nonlinear circuit term such as, for example, thermal effects. In addition, Fig. 9 shows asymmetries between the lower and upper (and ) intermodulation products. This behavior is also characteristic of self-heating effect as shown in [23] which is consistent with the low quality factor of the device. Note that as pointed out before, the stiffened elasticity (10) used in this model fails to predict these thermal effects. VI. CONCLUSION In this paper, we present a novel nonlinear distributed equivalent circuit based on the KLM model. Our new model is useful to account for the linear and nonlinear behavior of BAW resonators in a broad frequency band. We have used it to obtain closed-form expressions for the most relevant nonlinear effects: intermodulation distortion and second harmonic generation. We have checked these expressions with equivalent circuit simulations using harmonic balance techniques. We have shown that the new equivalent circuit can be easily modified to model the effects of the electrodes and other parasitic elements existing in FBAR. With this modified version of the equivalent circuit, we are able to fit the linear and nonlinear measurements in an FBAR (i.e., s-parameters, intermodulation distortion, and second harmonic generation). The measurements can be fitted to the closed-form expressions for intermodulation distortion and second harmonic generation derived in the paper. Once the model is fitted to the measurements, we can extract the nonlinear material parameters. The value obtained for is consistent in order of magnitude with the ones reported in the literature, whereas the gives a larger value than the ones reported. We suggest that this disagreement on is because there are other nonlinear effects contributing to the IMD3 generation such as self-heating mechanisms. The work presented in this paper may now be used in the following aspects. 1) The equivalent circuit could also be used as a basic building block to model more complex devices, such as filters containing several resonators. 2) Evaluation of additional nonlinear effects, such as saturation or detuning, which may also be performed by simulating the equivalent circuit proposed. 3) Inclusion of other nonlinear sources in the model. For example, a nonlinear viscosity could be considered adding a shunted nonlinear distributed conductance to the nonlinear transmission line in the model. 4) Extension of the model to consider self-heating mechanisms. Although the proposed circuit model has been particularly developed for FBARs with longitudinal propagating wave, this can be generally used for modeling the nonlinear performance in other BAW devices using other propagating modes, for example, quartz crystal resonator operating at the shear mode or even surface acoustic wave resonators, being therefore useful in other applications beyond the scope of this paper. APPENDIX I IMD AND 2H POWER COUPLED TO THE LOAD The power coupled to the load connected at port 2 (Fig. 1) at , , can be written as (41) where [24] and is the dissipated power into the resonator. Then, (41) can be written as a function of the unloaded quality factor and the normalized stored energy as (42) The dissipated power into the resonator at the fundamental frequency is given by (43) where is the incident power to the resonator. The maximum voltage at results in (44) For second-order nonlinearities , replacing (44) into (26) and the resulting into (42), the IMD3 load power gives (45) The IMD3 coupled to the load for first-order nonlinearities is calculated using (44), (42), and (36) (46) Note that this formulation is based on the definition of the quality factor and the coupling coefficient and it does not depend on the origin of the losses. Therefore, it can be used to account for losses modeled as lumped resistances at the ends of the acoustic transmission line, or for acoustic distributed losses which could be modeled adding a series distributed resistance and/or a shunted distributed conductance in the elemental segment of Fig. 2. The maximum voltage at the center of the acoustic transmission line is coupled to the electrical part of the circuit through the transformer 1: (see Fig. 1) resulting in a electrical voltage , thus the current at flowing to the source impedance and load will be (47) where corresponds to (4) evaluated at and (48) 55 3028 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 12, DECEMBER 2009 with obtained from (6) at . Therefore, the dissipated power at the load will be (49) which, by using (49), (44), and (31), can be written as (50) APPENDIX II PARASITIC AND ELECTRODE EFFECTS The parasitic elements may be included as part of the resonator, by using as a quality factor the one obtained from measurements instead of the inherent quality factor of the resonator. The term accounts for the acoustic power dissipated in the resonator plus the power dissipated in the parasitic resistances. Note that this does not affect (45) and (46) since the parasitic elements are included in the coupling coefficients and quality factor of the whole device. However, for the second harmonic power calculation, we need to consider the parasitic elements in the circuit analysis. The electrical voltage causes a current flowing to the load impedance following (47), although in this case, the impedance does not follow (48) and it is obtained by conventional circuit analysis of the electrical part of the device including the electrodes (see Fig. 6). This results in (51) where . In addition, the electrodes may have a significant impact in the spatial distribution of the voltage and current inside the resonator as shown in Fig. 8, so these spatial current and voltage distributions must be considered for the calculation of the normalized energy . The electrical energy corresponding to (25) for an ideal acoustic transmission line will be in this case (52) where and are the distributed conductance, normalized voltage distribution and layer thickness, respectively, and the sub index “ ” and “ ” denote piezoelectric layer and electrode, respectively. The normalized magnetic energy will be (53) where indicates the normalized spatial current distribution. is the ratio of maximum voltage and maximum current in a acoustic line loaded with electrodes. This constant can be calculated using conventional microwave analysis of the loaded transmission lines (54) where and are the characteristic impedance and electrical length, respectively. The normalized energy will be then (55) ACKNOWLEDGMENT The authors would like to thank H. Campanella from the Centre National de Microelectrónica (CNM-CSIC) for providing the FBAR used in this work. REFERENCES [1] K. M. Lakin, “Thin film resonator technology,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 52, no. 5, pp. 707–716, May 2005. [2] M. Planat and D. Hauden, “Nonlinear properties of bulk and surface acoustic waves in piezoelectric crystals,” Ferroelectrics, vol. 42, pp. 117–136, 1982. [3] D. A. Hall, “Nonlinearity in piezoelectric ceramics,” J. Mater. Sci., vol. 36, pp. 4575–4601, 2001. [4] H. F. Tiersten, “Analysis of intermodulation in rotated Y-cut quartz thickness-shear resonators,” in Proc. 28th Annu. Symp. Freq. Control, 1974, pp. 1–4. [5] J. Nosek and L. Burianova, “About nonlinear effects in the quartz homeotypes single crystals,” in Proc. IEEE Freq. Control Symp., May 2008, pp. 586–591. [6] J. Nosek, “A precise measurement of some nonlinear effects and its application to the evaluation of nonlinear elastic constants of quartz and GaPO4,” Rev. Sci. Instrum., vol. 68, no. 8, pp. 3143–3149, Aug. 1997. [7] L. Dworsky and R. G. Kinsman, “A simple single model for quartz crystal resonator low level drive sensitivity and monolithic filter intermodulation,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 41, no. 2, pp. 261–268, Mar. 1994. [8] R. Aigner, N.-H. Huynh, M. Handtmann, and S. Marksteiner, “Behavior of BAW devices at high power levels,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 12–17. [9] M. Ueda, M. Iwaki, T. Nishihara, Y. Satoh, and K.-Y. Hashimoto, “A circuit model for nonlinear simulation of radio-frequency filters using bulk acoustic wave resonators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 55, no. 4, pp. 849–856, Apr. 2008. [10] K. S. Van Dyke, “The piezo-electric resonator and its equivalent network,” Proc. IRE, vol. 16, no. 6, pp. 742–764, Jun. 1928. [11] E. Rocas, C. Collado, J. Mateu, H. Campanella, and J. M. O’Callaghan, “Third order intermodulation distortion in film bulk acoustic resonators at resonance and antiresonance,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2008, pp. 1259–1262. [12] Y. Cho and J. Wakita, “Nonlinear equivalent circuits of acoustic devices,” in Proc. IEEE Ultrason. Symp., Nov. 1993, vol. 2, pp. 867–872. [13] E. Rocas, C. Collado, A. Padilla, J. Mateu, and J. M. O’Callaghan, “Nonlinear distributed model for IMD prediction in BAW resonators,” in Proc. IEEE Int. Ultrason. Symp., Nov. 2008, pp. 1557–1560. [14] R. Krimholtz, D. A. Leedom, and G. L. Matthaei, “New equivalent circuits for elementary piezoelectric transducers,” Electron. Lett., vol. 6, no. 13, pp. 398–399, Jun. 1970. [15] B. A. Auld, Acoustic Fields and Waves in Solids. Malabar, FL: Krieger, 1990, vol. I. [16] J. Enlund, V. Yantchev, and I. Katardjiev, “4E-6 electric field sensitivity of thin film resonators based on piezoelectric AlN thin films,” in Proc. IEEE Int. Ultrason. Symp., Oct. 2006, pp. 468–471. [17] C. Collado, J. Mateu, and J. M. O’Callaghan, “Analysis and simulation of the effects of distributed nonlinearities in microwave superconducting devices,” IEEE Trans. Appl. Supercond., vol. 15, no. 1, pp. 26–39, Mar. 2005. [18] T. F. Hueter and R. H. Bolt, Sonics: Techniques for the Use of Sound and Ultrasound in Engineering and Science. New York: Wiley, 1955. 56 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. COLLADO et al.: FIRST-ORDER ELASTIC NONLINEARITIES OF BAW RESONATORS Fig. 5. Power P delivered to port 4 (Fig. 4) by use of (18) (continuous line) at 2 ! (blue line in online version) and ! + ! (red line in online version) and nonlinear simulations (squares and circles, respectively). Step 2) Determination of a linear circuital model of the DUT by measuring its reflection coefficient. Step 3) Measurement of H2 and IMD2 over a broad frequency, keeping a constant difference between the two driving frequencies. Step 4) De-embedding the stray effects produced by the measurement setup. Step 5) Determination of the value of that best better fits the measurement and its corresponding value of the geometry-independent parameter . Step 1. Characterization of the Measurement Setup: The measurement setup, with a 90 broadband coupler (2–8 GHz), is shown in Fig. 2. The measurement of the 4 4 scattering matrix of the whole setup is the first step in our characterization process. Fig. 6 shows the scattering parameters of the setup. The insertion loss from the DUT to the spectrum analyzer ranges between 4–5 dB, and the return loss at the port, where the device is connected , is around 10 dB at frequencies around the second harmonic of the fundamental frequencies. This poor VSWR is due to reflections at the output of the low-pass filters in spite of having an isolator between the filter and the power combiner. Step 2. Linear Measurements of the DUT: The linear parameters of a DUT were measured with on-wafer thru-reflect-line calibrations [11]. The measured reflection coefficient was fit with a linear model (Fig. 1) by means of information, supplied by the manufacturers, on the electrodes and Bragg mirror. For example, we show the measured imaginary part of the reflection coefficient and circuit simulations of device A1 around the operation frequencies in Fig. 7. The inset in Fig. 7 shows its broadband response and simulations. As shown in Fig. 7, where we only show the imaginary part of the reflection coefficient for simplicity, the KLM model agrees very well with the response of the resonator. Note that a better fit of the linear model implies a more accurate prediction of the voltage inside the acoustic transmission line model at the fundamental frequencies, which is responsible for the harmonic generation. Step 3. H2 and IMD2 Measurements: The DUT is driven by two frequency tones and , Fig. 6. S -parameters of the setup for numbering in Fig. 4. (a) S (continuous black line), S (dotted blue line in online version), S (dashed green line in online version), and S (dashed red line in online version). (b) S (dotted black line), S (dashed red line in online version), S (black line), and S (green line in online version). Fig. 7. Imaginary part of the reflection coefficient from device A1 measured (black continuous line) and simulated (blue dashed line in online version). The minimums correspond to the characteristic resonance frequencies (series and shunt). The inset shows the broadband response. where kHz and the frequency is swept from 1.88 to 1.98 GHz while keeping constant. The input power to the device [Port 3 in Fig. 4] is 20 dBm at each fundamental frequency. The reflected signal and spurious signals generated inside the resonator are directed by the 90 coupler to the spectrum 63 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Fig. 8. Output power at the fundamental tones (blue and red line in online version), H2 (cyan line in online version and black line), and IMD2 (red line in online version). Horizontal axis shows the f values at each measurement point. analyzer. Before connecting the DUT, we measure an on-wafer open stub in order to set the levels of intermodulation and harmonics generated by the measurement setup and we make sure these are negligible throughout the whole measurement process. Fig. 8 shows the measured second harmonics and IMD2 for device A1. Continuous lines show nonlinear circuit simulations, such as those explained in Section III-C, but including the measured -parameters of the setup from Step 1 and the linear circuit model from Step 2. With a unitless value of , measurements and nonlinear simulations overlap. The output power at the fundamental tones shows a small ripple due to the imperfections of the measurement setup. This ripple is more significant at higher frequencies (H2 and IMD2) due to the poor mismatch at port 3, as previously discussed. Fig. 8 reveals the advantage of performing broadband measurements instead of single-frequency point measurements. For instance, if single-frequency measurements (as done in our previous work [12]) were performed at a frequency close to the series resonance (first dip in Fig. 7), erroneous results could be obtained because the nonlinear behavior is affected by spurious resonances. Therefore, broadband measurements enable us to avoid the issue of anomalous nonlinear responses arising from spurious modes. To this point, we have made use of nonlinear circuit simulations to consider the influence of the measurement setup on the experimental observables. In the next steps, we show how the nonlinear stiffened elasticity can be obtained from the measurements without having to use a nonlinear circuit simulator. Step 4. De-Embed the Effect of the Setup: As shown above, the effect of the imperfections of the measurement setup on the second-harmonic measurements is significant. We can, however, de-embed this effect by means of conventional microwave circuit theory. To completely remove this effect, we need to determine the output power that the device would deliver to a perfectly matched 90 coupler versus the input power to the device . We can then define the normalized magnitude that is independent of the measurement setup. Fig. 9. Left vertical axis: normalized measured H2 (squares for 2 ! and circles for 2 ! ), measured IMD2 (stars), simulated H2 (black continuous line) and simulated IMD2 (red continuous line in online version). Right vertical axis: Maximum voltage at the fundamental frequencies (green line in online version). The input power to the device, at each fundamental frequency (and ), is calculated with (17) where we follow the definitions in Fig. 4. The power , as a function of the measured power , at port 4 of Fig. 4 can be calculated by use of the transducer power gain [13] from port 3 to port 4 of Fig. 4. We then obtain (18) By use of (16)–(18), we can normalize versus , which has again been checked with simulations. Fig. 9 plots the simulated (continuous line) and measured (circles) normalized power for the device A1. Fig. 9 also plots the square voltage at the center of the line normalized to the input power for the fundamental signal obtained in the simulations. As expected, the maximum is reached at the mechanical resonance frequency of the resonator at which the stored energy is higher. From Fig. 9, we see that the measurements of the H2 and IMD2 data follow the same frequency pattern as . The data are therefore consistent with the initial hypothesis, which proposes a nonlinear stress-dependent parameter that is the origin of the second-harmonic generation. Step 5 Find That Better Fits the Measurement: In this section, we determine the best fit nonlinear parameters from the normalized measured data for each device with (10)–(18), and then, with these values, we extract the intrinsic parameter with (7). Fig. 10(a) and (b) shows the results for resonators A and B, respectively, and Table I shows the best fit values of . As shown in Table I, a value of the nonlinear first-order stiffened elasticity around 10.5 is in reasonable agreement for all the A resonators despite having different areas and shape (furthermore, unlike other samples, A3 has a border ring to eliminate 64 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. COLLADO et al.: FIRST-ORDER ELASTIC NONLINEARITIES OF BAW RESONATORS Fig. 10. Normalized H2 power. (a) A-devices: A1 (blue triangles in online version), A2 (green squares in online version), A3 (black diamonds), A4 (red circles in online version), A5 (magenta stars in online version). (b) B-devices: B1 (red circles in online version), B2 (blue triangles in online version), B3 (black diamonds). TABLE I EXTRACTED 1 C spurious modes), and 13.0 is in reasonable agreement for all the B resonators. A systematic error is found between the values obtained with the formulation and the values obtained by harmonic balance simulations. The derivation of (10) and (11) assumes that the acoustic transmission line is symmetrical with respect to (see Fig. 1) or, in other words, the electrodes connected at both ends of the transmission line are electrically equal [5]. This does not usually happen in BAW resonators with a Bragg mirror so the applicability of (10) and (11) is questionable in this type of device. Nevertheless, we have performed several simulations, sweeping the frequencies around the mechanical resonance frequency, with different electrodes, and the maximum error we Fig. 11. Comparison between (10) and simulations for the B2-device (continuous line) and the A4-device (dotted line). Horizontal axis represents the frequency variation with respect to the mechanical resonance frequency (2.124 GHz for B2 and 1.958 GHz for A4). have found is around 20%. For example, we have checked by means of simulations that (10) overestimates by a maximum 16% over the measured frequency range for the A devices and by 14% for the B devices, as shown in Fig. 11. These differences cause a mismatch smaller than 1.5 dB on the load power and agree well with the differences in the values of Table I. If the asymmetry of the transmission line were considered, the formulation and the simulations would be in an excellent agreement. This process would, however, require prior knowledge of the standing-wave pattern for the fundamentals and H2 signals, which implies nonlinear simulations, to achieve accuracy better than 20%. V. DISCUSSION AND CONCLUSIONS We have presented a systematic step-by-step broadband approach for the characterization of the first-order nonlinearities in BAW resonators. We have shown that broadband measurements are advantageous for avoiding the anomalous nonlinear behavior due to spurious resonances, instead of single-point frequency measurements especially close to the series resonance frequency. We have found consistent values of stress-dependent nonlinearity in the stiffened elasticity of the piezoelectric material (AlN) of a set of resonators having widely different geometries, configurations, and delivered by two different manufacturers. We have determined values of within a range 10.7 0.3 for the five samples of one manufacturer and 13.1 0.3 for the three samples of the other manufacturer. This consistency in the values of proves that the first-order stress-dependent nonlinearity in the stiffened elasticity of the AlN piezoelectric is the origin of second harmonics and IMD2 products in BAW resonators. Moreover, both values are quite close, despite the AlN layers having been fabricated following different processes. Purely mechanical hydrostatic pressure measurements of AlN, reported in [14], give a value of the first-order nonlinear elasticity , which is of the same order of magnitude as the values of obtained in this study. This shows that could be a significant contributor to the nonlinear term . 