First self-resonant frequency of power inductors based on approximated corrected stray capacitances

Received: 31 March 2020 Revised: 21 August 2020 Accepted: 24 August 2020 IET Power Electronics DOI: 10.1049/pel2.12030 ORIGINAL RESEARCH PAPER First self-resonant frequency of power inductors based on approximated corrected stray capacitances Ignacio Lope1,2Claudio Carretero1Jesus Acero3 1Department of Applied Physics, University of Zaragoza, Zaragoza, Spain 2BSH Home Appliances Group, Zaragoza, Spain 3Department of Electronic Engineering and Communications, University of Zaragoza, Zaragoza, Spain Correspondence Claudio Carretero, Department of Applied Physics, University of Zaragoza, 50012 Zaragoza, Spain. Email: [email protected] Funding information Ministerio de Ciencia e Innovación, Grant/Award Number: RTC-2017-5965-6; BSH Home Appliances Group, Grant/Award Number: -; DGA-FSE, Grant/Award Number: -; Ministerio de Economía, Industria y Competitividad, Gobierno de España, Grant/Award Number: PID2019-103939RB-100 Abstract Inductive devices are extensively employed in power electronic systems due to their magnetic energy storage and power transfer capabilities. The current trend is towards increasing the frequency of operation in order to reduce the size of the magnetic components, but the main drawback is that the parasitic capacitance effect can become significant, and degrade the performance of the system. This work analyses the influence of this stray capacitance, and considers how to improve the performance of the device. In general, the impact of the stray capacitance on a magnetic component can be reduced by two methods: reducing the parasitic capacitance between turns of the winding or, alternatively, modifying the arrangement of the connection between turns. To evaluate the last option, an approximated expression of the first self-resonant frequency of the magnetic device is proposed. This gives a rapid assessment of the performance of different devices maintaining the overall equivalent inductance. The proposed expression accounts for the influence of the connection between turns in the bandwidth of the component. Finally, some numerical results are verified with planar coils manufactured on two-layer printed circuit boards. 1 INTRODUCTION Planar coils are often selected for power electronics systems because these components are well adapted to the geometry of many applications and are convenient for integrated applications. Some examples include inductive power transfer systems (IPTs) [1–7], devices for medical applications [8–12]and domestic induction heating systems [13, 14]. With respect to the integrated applications, there is a tendency towards using planar constructions of inductors and transformers in switching-mode power supplies (SMPS) [15–17]. The most common manufacturing techniques of planar coils are conventional wiring, PCB implementations, as shown in Figure 1, and microfabrication [18]. The present tendency towards higher operation frequencies has also led to an interest in modelling parasitic capacitive effects because, among other reasons, this stray capacitance can be used as the resonant element in converters or determines the bandwidth of the magnetic component. Various proposals trying to capture the capacitive effects by means of equivalent circuits can be found in the literature. One approach is a This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2020 The Authors. IET Power Electronics published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology simple equivalent circuit composed of an equivalent RL circuit connected in parallel to an equivalent capacitance [19]. In other approaches, the value of the frequency-dependent impedance has been obtained by means of empirical methods based on impedance measurements and curve-fitting techniques [20–22]. Although the preceding methods are valuable by themselves, the applicability of the results is limited to the prototypes studied. In these simplified lumped equivalent circuits it is assumed that all turns are driven by the same current and depending on the device or the frequency, the accuracy could be affected and therefore some alternatives have been proposed. These alternatives include distributed circuits [23–29] and transmission line approaches [30], which are mainly focused on obtaining selfresonant frequencies associated with the parasitic capacitances and inductances of the system. These parasitic capacitors could appear, for example, between turns, winding layers or winding and magnetic cores [31–34]. In general, stray capacitances arise from the electrical coupling between elements of the device [35, 36]. However, the connection of the winding’s turns can be optimised to improve the bandwidth of the device with a minor impact in its performance [37–41]. IET Power Electron. 