On the Design of Voltage-Controlled Sinusoidal Oscillators Using OTA's

198 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 2, FEBRUARY 1990 On the Design of Voltage-Controlled Sinusoidal Oscillators Using OTA’s ANGEL RODR~GUEZ-VAZQUEZ, MEMBER, IEEE, BERNABE LINARES-BARRANCO, JOSE L. HUERTAS, MEMBER, IEEE, AND EDGAR SANCHEZ-SINENCIO, SENIOR MEMBER, IEEE Abstract -A unified systematic approach to the design of voltage-con- trolled oscillators using only gperational pansconductance amplifiers (OTA’s) and capacitors is discussed in this paper. Two classical oscillator models, i.e., quadrature and bandpass-based, are employed to generate several oscillator structures. They are very appropriate for silicon monolithic implementations. The resulting oscillation frequencies are proportional to the transconductance of the OTA and this makes the reported structures well-suited for building voltage controlled oscillators (VCO’s). Amplitude stabilization circuits using both automatic gain control (AGC) mechanisms and limitation schemes are presented which are compatible with the transconductance amplifier capacitor gscillator (TACO). Experimental results from bipolar breadboard and CMOS IC prototypes are included showing good potential of OTA-based oscillators for high frequency VCO operation. I. INTRODUCTION HE generation of sinewaves is a classical problem T with application in communication systems, instrumentation, measurement, etc. In particular, the yoltagecontrolled frequency-variable gscillator (VCO) has a number of important applications in communication circuits A great variety of RC-active circuits have been developed for generating sinewaves, most of them based on the use of the conventional operational amplifier (op amp) as the active component ([3]-[6] and included references). These circuits typically perform correctly in the audio frequency range ( G 20 kHz), but their performance becomes severely degraded as the frequency increases. Even by either using special design techniques that exploit the opamp reactive behavior [7] or resorting to composite amplifiers [8], [9], the useful frequency range of RC-active oscillators is below 100 kHz for general purpose 741 op amps. Together with these frequency limitations, op amp-based VCO’s exhibit other problems further restricting their use- [I13 PI. Manuscript received August 1, 1988; revised March 9, 1989. A. Rodriquez-Vkquez and J. L. Huertas were supported by the Spanish CICYT under Contract ME87-0004. This paper was recommended by Associate Editor C. A. T. Salama. A. Rodriquez-Vbquez and J. L. Huertas, and B. Linares-Barranco are with the Department of Electronics and Electromagnetics, Faculty of Physics, University of Seville, Seville 41012, Spain. E. Sinchez-Sinencio is with the Department of Electrical Engineering, Texas A&M University, College Station, TX 77843. [EEE Log Number 8932874. In bipolar and weak inversion MOS OTA’s g, = h,I,,,, and g,,, = h, for MOS OTA in the saturation region where h, and h, are ependent upon temperature, device geometry and the process. fulness. Different variable-frequency RC-active oscillators have been reported whose frequency of oscillation can be controlled by a single resistor without affecting the oscillation condition [6 and included references]. Some additional circuitry has to be added to achieve VCO operation using these oscillator structures. One possible method is obtained by substituting the controlling resistor by a FET working in the ohmic region [lo]. This consequently reduces the voltage swing across the simulated resistance. Besides, the tunable frequency range is somewhat reduced and switching among different resistors is needed what makes the design approach not readily compatible with monolithic integrated circuits. Most of these problems can be overcome by the use of the operational t_ransconductance amplifier (OTA) as the active building block for VCO’s. By interconnecting OTA’s and capacitors (TAC), oscillating circuits can be obtased whose frequency of oscillation is proportional to the transconductance gain g, of the OTA. Since the transconductance gain of the OTA can be varied or programmed by an external power supply,’ the VCO operation can be readily implemented. Thus fully integrated voltage-con- trolled TAC Qscillators (TACOs) can be obtained with a frequenFdjustable over wide ranges, avoiding the necessity of switchmg among several passive components as observed in opamp designs. TAC oscillator structures are also shown in this paper to be advantageous in terms of the maximum frequency of operation. As compared to the conventional opamp, transconductance amplifiers exhibit a higher potential for high-frequency applications [ll]. This is an inherent property due to the fact that no internal high-impedance nodes are included in a typical transconductance amplifier architecture. As a matter of fact, several TAC filter structures have been recently reported for operation at frequencies in the range of megahertz [12]-[14]. Here we show that by using TACOs it is possible to push the oscillation frequency well above the maximum opamp oscillator ratings. In this paper we present several TACO structures and discuss them by taking into account both linear and nonlinear effects. The herein reported TACOs are generated from two classical oscillator models, namely the quadrature and the bandpass oscillators. In order to make the paper self-contained, we first discuss the quadrature and the bandpass models in some detail in Section 11. Section 0098-4094/90/0200-0198$01.00 01990 IEEE Authorized licensed use limited to: Universidad de Sevilla. Downloaded on April 06,2020 at 13:49:55 UTC from IEEE Xplore. Restrictions apply. RODRIGUEZ-VAZQUEZ et al. : SINUSOIDAL OSCILLATORS 199 I11 is devoted to the generation of TACO's from the oscillator models. Section IV addresses the topics of amplitude control and distortion. Section V considers OTA parasitics and Section VI presents some experimental results. Most of the material in this paper can be used for either discrete bipolar or monolithc CMOS implementations. Experimental results covering these two cases are included. These results shows very high potential of TACO's for VCO hgh-frequency applications. 