65 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES We have used two alternative approaches to arrive at these conclusions, one based on simulation, and another based on closed-form equations that take into account the model of the BAW resonators and the effects of the measurement setup. We have checked for consistency between both approaches and have found that they diverge by at most 16% due to the simplifying assumptions taken in the closed-form approach. Despite these differences, the formulation gives a reasonable value of the nonlinear stiffened elasticity that is close to the uncertainties, around 0.5 dB, of the harmonic and intermodulation measurements. This first-order nonlinearity is known to partially contribute to other spurious signals, such as third-order intermodulation products, and thus it sets the baseline level of spurious signals generated within the BAW resonator. REFERENCES [1] F. Z. Bi and B. O. Barber, “Bulk acoustic wave RF technology,” IEEE Microw. Mag., vol. 9, no. 5, pp. 65–80, Oct. 2008. [2] D. A. Hall, “Nonlinearity in piezoelectric ceramics,” J. Mater. Sci., vol. 36, no. 19, pp. 4575–4601, 2001. [3] E. Rocas, C. Collado, J. C. Booth, E. Iborra, and R. Aigner, “Unified model for bulk acoustic wave resonators’ nonlinear effects,” in Proc. IEEE Int. Ultrason. Symp., Nov. 2009, pp. 880–884. [4] J. Blackburn and M. Cain, “Nonlinear piezoelectric resonance: A theoretically rigorous approach to constant I/V measurements,” J. Appl. Phys., vol. 100, no. 11, pp. 1–10, Dec. 2006, 114101. [5] C. Collado, E. Rocas, J. Mateu, A. Padilla, and J. M. O’Callaghan, “Nonlinear distributed model for bulk acoustic wave resonators,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 12, pp. 3019–3029, Dec. 2009. [6] B. A. Auld, Acoustic Fields and Waves in Solids. Malabar, FL: Krieger, 1990, vol. I. [7] K. M. Lakin, G. R. Kline, and K. T. McCarron, “HighQ microwave acoustic resonators and filters,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 12, pp. 2139–2146, Dec. 1993. [8] C. Collado, J. Mateu, and J. M. O’Callaghan, “Analysis and simulation of the effects of distributed nonlinearities in microwave superconducting devices,” IEEE Trans. Appl. Supercond., vol. 15, no. 1, pp. 26–39, Mar. 2005. [9] R. Krimholtz, D. A. Leedom, and G. L. Matthaei, “New equivalent circuits for elementary piezoelectric transducers,” Electron. Lett., vol. 6, no. 13, pp. 398–399, Jun. 1970. [10] Advanced Design System (ADS). Agilent Technol., Santa Clara, CA, 2007. [11] G. F. Engen and C. A. Hoer, “Thru-reflect-line: An improved technique for calibrating the dual six-port automatic network analyzer,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 12, pp. 987–993, Dec. 1979. [12] E. Rocas, C. Collado, A. Padilla, J. Mateu, and J. M. O’Callaghan, “Nonlinear distributed model for IMD prediction in BAW resonators,” in IEEE Ultrason. Symp., Nov. 2008, pp. 1557–1560. [13] D. Pozar, Microwave Engineering. New York: Wiley, 1998. [14] S. P. Łepkowski and G. Jurczak, “Nonlinear elasticity in III-N compounds: Ab initio calculations,” Phys. Rev. B, Condens. Matter, vol. 72, no. 24, pp. 1–12, Dec. 2005, Art. ID 245201. Carlos Collado (A’02–M’03–SM’10), photograph and biography not available at time of publication. Eduard Rocas (S’06), photograph and biography not available at time of publication. Albert Padilla, photograph and biography not available at time of publication. Jordi Mateu (A’02–M’04–SM’10), photograph and biography not available at time of publication. Juan M. O’Callaghan (SM’01), photograph and biography not available at time of publication. Nathan D. Orloff, photograph and biography not available at time of publication. James C. Booth, photograph and biography not available at time of publication. Enrique Iborra, photograph and biography not available at time of publication. Robert Aigner, photograph and biography not available at time of publication. 66 CHAPTER III - ELECTRO-THERMO-MECHANICAL NONLINEARITIES Thermal considerations are becoming more and more important in today’s microwave components due to increasing requirements for higher power handling. With high temperatures and high dynamic thermal oscillations, temperature-dependent material properties originate nonlinearities that can limit the devices performance. This chapter covers the interaction of the electromagnetic, mechanical and thermal fields and how this impacts the nonlinear behavior of transmission lines and BAW resonators. Both physical and phenomenological models are proposed, which can be used to predict the nonlinear indicators at the design stage. The first and second articles study the mechanism by which third-order intermodulation distortion is generated even in transmission lines made of materials whose properties do not depend on the electromagnetic fields. The third article assesses the impact of self-heating on the performance of limiters made of HTS transmission lines. Appendix A covers the nonlinearities in ferroelectric (Ba0.3Sr0.7TiO3) thin films, where the measurable observables are used to obtain a large-signal model of the transmission lines. The fifth article and Appendix B propose models to explain the electro-thermomechanical nonlinearities in BAW resonators from a phenomenological approach and a physically rigorous approach respectively. The articles, including Appendix A and B, in Chapter 3 are: 67  E. Rocas, C. Collado, N. Orloff, J. C. Booth, “Third-order intermodulation distortion due to self-heating in gold coplanar waveguides”, 2010 IEEE MTT-S International Microwave Symposium Digest, pp. 425-428, 23-28 May 2010  E. Rocas, C. Collado, N. D. Orloff, J. Mateu, A. Padilla, J. M. O'Callaghan, J. C. Booth, “Passive Intermodulation Due to Self-Heating in Printed Transmission Lines”, IEEE Transactions on Microwave Theory and Techniques, vol. 59, no. 2, pp. 311-322, Feb. 2011  E. Rocas, C. Collado, J. Mateu, N. Orloff, J. C. Booth, J. C., “Modeling of Self-Heating Mechanism in the Design of Superconducting Limiters”, IEEE Transactions on Applied Superconductivity, Accepted for publication and available online at IEEExplore.org as Early Access.  E. Rocas, C. Collado, J. C. Booth, E. Iborra, R. Aigner, “Unified model for Bulk Acoustic Wave resonators' nonlinear effects”, 2009 IEEE International Ultrasonics Symposium, pp. 880-884, 20-23 Sept. 2009 A. E. Rocas, C. Collado, J. Mateu, N. Orloff, J. M. O’Callaghan, J. C. Booth, “A large-signal model of ferroelectric thin-film transmission lines”, Submitted to IEEE Transactions on Microwave Theory and Techniques, March 2011 B. E. Rocas, C. Collado, J. Mateu, N. D. Orloff, R. Aigner, J. C. Booth, “Electrothermo-dynamic model for bulk acoustic wave resonators”, To be submitted to IEEE Ultrasonics, Ferroelectrics and Frequency Control, March 2011. 68 Third-Order Intermodulation Distortion due to Self-heating in Gold Coplanar Waveguides Eduard Rocas!.2, Carlos Collado!.2, Nathan Orloft, James C. Booth2 IUniversitat Politecnica de Catalunya, Barcelona, CAT, 08034, Spain 2National Institute of Standards and Technology, Boulder, CO, 80305, USA Abstract -We present measurements and modeling of a self heating mechanism responsible for third-order intermodulation distortion in coplanar waveguide transmission lines. Temperature variations, at the envelope frequency of the input signal, induce dynamic changes in the distributed resistance that, when mixed with the fundamental tones, give rise to intermodulation. Index Terms -Coplanar waveguides, nonlinearities, electro thermal effects, intermodulation distortion. I. INTRODUCTION The existence of a self-heating mechanism that induces third-order intermodulation distortion (IMD3) is a well-known process in power amplifiers, on which successful efforts have been done on its modeling [1]-[4]. In contrast, passive microwave devices have received little attention, where few lumped models exist for discrete components such as connectors, attenuators and terminations [5]. For purposes of characterization and prediction, a nonlinear distributed model of the transmission lines with self-heating mechanism involved is required, in which the experimental observables can be related with the material properties and their temperature derivatives. In this work, we present a distributed model to account for these nonlinearities induced by self-heating in transmission lines. The model introduces a thermal domain that simulates the heat flow along the transmission line and through the substrate, and is coupled to the electromagnetic domain. The model was tested by use of coplanar waveguides (CPW) made of gold on sapphire, and showed good agreement with measured results on transmission lines of different lengths. II. UNIFIED MODEL Despite temperature-induced changes in the material properties, models usually, for simplicity, make use of only the steady-state ambient temperature to predict the device's performance, without taking into account the dynamic temperature fluctuations [6][7]. In fact, as presented in this work, time-dependent temperature fluctuations can lead to undesired nonlinear effects that must be consistently modeled by an electro-thermal model. A. Generation Mechanism A scheme showing the generation process of IMD3 due to self-heating is shown in Fig. 1, in which the two-tones test represents a useful method in nonlinear analysis. A two-tone signal (h and j) can be interpreted as a sinusoidal modulated signal at a frequency �f=f2-f, centered at fo=f,+�fl2= f2-�fl2. In the CPWs under study, dissipation is largely due to conductor losses for a substrate with negligible loss, such as sapphire. In such a case, the quadratic relation between instantaneous dissipated power and current implies that heat fluctuations are generated at frequencies �f, 2f" f,+f2 and 2f2' However, the relation between dissipation (Pif)) and temperature (T(f)) [8], (I) states that only variations at the envelope frequency ,1f will produce substantial temperature changes due to the low-pass filter behavior of the thermal impedance Zif) that is related to the slow nature of thermal dynamics [8]. �" �" �'" I�rt I'rt '" I 2f11112f2 Temperature .I�I. Dissipation Zth(f) EM domain TH domain Fig. I. Process by which IMD3 is generated in a transmission line. Resistive losses are responsible for temperature fluctuations that in tum change the metal resistivity, giving rise to a dynamically generated nonlinearity. As seen in Fig. 1, temperature fluctuations change the metal resistivity P(T) and, therefore, the distributed resistance RlT) of the line. In fact, the temperature-induced changes can also be treated as an amplitude modulation of the input signals h and/;, by a modulating signal at frequency M, which gives rise 978-1-4244-7732-6/101$26.00 ©:a<Wb�rtialIy supported by U.S G�ment, not subject to U.S. copyright IMS 2010 69 to IMD3 at 2fl-f2 and 2f2-fl. So, in such a situation, measurements of IMD3 while sweeping the tones spacing will lead to a low-pass filter shape [5] that could be used to de embed the thermal impedance if an accurate model is used: (2) B. Model Implementation A consistent model to account for the aforementioned process needs the interaction of an electromagnetic and a thermal domain. Then, we build the electromagnetic domain as a cascade of sections of a certain length Ax, of lumped elements Rd,J/T)= RiT)-!Jx, LdA,= Ld·!Jx, CdA,= Cd·!Jx, Cd,Jx= Ci!Jx, where RiT), Ld, Cd' and Cd are the distributed parameters of the coplanar waveguide. For very thin conductor strips as used in our experiments, the distributed resistance changes with temperature as follows: RiT)=Rdi 1 + dI), where ex. is the temperature coefficient of resistivity. r--------------------------l I Thermal Domain L1x I I 4 • I I RmMR RmMR I i·' R,mIC"'Avm • I i I T T+dT ,1zl I Pd I I _ I ;��������� t ���������������� 1-- --I LdLlx I Rd(T)LlxI + v ,1x _ �1�c.!!"��a31�e�c_D...?�§:i� ____________ _ Fig. 2. Implementation of a section of transmission line in both electromagnetic and thermal domains. On the other hand, thermal modeling of the heat flow along the transmission line and through the substrate is implemented by means of the thermal resistance R'h=L1,/k'h and the volumetric heat capacity C'h=;CP' that can be separately defined for the metal and the substrate [8] (in the previous expressions, kth, q and Cp are the thermal conductivity, density, and heat capacity of the material, respectively, and ,1/1 is a length increment in the direction of the heat propagation that can either be Ax along the line or LIz through the substrate (see Fig. 2)). So, with the above-mentioned thermal distributed parameters, each section of transmission line can be thermally modeled along the metal and through the substrate with a series resistance and shunt capacitance in the respective 978-1-4244-7732-6/101$26,00 ©201 0 IEEE 426 directions. In fact, the shunt capacitances can be combined in parallel with Ceq/w(f) to represent an equivalent overall heat capacity of the metal and substrate L1 V in volume (Fig. 2). In Fig. 2, the horizontal axis corresponds to the direction along the line, while the vertical axis is perpendicular to the wafer. So the thermal domain model presented is distributed along the horizontal axis but concentrated on the vertical axis. This means that L1x represents a section of the line and L1z is the total substrate thickness. The latter means that the 3D heat dissipation process has to be consistently modeled by use of a 2D model, even when the heat diffusion area on the XY plane changes as a function of frequency [8]. Such a frequency dependent lateral diffusion area can be taken into account by use of a frequency-dependent equivalent heat capacity Ceq/,if), similar to that used in [5]. Moreover, the fact that a CPW consists of a center conductor between two ground planes implies that an equivalent width Weq in terms of dissipation (usually related to the physical width by a geometrical factor [8]), has to be considered if a 2D model is used. In Figure 2, Rm,Ll,=Rm,./(Weqtm) and Rs,Ll,= Rs,./(WeqL1x) are the thermal resistances for a section of metal and substrate respectively, where tm is the metal thickness. Heat radiation and convection are considered negligible, as well as heat flow through the probes, which are modeled using high value resistances at the terminations of the thermal domain. Finally, dissipation in the electromagnetic domain is coupled to the thermal domain as a current source (see Fig. 2). This makes use of the fact that, in such a thermal transmission line, current and voltage are analogous to heat and temperature, respectively [8]. III. MEASUREMENTS AND RESULTS Linear and nonlinear characterization of two coplanar waveguides was carried out at room temperature for purposes of model validation. A. Test Wafer and Linear Measurements Two 480 nm-thick gold CPWs, with the same geometry except for different length IA=4.200 mm and 19=9.933 mm, were fabricated using photolithographic techniques on a 430 f.l.m-thick sapphire substrate. A 20 nm-thick titanium intermediate layer was used for better adhesion between gold and substrate. The center conductor, ground planes and gap widths are 30 fAm, 200 fAm and 15 fAm, respectively. A procedure of multi-line TRL calibrations and impedance comparison was applied to extract the distributed parameters of the CPWs. Details on these extraction technique can be found in [9]. Extracted resistance and inductance per unit length from measurements is shown in Fig. 3. The extracted capacitance per unit length is Cd=162 pF/m, and Gd is negligible. IMS 2010 70 -7 5ooo r- --------------------------�� 1 0 2000 \-. ..... ....... ..... � Fig. 3. Resistance and inductance per unit length as extracted from measurements (dashed lines) and a polynomial fit (solid lines). Figure 3 also shows the polynomial fit for Rd and Ld, which smoothes the data from measurements and is used in the circuit model. B. Nonlinear Measurements As previously noted, the dependence of the intermodulation distortion level on the envelope frequency of the input signal is probably the convincing evidence for self-heating induced nonlinearities. To evaluate this and to validate the model with measurements, IMD3 has been measured on both CPW s for a wide range of tone spacings. The upper tone was set at f2=6 GHz, while the lower tone f,=f,-Llf varied so that Llf ranges Llf=2 Hz to Llf= 1 GHz at an input power of 20 dBm per tone. -70 � E co -80 � M 0-90 � -100 -110 -120 L- ---- � � ---- � � ---- � � -- � � -- � 10° 10 2 10 4 10 6 Frequency [Hz] Fig. 4. Measurements (2f,-f, and 2f,-f, are squares and circles respectively) and simulations (solid lines overlapped) of the generated IMD3 of the two CPWs presented in this work. The higher IMD3 level corresponds to the longer B CPW (18=9.933 mm), and the lower corresponds to A CPW (lA=4.2 mm). A special measurement setup intended to increase the measurement dynamic range at the spectrum analyzer by means of cancellation of the fundamental tones was used. 978-1-4244-7732-6/101$26.00 ©2010 IEEE 427 Further details on this measurement setup can be found in [9]. The measurement results can be seen in Fig. 4 in dashed lines for both CPWs A and B. As observed from the measurements, the baseline IMD3 level of the system itself is around -100 dBm for the specified input power, while the measurement noise floor is at -140 dBm. This makes the system nonlinearities predominant over the device for modulation frequencies of 1 MHz and above. Theoretical thermal conductivity values used in this work are ks,th = 42 W·m-'·K' and km,th = 318 W·m-'·K' for sapphire and gold, respectively [8]. A temperature coefficient of resistivity for gold of ex = 0.0037 K' is also used for both lines [8]. Equation (2) states that the measured intermodulation level is proportional to the square of the thermal impedance, and this is used next to extract the equivalent heat capacity Ccq,Llif) as a function of frequency, implemented as Ceq, if)= Cd,th(f)'WCq' z. The procedure to extract Ceq.LlV(f) from the measured results by means of (2) starts by obtaining the magnitude of the un scaled thermal impedance Zthu(f). Then, IZth.if)1 is extrapolated to obtain the theoretical DC value of the thermal resistance, given by Rs,Llz' Finally, Ceq,LlV(f) is obtained from the following relation: Ceq,LlV (f) = IZ'h,tI (1)1 R; 6z j2 (3) By means of the procedure above, an equivalent heat capacity can be described using a polynomial in log-log scale: Cd,lhC!) = lOCth,o+Cth,,]Og(f)+Cth,,]Og(f)', (4) in which Ctho=3.94, Cth,=-0.784 and Cth,=0.0186 are used in the model to'simulate the results shown in Fig. 4. A low-order polynomial is used to keep the tendency of the thermal impedance at frequencies above those used for the fitting. As can be seen in Fig. 4, the agreement between measurements and modeling is very good for both lines using Wcq=700 Ilm. Only a small deviation is observed at the modulation frequency of 2 Hz, which could be an indication of the probes effect. IMS 2010 71 I� .