2021;14:257–267. wileyonlinelibrary.com/iet-pel 257 258 LOPE ET AL. FIGURE 1 Planar PCB coils In this paper, a simple expression enabling a rapid calculation of the first self-resonant frequency from the distributed circuit elements of the component is derived. In the proposed expression, the circuit elements are intrinsic parameters associated with the geometry and physical properties of the media, but, the first self-resonant frequency is dependent on the turns’ arrangement. From this expression it is evident that the bandwidth of the coil can be extended by selecting the appropriate connection between the turns as different connections provide similar low frequency characteristics. The remainder of the paper is organised as follows: in Section 2, the distributed circuit model for inductors including stray capacitances is described; next, the behaviour of these system are explained in Section 3; in Section 4, the procedure to calculate the distributed circuit element is given; experimental validation of the proposed model is drawn in Section 5and Section 6concludes. 2ELECTRICAL EQUIVALENT OF THE INDUCTOR 2.1 Ideal coil Generally speaking, a planar inductor consists of the series connection of nconcentric turns. For the sake of simplicity, spiral turns will be considered in this paper, but the analysis is also valid for turns with alternative geometries, for example, elliptical or square. The turns of a planar coil are placed concentrically to increase the inductance value of the device while maintaining a reduced size. The inductance of this arrangement, Leq, is greater than the sum of the values of the self-inductances of the turns, Lii, due to the effect of the mutual coupling between them, Lij. Electrical characteristics of an ideal multi-turn coil can be described from the coupling inductances, Lij, which relate the FIGURE 2 Distributed circuit model of an ideal coil value of the induced voltage Viin each turn and sources of the magnetic fields, which are given by the current amplitude, Ij, circulating through each turn. In this description, the harmonic approach is assumed, that is, ij(t)=ℜe(Ij⋅ej𝜔t), where 𝜔is the angular frequency of the excitation. The induced voltage in the ith turn, Vi, is determined by the sum of the induced voltages Vij =LijIj. The induced voltage Vij can be easily carried out by integrating, along the trajectory of the ith turn, the electric field, Ej, created by the current flowing through the jth turn, as stated in [42]. As is commented above, inductors are normally built by connecting the turns in series. Therefore, the current of each turn, Ij, is equal to the source’s current, IS, injected through coil terminals. In that case, the voltage in each turn is therefore Vi=∑n j=1j𝜔LijIS, and the total voltage in the coil is VS= ∑n i=1Vi=j𝜔IS∑n i=1∑n j=1Lij. Thus, the equivalent inductance of the ideal inductor, Leq, is the well-known expression Leq =∑n i=1∑n j=1Lij, which is independent of the excitation frequency, according to the above assumptions. Moreover, the equivalent inductance, Leq, does not depend on the sequence of the series connection among turns. Figure 2represents the circuit diagram of distributed elements corresponding to an ideal winding. Similarly, a circuit of distributed elements can be constructed by dividing the coils into several segments and applying the definition of coupling inductance based on the integration of the electric field along the path of the element. This extended model does not provide additional data but it can be the starting point for a circuit model that includes capacitive effects. 