Design of RC-active oscillators using composite amplifiers

DESIGN OF RC-ACTIVE OSCILLATORS USING COMPOSITE AMPLIFIERS B. Pbrez-Verdfi, J.L. Huertas and A. Rodriguez-Vhzquez Dept. of Design of Analog Circuits, Centro Nacional de Microelectr6nica, Sevilla, SPAIN Abstract The design of composite opam Wien-Bridge oscillators is systematically approac!ed by using a general model including amplitude control issues. Two different design criteria are presented and their main features summarized. A general composite opamp topology is presented from where a catalog of structures can be obtained in a systematic way. Experimental data are included illustrating the performance of the proposed design criteria. Introduction It is well known that the finite gain-bandwidth (GB) roduct of o erational amplifiers (opamp) degracfes the high-Eequency behavior of active RC circuits. One of the solutions proposed in the literature is the use of active compensation techniques. Many papers have dealt with the topic of proposing active-compensated amplifiers (henceforth called composite amplifiers), see for instance [1,21, and a systematic study have been reported recently [31. Composite amplifier techniques are not directly applicable to the design of sinusoidal oscillators. Although some circuit structures have been 'ust proposed [4-81, there is a lack of systematic, this jack preventing from a meaningful comparison among the reported solutions,oand among them and noncompensated oscillators. Besides some approaches are based on oversimplified linear models, which are not able to correctly explain data measured on actual proto types. In this communication we will undertake a s stematic study of the use of composite amplifiers for tie Wien-Bridge famil of oscillators. Models including issues relatei to the the control and stabilization of the amplitude are used to this purpose. Single-opamp WienRridge Family Fig.1 is the block diagram for the Wien-Bridge family of opamp based RC-active oscillators. Fig.2 shows a complete set of canonical RC structures for the passive block (see references in [9]). The function of this block is to make the phase around the loop to be zero at a frequency wo=w, being nonzero at any other frequency. The function of the amplifier is, on the other hand, twofold. First, it has to provide si nal gain to make the loop transfer function magnitufe to be 1 at w,. Second, it has to include an adaptive mechanism making the amplifier gain to depend on the signal amplitude A in such a way that the critical gain value is obtained just for a single amplitude value A,. Foi A>A, the am lifier gain must be smaller than the critical value king larger otherwise. Whether the amplifier fulfills previous requirements, Fi .1 would generate a quasi-sinusoidal signal of ampfitude A, and frequency 0,. passive network amplifier Fig.1: Block diagram for the Wien-Bridge family of oscillators. I I I I I I I L_______-__-___________________----------------------------l j w,=lI(RC) Fig.2 Canonical RC structures for the WienRridge family. Fig.3(a) shows a conventional one-opamp implementation for the amplifier of Fig.1. Opamp nonlinearities can be exploited to get an adaptive gain and hence to stabilize the amplitude. For lower distortion, it may be however more convenient to use an AGC circuit controlling the value of resistor R as a function of the opamp signal amplitude. No matter how the amplitude control is made, if ideal opam s were available Fig.3(a) would allow us get a,= w,, 8r any value of wi, by just making k = 3 +e (0 < e < < 1). Let us now consider a real opamp and use the model of Fig.3(b) to describe its corresponding smallsignal behavior. After some calculations the following result for the oscillation frequency and the oscillation condition, respectively: CH 3006-4/91Kxxx) - 2589 $1 .oO 0 IEEE I I I I I I 1: V,(S) = VJS) - I (k - l)R I m1 I I I I - composite opamp I I I I L~~~~~~___~~~~___~~_~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~l I (a) (b) Fig. 3: (a)One-opamp amplifier, (b)First-order These e uations rovide a linear view on the operation o? the oscilrator. They have to be slightly modified to account for the amplitude stabilization mechanism. Let us assume to this purpose that parameter b above is controlled by an amplitudedependent adaptation parameter C(A) which we define to be a function of the signal amplitude as follows: opamp small-signal model. single opamp where A, is the critical amplitude value (i.e., the stable oscillation amplitude). For stable self-starting oscillations to exist parameter b must fulfill the following set of conditions: b(Q<O for A<Ao b(Q=O for A=Ao First one guarantees that for low amplitude there will be a pair of imaginary roots on the right-half of the complex frequency plane and hence that oscillations will self-start. Second one allows to calculate the amplitude value for'which the roots are on the imaginary axis. Finally, the third one ensures the roots will cross the imaginary axis from right to left for the amplitude increasing, as it is required for the oscillations to be stable. In summary, (2c) is a much more realistic oscillator condition than (2a). Equations (1) and (2) are the basic design equations for the one-opamp Wien-Bridge oscillators. In actual circuits there are two ways to achieve b to depend on 5. One pssibilit is to control the amplifier DC gain k by making kat,. &her podsibility is to control the opamp time constant T by making ~al/t,. In both cases nonlinear limitation (for instance the slew-rate for the time constant or diodes for k) as well as AGC circuits can be used. However no matter which technique is used circuit operation is described with reasonable accuracy by (1) and (2) for frequencies up to about. 