-----�----�----�------�----� c'C: 10' -= � 2-10' .., u = .� 10' u ., 0. J 10° 'cO .., :c W' w' '--:- -�-::-- -� --,----- -� -::-- -� :--- ----' 10° 10' 10 ' 10 ' 10 ' 1010 Frequency (Hz) Fig. 5. Extracted heat capacitance Cd.,h(f). IV. DISCUSSION Further research could be performed to relate the extracted frequency-dependent equivalent heat capacity to the heat diffusion area. This may result in a better description of the 3D dissipation process. The model presented in this work is useful not only for use in nonlinear prediction but also also for simulating the steady state temperature distribution along the transmission lines. Figure 6 shows the current and temperature distribution for CPW B (1B = 9.933 mm) with a clear cause-effect relation between them. In fact, the nonlinear generation mechanism presented has some similarities with the 3(0 technique [10], in which the generated third harmonic is used to extract the thermal properties of the substrate at low frequencies. 0.8 ,-------------------, 0.07 Q' 0.7 '-' Q) .3 0.6 c<:S  Q) E 0.5 � <l 0.4 o 2 4 6 Position (mm) 0.065 (J s:: 0.06 =i CD ;:;. 0.055 > '-' �-e _ 4\ 0.05 8 Fig. 6. Simulation of the steady state temperature increment distribution (circles) and current distribution (squares) along CPW B (lB = 9.933 mm). Future research will consider metals other than gold, with higher and lower temperature coefficients of resistivity. A thicker metal layer should also be evaluated to check the validity of the model. Substrates with better and poorer thermal conductivities could also be addressed for a better understanding of how the self-heating impacts the IMD3 level. 978-1-4244-7732-6/101$26.00 ©2010 IEEE 428 ACKNOWLEDGEMENT This work was partially supported by the Spanish Government (CICYT) under Grant TEC-2006-13248-C0402ITCM. E.R. thanks the support from BES-2007-16775. C.C thanks the support from Generalitat de Catalunya, grant 2008BE2-00l96. The authors thank David Novotny for his help in performing this work. REFERENCES [I] A. E. Parker and 1. G. Rathmell, "Self-heating process in microwave transistors", in URSI Commission C Appl. Radio Sci. Workshop, P. Wilkinson, Ed., Hobart, Australia, 2004, pp. 1-8 [2] 1. Vuolevi, T. Rahkonen, "Analysis of third-order intermodulation distortion in common-emitter BJT and HBT amplifiers," IEEE Trans. on Circuits and Systems II: Analog and Digital Signal Processing, vo1.50, no.12, pp. 994-100 I, Dec. 2003 [3] V. Camarchia, F. Cappelluti, M. Pirola, S. D. Guerrieri, G. Ghione, "Self-consistent electrothermal modeling of class A, AB, and B power GaN HEMTs under modulated RF excitation", IEEE Trans. on Microwave Theory and Techniques, vol. 55, pp. 1824-1831,2007. [4] 1. Vuolevi, T. Rahkonen, 1. Manninen, "Measurement technique for characterizing memory effects in RF power amplifiers", IEEE Trans. on Microwave Theory and Techniques, vol. 49, no 8, August 2001, pp. 1383-1389 [5] 1. R. Wilkerson, K. G. Gard, A. G. Schuchinsky, M. B. Steer, "Electro-thermal theory of intermodulation distortion in lossy microwave components", IEEE Trans. on Microwave Theory and Techniques, vol. 56, no. 12, Part I, pp. 2717-2725, Dec. 2008. [6] N. B. Hassine, D. Mercier, P. Renaux, D. Bloch, G. Parat, B. Ivira, P. Waltz, C. Chappaz, R. Fillit, S. Basrour, "Self heating under RF power in BA W SMR and its predictive I D thermal model", Frequency Control Symposium, 2009 Joint with the 22nd European Frequency and Time forum. IEEE International, vol., no., April 2009, pp.237-240. [7] B. Ivira, P. Benech, R. Fillit, F. Ndagijimana, P. Ancey, G. Parat, "Modeling for temperature compensation and temperature characterizations of BA W resonators at GHz frequencies", IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 55, no. 2, pp. 421-430, Feb. 2008 [8] F. P. Incropera and D. P. de Witt, Fundamentals of Heat and Mass Transfer, 5th ed., 1. Wiley & Sons N.Y., 2002. [9] 1. Mateu, 1. C. Booth, S. A. Schima, "Frequency tuning and spurious signal generation at microwave frequencies in ferroelectric SrTi03 thin-film transmission lines", IEEE Trans. on Microwave Theory and Techniques, vol. 55, no. 2, pp. 392396, Feb. 2007. [10] D. G. Cahill, "Thermal conductivity measurement from 30 to 750 K: The 3m method", Review of Scientific Instruments, vol. 61,pp. 802-808,1990. IMS 2010 72 ROCAS et al.: PASSIVE INTERMODULATION DUE TO SELF-HEATING IN PRINTED TRANSMISSION LINES Fig. 10. Steady-state temperature rise along the 9.93-mm gold line. Solid line is using (13) and dashed line represents the results from simulation. Fig. 11. Temperature rise at z =9 : 93 mm of a 9.93-mm gold line for a wide range of envelope frequencies. Squares and solid line are simulation and equation, respectively. Fig. 12. Power delivered to a matched load at the end of the 9.93-mm gold line. Circles and solid line are simulation and equation respectively. lustrate this with a real case, Fig. 10 shows the results for (13) and the circuit model implementation of a 9.93-mm gold line, and Hz, where simulations give a small ripple due to the imaginary part of . For the real 9.93-mm gold line circuit implementation, distributed parameters extracted from measurements are used to construct the transmission line as a cascade of cells. Additionally, temperature rise oscillations are validated, by use of (12) and simulations, at different envelope frequencies at mm (Fig. 11). Once the equations of the temperature rise have been validated, the nonlinear signal, generated as a consequence of temperature oscillations, can be evaluated. Fig. 12 shows the result of (25) and simulations of the intermodulation power delivered to a matched load at mm for a wide range of envelope frequencies. Fig. 13. Extracted distributed resistance (symbols) and polynomial fit (lines) for the different coplanar waveguides. Circles and solid line represent gold, squares and dashed line represent platinum, and triangles and dotted line represent palladium-gold. V. MEASUREMENTS AND RESULTS A test wafer has been constructed to validate the model with measurements. Metals with different temperature coefficients of resistivity have been selected to construct coplanar waveguides of different lengths. To check the model, measurements of the resistivity at different temperatures have been performed. Thirdorder intermodulation distortion measurements for a wide range of envelope frequencies have also been used. A. Test Wafer Coplanar waveguides, with the cross-section geometry shown in Fig. 3, have been constructed following standard fabrication techniques on a sapphire substrate, chosen for its low dielectric losses at microwave frequencies. Different metals have been deposited, including gold (Au), platinum (Pt), and palladium-gold (PdAu) with a 55% gold content. The reason for choosing these metals is that they offer different combinations of resistivity values and resistivity change with temperature, which translates into different levels of intermodulation distortion. Three transmission lines, A, B, and C for each type of metal, have been measured. Their lengths are mm, mm, and mm, to demonstrate the distributed effects. B. Linear Measurements and Model The first step in constructing the linear part of the circuit model is to obtain the distributed parameters of the fabricated transmission lines. The procedure used in this work consists in performing a multiline thru-reflect-line (TRL) calibration to obtain the propagation constant. It is then used, along with an impedance comparison method, to obtain the distributed parameters , , , and . Details on this procedure can be found in [28]. The linear measurements have been performed at a low power, 5 dBm, to ensure the linear regime of the devices. Results for the extracted distributed resistance and inductance are shown in Figs. 13 and 14, respectively. The extracted values of the distributed conductance are below the sensitivity of the measurements; its impact is, there79 318 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 2, FEBRUARY 2011 Fig. 14. Extracted distributed inductance (symbols) and polynomial fit (lines) for the different coplanar waveguides. Circles and solid line represent gold, squares and dashed line represent platinum, and triangles and dotted line represent palladium-gold. Fig. 15. Measured and simulated S -parameters for line B, made of palladiumgold. The line length is 4.2 mm. Squares and triangles represent measurements. Solid lines represent simulation. fore, considered negligible. On the other hand, the extracted distributed capacitance, which is constant over frequency due to the nondispersive nature of sapphire, has a value of pF/m. The presented extracted distributed resistance and inductance are fitted to polynomials so that they can be easily used in the circuit model implementation. Next, the calibrated -parameters from measurements are compared to the simulated -parameters of the circuit implementation for each line to check that the extraction procedure is correct. For example, Fig. 15 illustrates both measured and simulated -parameters for PdAu line B. Characteristic impedances obtained at 6 GHz are , , and . C. Nonlinear Measurements and Model Once the linear measurements and modeling are complete, the thermal impedance that correctly predicts the temperature along the line is obtained. An accurate temperature simulation is crucial to properly simulate its effect on the metal resistivity and also to predict the third-order intermodulation generation. In the model presented, the thermal resistance and the thermal capacitance per unit length represent the resistance to heat flow at steady state and its frequency dependence, respectively. These can, therefore, be solved separately. As stated TABLE II ELECTRIC AND THERMAL PARAMETERS OF THE CENTER CONDUCTOR AT z =0 USING TWO TONES AT 20 dBm EACH in (2), the third-order intermodulation measurements unveil the frequency dependence of the thermal impedance, which is used in this work to obtain . On the other hand, finite-element thermal simulations are used to obtain . 1) Finite-Element Thermal Simulations: The thermal resistance sets the steady-state temperature, given a certain heat dissipation in the center conductor. It depends not only on the material properties of the metal and substrate, but also the geometry of the strip. Finite-element simulations can be used to obtain the thermal resistance with (1) as the division of temperature over heat. We look at the heat density through the line, shown in Fig. 4, and perform the surface integral at the center conductor. A steady-state thermal simulation is done with [23] to obtain the temperature profile on the cross section of the line. Table II summarizes the current and heat flux at for all types of metal transmission lines, the average temperature rise on the center conductor , and the thermal resistance obtained. As can be seen from Table II, the change in the thermal resistance is around 5% for the three metals. This implies that the upper metal layer plays a negligible role in how the heat flows to the substrate. Approximate values for might be obtained with closed-form expressions such as those found in [13] and [14] to get K m/W and K m/W, respectively, neglecting the titanium adhesion layer. The circuit model can also be used to simulate the temperature distribution along the line. To check this, we performed 3-D steady-state thermal finite-element simulations with specifically implemented 3-D electrothermal finite-elelent software. This is done by use of the current distribution previously obtained with the circuit model, as shown in Fig. 16. Results in Fig. 16 show the contribution to temperature rise due to dissipation in the center conductor for the circuit model and finite-element simulations. There is total agreement between them. Moreover, Fig. 16 also shows the total temperature rise in the center conductor as a consequence of dissipation in the center conductor and the ground planes obtained with finite-element simulations. The simulations for the circuit model have been performed with and without the series thermal impedance , which totally overlap. These results confirm that, for the given geometry, the heat propagation along the axis is negligible and the circuit model can be used to estimate the total temperature rise in the center conductor. 2) Third-Order Intermodulation Distortion: Forward thirdorder intermodulation distortion measurements, of and , have been performed on all lines by means of the 80 ROCAS et al.: PASSIVE INTERMODULATION DUE TO SELF-HEATING IN PRINTED TRANSMISSION LINES Fig. 16. Steady-state temperature rise profiles along the C lines for the different metals. Solid and dashed lines represent the contribution to temperature rise in the center conductor, as a consequence of its dissipation, from using the circuit model and finite elements, respectively. The dotted lines represent the total temperature rise in the center conductor, as a consequence of dissipation in both the center conductor and the ground planes, obtained with finite-element simulations. Simulations using the circuit model, with Z ( ! ) and with an open circuit instead of Z ( ! ) , overlap. Fig. 17. Measured and simulated third-order intermodulation in A line, l = 1 mm. 2 f 0 f (dashed line) and 2 f 0 f (solid line) overlap for all the simulations. Unfilled circles and crosses represent 2 f 0 f and 2 f 0 f for Au. Triangles and filled circles represent 2 f 0 f and 2 f 0 f for Pt. Stars and diamonds represent 2 f 0 f and 2 f 0 f for PdAu. two-tone test [10]. A special measurement setup [28] consisting of a configuration that cancels the fundamental signals and after the device-under-test is used to achieve a high-dynamic range at the spectrum analyzer. For such measurements, the input power has been fixed at 20 dBm, while the tones spacing has been taken from 2 Hz up to 1 GHz. Results can be seen in Figs. 17–19. High separation between tones measurements are limited by the baseline intermodulation level of the measurement setup. On the other hand, small separations between tones measurements can be limited by the phase noise of the sources. The shortest Au line represents the worst scenario for this type of measurements because it shows a low intermodulation level. System nonlinearities easily dominate at intermediate and high separation between tones Hz and phase noise Fig. 18. Measured and simulated third-order intermodulation in B line, l = 4 : 2 mm. 2 f 0 f (dashed line) and 2 f 0 f (solid line) overlap for all the simulations. Unfilled circles and crosses represent 2 f 0 f and 2 f 0 f for Au. Triangles and filled circles represent 2 f 0 f and 2 f 0 f for Pt. Stars and diamonds represent 2 f 0 f and 2 f 0 f for PdAu. Fig. 19. Measured and simulated third-order intermodulation in C line, l = 9 : 93 mm. 2 f 0 f (dashed line) and 2 f 0 f (solid line) overlap for all the simulations. Unfilled circles and crosses represent 2 f 0 f and 2 f 0 f for Au. Triangles and filled circles represent 2 f 0 f and 2 f 0 f for Pt. Stars and diamonds represent 2 f 0 f and 2 f 0 f for PdAu. dominates for small separation between tones Hz . The results in Figs. 17–19 unveil the low-pass filter behavior of the thermal impedance, showing similar results as those found in [12], which can be used to extract the frequency-dependent thermal capacitance . Measurements of any of the metals could be used to extract , as long as the thermal impedance is dominated by the substrate. The platinum B line is preferable, though, because it shows the highest dynamic range above the baseline intermodulation level of the measurement setup, which is around 100 dBm. The extracted phenomenological , for the procedure explained in Appendix II, can be seen in Fig. 20, where its frequency-dependence embeds the 3-D dissipation effect. Once the thermal model is completely implemented, the last step consists of determining the nonlinear variable . This sets how the distributed resistance of the center conductor changes with temperature. To do that, we use the relation between and temperature, presented in Appendix I, so that the temperature coefficient of resistivity of the metal is used. Since the material properties are process dependent, we performed resistance measurements at dc of the fabricated coplanar waveguides at several temperatures. 