2.2 Distributed model of the coil with stray capacitances The model described in the preceding subsection only considers inductive effects arising from currents. However, a complete electrical model of the component also needs to consider the inclusion of capacitive effects. As a consequence, an accurate evaluation of the component considers both selfand coupling inductances as well as stray capacitances, as shown in Figure 3. The stray capacitances are strongly dependent on the geometry and relative distances between segments of the conductors. In particular, multi-layered spiral coils exhibit a highly variable stray capacitance between facing segments of different turns associated with the rapid variation LOPE ET AL.259 FIGURE 3 Coil with stray capacitances between pair of segments FIGURE 4 Distributed circuit model of a three-turn coil on their relative positions. Stray capacitances can be found from analytical models [43, 44] as well as from numerical simulations [45–47]. The stray capacitances can also be found by considering segments of conductors at different electric potentials [48]. Stray capacitance can reach high values when the distance between turns is small or facing surfaces are large. In addition, the influence of the dielectric properties of the media, for example substrate of printed circuit boards, play an important role in this characteristic. Stray capacitance provides electrical connection between turns at different voltage levels. The distributed circuit model of an ideal coil depicted in Figure 2 can be easily extended to include capacitive effects by inserting stray capacitances, Cij, connected to the middle points of the ith and jth turns, as shown in Figure 4. As it occurs with the coupling inductances, Lij, the number of stray capacitances depends on the square of the number of turns. However, only parasitic capacitances between nearby turns have a considerable influence in the frequency response of the coil due to the strongly dependence of the stray capacitances with the relative distance between segments of the turns. The model described in this paper does not consider the effect associated with power losses in the winding, because it would compromise simplicity in describing the behaviour of the system. In order to extend the analysis to include winding losses, on the one hand, it would be necessary to include in the FIGURE 5 Ideal single-turn coil with long terminals. (a) Schematic of the geometry. (b) Distributed element model model represented in Figure 4resistive type elements connected in series with the inductive elements, in order to consider both the current conduction losses and the proximity losses associated with the variable magnetic field in which the conductors are immersed [49, 50]. On the other hand, resistive type elements connected in parallel with the parasitic capacitors should be inserted to model the conduction and dielectric losses in the substrate [51]. However, the preceding elements have a limited impact in the equivalent impedance of the whole system, thus, their influence in the first self-resonant frequency value is moderate. 3ELECTRICAL BEHAVIOUR OF COIL WITH STRAY CAPACITANCES 3.1 LC-parallel resonant circuit Stray capacitances in coils introduces additional circuit paths for the current which increase the complexity of analysis of the magnetic device’s electrical behaviour. For illustration purposes, a simple reference system composed of a loss-less single-loop coil with long parallel terminals is selected, as shown in Figure 5a. The distributed electrical equivalent of the reference systems is also represented in Figure 5b. As can be seen in the schematic, the equivalent of the single loop is the inductance, Leq, and the long parallel terminals are transformed in parallelconnected stray capacitances, Ci, whose addition provides the equivalent capacitance Ceq. At the endpoints of the coil’s terminals, a voltage source, vS(t), is connected driving the current iS(t). The simplified electrical equivalent of the preceding system is shown in Figure 6a which corresponds to the LCparallel circuit fed by a voltage source. The current, iS(t)canbe decomposed into two contributions represented in Figure 6b: the capacitor’s current or recirculating current, iC(t), and the inductor’s current, iL(t). Applying the Kirchhoff’s current law 260 LOPE ET AL. FIGURE 6 Currents in parallel LC resonant tank. (a) Source’s current flow. (b) Distributed element model condition, the relationship iS(t)=iC(t)+iL(t) can