0.21~ [91. From (1) it can be seen that the actual oscillation frequency deviates with respect to the ideal one, the deviation increasing as (o~T) increases. This drawback can be however partially overcome in case amplitude control is made via T, parameter k remainin fixed. From (2) it can be seen that the critical oit vake (the one making b =O) fur this case does not depend on U,. Hence the actual and the ideal frequency are related in a linear way (w,=pwi), the ratio fl depending only on k and hence being independent on the amplifier characteristics. This is a very interesting feature of single opamp whose only drawback comes from the fact that parameter b determining the oscillator condition is strongly dependent on wit. Hence, in case k is selected to ensure b>O (self-starting operation) in wide frequency ranges, 5, can be shown to exhibit large vanations from one extreme to the other of the range. This is not convenient in practice as it can be understood from Table 1. Data on this table have been measured from oscillators using the slew-rate as the controllin mechanism. The two rightmost columns corresponfi the single-opamp case. Parameter dist. refers to the ratio between the first and the third harmonics. k was selected to ensure self-starting operation in the range shown, which resulted in 5, being very close to unity for the higher frequency and increasing as the frequenc decreases. Since changes in Z, have to be absorbed 8y the nonlinearity, large distortion may be expected for low frequencies, as it is confirmed from the measured data. Composite Opamp Oscitlators Let us assume the amplifier in Fig.1 is made by using two opamps and some linear resistors (see Fig.4). In the more general case and usin the opamp model of Fig.3(b) the following transfer knction can be obtained for the amplifier: (3) where k, pl, p2 and p3 are controlled by resistor ratios (they are not typically separately controllable) and a(rl, TZ) can be either TI or ~2 depending on the actual composite opamp structure. After some analysis, the following can be obtained for parameters woand b of the composite opamp oscillator: 3-k 1+- 2590 b=04x zzp3-ot ~~~~~~~~+",~~(3p~-kp,~~+~~+ independent of the opamp time constants, which is a I TZ (4b) very appealing feature deserving further attention. From (4b) it can be shown that by making, + a; 01 As for the one-opamp case, an adaptation 3p2--kpl1=0 (6) P the oscillation frequency exactly coincides to ai providing the following is fulfilled for the Critical value of the adaptation parameter, arameter E, fulfillin (2b) is assumed to be the vehicle For amplitude controfvia either one of the opamp time constants or the amplifier parameters k and p,. Equations (2b-C) hence also holds as the oscillator condition, parameter b for this case being that in (4b). Ideal gain design criterion. Different design criteria or com site o amp Wien-Bridge can tzSfiised by f,arefuG anafyzing (4). Since the pur ose of Cbm osite o amp is to approach the pedrmance of i&al ampifiers, one possible criterion is to try to fix k to the ideal value 3. As it can be seen from (4a) this yields frequency deviations which are very similar to the ones for the single-opamp case. The advantage of the composite-opamp case comes from the oscillator condition. After some analysis, we get, a '1 =2 3 3pz(p2-p1-)- -3pzp3 (5) where it can be seen that parameter b does not depend, to a firsborder, on ai. Hence and no matter the amplitude control be made via r, k or an of the p, parameters, the critical value of the aIaptation Farameter will change only slightly over wide requency ranges. This is advanta ous in comparison to the one-opamp case as it can ye confirmed from Table 1. In the two leftmost columns measured data have been collected for a composite amplifier structure withpl=1.55,p~=k=3 andp3=kpl and for the slew-rate of amplifier #2 being the amplitude control mechanism. LM747 dual opamps were used. As it can be seen, the distortion remains practically unchanged in the whole oscillation frequency range. There are several observations to be made concerning the use of this ideal gain design criterion: 1) According to the first-order expression of b, it is not possible to get stable oscillations by controlling via the slew-rate of o amp #1 (rlarl'/o. For slew-rate based stable oscilfations the opamp #2 should be the controlling device. 2) It is possible to make pI =O (no phase compensation allowed) without degrading the criterion features. 