81 320 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 2, FEBRUARY 2011 Fig. 20. Extracted C ( ! ) from measurements on the platinum B line. The measured values for the temperature coefficients of resistivity are K , K , and K . The resistivity values at room temperature are shown in Table I. The complete electrothermal model can now be constructed, making use of the measured values and the extracted thermal resistance and capacitance. Simulations of the third-order intermodulation distortion with the model are presented, along with the measurements in Figs. 17–19. These show good agreement and the predicted dependence on the line length. VI. DISCUSSION The good agreement obtained between measurements and modeling indicates that scaling with the line length is correctly predicted. From the results above, the high difference between nonlinear levels on lines made of different metals can also be observed. The results show that there is not a simple relation between length and intermodulation level, and this is because the distributed nonlinearity increases with length, but also gets attenuated by it. Therefore, depending on the length, a high attenuation transmission line can show more or less nonlinearity than a lower loss transmission line. In addition, the fact that the platinum and the palladium-gold lines are highly mismatched translates into a variable relation between length and intermodulation level [27]. However, several considerations to minimize the intermodulation can be made from (25), in particular that referred to the correct choice of materials. In this sense, metals and dielectrics with low losses and weakly temperature-dependent properties are preferable. Additionally, high thermal conductivity dielectrics provide the right path for heat, minimizing temperature rise. To check that the ground planes have a negligible contribution to the intermodulation distortion generation, we use (25). We consider that each ground metal strip has the same thermal impedance of the center conductor, which is a reasonable approximation. With this assumption, we get a much lower contribution coming from the ground planes of dB when compared with that coming from the center conductor in the gold lines. For the platinum and palladium-gold lines, we get dB and dB. These results confirm the validity of the simplified model. From the closed-form expressions obtained, we observe that the third-order intermodulation distortion generation process might also be described, for a specific separation between tones, by a phenomenological model of a quadratic current-dependent distributed resistance of the form (27) Several authors have suggested the use of (27) to explain the relation between losses and intermodulation [16]. However, this phenomenological model is incorrect, and predicts a nonexistent third harmonic. VII. CONCLUSIONS We have presented the mechanism by which third-order intermodulation distortion is generated in transmission lines and its circuit model. Additionally, closed-form expressions have been obtained and validated along with the model. The model has been properly checked with measurements of lines of different lengths and made composed of different metals. This work also reveals the negligible impact of the ground planes when compared to that of the center conductor and in terms of nonlinear behavior due to self-heating. Additionally, the envelope frequency-dependent intermodulation cannot be described by a constant slope of a certain decibel/decade. The advantage of having such a distributed circuit model for a nonlinear transmission line is that it can be used for prediction purposes of any device in which transmission lines are used such as filters or directional couplers. Moreover, its usefulness is not restricted to a specific kind of input signal and can predict the nonlinear effects on complex modulated signals like global system for mobile communication (GSM) or code division multiple access (CDMA) for example. APPENDIX I DISTRIBUTED RESISTANCE TEMPERATURE DEPENDENCE The distributed resistance at a specific frequency can be described as a function of the resistivity increment as follows: (28) where is the distributed resistance at ambient temperature. The resistivity increment due to temperature rise can be identified in the following relation: (29) is the local derivative, at the room temperature resistivity value, of the function that relates the distributed resistance with the metal resistivity. Therefore, is a factor that relates the increment in the distributed resistance, as a consequence of an increment in the resistivity, for a specific geometry and metal (30) By using the Weeks method [29], we obtain the factor for the center conductor, for the given geometry at 6 GHz to be 82 ROCAS et al.: PASSIVE INTERMODULATION DUE TO SELF-HEATING IN PRINTED TRANSMISSION LINES m , m , and m . APPENDIX II THERMAL CAPACITANCE EXTRACTION PROCEDURE The procedure starts by converting the intermodulation measurements to a linear scale to get the unscaled magnitude of the thermal impedance . is then fitted to a polynomial and scaled to a value previously obtained using the circuit model to get . A frequency-dependent thermal capacitance of the form (31) is used so that its coefficients can be obtained from the relation that follows from a simple circuit analysis of Fig. 8: (32) The extracted phenomenological has coefficients , , and . ACKNOWLEDGMENT The authors thank Dr. Y. Wang and Dr. T. M. Wallis, both with the National Institute of Standards and Technology (NIST), Boulder, CO, for their help with this work. Work partially supported by the U.S. Government not subject to U.S. copyright. REFERENCES [1] C. Collado, J. Mateu, and J. M. O’Callaghan, “Analysis and simulation of the effects of distributed nonlinearities in microwave superconducting devices,” IEEE Trans. Appl. Superconduct., vol. 15, no. 1, pp. 26–39, Mar. 2005. [2] J. Mateu, J. C. Booth, S. A. Schima, C. Collado, D. Seron, and J. M. O’Callaghan, “Measurements and analysis of microwave nonlinearities in ferroelectric thin film transmission lines,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 11–16, 2006, pp. 1622–1625. [3] A. P. Shitvov, D. E. Zelenchuk, A. G. Schuchinsky, and V. F. Fusco, “Passive intermodulation in printed lines: Effects of trace dimensions and substrate,” IET Microw., Antennas, Propag., vol. 3, no. 2, pp. 260–268, Mar. 2009. [4] D. E. Zelenchuk, A. P. Shitvov, and A. G. Schuchinsky, “Effect of laminate properties on passive intermodulation generation,” in LAPC Antennas Propag. Conf., Apr. 2–3, 2007, pp. 169–172. [5] P. L. Lui, “Passive intermodulation interference in communication systems,” Electron. Commun. Eng. J., vol. 2, no. 3, pp. 109–118, Jun. 1990. [6] P. N. Radha, “Predicting junction temperature and MTTF for MMIC devices,” RF Design, vol. 28, no. 8, pp. 68–70, 2005. [7] R. Hopper and C. Oxley, “Thermal measurement a requirement for monolithic microwave integrated circuit design,” in Proc. ARMMS, Apr. 2008, pp. 385–390. [8] V. Camarchia, F. Cappelluti, M. Pirola, S. D. Guerrieri, and G. Ghione, “Self-consistent electrothermal modeling of class A, AB, and B power GaN HEMTs under modulated RF excitation,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 9, pp. 1824–1831, Sep. 2007. [9] A. E. Parker and J. G. Rathmell, “Self-heating process in microwave transistors,” presented at the Appl. Radio Sci. Workshop, Hobart, TAS, Australia, Feb. 18–20, 2004. [10] J. H. K. Vuolevi, T. Rahkonen, and J. P. A. Manninen, “Measurement technique for characterizing memory effects in RF power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 8, pp. 1383–1389, Aug. 2001. [11] R. S. Timsit, “High speed electronic connectors: A review of electrical contact properties,” IEICE Trans. Electron., vol. ESS-C, no. 8, pp. 1532–1544, 2005. [12] J. R. Wilkerson, K. G. Gard, A. G. Schuchinsky, and M. B. Steer, “Electro-thermal theory of intermodulation distortion in lossy microwave components,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 2717–2725, Dec. 2008. [13] R. Wilcoxon, “The effects of geometry and dielectric material on stripline and microstrip internal temperatures,” IEEE Trans. Compon. Packag. Technol., vol. 28, no. 4, pp. 674–679, Dec. 2005. [14] J. Adam, “New correlations between electrical current and temperature rise in PCB traces,” in 20th Annu. IEEE Semiconduct. Thermal Meas. Manag. Symp., Mar. 9–11, 2004, pp. 292–299. [15] D. G. Cahill, “Thermal conductivity measurement from 30–750 K: The 3 ! method,” Rev. Sci. Instrum., vol. 61, pp. 802–808, 1990. [16] D. E. Zelenchuk, A. P. Shitvov, A. G. Schuchinsky, and V. F. Fusco, “Passive intermodulation in finite lengths of printed microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 11, pp. 2426–2434, Nov. 2008. [17] J. Z. Wilcox and P. Molmud, “Thermal heating contribution to intermodulation fields in coaxial waveguides,” IEEE Trans. Commun., vol. COM-24, no. 2, pp. 238–243, 1976. [18] G. H. Strauss, “Intrinsic sources of IM generation,” Naval Res. Lab., CITY, STATE, Memo. Rep. 4233, 1980, ch. 5. [19] E. Rocas, C. Collado, N. D. Orloff, and J. C. Booth, “Third order intermodulation distortion due to self-heating in gold coplanar waveguides,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2010, pp. 425–428. [20] F. P. Incropera, D. P. DeWitt, T. L. Berqman, and A. S. Lavine, Fundamentals of Heat and Mass Transfer. New York: Wiley, 2002. [21] C. Collado, J. Mateu, O. Menendez, and J. M. O’Callaghan, “Nonlinear distortion in a 8-pole quasi-elliptic bandpass HTS filter for CDMA system,” IEEE Trans. Appl. Superconduct., vol. 15, no. 2, pp. 992–995, Jun. 2005. [22] D. Pozar, Microwave Engineering. New York: Wiley, 1998. [23] QuickField. Tera Software, Svendborg, Denmark, 2010. [24] “Material property data,” MatWeb, Blacksburg, VA, 2010. [Online]. Available: http://www.matweb.com [25] “Characteristics of single crystal sapphire,” KYOCERA, Kyoto, Japan, 2010. [Online]. Available: http://global.kyocera.com/prdct/fc/product/ pdf/s_c_sapphire.pdf [26] C. Y. Ho, “Thermal conductivity of ten selected binary alloy systems,” J. Phys. Chem. Ref. Data, vol. 7, no. 3, pp. 959–1177, 1978. [27] J. Mateu, C. Collado, N. D. Orloff, J. C. Booth, E. Rocas, A. Padilla, and J. M. O’Callaghan, “Third-order intermodulation distortion and harmonic generation in mismatched weakly nonlinear transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 1, pp. 10–18, Jan. 2009. [28] J. Mateu, J. C. Booth, and S. A. Schima, “Frequency tuning and spurious signal generation at microwave frequencies in ferroelectric SrTiO3 thin-film transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 2, pp. 391–396, Feb. 2007. [29] W. T. Weeks, L. L. Wu, M. F. McAllister, and A. Singh, “Resistive and inductive skin effect in rectangular conductors,” IBM J. Res. Develop., vol. 23, no. 6, pp. 652–660, Nov. 1979. Eduard Rocas (S’07) received the Telecommunication Engineering degree from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 2005, where his final project was associated with the creation of the Intelligent Communications and Avionics for Robust Unmanned Aerial Systems (ICARUS) Research Group, and is currently working toward the Ph.D. dgree at UPC. From September 2005 to July 2006, he was involved in the simulation and modeling of advanced SONARs with the Computer Vision and Robotics Group (VICOROB), University of Girona, as an Formación del Profesorado Universitario (FPU) Grant Holder. Since November 2006, he has been with UPC as a doctoral student (Formación de Personal Investigador (FPI) Grant Holder), where his research is focused on new materials and structures for novel RF/microwave (MW) devices. Since January 2009, he has been a Guest Researcher with the National Institute of Standards and Technology (NIST), Boulder, CO. 83 322 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 2, FEBRUARY 2011 Carlos Collado (A’02–M’03–SM’10) received the Telecommunication Engineering degree, Ph.D. degree, and M.S. degree in biomedical engineering degree from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 1995, 2001, and 2002, respectively. In 1998, he joined the faculty of UPC, and became an Associate Professor in 2005. From November 2005 to January 2008, he was Vice-Dean of the Technical School of Castelldefels (EPSC), UPC, where he was responsible of the telecommunication and aeronautic engineering degrees. In 2004, he was a Visiting Researcher with the University of California at Irvine. From 2009 to 2010, he was a Guest Researcher with the National Institute of Standards and Technology (NIST), Boulder, CO. His primary research interests include microwave devices and systems with focus on nonlinearities of passive devices. Dr. Collado was the recipient of a 2001 prize for the best doctoral thesis in electronics and telecommunications from UPC. Nathan D. Orloff was born in Columbia, SC, on August 10, 1981. He received the B.S. degree in physics (with high honors) and Ph.D. degree from the University of Maryland at College Park, in 2004 and 2010, respectively. His doctoral research focused on the development of a broadband on-wafer calibration and characterization approach that has been applied to number of novel material systems. He is interested in novel dielectric characterization techniques, complex-nonlinear device modeling, on-wafer measurement, and the integration of microelectronics and microfluidics. Dr. Orloff was the recipient of the 2004 Martin Monroe Undergraduate Research Award, the 2006 CMPS Award for Excellence for Teaching Assistants, the 2010 Michael J. Pelczar Award for Excellence in Graduate Study, and an honorable mention for the 2010 Block Award for Best Poster Presentation from the Aspen Center for Physics. Jordi Mateu (M’03–SM’10) received the Telecommunication Engineering and Ph.D. degrees from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 1999 and 2003, respectively. In October 2006, he joined the Signal Theory and Communications Department, UPC, initially as a Ramon-y-Cajal Research Fellow, where since March 2009, he has been an Associate Professor. Since 2007 he has also been an Associate Researcher with the Centre Tecnològic de Telecomunicacions de Catalunya (CTTC), Castelldefels, Spain. From May to August 2001, he was a Visiting Researcher with Superconductor Technologies Inc., Santa Barbara CA. From October 2002 to August 2005, he was a Research Associate with CTTC. Since September 2004, he has held several guest researcher appointments with the National Institute of Standards and Technology (NIST), Boulder, CO, where he was a Fulbright Research Fellow from September 2005 to October 2006. In July 2006, he was a Visiting Researcher with the Lincoln Laboratory, Massachusetts Institute of Technology (MIT). From September 2003 to August 2005, he was a Part-Time Assistant Professor with the Universitat Autònoma de Barcelona. In Summer 1999, following graduation, he was a Trainee Engineer with the Investment Technology Department, Gillette, Isleworth, U.K. He has authored or co-authored over 40 papers in international journal, over 55 contributions in international conferences, and three book chapters. He holds two patents. He has collaborated and led several research projects for national and international public and private organizations and companies. He is reviewer of several journals and international conferences. His primary interests include microwave devices and system and characterization and modeling of new electronic materials including ferroelectrics, magnetoelectric, superconductors, and acoustic devices. His recent research interests also include the synthesis, design, and development of novel microwave filtering structures. Dr. Mateu was the recipient of the 2004 Prize for the best doctoral thesis in Fundamental and Basic Technologies for Information and Communications by the Colegio Oficial de Ingenieros de Telecomunicación (COIT) and Asociación Española de Ingenieros de Telecomunicación (AEIT). He was also the recipient of a Fulbright Research Fellowship, an Occasional Lecturer Award for visiting MIT, and a Ramón y Cajal Contract. Alberto Padilla was born in Barcelona, Spain, in 1984. He received the Telecommunication Engineering degree from the Technical University of Catalonia (UPC), Barcelona, Spain, in 2008. His final project concerned the mitigation of the nonlinear behavior of high-temperature superconducting planar devices. Since March 2008, he has been with the Department of Signal Theory and Communications, UPC. He is currently a Trainee—Stagiaire Researcher with the European Space Agency (ESA)–European Space Research and Technology Centre (ESTEC), where his focus is focused on the synthesis and design of a new class of receiver filters for satellite communications. Mr. Padilla was the recipient of a Ph.D. grant (FPU) from the Spanish Ministry of Education in January 2009. Juan M. O’Callaghan (SM’01) received the Telecommunication Engineering degree from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 1987, and the Ph.D. degree from the University of Wisconsin–Madison, in 1993. In 2003, he became a Full Professor with UPC. In 1989, he was with the Systems Research Center, Honeywell, Bloomington, MN, where he was involved with noise measurement methods for field-effect transistors (FETs) at Ka -band. In 1993, he became a tenure-track faculty member with UPC. From 2003 to 2006, he was a Manager for MERIT, a consortium of European universities delivering a joint Master’s program in Information Technologies within the Erasmus Mundus Program (see www.meritmaster.org). From 2006 to 2009, he was Vice-Dean of academic affairs with Telecom BCN, the telecommunication engineering school of UPC (see www.telecombcn.upc.edu). He has coauthored over 55 papers in peer-reviewed international journals and 90 contributions to conference proceedings. He holds four patents. His research interests include microwave devices and materials and microwave photonics. He has been involved with noise characterization, large-signal properties of GaAs FETs, and advanced microwave materials such as superconductors and ferroelectrics. Dr. O’Callaghan was the recipient of the 2001 prize to the most outstanding research project from the Catalan Association of Telecommunication Engineers (ACET). In 2004, one of the theses he supervised received the prize for the best doctoral thesis in fundamental and basic technologies for information and communications by the Colegio Oficial de Ingenieros de Telecomunicación (COIT) and Asociación Española de Ingenieros de Telecomunicación (AEIT). James C. Booth received the B.A. degree in physics from the University of Virginia, Charlottesville, in 1989, and the Ph.D. degree in physics from the University of Maryland at College Park, in 1996. His doctoral dissertation concerned novel measurements of the frequency dependent microwave surface impedance of cuprate thin film superconductors. Since 1996, he has been a Physicist with the National Institute of Standards and Technology (NIST), Boulder, CO, initially as aNational Research Council (NRC) Postdoctoral Research Associate (1996–1998) and currently as the leader of the High Frequency Devices and Characteristics Project. His research with NIST is focused on exploring the microwave properties of new electronic materials and devices, including ferroelectric, magneto-electric, and superconducting thin films, as well as the development of experimental platforms integrating microfluidic and microelectronic components for RF and microwave frequency characterization of liquid and biological samples. 