be straightforwardly established, and, rearranging the preceding expression, iL(t)=iS(t)−iC(t). Equivalently, the current flowing through the loop turn can also be decomposed into a recirculating current, iC(t), and a source current iS(t), as can be seen in Figure 6. Mesh defined by the recirculating current, iC(t), obeys the Kirchhoff’s voltage law, that is, the voltage in the equivalent capacitance, vC(t), is equal to the voltage in the equivalent inductance, vL(t). Voltage vC(t) is totally determined by the product between the recirculating current iC(t) and the equivalent capacitive impedance, whereas, voltage vL(t) is given by both the recirculating current, iC(t), as well as the source current, iS(t). At low frequency, the recirculating current is negligible and the current through the equivalent inductance is almost totally equal to the source current, whereas at higher frequencies, the recirculating current increases. Considering the harmonic source’s voltage vS(t)=VSej𝜔t, the recirculating current amplitude is IC=j𝜔CeqVS, and the inductance’s current amplitude is IL=VS∕j𝜔Leq. The resonant frequency is reached when the inductance’s voltage is totally due to the recirculating current, given the source current, iS(t), equal to zero at this point. The resonant angular frequency is given by the well-known expression 𝜔0=1∕√LeqCeq, being infinite the equivalent impedance of the LC-tank circuit. The preceding LC-tank circuit is selected as the reference system to understand the influence of the parasitic capacitances in n-turns coils. Reduction of the distributed circuit model represented in Figure 4to the concentrated circuit model shown in Figure 6b is a challenging task because the total current carried by each turn is the composition of multiple recirculating currents. As was mentioned, at low frequencies, the recirculating currents are negligible. However, the current distribution changes depending on the turn position when the operation frequency increases. At frequencies close to resonance, the current in the coil is almost totally associated with the recirculating currents. The current reaches its maximum value in the intermediate turns and decays to zero in the turns near the coil terminals. The imaginary part of the current into the terminals is negative or positive, that is inductive or capacitive, respectively, depending on whether the excitation frequency is above or below resonance, respectively. The resonant frequency of the distributed circuit model will be estimated by weighting the contribution of each parasitic capacitance to the total current in the inductor. At the resonant frequency, the voltage drop between the turns can be determined from the inductive elements’ voltages, whereas the effect of the parasitic capacitors comes from the creation of closed paths for the recirculating currents because the current driven by the external voltage source tends to zero at this operating point. The expression to estimate the first resonance frequency is obtained by performing two different steps: first, the approximate value of the recirculating current can be determined assuming that the same current circulates, ILthrough all the turns of the coil as well as recirculating current, iCij is proportional to the potential drop between the points that present the Cij parasitic capacitance; second, from the recirculation currents, and considering that the current of the source feeding the winding tends to zero at the resonance frequency, that is, the currents in each loop are due to the sum of the recirculation currents, the dependence of the potential drop with respect to the frequency in the complete coil is calculated, being the expression of the approximated self-resonance frequency straightforwardly derived from the preceding relationship. 3.2 Approximated recirculating currents at resonance To obtain a simplified expression for the first resonant frequency of a multiple turn coil, the recirculating currents iCij will be estimated. First, the currents’ through the inductive elements at frequencies close to the resonance are approximated to the estimated mean current, Ii≅I′ L. The voltage drop amplitude in the ith turn, Vi, can be now given by the following expression: Vi= n ∑ l=1 j𝜔Lil Il≅j𝜔I′ L n ∑ l=1 Lil ,(1) where Ilis the current amplitude through the lth turn. The addition of the voltage drop amplitudes over all the turns of the coil gives the total voltage drop in the coil, Vcoil which is equal to the source’s voltage amplitude, VS: VS= n ∑ i=1 Vi≅j𝜔I′ L n ∑ i=1 n ∑ l=1 Lil .