3) Controlling via p3 (p3a 5) yields the oscillator frequency to be insensitive to simultaneous proportional changes in the opamp time constants. Random changes in zz cannot however be absorbed by the adaptation mechanism and will hence influence the oscillation fkequency. Notice finally that, since ai does not influence the sign of b, it is not possible to get a linear relationship between w, and w,, in opposition to what can be achieved for the one-opamp case. Ideal frequency design criterion. Some interesting properties can be observed in case k is not fixed to the ideal value. As a matter of fact, in [71 the authors proposed a composite amplifier structure were by properly selecting k it was possible to get w,=ai for frequencies up to about 1.5/~. Tt means that the osci1l;ih)n frequency can be madc to be completely (7) Regarding the parameter determining the oscillation condition, it is convenient to separately consider two different cases. One where PI, pz, and the time constants are fixed according to (6) and the control is made either via k or p3. For this case It I esults, 3n the other hand if k and p3 are fixed using (7) and the control is made via any of the other parameters, the following can be obtained: There are some observations applying to this design criterion: 1) Stable operation controlling via the slew-rate is only possible for those structures where a=q, and for the controlling device being the opamp #2 (rzaT;2'/5). Besides, the oscillation condition for this case is insensitive to proportional changes in the opamp time constants. 2) Amplitude control via ~2, pl or p2 (based on (8b)) requires k to be selected to ulfill(7) which depends on frequency. Hence this approach is only appropriate for fixed frequency applications. 3) For amplitude control via k (kak E,), the oscillation condition can be made insensitive to both random and simultaneous changes in the opamp time constants by resorting to structures having a=xZ. Otherwise, this condition will only be insensitive to simultaneous proportional changes. 4) Observe for amplitude control via k, the oscillation condition (8a) depends on frequency. As a consequence, large variations of the critical values result in case k'(kak'5) is selected to ensure self-starting operation in a wide frequency range. In case the the adaptation process is implemented by resorting to nonlinear limitation, lar e distortions are thus expected for the hig%- frequency edge of the range. Performance of the ideal frequency criterion is illustrated in Table 2 where we show experimental results for a composite opamp structure with p1=3.105, p2=k and p3=kpI. LM747 dual opamps were used, the controlling device being the slew-rate of the opamp #2. As it can be seen, deviations in frequency are lower than 2% for frequencies up to 1 OOKhz. 2591, Ideal frequency, kHz 22.7 49.1 88.4 107.8 105.5 Real frequency, kHz 22.5 492 88.3 Discussion on Composite oDamD structures Fig.4(a) shows a general block diagram for a two opamps composite amplifier. Triangular blocks correspond to opamps while rectangular blocks represent weightmg by resistors as it is illustrated in Fig.qb). Switches SI and Sp have not be actually im lemented but have been used to indicate that two di8erent points a? available to be used as the actual amplifier output. ,------------------------------------------------- I 1 I I ! * ! I I I I I I I I I I I I I I I I I I I I I 1 I I I I I I I I I I I I I I I I I I IF I 1 I I I I I I I I I------___----_-____----- Fig.4: General block diagram for a two Table 3 shows expressions for k, PI, p2, p3 and a as functions of C, and yi for the two possible amplifier outputs. For proper operation of the composite amplifier Wien-Bridge, actual structures derived from Fig.4 would fulfill the following constraints, opamps composite amplifier. Table 3 "aa ensures input to the composite opamp will be via high impedance nodes. from Fig.4 with (9) and taking into account Ta%le 3 it is not a complicated task to derive com site amplifier structures for two composite am ffier design criteria presented in the aper. As it is gmonstrated by the experimental resurts included in the communication, structures can be found fulfilling the criteria for frequencies up to about 0.9~. References [ 11 A. Soliman: "Classification and Generation of Active Com ensated Non-inverting VCVS Building BlmLs". Znt. Journal of Circuit Theory and Applic., Vo1.8, pp 395-405, Oct. 1980. [21 J.L. Huertas and A. Rodriguez Vbz uez: "On the Active Com ensation of Operational AmplifiersBased VCV&. ZEEE Trans. Circuits and Systems. Startin CAS-29, p 497-506, July 1982. W.B. Milhael and S. Michael: "Composite Operational Amplifiers: Generation and FiniteGain Applications". ZEEE Trans. on Circuits and Systems, CAS-34, nQ5, pp. 449-459, May 1987. M.A. Redd : "Operational-Amplifier Circuits with Variable $has, Shift and their Application to High-Q Active RC-filters and RC%scillators". ZEEE Trans. Circuits and Systems, CAS-23, pp __ 384-389, June 1976. [51A. Budak and K. Nay: "Operational Amplifier Circuits for the Wien-bridge Oscillator". ZEEE Trans. Circuits and Systems, CAS-28, pp 930-934, September 1981. [6] S. .Awad: "Extending .the Frequency Range of a Wien-Bridge Oscillator using Composite Operational Amplifiers". ZEEE Trans. on Inst. and Measurement, vo1.38, nQ3, pp. 740-744, June 1989. [7] A. Rodriguez VBzquez et al.: "High-Frequency Design of the Wien-Bridge Oscillator using Composite Amplifiers". ZEEE Trans. Circuits and Systems, CAS-34, pp 441-443, April 1987. [8] A. Carlosena et al. : "An Improved Wien Bridge Oscillator". IEEE Trans. Circuits and Systems, CAS-37, pp 543-546, April 1990. [9] J.L. Huertas et al.: "Analysis and Design of SelfLimiting Single-Opamp RC Oscillators". Int. Journal of Circuit Theory and Applications, ~~ Vo1.18, pp.53-69,1990. [lo]B. Perez-Vedu: "Nonlinear Modeling of Operational Amplifiers Based RC Oscillators". PhD dissertation, University of Seville 1985. first one ensure no parasitic poles will be allocated in the right half of the complex frequency plane (according to the one pole opamp model). Second one