84 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY Modeling of Self-Heating Mechanism in the Design of Superconducting Limiters Eduard Rocas, Student Member, IEEE, Carlos Collado, Senior Member, IEEE, Jordi Mateu, Senior Member, IEEE, Nathan Orloff, and James C. Booth Abstract—We propose a modeling method to simulate how the local temperature rises, due to power dissipation, and how this affects the performance of a high temperature superconductor (HTS) limiter. For given material properties, power, and frequency we determined, by use of both electromagnetic and thermal modeling, the spatial distribution of the temperature rise across an HTS transmission line. This temperature rise in turn affects the local description of the superconductor nonlinearities. To model this effect, we use an iterative technique that combines the Weeks-Sheen method to calculate the current-density distribution with a finite-element method to calculate temperature rise at each point of the transmission line. Simulations of coplanar waveguide HTS limiters on sapphire and quartz are presented. Index Terms—HTS, limiter, self-heating, superconductor. I. INTRODUCTION HIGH temperature superconductor (HTS) materials can undergo a transition from the superconducting state to the non-superconducting state when the current density exceeds a critical level. At this point, pair-breaking begins to occur, and the material gradually loses its superconducting properties. This implies higher losses, and therefore higher dissipation, which produce a temperature rise, which in turn increases the fraction of carriers in the normal state. These transient effects end when electrothermal equilibrium is reached. The above-mentioned current-density dependence of superconductor material properties has been exploited to develop power limiters based on HTS materials, which is an emerging microwave application for high temperature superconductors [1]. The short switching times, one nanosecond or less, ensure that very little energy is transmitted through the limiter prior to switching, and so guarantees a high degree of protection to the downstream electronics from high-power transient signals. This characteristic makes HTS technology a viable candidate compared to conventional limiting technologies, such as diodes. Manuscript received August 03, 2010; accepted October 07, 2010. This work was partially supported by the Spanish Ministry of Science and Innovation under Grant TEC-2009-13897-C03-01/TCM and Grant MAT-2008-06761-C03-02; by the AGAUR, Generalitat de Catalunya (2008-BE2-00196), and by Spanish Ministry of Education through PhD fellowship for E. Rocas (BES-2007-16775). E. Rocas is with Universitat Politècnica de Catalunya (UPC), Barcelona 08034, Spain. He is also with the National Institute of Standards and Technology, Boulder, CO 80305 USA. C. Collado and J. Mateu are with Universitat Politècnica de Catalunya (UPC), Barcelona 08034, Spain (e-mail: [email protected]). N. Orloff and J. C. Booth are with the National Institute of Standards and Technology, Boulder, CO 80305 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TASC.2010.2090449 The origin of this switching behavior is a nonlinear process driven by the current-density-dependent surface impedance, where the strength of the nonlinearity is described by a temperature-dependent material parameter related to the critical current density . In this situation, and taking into account that the local temperature can increase substantially when the device switches to the non-superconducting state, it is necessary to consider electro-thermal effects when designing HTS limiters. For this purpose, we implement an electro-thermal simulation method that combines the Weeks-Sheen technique [2], [3] to determine the current density distribution with a two-dimensional thermal domain equivalent circuit to obtain the local temperature increase. Both electromagnetic and thermal simulations are coupled and solved concurrently by an iterative convergence algorithm, where the dissipated power is used as the input to the thermal-domain simulation, and the obtained temperature rise is used to change the material properties in the electromagnetic domain simulation. Based on this approach, we present simulations comparing the limiting performance of coplanar waveguides on substrates with different thermal properties: sapphire and quartz. The results confirm the importance of the thermal material properties, and consequently the temperature rise, on the performance of the superconducting limiters. II. ELECTROMAGNETIC MODEL We use the method described in [2]–[4] to calculate the current density in the cross-section of a planar transmission line. The method described in [2] involves meshing the cross-section in small unit cells that effectively act as transmission lines, so that a uniform current density is considered at each cell. The mutual inductances between the small transmission lines are then calculated and used to obtain the current density at each point of the waveguide cross-section, and also to obtain the distributed resistance , and the distributed inductance . Reference [3] extended this method to calculate the distributed parameters of superconductor transmission lines by use of a complex conductivity, and [4] completed the analysis with an iterative algorithm that takes into account the fact that the superfluid current density depends on the ratio between the current density at each point , and the critical current density . In this work, we apply an improved convergence algorithm that allows for simulations at higher power levels, compared to the algorithm used previously for the analysis of weak nonlinearities in a low-power regime [5]. 85 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY We begin with the nonlinear London penetration depth [4], with a temperature dependence based on the two-fluid model (1) where and represent the normalized temperature and current density values relative to their critical values and respectively: , and . The parameter is the penetration depth at and, at low values of , and is given by (2) where is a temperature-dependent parameter that provides useful information about the d-wave or s-wave nature of a superconducting material [4]. In this work, we will use the values extracted from measurements of YBCO in [6]. As seen in (1), an increase in both the temperature and the current density modify the London penetration depth, implying a decrease in the superconducting carrier density for . When the current density increases, the losses increase with a resulting temperature increase, which in turn further increases the losses, etc. The electromagnetic model makes use of a complex conductivity that includes the above-defined nonlinear London penetration depth, which is defined as (3) where the real part of the complex conductivity is (4) where is the normal state conductivity at . The imaginary part of the conductivity is (5) The electromagnetic domain simulation calculates the current density at each point of the transmission line cross section and then updates the complex conductivity, locally at each point, by use of (3)–(5). A convergence algorithm is used to recalculate the current density, which is compared to the previously obtained value. The iterative procedure ends when convergence is achieved. As an example, Fig. 1 shows the current density distribution obtained for the cross section of a coplanar waveguide with a 10 wide center strip and a 5 gap. The HTS film is 150 nm thick and the ground planes are 100 wide. The simulation method for the electromagnetic domain described in this section does not consider temperature changes due to self-heating mechanisms. At low powers, the temperature rise due to self-heating is negligible because the losses are very small. But at higher powers, when the current density is comparable to the critical current density, the self-heating can have an important impact that is strongly dependent on the thermal properties of the substrate. Fig. 1. Current density in a coplanar waveguide with width 10  m and gap 5  m . The HTS thickness is 150 nm and the input power is 0 dBm. Fig. 2. Equivalent circuit implementation of an elemental cell to model heat propagation in a body. III. THERMAL MODEL The heat propagation in a body, with internal heat generation, follows the heat equation defined as [7] (6) where is the thermal conductivity, is the mass density, is the heat capacity, and is the internally generated heat per unit volume. Eq. (6) is fundamental in determining the temperature distribution in a body and is consistent with a three-dimensional matrix of thermal resistances, in which a thermal capacitance is connected to ground at each node [7]. In such an analog circuit representation of the heat equation, heat is represented by current, and temperature is represented by voltage. Therefore, a current source of value equal to is connected at each node, which models the internally generated heat in a volume ; that is, the power dissipation by the Joule effect calculated in the electromagnetic domain. Fig. 2. shows an elemental three-dimensional cell of the circuit model. The thermal resistance for any direction , where ,,or and heat capacitance of the cell can be defined as (7) (8) 86 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ROCAS et al.: MODELING OF SELF-HEATING MECHANISM IN THE DESIGN OF SUPERCONDUCTING LIMITERS Fig. 3. Temperature rise distribution in the cross-section of a coplanar waveguide of YBCO on a 0.5 mm-thick sapphire substrate. where is the area perpendicular to . The circuit model can be understood as a three-dimensional matrix representing the thermal impedance . Use of the thermal impedance, in K/W units, implies that the temperature rise produced at each node by an amount of heat can be calculated by means of conventional circuit analysis, stored, and used to obtain the temperature at each point for a given heat source distribution, with: (9) The three-dimensional model can be reduced to a two-dimensional problem if only the cross-section of the transmission line is analyzed. IV. ELECTRO-THERMAL MODEL Electro-thermal simulations to reproduce the interactions between electromagnetic and thermal domains can be performed by combining the numerical techniques described in the last two sections. To couple both domains we employ the following procedure. First, the current density distribution is calculated. The dissipated power (heat) is also obtained at each point. Then the dissipated heat is used as the input for the thermal domain model and the temperature rise at each point is calculated by use of (9). The resultant temperature rise is used to recalculate the complex conductivity in the electromagnetic domain, by use of (3)–(5). Steps 1–3 are then repeated until acceptable convergence is achieved. The identical meshing is used in both techniques to ensure proper coupling. An adaptive meshing is used in the HTS thin film to accurately compute the higher current density at the edges of the strips. In addition, the substrate is also adaptively meshed to reproduce the temperature rise in detail near the HTS thin film. The reference temperature is set as the boundary condition at the bottom of the substrate. This is a reasonable consideration in real experiments, where the sample is usually in very good thermal contact with a copper chuck to keep the sample thermally stable at a desired temperature. Fig. 3 shows the temperature rise, obtained by electro-thermal simulations, at the cross-section of a YBCO thin-film transmission line on a 0.5 mm-thick sapphire substrate of thermal conductivity . The geometry of the transmission line is the same as that presented in Section II. We used Fig. 4. Distributed inductance and resistance versus input power for two different samples, one having a sapphire substrate (circles, dashed line) and the other having a quartz substrate (squares, continuous line). the YBCO material parameters described in [6]. The reference temperature is fixed at 77 K and the input power is 0 dBm. As can be seen in Fig. 3, the temperature change is very small at that input power. V. LIMITER COMPARISON FOR DIFFERENT SUBSTRATES We used the electro-thermal model developed above to compare the performance of a coplanar waveguide power limiter on two different substrates, sapphire and quartz, to demonstrate the impact of the thermal properties on the limiter performance. A thermal conductivity of is used for quartz, compared to 42 for sapphire. We obtain the distributed inductance and resistance as a function of the input power as described previously for identical coplanar waveguide devices. Fig. 4 shows the distributed inductance and resistance, for devices on a sapphire and a quartz substrate, for different input powers. A reference temperature of 83 K is considered along with a critical temperature of and a critical current density of . As seen in Fig. 4, the distributed inductance increases abruptly when the input power approaches a certain threshold. At that point, the HTS material in the center conductor switches to the normal state and the inductance drops. After that, a second transition occurs when the ground planes switch to the normal state. The distributed resistance follows a similar behavior and the transition of the ground planes can be more easily observed. Note that, for both substrate materials quartz 87 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY Fig. 5. Temperature rise at the center conductor of two different samples made of a sapphire substrate (circles) and a quartz substrate (squares), for different input powers. Fig. 6. Temperature rise and upper power threshold in two different samples made of a sapphire substrate (circles) and a quartz substrate (squares). and sapphire, the threshold is approximately the same, although the quartz sample shows a more abrupt transition. The impact that the thermal properties of the substrate have on the limiter performance is more noticeable if we examine the temperature rise. Fig. 5 shows the temperature increase at the center conductor for the two different substrates. The temperature increases much more abruptly if a quartz substrate is used, as it fails to propagate the dissipated heat effectively due to its lower thermal conductivity. This sudden temperature rise, over 1000 K as shown in Fig. 5, would destroy the device. This analysis also determines the upper power threshold at which the limiter can still recover its superconducting state instantaneously. For high-power continuous-wave signals, in which the temperature rise can be higher than the critical temperature, the thermal inertia of the system requires some time to cool back down to its nominal temperature. Fig. 6 shows the power threshold at which the transition is still reversible. If the reference temperature is fixed at 83 K, a temperature increment of 7 K implies reaching the critical temperature of 90 K. Therefore, the 7 K temperature increment sets the upper power threshold for fast-switching behavior between the limiting and the superconducting state. As shown in Fig. 6, the upper threshold is around 35 dBm for the sapphire sample and 22 dBm for the quartz sample. VI. CONCLUSION This work presents a method for simulating the interaction between the electromagnetic and the thermal domains in predicting the performance of HTS microwave circuits. We showed that, for a continuous wave on a HTS transmission line, the temperature rise due to self-heating enhances the switching from the superconducting state to the lossy normal state. Moreover, the correct choice of substrate, in terms of its thermal properties, plays a fundamental role in spreading the heat and consequently in setting the limiting power. Otherwise, device destruction due to overheating can occur. Additionally, the numerical method presented allows for prediction of the upper threshold power, above which the device exceeds the critical temperature . REFERENCES [1] J. C. Booth, D. A. Rudman, and R. H. Ono, “A self-attenuating superconducting transmission line for use as a microwave power limiter,” IEEE Trans. Appl. Supercond., vol. 13, no. 2, pp. 305–310, Jun. 2003. [2] W. T. Weeks, L. L. Wu, M. F. McAllister, and A. Singh, “Resistive and inductive skin effect in rectangular conductors,” IBM J. Res. Dev., vol. 23, pp. 652–660, Nov. 1979. [3] D. M. Sheen, S. M. Ali, D. E. Oates, R. S. Withers, and J. A. Kong, “Current distribution, resistance, and inductance for superconducting strip transmission lines,” IEEE Trans. Appl. Supercond., vol. 1, pp. 108–115, Jun. 1991. [4] T. Dahm and D. J. Scalapino, “Theory of intermodulation in superconducting microstrip resonator,” J. Appl. Phys., vol. 81, no. 4, pp. 2002–2012, Feb. 1997. [5] C. Collado, J. Mateu, and J. M. O’Callaghan, “Analysis and simulation of the effects of distributed nonlinearities in microwave superconducting devices,” IEEE Trans. Appl. Supercond., vol. 15, no. 1, pp. 26–39, Mar. 2005. [6] K. T. Leong, J. C. Booth, and S. A. Schima, “A current-density scale for characterizing nonlinearity in high-Tc superconductors,” IEEE Trans. Appl. Supercond., vol. 15, no. 2, pp. 3608–3611, Jun. 2005. [7] F. P. Incropera, D. P. DeWitt, T. L. Berqman, and A. S. Lavine, Fundamentals of Heat and Mass Transfer. Hoboken: Wiley, 2002. 