(2) LOPE ET AL.261 Rewriting the preceding equation, the current amplitude I′ L can be approached by the following equation: I′ L≅VS j𝜔 n ∑ i=1 n ∑ l=1 Lil =VS j𝜔Leq .(3) Similarly, the voltage drop between the middle point i′+1 2of the ith turn and the middle point of the next turn is given by: V(i′+1 2)(i′+1 2+1)≅j𝜔I′ L(n ∑ l=1 Lil 2+ n ∑ l=1 L(i+1)l 2).(4) And the voltage drop between the terminals of the parasitic capacitor, VCij =V(i′+1 2)(j′+1 2), can be approached by: VCij ≅j𝜔I′ L j−1 ∑ m=i(n ∑ l=1 Lml 2+ n ∑ l=1 L(m+1)l 2).(5) Substituting the value I′ Lgiven by (3), we obtain: VCij ≅ j−1 ∑ m=i n ∑ l=1 Lml +L(m+1)l 2Leq VS.(6) The last expression establishes the relationship between the voltage drop, VCij, in the middle points of the ith and jth turns, respectively, and the source’s voltage, VS. To a certain extent, expression (6) could be considered as the formula of a voltage divider. The factor of proportionality, F Cij, defined from the identity VCij =F CijVS, can be expressed as: F Cij = j−1 ∑ m=i n ∑ l=1 Lml +L(m+1)l 2Leq .(7) The factor of proportionality, F Cij, ranges from 0 to 1, because it is positively defined and the maximum value is reached for the middle points of the turns connected to the coil’s terminals. For the maximum value configuration, the voltage drop VC1nis lower to VSdue to the non-zero voltage drops between the coil’s terminals and the middle points of the extreme turns, V1′(1+1 2)′and V(n+1 2)′(n+1)′, respectively. The factor of proportionality, F Cij, is independent of the frequency, as it also occurs for the stray capacitances, but presents dependence on the selfand mutual inductances of the turns, that is, it depends on the geometry and magnetic properties of the coil. Note that the factor of proportionality, FCij, can be modified by changing the arrangement of connections between turns but the equivalent inductance of the coil, Leq, remains the same. Equation (6) can be rewritten as: VCij ≅F CijVS.(8) Capacitive recirculating current ICij is proportional to the voltage drop between the stray capacitor terminals placed in the middle points of the ith and jth turns, as it is given by ICij =j𝜔CijVCij. Thus, considering Equation (8), the recirculating current ICij can be expressed as: ICij ≅j𝜔F CijCijVS.(9) By defining the frequency-independent corrected capacitance between the ith and jth turns, C∗ ij,as: C∗ ij =F CijCij.(10) Thus, the recirculating current amplitude can be rewritten as: ICij ≅j𝜔C∗ ijVS.(11) As can be appreciated in the preceding expression, the corrected capacitance C∗ ij represents the effective capacitance associated with the stray capacitance. As a consequence, the influence of the physical stray capacitance can be altered by modifying the factor of proportionality, F Cij, that is. changing the turns between terminals of the stray capacitance. 3.3 Approximated first self-resonant frequency Once the recirculating current amplitude is estimated, the overall current carried by the inductive elements of the distributed model at resonance can be calculated by the addition of the recirculating currents because the source’s current is zero at this operating point. The estimated current, I′ i′+1, at resonance through the i′+1th point is: I′ i′+1≅ i′ ∑ k=1 n ∑ m=i′+1 −ICkm ,(12) which corresponds to the current flowing through inductance element with terminals at the i′+1 2th and i′+1th endpoints as well as the element at the i′+1th and i′+1 2+1th endpoints. The estimated voltage drop in the ith turn, V′i,canbe calculated as: V′ i≅j𝜔0(n−1 ∑ l=1 Lil 2I′ (l+1)′+ n−1 ∑ l=1 Li(l+1) 2I′ (l+1)′).(13) Including (12) in the preceding expression, yields: V′ i≅−j𝜔0 n−1 ∑ l=1 l ∑ k=1 n ∑ m=l+1 Lil +Li(l+1) 2ICkm .(14) 262 LOPE ET AL. Thus, the total voltage amplitude in the coil is the addition of the voltages’ amplitudes, V′iover all the turns: V′ coil ≅−j𝜔0 n ∑ i=1 n−1 ∑ l=1 l ∑ k=1 n ∑ m=l+1 Lil +Li(l+1) 2ICkm .(15) The coil’s voltage amplitude is VS=V′ coil, and the recirculating current is given by (11), thus: VS≅𝜔 2 0 n ∑ i=1 n−1 ∑ l=1 l ∑ k=1 n ∑ m=l+1 Lil +Li(l+1) 2C∗ kmVS.