88 Abstract— This work evaluates the microwave nonlinear properties and tuning at RF frequencies of ferroelectric Ba0.3Sr0.7TiO3 thin-films by on-wafer measurements of the second and third-order harmonics and intermodulation products of several coplanar transmission lines as a function of DC bias. We analyze these data by use of a large-signal circuit model to obtain the nonlinear distributed capacitance as a function of voltage. Whereas the model with an electric field-dependent dielectric constant for the ferroelectric material is able to predict the tuning when a DC voltage is applied and also the generated harmonics, it disagrees with the measured third-order intermodulation products at frequencies 21-2 and 22-1. Nonlinear effects due to self-heating mechanisms are also considered, along with an additional phenomenological source of nonlinearities, to model all the observables. Interactions between the ferroelectric and self-heating components give rise to a deep null in both the measured and modeled IMD component. Index Terms— Circuit modeling, ferroelectric films, largesignal model, microwave measurements, nonlinear response, thermal effects. I. INTRODUCTION ERROELECTIC thin-film materials are attractive for many microwave applications, such as those requiring frequency agility, phase shifting, harmonic generation or pulse shaping [1], due to the possibility of changing the material’s permittivity with an applied electric field. A detailed understanding of this tuning mechanism at DC, as well as at RF frequencies, is crucial to assess critical issues such as maximum tuning speed or spurious signal generation in ferroelectric-based devices. In our previous work [2], we presented measurements and analysis of the microwave nonlinear response in a Ba0.3Sr0.7TiO3 (BSTO) thin-film on a LaAlO3 (LAO) substrate. Those measurements involved broadband microwave characterization of the thin film up to 40 GHz with an applied 0 to 40 V DC bias voltage. Such measurements yielded the distributed capacitance of the structures and the tuning (change in permittivity with DC bias voltage) properties for a DC bias. In [2], we also reported on measurements of the third harmonic for a fixed fundamental frequency at 2 GHz and zero DC voltage, yielding a small-signal model that was able to model the third-harmonic generation around zero applied DC voltage, demonstrating the tuning capability of the device on a nanosecond time scale. These results gave limited insight into the time response and nonlinear effects of the ferroelectric transmission lines, and as discussed in [2], further experiments were required for a complete characterization of the ferroelectric thin-films. For example, measurements of the IMD products in addition to third harmonic provide information about the dynamics of the system. This work aims to provide a full characterization and modeling of the nonlinear effects produced in tunable ferroelectric devices. To that end, we measured the complete set of second-order and third-order spurious signals generated in several BSTO transmission lines, in the paraelectric state, at room temperature. A large-signal model has been proposed to explain all observed spurious signals. The full characterization requires driving the BSTO transmission lines with a two-tone signal at frequencies f1 and f2 and measuring, at the output, the second harmonics (2H), third harmonics (3H), second-order intermodulation products (2IMD) at frequencies f1+f2 and the third-order intermodulation products (3IMD) at frequencies 2f1+f2, 2f2+f1, 2f1-f2 and 2f2-f1. These measurements have been carried out for several values of applied DC bias voltage, and also the values of f1 and f2 have been varied to sweep the frequency difference between the two drive tones (  f = f2 - f1) between 2 Hz and 1 GHz. To describe the measured results, we built a large-signal circuit model in incremental steps to include all the experimental results. We started by using a simple nonlinear equivalent circuit, which models a nonlinear transmission line by cascading many resistance, inductance, capacitance and conductance (RLCG) elemental segments, where the distributed capacitance C(v) depends on the voltage dropped at each section. This circuit model, or a variation of it, has been used to explain the nonlinear distributed effects in superconductors [3] and ferroelectrics [4], and has been successfully applied to generate nonlinear models of more sophisticated devices, such as filters [5]. This model is able to predict the second and third harmonic, the second-order intermodulation products, and the third-order intermodulation products at higher frequencies 2f1+f2 and 2f2+f1. However, it fails to predict the in-band thirdorder intermodulation products 2f1-f2 and 2f2-f1. These spurious signals are significantly higher in amplitude than the low bound given by the large-signal model. This anomalous imbalance, which depends on the tone spacing  f, reveals the existence of additional sources of nonlinearities due to selfAPPENDIX A TO CHAPTER III - A Large-Signal Model of Ferroelectric Thin-Film Transmission Lines Eduard Rocas, Student IEEE Member, Carlos Collado, Senior IEEE Member, Jordi Mateu, Senior IEEE Member, Nathan Orloff, J.M. O’Callaghan, Senior IEEE Member, James C. Booth F 95 heating effects [6], [7]. The circuit model is then generalized in the same way as recently done, for metallic transmission lines on dielectric substrates [7], to account for the self-heating-induced nonlinearities. This is accomplished by defining a nonlinear voltage and temperature-dependent distributed capacitance. The resultant circuit models predict, for first time, all the nonlinear manifestations occurring in the nonlinear ferroelectric transmission line, including asymmetries between 2f1-f2 and 2f2-f1 due to an interaction of two sources of nonlinearity. Therefore, it can help to provide a better understanding of the nonlinear electro-thermo-dynamics inherent in ferroelectric materials. In addition, this model, as used with superconductors in [8] may be useful for predicting the nonlinearities of more complex ferroelectric devices under DC voltages or even devices that are driven by complex modulated signals with high peak power levels in their envelopes, like those in Wideband Code Division Multiple Access (WCDMA) systems [5]. II. LARGE-SIGNAL MODEL OF FERROELECTRIC TRANSMISSION LINES In this section, we analyze the harmonics and intermodulation distortion occurring in a ferroelectric transmission line due to its inherent tunability, by analyzing the equivalent circuit model of Fig. 1 (self-heating effects will be considered in Section V). The circuit model corresponds to an elemental segment of a whole transmission line, where L, R, C and G are the inductance, resistance, capacitance and conductance per unit length [9]. A. Inherent nonlinear response of ferroelectrics In a transverse electric and magnetic mode (TEM) ferroelectric transmission line, where the dielectric permittivity depends on the applied electric field,  (E), the C and G become voltage-dependent; i.e., C(v) = C0+  C(v) and G(v) = G0+  G(v). Most of the previously reported work [2],[10], based on nonlinear experiments around zero DC voltage, suggest a quadratic voltage-dependent capacitance, which is appropriate for a ferroelectric material in the paraelectric state. This work considers a slightly more complicated function in order to describe the voltage-dependent capacitance for a wide range of applied DC voltage as 0 2 0 0 2 20 )( )( vvifvCvCCvC vvifvCCvC ba   , (1) where C0 is the linear part of the distributed parameters,  C2 is a scaling constant that sets the strength of the quadratic dependence at zero DC bias and 0 2v C CC a b   (2) is set to keep the continuity of C(v) at voltage v0. The proposed function represents a smooth transition, captured by v0, from a square dependence at low voltages to first-order and second-order dependence at higher voltages. Although the voltage dependence of the distributed conductance would also produce spurious signals, previous work demonstrates that the predominant contribution for the nonlinear response is due to the voltage-dependent capacitance [11]. Therefore, the value  C2 may be related to the harmonics generated at zero DC voltage, and the parameters Ca and v0 must fit the second harmonic and third harmonic when a certain DC bias voltage v > v0 is applied. B. Harmonics and intermodulation at given DC-bias voltage Harmonics and intermodulation products can be calculated in passive transmission lines by use of the nonlinear telegrapher’s equations [3]. From the equivalent circuit of Fig. 1, in which we show the general case of a ferroelectric material with nonlinear capacitance and conductance, the telegraphers’ equations may be written as:       . , )( Ri dt Lid dz dv vvG dt vvCd dz di   (3) The nonlinear contribution due to the voltage-dependent capacitance can be modeled as a nonlinear current as follows:   dt tzvtzvCd dz dinl ),()),((  , (4) and by use of the procedure detailed in [3], the current generated by the nonlinear effects in the frequency domain for a given frequency fi can be written as   ii fnl ftzvtzvCFj dz dI i),,()),(( ,  , (5) where v(z,t) is the voltage distribution in the transmission line and F( g(t) , fi ) is the Fourier transform of function g(t) at fi. For intermodulation experiments, where the transmission line is driven with a two-tone signal at frequencies f1 and f2, the voltage distribution along the line may be written as       12 12 , Re j t j t rf z t V z e V z ev      , where V1(z) and V2(z) are the voltage distribution along the line at frequencies f1 and f2, respectively. At zero DC bias, vDC=0, the first equation in (1) generates only third harmonics and third-order intermodulation products, according to (5):   irfi fnl ftzvCFj dz dI i),,( 3 2 ,  , (6) which, for third harmonics and third-order IMD, results in Fig.1. Telegrapher’s model of an elemental segment of a ferroelectric transmission line of length dz. 96       ),()( 4 3 |2),,( )()( 4 3 |2),,( )( 4 1 |3),,( 2 20 3 2 *2 20 3 2 3 20 3 2 zVzVCfftzvCF zVzVCfftzvCF zVCftzvCF jkvjkrf jkvjkrf kvkrf DC DC DC       (7) where k,j = 1,2 and k  j. By use of the nonlinear current generators (6) at each spurious frequency, the dissipated power at the load can be found as shown in [11]. For example, the power delivered to the load at 2ω1+ω2 is:     212 2 2 2 1 2 2 2 2102 ,2 16 9 |2121 ffKlPPCP ffvff DC    , (8) where P1 and P2 are the available power of the incident sources, l is the length of the transmission line and K2f1+f2(f1, f2) accounts for the linear effects of the line, including losses, dispersion and impedance mismatch. The K2f1+f2 (f1, f2) term can be obtained from the linear distributed circuit parameters as shown in [12]. In analogy to (7), we can use (5) and (6) to obtain the output power for other spurious signals at IMD and third-harmonic frequencies. When a DC voltage vDC is applied, the 3IMD and 3H can be found following the previous procedure by use of vDC+vrf (z,t) in (5). Note as well that for vDC > v0, the second equation in (1) should be used instead of the first. Additionally, in this case a second harmonic and secondorder intermodulation term will appear as well. The resulting nonlinear Fourier transform coefficients at frequencies 2fk and f1+f2 for vDC < v0 is         ),()(3|,),( )( 2 3 |2,),( 21221 3 2 2 2 3 2 0 0 zVzVVCfftzvVCF zVVCftzvVCF DCvvrfDC kDCvvkrfDC DC DC     (9) where k = 1,2, whereas for vDC > v0 the previous equation can be read as                 ).()(3|,,, )(3 2 1 |2,,, 2121 32 2 32 0 0 zVzVCVCfftzvCtzvCF zVCVCftzvCtzvCF aDCbvvba kaDCbvvkba DC DC     (10) The expressions above represent the DC and RF voltage dependence of the spurious signals generated in a ferroelectric transmission lines as a function of the circuit parameters  C2,  Ca ,  Cb and, consequently, as a function also of the electricfield dependence of the permittivity. III. SAMPLES AND BROADBAND MICROWAVE MEASUREMENTS Extraction of the nonlinear terms  C2,  Ca ,  Cb from the measurements of the spurious signals requires characterization of all linear distributed circuit parameters defining the circuit model of Fig. 1, that is, C0, G0, L and R. In what follows we present the samples and test structures we have measured, and outline the characterization procedure to obtain C0, G0, L and R. A. Samples and structures Our measurement structures consist of 300 nm thick Au coplanar waveguide transmission lines patterned on a 400 nm thick BSTO thin-film grown by pulsed-laser deposition on a 16x16 mm2 LAO substrate. The center conductor line width of the transmission lines is 20 m, and the gap spacing is 10 m. Measurements of different-length transmission lines, L1 = 4.20 mm and L2 = 9.93 mm, will verify the distributed nature (length dependence) of the nonlinear effects predicted by (8). Identical structures are also patterned on a bare LAO substrate. The BSTO has a curie temperature (TC) of 230 K, which for room temperature measurements the material is in the paraelectric state. B. Linear distributed circuit parameters We performed an initial multiline thru-reflect-line (TRL) calibration [13] in a reference set of CPW transmission lines fabricated on a non-dispersive and low-loss substrate. This reference calibration set consisted of gold lines on a sapphire substrate, along with embedded on-wafer lumped resistors to obtain the transmission line capacitance per unit length, for the reference calibration set, by use of the procedure detailed in [14]. From the complex propagation constant obtained and the distributed capacitance of the reference calibration set, we determined the characteristic impedance of the transmission lines in the reference sample [15]. Then, we can use the calibration comparison method [16] to estimate the characteristic impedance of transmission lines fabricated on lossy and/or dispersive dielectrics. We used this method to obtain the characteristic impedances of the transmission lines on the bare LAO substrate and on the BSTO thin-film. By combining the characteristic impedance with the propagation constant obtained from the multiline TRL calibration performed on the sample and on the bare substrate, we extracted the linear, frequency-dependent, distributed parameters C0, G0, L and R, for the lines on BSTO thin-film and on the bare LAO substrate. To verify the results we measured the DC resistance of the transmission lines on bare LAO and BSTO thin film and obtained the conductivity of the gold conductors by use of the measured cross-section of the center conductor. The conductivity obtained was used to calculate the frequencydependent inductance and resistance per unit length for the geometries under consideration, by means of a commercial quasi-static finite element analysis. The simulated L(f) and R(f) showed good agreement with the L(f) and R(f) obtained above over the entire frequency range. IV. IMD AND HARMONICS MEASUREMENT SET-UP As outlined in the introduction, a full characterization of the ferroelectric nonlinear response requires accurate measurements of all second-order and third-order spurious signals generated when a two-tone signal feeds the test transmission line. The measurement set-up used to achieve this is shown in Fig. 2, and has been previously used to characterize the nonlinear response of superconductors [17], STO ferroelectric films [11] and self-heating effects in copper and gold transmission lines [7]. 97 This measurement setup consists of two sources at f1 and f2. Each source feeds a cascaded amplifier, isolator and low-pass filter (LPF) before the signal is split equally into two branches by use of a power divider. The LPF plays an important role in the system performance, and is discussed in detail below. Two of the branches having f1 and f2 are combined together through a two-port combiner to feed the device under test (DUT) with the two-tone signal. The two other branches of the measurement system go through a variable attenuator and a phase-shifter. Then the output of the DUT, which contains the fundamental tones as well as the nonlinear products to be measured, is combined with the reference branches through a three-port combiner. Each reference branch is manually adjusted to feed the three-port combiner (indicated by port 3 and port 4 of Fig. 2) with the same amplitude and opposite phase as the fundamental tones coming from the DUT (through port 2, in Fig. 2). By use of this technique, we may attenuate the fundamental tones by up to 60 dB, which dramatically increases the measurement dynamic range at the spectrum analyzer. This measurement configuration also reduces the phase noise of the fundamental tones, which otherwise could mask the spurious signals at 2f1-f2 and 2f2-f1, for small tone spacing. Since the spurious signal level is usually much below that of the fundamental signal, a very linear measurement system is required to minimize the effects of baseline nonlinearities. Besides the linearity of the spectrum analyzer, which is achieved by cancelling the fundamentals, we need to feed the DUT and the reference branches with low-distortion tones. This is particularly significant in this setup, where the impedance mismatch of the reference branches may produce reflections of the incident and spurious signals. We achieve this low distortion by placing the fundamental tones just below the cut-off frequency of a high-performance LPF to suppress the second and third harmonics generated by the source, amplifier and/or isolator. The measurement setup described above operates over the frequency range from 2 GHz to 8 GHz for the fundamental signals. We characterize the linear and nonlinear response of the system itself, within the frequency range, by connecting port 1 directly to port 2 (see Fig. 2), omitting the probes and the sample. The system itself shows no spurious signal above noise level, which is around –100 dBm, for incident powers up to +20 dBm. Characterization of the measurement system, including the probes, requires measurements of linear transmission lines or well-characterized nonlinear transmission lines as a reference. Measurements of CPW lines on a bare LAO substrate are detailed in the following section. V. MEASUREMENT RESULTS. LARGE-SIGNAL MODEL This section presents the spurious signal measurements outlined in the introduction. We start by performing measurements on the bare LAO transmission lines. The aim of measuring these transmission lines is twofold: first, it demonstrates the high dynamic range of the measurement setup of Fig. 2. Second, as the transmission line geometries are the same as the ones on BSTO thin film, measurements of the 3IMD at 2f1-f2 and 2f2-f1 will allow us to explore the existence of heating effects produced due to conductor losses [7]. These measurements will then be used to complement the equivalent circuit model of Fig. 1 to model heating effects. The new circuit model will allow us to explain all the spurious signal manifestations occurring on the BSTO transmission lines. A. Nonlinear effects in LAO substrate No measurable harmonic signals have been measured down to the noise floor, around –100 dBm, neither IMD signals at frequencies f1+f2 and f2+f1, 2f1+f2 and 2f2+f1, when the patterned transmission lines are fed with two tones centered at 6 GHz over a wide range of tone spacing. Fig. 3, on the other hand, shows the measured IMD at frequencies 2f1-f2 and 2f2-f1 as a function of the tone spacing  f, which clearly displays self-heating effects due to transmission line conductive losses [7]. This occurs because the dissipated power due to the finite metal conductivity (dielectric losses in the LAO substrate are negligible) causes the temperature of the transmission line to rise according to the thermal impedance of the structure. Therefore, the distributed resistance R changes due to effect of the temperature variations on the temperature-dependent resistivity, which can be modeled as shown in the equivalent circuit of Fig. 4 [7]. This self-heating mechanism generates 3IMD, which is higher as the envelope frequency of the combined input signal decreases, due to the slow dynamics of heat propagation through the substrate. The frequency dependence of the 3IMD Fig. 2. Diagram of the nonlinear measurement setup. Fig. 3. Simulations (continuous line) and measurements of third-order intermodulation products 2f1 - f2 (circles), 2f2 - f1 (squares) of L2 = 9.33 mm transmission line on LAO substrate. For  f >1 MHz simulation and measurements disagree because the baseline intermodulation level of the system is around -100 dBm for this measurement. 