(16) Rearranging the preceding expression: 𝜔2 0≅VS n ∑ i=1 n−1 ∑ l=1 l ∑ k=1 n ∑ m=l+1 Lil +Li(l+1) 2C∗ kmVS .(17) And, finally, the first self-resonant angular frequency can be approximated by the following relationship: 𝜔0≅1 √n ∑ i=1 n−1 ∑ l=1 i ∑ k=1 n ∑ m=i+1 Lil +Li(l+1) 2C∗ km .(18) Equation (18) can also lead to an estimation of the parallel resonant capacitor of the lumped circuit equivalent shown in Figure 6a. From the definition, 𝜔0=1∕√LeqCeq , the equivalent capacitance is provided by: Ceq ≅ n ∑ i=1 n−1 ∑ l=1 l ∑ k=1 n ∑ m=l+1 Lil +Li(l+1) 2Leq C∗ km.(19) This means that 𝜔0can be increased by reducing the equivalent capacitance applying the correct arrangement of the turns which is useful because it increases the bandwidth of the inductive device. 4 CALCULATION OF THE DISTRIBUTED CIRCUIT ELEMENTS 4.1 Inductive parameters The analytical expression of the inductance for a coil made of filiform concentric spiral loops is already found in Maxwell’s treatise [52]. Despite, analytical expressions for alternative inductor structures are not easily available, the selfand mutual inductance assuming linear conditions can be easily calculated with numerical tools. Simplified geometric structures provide FIGURE 7 Two series-connected spiral turns placed in parallel planes (dark grey: upper layer turn, light grey: lower layer turn) connected in the inner radius common point accurate results due to the low sensitivity of the mutual coupling between turns. Consequently, mutual inductance can be calculated by taking advantage of an imperfect symmetry, as it occurs for spiral winding which can be accurately modelled with 2D axisymmetric models. Inductive elements, Lij, can be easily extracted from the relationship between the induced voltage in the jth turn, Vj, and the current amplitude, Iithrough the ith turn, as is given in [42]. 4.2 Capacitive parameters The total capacitance between two turns corresponds to the total effect of several distributed capacitances. Stray capacitances can be also evaluated from analytical expressions in simplified structures [53], but analytical expressions are not available for common arrangements or are inaccurate [54]. However, the numerical extraction of the parasitic capacitances can be performed in an easy way because they are physically decoupled from the inductive effects. Parasitic capacitance Cij can be calculated by addition of the capacitance for facing segments of the turns. The numerical extraction of stray capacitance can be performed by applying Cij =Qj∕Vi, where Qj is the charge accumulated in the jth turn when a voltage Viis applied in the iturn, and the remaining turns grounded. 4.3 Example of parameter calculation The electrical parameter extraction of the two-turn coil represented in Figure 7is performed with COMSOL®. Computational extraction can determine both inductive and capacitive elements of the device [55, 56]. Turns are built of standard PCB copper tracks with thickness, h=70 µm, and width, w=500 µm, placed in each side of the board with thickness, t=1.55 mm, respectively. The middle points of the internal terminals are at radial distance, rint =30 mm, and the external terminals are at rext =31 mm, that is, the separation between tracks is d= 500 µm. Selfand mutual inductances can be easily calculated LOPE ET AL.263 FIGURE 8 Schematic of the PCB-coil (side view including axis of revolution) by using the magneto-quasi-static approach. Turns are modelled by two rectangular cross-section turns with axial 2D symmetry of mean radius ri=rj=30.5 mm facing at a distance t.Magnetic field arises from the current Iiuniformly distributed in the cross-section area of the ith turn. The induced voltages Viand Vjof the ith and jth turns have been derived by integrating the electric field. Finally, self-inductance, Lii =211 nH, and mutual inductance Lij =115 nH, are obtained from numerical simulations, respectively, being a good approach of the exact values. In case of inserting magnetic materials, parameter extraction can be performed in a similar fashion because numerical tools easily include the influence of these elements by considering an object with a high relative permeability. On the other hand, stray capacitance, Cij, was obtained by applying the electro-quasi-static approach, because the frequency