98 level is due to the thermal impedance of the transmission line. This allows us to obtain the frequency-dependent thermal impedance, and to relate dissipation to temperature rise for the given geometry and substrate. This thermal impedance will then be also used in the model for the BSTO lines, where we will assume that the heat flow through the substrate is not significantly affected by the very thin BSTO layer. Note that for  f >1 MHz, simulation and measurements disagree because the baseline intermodulation level of the system is around -100 dBm for this measurement. Fig. 3 also shows, as solid lines, the envelope frequency dependence obtained by use of the circuit model of Fig. 4, showing very good agreement between measurements and simulation. B. Measurement of 2H and 3H as a function of the DC bias voltage on BSTO transmission line These experiments measure all spurious signals occurring in the L1 BSTO transmission line at 2H, 3H, 2IMD and 3IMD, when a two-tone signal centered at 6 GHz with a fixed tone spacing of  f = 14 kHz feeds the transmission line. The input power of the two tones is balanced and swept from -8 dBm to +22 dBm. This experiment has been performed for several DC bias voltages applied: 0.005 V, 10 V, 20 V and 30 V, selected in accordance with previous measurements on ferroelectrics [11] to capture the dependence of the distributed capacitance on voltage. Figures 5 and 6 show the second-order distortion measurements, at 2f1, 2f2 and f1+f2, and third-order distortion, at 3f1, 3f2, 2f1+f2 and 2f2+f1 respectively, for the DC bias voltages mentioned above. Fig. 5a shows 2H for 0.005 V and 10 V DC bias and demonstrates that the second harmonic signals dramatically increase as the bias voltage is increased from zero, which is consistent with the circuit model proposed in section II. Also, note however that a very small deviation of the bias voltage from zero (0.005 V) produces measurable second harmonic signals. Note that the baseline nonlinear level of the system, in Figs. 5 and 6, is determined by the input power at the spectrum analyzer, and thus depends on the line loss and length. Moreover, a small tendency to saturation is observed at high powers which is caused by the power amplifiers of the measurement system. We use the measurements above to extract the nonlinear parameters  C2,  Ca and v0 along with the previous obtained linear parameters L, R, C0 and G0. These parameters are:  C2 = 1.610-14 [F/V2],  Ca = 410-14 [F/V], v0 = 5 V, which describe a voltage-dependent capacitance. The simulated results for these values are also plotted as solid lines in Figs. 5 and 6, showing good agreement between simulations and measurements for the entire power range. Fig. 7 uses these values to plot the relative capacitance deviation  C/C0 as a function of the voltage, which is consistent with previous results [2], and indicates the degree of tunability in ferroelectrics that can be achieved on nanosecond time scales. At this point, we should mention that the nonlinear circuit model obtained from Figs. 5 and 6, fails, as expected, to explain the measured 3IMD at 2f1-f2 and 2f2-f1, primarily due to the existence of self-heating effects in the BSTO sample. Fig. 4 Equivalent circuit model considering the self-heating effects. The temperature oscillations produce dynamic changes in the metal conductivity, and give rise to third-order intermodulation distortion. Fig. 5 Output power measured at 2f1 (squares), 2f2 (circles) and f1+f2 (diamonds) and simulations of second harmonics (continuous lines) and f1+f2 (dotted line) for several DC voltages; Fig 5a plots 0.005 V (blue) and 10 V (red) and Fig. 5b plots 20 V (blue) and 30 V (red). (a) DC=0.005 V and DC=10 V (b) DC=20 V and DC=30 V Separation between tones (Hz) 99 C. 3IMD at 2f1-f2 and 2f2-f1: self-heating effects In order to complete our circuit model, we include selfheating effects by merging the circuit of Fig.1 with the circuit of Fig. 4. We performed IMD and harmonics measurements in lines L1 and L2, keeping the central frequency constant and sweeping the envelope frequency  f from 2 Hz to 1 GHz for vDC = 0. This broad envelope frequency range gives enough information to discern between memory effect nonlinearities, caused by self-heating [6][7], and non-memory effects caused by inherent, ferroelectric-type, nonlinearities. As done in [11], we have verified that a nonlinear capacitance scales better with frequency, for the inherent nonlinearities, than a nonlinear conductance. Fig. 8 shows the measured harmonics and intermodulation distortion as a function of the envelope frequency  f. These measurements reveal that, for high envelope frequencies,  f > 106, the 3IMD at 2f1-f2 and 2f2-f1 is almost 10 dB higher than the 3IMD at 2f1+f2 and 2f2+f1, and the 3IMD at 2f1-f2 and 2f2-f1 dramatically increases as  f decreases. In addition, there is a dramatic attenuation of the lower side-band IMD signal for envelope frequencies just below 105 Hz. As an initial step, we used the temperature-dependent resistance of the metallic electrodes obtained from the measurements on bare LAO substrate transmission lines in our combined model. Note that this assumption makes sense, because the conductors for the BSTO sample are identical to Fig. 6 Output power measured at 3f1 (squares), 3f2 (circles) , 2f1+f2 (diamonds) and 2f1+f2 (triangles), and simulations of third harmonics (continuous lines) and high frequency third-order products (dotted lines) for several DC voltages; Fig 6a plots 0.005 V (blue) and 10 V (red) and Fig. 6b plots 20 V (blue) and 30 V (red). (diamonds). Fig. 7. Relative change in the distributed capacitance obtained by use of (1), from the nonlinear measurements shown in Figs. 5 and 6. Fig. 8. Measured third-order IMD products at 2f1-f2 (squares), 2f2-f1 (circles), 2f1+f2 (diamonds) and 2f2+f1 (triangles) of the L1 tranmission line. Simulations are plots with blue continuous line, dashed-dotted red line, dotted green line and dashed magenta line respectively. Fig 8a shows simulations considering only a temperature-dependent distributed resistance. Fig 8b considers a temperature-dependent distributed capacitance along with the temperature-dependent resistance. (b) (a) (b) DC=20 V and DC=30 V (a) DC=0.005 V and DC=10 V 100 the LAO transmission lines. However, dielectric losses are included in the model, and they also contribute to the temperature rise due to dissipation. Because of this, the spatial distribution of the generated heat along the LAO-BSTO coplanar waveguide differs from that along the bare LAO coplanar line. In spite of this, we can use the same thermal impedance for the two transmission lines because the BSTO film is very thin with negligible thermal resistance and the thermal properties of the LAO substrate dominate the heat diffusion irrespective of the origin of the dissipation. Therefore, the negligible impact of the BSTO thin-film on the heat path does not affect the obtained temperature derivative for the capacitance. Fig. 8a shows calculations of the low-frequency 3IMD along with measured data arising from a temperaturedependent gold resistivity. The agreement could not be improved by changing the thermal impedance. By means of simulations, we see that a specific length of transmission line could be fitted, but the resultant model does not scale properly with line length. The two lines are impedance-mismatched, and the voltage standing-wave ratio is high, which implies different standingwave patterns for lines of different lengths. We have verified that the measured 3IMD, for the shortest line L1 in Fig. 8a, can only be explained by means of a temperature-dependent capacitance or conductance. In order to improve the agreement between the circuit model and data of Fig. 8, we must include in the circuit model a nonlinear term describing the change in the distributed capacitance and/or conductance with temperature. Additional measurements of the scattering parameters at 40 ºC allows us to neglect the temperature-dependent conductance as a source of nonlinearity, so we include in the model an additonal term  CT ·T to the capacitance described in (1), where )()()( vTCvCvC TT  . (11) The complete model, along with an additional phenomenological nonlinearity explained in the following subsection, is shown in Fig. 9. The transmission line circuit model is implemented, in a distributed manner, as a cascade of cells. Each cell includes the electromagnetic domain, which contains the linear and nonlinear distributed circuit parameters, and the thermal domain, which models the thermal impedance. The dissipation ocurring in the electromagnetic domain as a result of conductive and dielectric losses is coupled to the thermal domain to obtain the temperature rise. By doing this, the temperature-dependent capacitance can be properlly modeled, along wth the temperature-dependent resistance. The simulated results with this new term set to  are shown in Fig. 8b. This temperaturedependent term allows us to describe the 3IMD at 2f1-f2 and 2f2-f1 for  f < 1 KHz. However it does not explain the cancellation point around  f = 10 KHz, with a cancellation of more than 40 dB, nor the IMD values for  f > 10 KHz. This fit therefore suggests the existence of an additional mechanism of nonlinearity. D. 3IMD at low frequencies: Phenomenological model This subsection introduces an additional new term for the voltage-dependent capacitance. Although the origin of this mechanism is not yet fully understood, we suggest introducing the frequency-dispersive permittivity phenomena [18]. This new term is required to be low-pass filtered and quadratically dependent on the voltage, as indicated below: filtered mTmT vCTCvCvC 2 ,)()(  (12) The complete circuit model of Fig. 9 is then used to fit all spurious signals generated in the BSTO transmission line. Fig. 10 displays the measured and simulated results for the measured IMD in the two transmission lines L1 and L2, showing very good agreement for  Cm=310-3 F/V2. These results demonstrate that the self-heating mechanisms and the thermal model reproduce the asymmetry between 2f1-f2 and 2f2-f1. We have also verified that the cancellation does not change when changing the input power, which means that the unknown effects and the thermal effects follow the same power-law dependence. In addition, when no heating effects exist, for instance when  f is large enough or in cryogenic measurements on STO transmission lines [11], this new term also successfully models both the spurious signals at 2f1-f2 and 2f2-f1 and the spurious signals at 2f1+f2 and 2f2+f1. Table I summarizes the nonlinear coefficients of the obtained model: Fig. 9. Complete circuit model implementation of an electro-thermal cell. The electromagnetic domain contains the linear and nonlinear distributed parameters, and the thermal domain contains the thermal impedance. TABLE I NONLINEAR COEFFICIENTS OF THE LARGE-SIGNAL MODEL Nonlinear term Value  C2 1.610-14 F/V2  Ca 410-14 F/V v0 5 V  CT -4.410-12 F/K  Cm 310-3 F/V2 101 VI. CONCLUSION This work has demonstrated a comprehensive procedure for evaluating and modeling the nonlinear response of ferroelectric transmission lines. 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Duffo, J.M O'Callaghan, “General electromagnetic simulation tool to predict the microwave nonlinear response of planar, arbitrarily-shaped HTS structures”, IEEE Transactions on Applied Superconductivity. vol. 11, no. 1, pp. 399-402, Mar 2001 [9] D. M. Pozar, Microwave Engineering, John Wiley & Sons, Inc, 1998 [10] J.C. Booth, R. H. Ono, I. Takeuchi, and K.-S. Chang, “Microwave frequency tuning and harmonic generation in ferroelectric thin film transmission lines,” Appl. Phys. Lett., vol. 81, no. 4, pp718-720, 2002. [11] J. Mateu, J.C. Booth and S.A. Schima,”Frequency Tuning and Spurious Signal Generation at Microwave Frequencies in Ferroelectric SrTiO3 Thin-Film Transmission Lines”, IEEE Trans. Microwave Theory and Tech, Vol. 55, No. 2, pp. 391 – 396, Feb. 2007 [12] J. Mateu, C. Collado, N. Orloff, J.C. Booth, E.Rocas, A. Padilla and J. M. O'Callaghan, “Third order intermodulation distortion and harmonic generation in mismatched weakly nonlinear transmission lines”, IEEE Trans. Microwave Theory and Tech. Vol. 57, No. 1, pp. 10-18, 2009. [13] R.B. Marks, “A Multiline method for Network Analyzer calibration”, IEEE Trans. Microwave Theory and Tech. vol. 39, no 7, pp. 1205-1215, July 1991. [14] D. F. Williams and R. B. Marks, ”Transmission line capacitance measurements”, IEEE Microwave and Guided wave Lett., vol., 1. no.9, pp. 243-245, Sept. 1991. [15] R. B. Marks, D. F. Williams ”Characteristic Impedance Determination Using Propagation Constant Measurement”, IEEE Microwave and Guided wave Lett., vol., 1. no.6, pp. 141-143, June 1991. [16] D. F. Williams and R. B. Marks, ”Accurate Transmission Line Characterization”, IEEE Microwave and Guided wave Lett., vol., 3. no.8, pp. 247-249, Aug. 1993. [17] J. Mateu, J. C. Booth, B. H. Moeckly, “Frequency Dependence of the Nonlinear Response in YBaCuO transmission lines”, Applied Physics Letters, Vol. 90, No. 12512, pp 1-3, Jan. 2007. [18] D. Damjanovic, S. S. N. Bharadwaja, N. Setter, “Toward a unified description of nonlinearity and frequency dispersion of piezoelectric and dielectric responses in Pb(Zr, Ti)O3”, Materials Science and Engineering B, Volume 120, Issues 1-3, The 8th International Symposium on Ferroic Domains (ISFD-8, 2004), 15 July 2005, Pages 170-174. Fig. 10 Measured third-order IMD products at 2f1-f2 (squares), 2f2-f1 (circles), 2f1+f2 (diamonds) and 2f2+f1 (triangles) of the L1 (a) and L2 (b) transmission lines. Simulations are plot with blue solid line, dasheddotted red line, dotted green line and dashed magenta line respectively. (a) Line L1 (b) Line L2 102  Abstract—In this work we present the electro-thermomechanical constitutive relations, expanded up to the thirdorder, for a BAW resonator. The obtained relations are implemented into a circuit model, which is validated with extensive linear and nonlinear measurements. The mathematical analysis, along with the modeling, allows one to explain, for the first time, all observable effects in a BAW resonator by use of a unified physical description. Moreover, the term that is responsible for the second harmonic and the frequency shift with DC voltage is showed to be the same. Finally, we unveil the impact of an applied DC voltage on the acoustic losses. Index Terms—Nonlinearities, Self-Heating, Temperature, Thermal Effects, Intermodulation Distortion, Second Harmonic, Bulk Acoustic Wave Resonators. I. INTRODUCTION ULK Acoustic Wave (BAW) technology is accomplishing its market expectations with a promising future in the field of microwave filters for mobile applications. Several advantages, including miniaturization, power handling and frequency operation, make BAW filters preferable among other existent technologies [1][2][3]. But, in spite of the current success of this technology, there is still a lack of a complete understanding of all observable effects in BAW resonators, what could eventually be crucial to push this technology beyond its current limits. In particular, harmonic generation due to intrinsic effects, intermodulation distortion and detuning due to temperature changes are being a matter of active research [4]. The intrinsic nonlinear behavior of the piezoelectric phenomena causes harmonics generation, intermodulation distortion, and frequency detuning with an applied DC voltage. Although recent work has shed light to the possible origin of these nonlinearities [5], none is concluding, neither none successfully models all those undesired effects with a unified nonlinear description. On the other hand, temperature has always been a matter of special attention with regard to BAW resonators. Temperature induces changes to the material properties and dimensions that should ideally be compensated, generally by use of additional compensation layers, if the frequency drift is to be minimized [6]. An accurate understanding and modeling of the material temperature derivatives is crucial to resize properly the compensation layers. Several authors have faced this problem [6][7], but none of them has considered the clamping conditions that force one to use the Poisson’s ratio to properly account for the thickness thermal expansion in Solidly Mounted Resonators (SMR). In addition, temperature concerns not only involve the frequency drift in BAW devices, but also the temperature rise due to self-heating, which can limit the power handling of, for example, Film Bulk Acoustic Resonators (FBAR) [8], and/or contribute to the third-order intermodulation distortion [ref]. Reference [10] was the first distributed nonlinear modeling approach to cover intrinsic effects through the integration of nonlinearities in different sections of the circuit model. A similar approach is followed in [5][11], but with a grouping of the nonlinear sources. On the other hand, our previous work [9] proposed a phenomenological approach to cover intrinsic and thermal effects. In this article we propose a rigorous electro-thermomechanical study, based on the expansion of the Gibbs electrical energy function, capable of explaining most of the observable effects within a BAW resonator with a unified physical description. We derive a circuit model from the obtained constitutive equations and validate it with linear and nonlinear measurements on BAW resonators, what allows identifying the dominant terms that produce the frequency shift in temperature with an applied DC voltage, and the generation of harmonics and intermodulation distortion. II. CONSTITUTIVE RELATIONS IN A BAW RESONATOR The working mechanism of a BAW resonator lies on several physical principles that can be explained with the electro-thermo-mechanical constitutive equations and the Newton’s second law of motion. The first group of equations describes the energy coupling between fields associated to each physical domain (electrical, mechanical and thermal as shown in the Heckmann’s diagram in Fig. 1), whereas Newton’s second law states the relation between particle velocity and stress in a medium. A rigorous nonlinear analysis of those constitutive relations allows modeling most of the experimental observables in BAW resonators. APPENDIX B TO CHAPTER III - Electro-thermo-mechanical Model for Bulk Acoustic Wave Resonators Eduard Rocas, Student Member, IEEE, Carlos Collado, Senior Member, IEEE, Jordi Mateu, Senior Member, IEEE, Nathan D. Orloff, Robert Aigner, James C. Booth B 103 We start obtaining the nonlinear electro-thermo-mechanical relations for a piezoelectric material, which can be simplified then for non-piezoelectric dielectrics and metals. We make the choice of independent variables to be strain, S, electric field, E, and temperature,  , in order to obtain equations describing the stress, T, and electric displacement, D, which have a direct representation in the Mason model [12]. Moreover, this selection of independent variables allows having the field derivatives of the material properties on magnitudes that can be easily changed when performing measurements, such as the electric field and the temperature. Appendix I describes the procedure we have followed to obtain the nonlinear constitutive equations by use of a thirdorder Taylor series expansion of the relations between fields plotted in Fig. 1, in a uni-axial notation. The procedure detailed there does not assume approximations other than the series truncation and results in NL EE TEeScT    (1) NL SSE DESeD    (2) NL ESSE rES   , (3) where the nonlinear contribution, with all the constants defined in the Appendix I, are described in (4)-(6)            2 12 2 11 2 10 2 9 2 8 2 7 6 3 5 3 ,3 3 3 765 2 4 2 3 2 2 2 1 ... 6 1 ... 2 1 EESESSSE SEEeSc ESSEEScT E E E NL    (4)            2 3 2 2 2 6 2 7 2 11 2 4 12 3 1 3 9 3 3 273 2 1 2 5 2 2 2 1 ... 6 1 ... 2 1 EESESSSE SESE ESSESED S S NL    (5)          2 2 2 1 2 8 2 6 2 5 2 12 11 3 10 3 3 3 3 147 2 6 2 2 2 2 2 1 ... 6 1 ... 