dependence of the capacitance associated with the skin effect can be neglected for conductors of small cross-sectional area, as it occurs in the coils analysed in this work. The dielectric constant of the FR4 substrate is 𝜖r=5. First, the turns are divided into Nangular segments. Partial axial symmetry is exploited for each pair of facing segments, but can be also considered as straight segments with a similar outcome. Partial parasitic capacitances ranges from 1.04 to 1.20 pF per radian, due to the variation of the distance between segments with respect to the angular position. The calculated two-turn stray capacitance is Cij =7.18 pF. 5EXPERIMENTAL RESULTS Three prototypes of two-layer PCB coils with n=30 equally distributed turns were built to validate the numerical values using standard copper thickness, h=70 µm, and dielectric thickness, t=1.55 mm, of FR4 material (𝜖r=5). The remaining common characteristics of the prototypes, as shown in Figure 8, are the following: external radius, rext =60 mm, internal radius, rint =30 mm, and trace width, w=500 µm, respectively. Furthermore, a distance from the coil to the ferrite equal to df=1 mm was set. The first and the second prototypes were tested with terminals placed at different turns, but in both cases the connection is performed from outer turn to inner turn in the bottom layer and in the opposing sense for the top layer. The first configuration is defined by the coil’s terminals placed at outer turns, that is first and nth turns, as can be seen in Figure 9a, and the second one corresponds to the terminals FIGURE 9 Reference winding arrangements of n-turns (side view including axis of revolution): (a) outer connection; (b) intermediate connection; (c) alternating-layers connection FIGURE 10 Experimental set-up for measuring PCB coil with ferrite at seventh and eighth turns of the top layers, as shown in Figure 9b. In addition, the last prototype was built by series connecting the turns from outer-to-inner radius turns, but, sequentially exchanging the layer by means of n-1 vias. Measurements were performed with a high precision LCR-meter HP 4285A (measurement frequencies range from 75 kHz to 30 MHz and the basic accuracy of the measurements is 0.1 %), as shown in Figure 10. Figure 11a shows the reactance-to-angular frequency ratio, equivalent to the inductance at low frequencies, for the coil in air configurations. Model-based values are calculated from the same distributed circuit parameters, thus, the equivalent inductance of both configurations are the same but the equivalent capacitance differs because different corrected capacitances are obtained. The change in the values of the corrected capacitances for each configuration is due to the variation in the values of the proportionality factors, FCij, defined in (7) because the parasitic capacity between the ith and the jth turns, Cij, remain constant in all configurations. Therefore, when applying a different connection of the turns, the equivalent capacitance, Ceq can be 264 LOPE ET AL. FIGURE 11 Ratio between reactance and angular frequency experimental (circles) and model-based (solid lines) values applying the outer (black) and intermediate (red) connection-modes. (a) Two-layer PCB coil in air. (b) Twolayer PCB coil with ferrite reduced if the configuration reduces the proportionality factors, F Cij, between the turns with the highest parasitic capacitance between them. Moreover, Figure 11b shows the results when a ferrite layer of magnetic permeability, 𝜇r,f=2000, is inserted at a distance of df=1 mm below the coil, with a good agreement between experimental and modelled data because the mismatch between them arise from a slight difference on the first selfresonant frequencies in the prototype and the simulation results but a high correlation in the frequency dependence between both types of values is observed for the different configurations. As can be seen in the graphs, configurations with the intermediate turns’ terminals extend the bandwidth of the coil compared to configurations with the outer turns’ terminals but better results are achieved when facing turns located at different layers are connected between them, as it occurs in the last prototype. However, the construction of the last configuration exhibits two main drawbacks: including vias in the coil would impact in the power losses in the coil and the terminal are placed into distant points. A comparison between the simulated values and the measured first resonant frequencies FIGURE 12 Numerical results for coil’s terminals at different consecutive positions (black: coil in air, red: coil with ferrite plane). (a) First self-resonant frequency, f0. (b) Equivalent inductance, Leq. (c) Equivalent capacitance, Ceq for the tested arrangements is given in Table 1. Experimental first self-resonant frequencies, f0, are obtained by fitting the measured impedances to the frequency response of parallel LC resonant tanks. Figure 12 presents the expected first self-resonant frequency calculated for the equivalent inductance, Leq and equivalent LOPE ET AL.265 TABLE 1 Self-resonant frequency and electrical parameters of 30-turn coil Additional Connection Measured parameters Simulated parameters media mode f0(MHz) Leq (µH) Ceq (pF) f0(MHz) Leq (µH) Ceq (pF) Air Outer turns 2.09 103.84 55.84 2.01 103.19 60.75 Intermediate turns 3.21 103.84 23.64 3.44 103.19 20.74 Alternating turns 6.31 102.70 6.19 6.77 103.19 5.36 Ferrite plane Outer turns 1.44 189.30 64.89 1.48 189.20 61.30 Intermediate turns 2.54 187.40 21.03 2.39 189.20 23.36 Alternating turns 3.63 180.90 10.60 3.33 189.20 12.07 TABLE 2 Self-resonant frequency of two-layer PCB coil Turns Measured f0(MHz) Simulated f0(MHz) 18 6.17 5.70 26 3.61 3.43 34 2.56 2.47 capacitance, Ceq of the tested arrangements but varying the position of the coil’s terminals, starting from the outer connection, that is 1st and 30th turns, and varying the connection sequentially. Outer connection and inner connection exhibit a similar performance because turns with large stray capacitances are connected through a high number of intermediate turns, implying a high value of corrected stray capacitances. Maximum self-resonant frequency is achieved for coils with terminals at intermediate turns, but the upper-layer connection is weakly preferred for the coil with ferrite plane. Alternative complex connections can increase the bandwidth of the inductor, that is, first, up-to-down or down-to-up turns’ connection and, next, outer-to-inner connection gives resonant frequencies at f0=6.77 MHz for coil in air and f0=3.33 MHz for coil with ferrite plane. As can be seen in Figure 12c, the equivalent capacitance is clearly dependent on the arrangement of the turns, but the inclusion of the ferrite plane slightly increases its value. Additionally, three prototypes of two-layer PCB coils with n=18, n=26 and n=34 equally distributed turns were also built using copper thickness, h=400 µm, and dielectric thickness, t=1.55 mm, of FR4 material (𝜖r=5). The remaining common characteristics of the prototypes are the following: external radius, rext =60 mm, internal radius, rint =12 mm, and trace width, w=2000 µm, respectively. A comparison between the measurement data and the model results are shown in Figure 13, and the self-resonant frequency extracted from the measurements and the calculated self-resonant frequency, f0,are listed in Table 2wherethe good agreement between them can be seen. Calculated self-resonant frequencies are obtained from self-inductances of 18.20, 38.01 and 65.21 µH, and corrected capacitances of 42.82, 56.57 and 65.54 pF, respectively. Finally, an arrangement composed of two PCB’s was made to double the number of turns by applying a series connection between them in order to test the proposed model for multiFIGURE 13 Ratio between reactance and angular frequency experimental measurements (circular symbols) and model-based values (solid lines) for twolayer PCB coils of n=18 turns, n=26 turns and n=34 turns FIGURE 14 Ratio between reactance and angular frequency experimental measurements (circular symbols) and model-based values (solid lines) for fourlayer PCB coils of n=36 turns, n=52 turns and n=68 turns layered coil configurations. The new configuration was built by using two PCBs with the same characteristic of the preceding arrangement. The distance between PCB’s in that case is fixed to 1.55 mm. These configurations verify the generality of the proposed model. In that case, the comparison between the experimental and model results are shown in Figure 14 verifying the correctness of the proposed model because, as can be seen