2 1 EESESSSE SESEr ESSESEr ES ES NL    (6) What follows is the simplification of the constitutive relations for the specific types of materials used in commercial SMR type resonators, which usually comprise Aluminum Nitride for the piezoelectric layer, metals for the electrodes and non-piezoelectric dielectrics for passivation, temperature compensation and Bragg mirror implementation. In the following sections the thermal expansion coefficient, cE  , will be used instead of the thermal pressure,  E, through the  E= - cE  E relation, because it is a more commonly used material property. A. Thermal considerations Although we have described the entropy with (3) and (6) for completeness, we will not use those equations because we assume that the electrocaloric effect, the piezocaloric effect, the heat of polarization, and heat of deformation are negligible in the materials under consideration [13]. We will deal instead with the heat equation for the thermal domain, which takes into account the temperature rise and heat propagation through a medium [14]. The heat equation can be written as:              2 2 z c k tp    , (7) where k, cp and  are thermal conductivity, specific heat and density respectively. The change of in-plane dimensions in SMR resonators is dominated by the substrate and this imposes a lateral-clamped condition that especially impacts how the device expands with temperature. For this reason we will consider the change in area negligible [15], and the model will only account for dimensional changes in the vertical direction by use of the Poisson’s ration. B. Constitutive relations for Aluminum Nitride For the specific case of Aluminum Nitride (AlN) several simplifications can be made by use of well-known properties of this material and comparisons between nonlinear measurements and simulations. 1) Assumptions related to the AlN properties We assume that the electrocaloric and pyroelectric effects are negligible [16], what means there is no direct energy exchange, other than dissipation, between the electric and thermal domains. This assumption translates into  S = 0. The AlN properties around room temperature change linearly with temperature [17], so we consider that the second and third-order temperature-dependent coefficients in (4)-(6) are zero, that is  1 =  4 =  1 =  2 =  5 =  8 =  11 = 0. The expansion with temperature has to be taken into account. The BAW resonator is made of thin layers, deposited on a thick Silicon substrate, that expand laterally in Electrical ThermalMechanical Electrothermal effects Electroelastic effects Thermoelastic effects Thermal pressure Thermal expansion Heat of deformation Permittivity Piezocaloric effect Elasticity Heat capacity Piezoelectricity Heat of polarization Piezoelectricity Pyroelectricity Direct piezoelectric effect Converse piezoelectric effect Electrocaloric effect Pyroelectric effect E D S T   Fig. 1. The Heckmann’s diagram shows the electro-thermo-mechanical relations on a crystal. 104 properties provided by the manufacturer and couple the dissipation in their corresponding position of the thermal domain. Fig. 10 shows measurements and simulations of the 3IMD for a tone spacing of 1 KHz. It can be observed that the resulting 3IMD is the result of dissipation in the electric and acoustic domains, both having different frequency dependences. The second measurement we perform is shown in Fig. 11 along with the simulations by use of the model. Measurements and simulations agree to the expected low-pass filter behavior, which is also a good way to observe the thermal time constant of the device, in the order of milliseconds. The disagreement for tone spacing above 100 KHz is because intrinsic thirdorder nonlinearity has not been introduced in the model so far, and is explained in the following subsection. An accurate modeling and understanding of the 3IMD due to self-heating is important because it is responsible for inband signal integrity degradation, resulting, for example, in an increased Error Vector Magnitude (EVM). 2) Intrinsic third-order nonlinearity The constant 3IMD level observed above 100 KHz in Fig. 11 reveals the existence of a contribution arising from intrinsic third-order nonlinearity. Intrinsic nonlinearity, defined as the dependence of material properties on fields other than temperature, generates harmonics and intermodulation distortion levels that do not depend on the envelope frequency. Although one might think that the lower level of intrinsic nonlinearity, when compared to self-heating 3IMD, makes it a negligible effect, the reality is that it can result in important signal quality degradation in a BAW filter. The main nondesired effect of intrinsic nonlinearity is receiver desensitization, which occurs when an intermodulation product, generated by the mixing of the transmitted signal with a jammer signal, falls within the receiver band [29]. This makes the understanding and modeling of intrinsic third-order nonlinearity an important aspect of BAW technology. Third-order nonlinearities in BAW resonators have been less explored than second-order nonlinearities. The BAW resonators used in this work show no measurable thirdharmonic with the current measurement setup, therefore only the frequency dependence of the intrinsic 3IMD can be used to unveil the dominant third-order terms [9]. The candidate terms to dominate on the 3IMD level are , , ,  7 and  9. We perform two-tone measurements with tone spacing of  f=1 MHz, to ensure that intrinsic nonlinearity dominates above self-heating effects, and sweep the central frequency as previously done in Fig. 10. It is found by simulation that the contribution of ( )=-18.5 is needed to reproduce the frequency dependence of the measurements, as show in Fig. 11. The obtained nonlinear term is the third-order elasticity. With the nonlinear terms obtained from the frequency dependence for  f=1 MHz we expect the model to be complete and able to reproduce the 3IMD level stabilization observed in the measurements. Figure 13 shows the agreement between measurements and simulations when intrinsic thirdorder nonlinearity is introduced in the model. 1.75 1.8 1.85 1.9 1.95 -90 -85 -80 -75 -70 -65 -60 -55 Frequency (GHz) Intermodulation distortion (dBm) Fig. 12. Measurements and simulations of the low-frequency third-order intermodulation distortion response for a tone spacing of  f=f2-f1=1 MHz. For this tone spacing, the 3IMD is dominated by intrinsic nonlinearities. Dashed lines with rectangles and circles represent measurements of 2f1-f2 and 2f2-f1 respectively. Solid line and dashed line represent simulations of 2f1-f2 and 2f2-f1 respectively. -65 -60 -55 -50 -45 -40 Intermodulation distortion (dBm) 103104105106 Separation between tones (Hz) Fig. 11. Measurements and simulations of the low-frequency third-order intermodulation distortion response for a tone spacing ranging from 300 Hz to 1 MHz. The central frequency is kept constant at f0=f1+  f/2= f2-  f/2=1.845 GHz. The simulations include the self-heating effects. Dashed lines with rectangles and circles represent measurements of 2f1-f2 and 2f2-f1 respectively. Solid line and dashed line represent simulations of 2f1-f2 and 2f2-f1 respectively. 111 V. DISCUSSION AND CONCLUSION We have presented a unified electro-thermo-mechanical description of the constitutive relations in a BAW resonator. The study lies on the expansion of the Gibbs electrical energy function and provides useful relations that allow one to truly relate all the observables to their physical origins. To do that we translate the obtained equations into a circuit model that is used, along with measurements on BAW resonators, to characterize the linear and nonlinear terms. The obtained model is able to predict the BAW resonator behavior on temperature, its self-heating induced nonlinearities, and the harmonics and intermodulation distortion generation, as well as the frequency shift with DC voltage. In fact, the results point that the frequency shift with DC voltage and the second harmonic generation arise from the same nonlinear term  5. The mathematical derivation in Appendix I states that  5 can be either understood as a strain-dependent piezoelectric coefficient or an electric-field dependent elasticity, due to the Maxwell Relations for energy conservation. This can be explained from an atomistic perspective, because on the zero electric field and zero strain situations the atoms are in electrostatic-mechanical equilibrium. In other words, the position of each atom is given by the charge attraction and mechanical elasticity forces, given an isothermal situation. Any variation from the equilibrium, by an applied strain or electric field, translates into an equivalent strain-dependent piezoelectric coefficient or an electric-field dependent elasticity. This means that when a deformation occurs, the atoms gradually lose their relative distance between them, because of internal electrostatic-mechanical forces redistribution. In other words, the dipole moment depends on the deformation in the vertical direction because the relative distances between atoms change with the deformation. This translates into a strain-dependent piezoelectric constant. The principle above also gives rise to an electric fielddependent elasticity. The elasticity is the material property that relates a stress with a deformation, so as we gradually apply an electric field we change the electrostatic forces between atoms. This fact redistributes the forces equilibrium, so that the effective elasticity changes. This explanation is reinforced by the fact that not only the frequency shift with DC voltage is consistent with the change on the real part of the elasticity, but the DC voltage also induces a change in the imaginary part of the elasticity, which is related to the acoustic loss. For this reason, we have also addressed the impact of DC voltage on acoustic losses. With all these considerations, the obtained nonlinear relations are as follows:               E c Kj c E c S c c cc E D EEE E E NL          2 1 ... 26 1 5 65 2 3 (37)           e S e eeNL 75 2 1 (38)             S S NL 2 1 , (39) A circuit model that implements a unified physical description and accounts for the thermal, acoustic and electrical domains and all their observable effects, is of crucial need to predict the linear and nonlinear behavior of BAW resonators at the design stage. Such a model can be used to accurately predict the linear figures of merit and nonlinear indicators like the Error Vector Magnitude, and to design resonators with mitigated nonlinear effects. APPENDIX I ELECTRO-THERMO-MECHANICAL RELATIONS In connection with Fig. 1 and by virtue of the first law of thermodynamics, one can define the differential of internal energy of an adiabatically insulated body as the sum of mechanical, electrical and thermal energy [29]: ,  dEdDTdSdU  (40) where T, S, E, D,  and  are stress, strain, electric field, displacement current, temperature and entropy respectively. The natural independent variables of (40) are S, D, and  , but we are interested in using S, E and  as the independent variables. Therefore we use the electric Gibbs free energy function G(S,E,  ), obtained through the appropriate Legendre transformation 0: G=U-ED-  , whose differential form can be obtained by use of (40):  dDdETdSdG  . (41) Therefore, the dependent variables T, D and  are related to the Gibbs function as: ESSE G E G D S G T ,,, ;;             (42)   GGDGT ES  ;; , (43) -65 -60 -55 -50 -45 -40 103104105106 Separation between tones (Hz) Intermodulation distortion (dBm) Fig. 13. Measurements and simulations of the low-frequency third-order intermodulation distortion response for a tone spacing ranging from 300 Hz to 1 MHz. The central frequency is kept constant at f0=f1+  f/2= f2-  f/2=1.845 GHz. The simulations include the self-heating effects and the intrinsic thirdorder intermodulation distortion. Dashed lines with rectangles and circles represent measurements of 2f1-f2 and 2f2-f1 respectively. Solid line and dashed line represent simulations of 2f1-f2 and 2f2-f1 respectively. 112 where yx represent the partial derivative of y with respect to x. The relations in (42) can be expanded into a Taylor series, around the zero fields point, as a three-variable function as follows:   !!! ,, 0 0 0 pq E n S ES f ESf pqn n q p pqn pqn               , (44) which up to the third-order can be written as: 222222 33 3222 ... ... EfSfEfESfSfSEf SEfEfSfSEffEf SffEfSffEfSff EESSEESSSSEE SEESSEEEE SSSEESSES          , (45) where f is T, D, or  . If we replace (43) in (45) the partial derivatives can be written in terms of G, what allows identifying the equivalent partial derivatives, for example, TE=GSE=-DS. Table II, shows the nomenclature we will follow for the material constants. TABLE II MATERIAL CONSTANTS TS  E c Elasticity DEEE   S 3 3nd-order permittivity TSS  E c2 2nd-order elasticity  ES r Heat capacity TSSS  E c3 3rd-order elasticity  ES r2 2nd-order heat capacity DE   S Permittivity  ES r3 3rd-order heat capacity DEE   S 2 2nd-order permittivity TE=GES=GSE=-DS  e Piezoelectric constant T  =GS  =G  S=-  S E   Thermal pressure D  =GE  =G  E=-  E S   Electro-caloric constant Note that the subscript denotes the order and the superscript denotes the independent variables that are kept constant. Table III shows the second-order material constants which are also written in the first column as derivatives of the material constants of Table II. The second column is the nomenclature that indicates the order and, as an example, the first row should be read as “E-dependent heat capacity is equal to - dependent electro-caloric coefficient”, which allows to clarify the equivalent second-order interactions between physical magnitudes in Fig. 1. The third column shows the nomenclature we will use. TABLE III 2ND ORDER MATERIAL CONSTANTS ES E S ErD    SSE r   ,2,2    1    SS EEEE D SSE   ,2,2    2   SESEEE eDT    SE e,2,2    3 ES S E SrT    ES Er,2,2      4  S E E C SSSE eDT   SE ec ,2,2    5    EE SSSScT  EESc   ,2,2    6 S S E ESESE eDT     SE e,2,2,2      7 ES E S ErD    SSE r   ,2,2    1 Table IV shows the third-order material constants, which are also related to the linear material properties of Table II. TABLE IV 3RD ORDER MATERIAL CONSTANTS S E D    S   ,3   1 ES EE S EEErD     SE Sr,3,3      2 S EEEEEEE D    SE,3    3  EESEEEEE eDT   E e,3  -  4 E S T    E   ,3  -  5 S SSSSSE DT    S,3   -  6   S SS E EESSESEE cDT    SE c,3,3  -  7 SE SS E SSSrcT ,     E SE rc ,3,3     8  SSSSSSSE eDT   S e,3  -  9 E SSSSSSS T    ES,3   -  10    eDT SESE   ,3 e -  11 E EESEESEEE DT    E,3   -  11 The complete nonlinear constitutive equations up to thirdorder read: NL EE TEeScT    (46) NL SSE DESeD    (47) NL ESSE rES   (48) where the nonlinear contribution is:            2 12 2 11 2 10 2 9 2 8 2 7 6 3 5 3 ,3 3 3 765 2 4 2 3 2 2 2 1 ... 6 1 ... 2 1 EESESSSE SEEeSc ESSEEScT E E E NL    (49)            2 3 2 2 2 6 2 7 2 11 2 4 12 3 1 3 9 3 3 273 2 1 2 5 2 2 2 1 ... 6 1 ... 2 1 EESESSSE SESE ESSESED S S NL    (50)          2 2 2 1 2 8 2 6 2 5 2 12 11 3 10 3 3 3 3 147 2 6 2 2 2 2 2 1 ... 6 1 ... 2 1 EESESSSE SESEr ESSESEr ES ES NL    (51) REFERENCES [1] R. 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In chapter I, nonlinear transmission lines are analyzed, with an emphasis on High Temperature Superconductors, and models are derived to account for the nonlinear behavior. Such models can be used as building blocks for devices that make use of transmission lines, like filters or couplers, to predict their performance under large-signal conditions. Moreover, circuit modeling allows testing the designs with real complex signals, like WCDMA, and obtaining performance indicators that are of crucial importance to fulfill the requirements of today’s communications industry regulations. As an application example, the use of High Temperature Superconductor transmission lines as power limiters in a multiplexer filter bank is considered, and its nonlinear model allows for its behavior prediction. Additionally, the analytical expressions of impedance mismatched nonlinear transmission lines are provided. In chapter II lumped elements models and distributed models, both from a phenomenological perspective, are presented for Bulk Acoustic Wave resonators. The phenomenological models represent a fast way to take into account the nonlinear response of BAW resonators, and their correct scaling with area allows using them to 115 predict the filter’s response. Moreover, chapter II presents advanced measurement configurations and techniques to unveil the response of the devices under test without taking into account the effects introduced by the measurement setup, which allows for an accurate characterization of the device. Chapter III sheds light on the impact of temperature rise in passive microwave devices performance. More specifically, transmission lines made of regular metals and dielectrics are considered, as well as transmission lines with thin-film ferroelectrics and Bulk Acoustic Wave devices. Physical electro-thermo-mechanical models are obtained, which can be used to predict the performance of the devices at different temperatures and assess the impact of self-heating on passive intermodulation distortion. As a result, electromagnetic and electro-acoustic devices have been properly characterized and their nonlinear material properties identified. The presented circuit models can be used to optimize the linear and nonlinear response, which is of extreme importance in proposing designs with reduced nonlinearity or even total cancellation. Additionally, the presented circuit models allow one to use them in integration into more complex systems and obtain performance indicators. Two of these works, Appendix A and B, are a submitted article which is pending for acceptation to be published and an article to be submitted, respectively. 116 IMPACT FACTORS OF THE PUBLICATIONS According to the Journal Citation Reports® (JCR) database of Thomson Scientific and the SciVerse Scopus® SJR index of Elsevier B. V., the impact factors, on March 2011, are as follows: Journal/Proceedings JCR* SJR* IEEE Transactions on Applied Superconductivity 1.310 0.075 IEEE Transactions on Microwave Theory and Techniques 2.076 0.217 Proceedings of the IEEE MTT-S International Microwave Symposium - 0.071 Proceedings of the IEEE UFFC-S International Ultrasonics Symposium - 0.036 *The JCR and SJR indexes have different scales, according to their calculation method. 117 VITA Eduard Rocas received the Telecommunication Engineering degree from the Universitat Politècnica de Catalunya (UPC) in 2005, where his final project was associated with the creation of the Intelligent Communications and Avionics for Robust Unmanned Aerial Systems (ICARUS) research group. From September 2005 to July 2006, he was involved in the simulation and modeling of advanced SONARs at the Computer Vision and Robotics Group (VICOROB - University of Girona) as a FPU Grant Holder. Since November 2006 until June 2011 he stayed with UPC as a PhD Student (FPI Grant Holder) focusing his research on new materials and structures for novel RF/MW devices. Since January 2009 until June 2011 he stayed as a Guest Researcher at the National Institute of Standards and Technology (NIST), Boulder, CO. Eduard Rocas may be reached at Universitat Politècnica de Catalunya (UPC). His email is eduard.r[email protected]. 118 119