11. THE QUADRATURE AND BANDPASS-BASED OSCILLATOR A, The Quadrature Oscillator Model An ideal quadrature oscillator consists of two lossless integrators (one inverting and one non-inverting) cascaded in a loop, what results in a characteristic equation with a pair of roots lying on the imaginary axis of the complex frequency plane. In practice, however, parasitics may cause the roots to be inside the left half of the complex frequency plane, hence avoiding the oscillation to begin. Any practical oscillator must include some form of regeneration to ensure that the roots are initially located in the right half plane and hence that the oscillation is created [l]. Besides, some amplitude stabilization mechanism must be added to pull the roots back towards the imaginary axis until a stable value for the amplitude is obtained. A practical quadrature model including an amplitude stabilization scheme of the limiting type [l] is shown in Fig. 1, where f(.) represents a weakly nonlinear limiter. Analysis applying the chain differentiation rule gives, for Fig. 1: d2x dx df(x) --p - [ b,? - b,] + s2ix = 0 (1) where b,, b,, and s2, A ( wolwo2)1/2 are positive numbers. The following equations have to be fulfilled for the existence and uniqueness of stable quasi-sinusoidal oscillation in the weakly nonlinear system of Fig. 1 [15]: b, dfo 1 - b, < 0, for some x, > 0 (2b) dx x=xo where f(x) is assumed to be odd-symmetric and df/dx is assumed to be decreasing for x > O., Imposing the condition in (2a) means that the roots of the characteristic equation of the linearized model of Fig. 2 are initially inside the right-half plane; it ensures that the system is self-starting. On the other hand, (2b) means that these roots move towards the left-half plane once the oscillation amplitude increases. 2Buonomo et ul. give conditions for an oscillator composed by a nonlinear element and a passive linear network. Although in this paper the linear network is active, the conditions are still valid providing the oscillator is represented by the block-diagram of Fig. 1. Also, we assume the nonlinear function f( ') is odd-symmetric for simplicity, but this condition is not required in the paper by Buonomo et al. Fig. 1. Block diagram for a practical quadrature oscillator including a limiter. The oscillation frequency and amplitude can be calculated by using the describing function approach [16]. The input of the nonlinearity is assumed to be an undistorted sinusoidal signal, x(t) = Asin(wt). The output of the nonlinear block is first expanded into a Fourier series and then this series is truncated to include the fundamental harmonic component. The describing function3 is the gain relating the amplitude of this fundamental component to that of the input, N( A) = -?-. /*"@f [A sin (ut)] sin (at) dt nA o (3) The following characteristic equation results for Fig. 1 by using the describing function: s2 - s [ b,N( A) - b2] + s2i = 0. (4) The oscillation frequency w, and amplitude A, can be obtained by first making s = jw and then considering separately the real and imaginary parts of (4). The resulting limit cycle is stable provided that the coefficient of the s-term in (4) is positive when A > A, and negative otherwise. It is fulfilled in case the nonlinear function f( -) has a decreasing derivative. All the previous discussions can be summarized in the following design equations: where an odd-symmetric nonlinearity have been assumed. The upper equations give a first-order approximation of the frequency and amplitude, while the lower ones ensure that oscillation exists, is stable, and unique. In Fig. 1 the amplitude is controlled by a limiting function. Automatic gain gontrol (AGC) is a more sophisticated scheme which-usually reduces distortion [ 11. Fig. 2 shows the block diagram for an AGC stabilized quadrature oscillator. Note that the block for f( . ) is no longer present. 30bviously an error results as a consequence of the use of the describing function. For quasi-sinusoidal oscillators ths error is expected to be small since low-distortion signals are produced. A method for the error analysis of the describing function approach can be read from [17]. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on April 06,2020 at 13:49:55 UTC from IEEE Xplore. Restrictions apply. 200 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 2, FEBRUARY 1990 h=A-E J Fig. 2. Block diagram for a practical quadrature oscillator including an AGC mechanism. (b) Fig. 3. The Bandpass-based oscillator model. (a) With a limiting type stabilization scheme. (b) With an AGC scheme. Observe in Fig. 2 that the amount of positive feedback around the noninverting integrator depends on the difference, denoted by A, between the oscillator output amplitude, A, and an external reference level, E. In steady-state both the positive and the negative feedback loops are linear and the system can be described by the following equation; d2x dx dt2 dt [ b,( A - E) - b2] + Q~x = 0 (6) where b,( .) is used to denote a functional dependence. The design equations for this case are I I WO = 52,; A, = E + bF1( b,) 1 V+ V- V+ V- b --+ ---o Fig. 4. Symbolic and ideal model for the OTA B. Bandpass-Based Oscillator Model The bandpass-based oscillator consists of a bandpass filter structure with positive feedback. Fig. 3(a) shows the block diagram for a bandpass oscillator including a stabilization mechanism of the limiting type. Fig. 3(b) shows the corresponding block diagram for a general AGC mechanism. Analysis allows us to derive the following differential equation from Fig. 3(a): +Q~X=O (8) dt2 dt which corresponds to (1) by defining kh 4 b, and Q,/Q A b,. In a similar way, the characteristic equation for Fig. 3(b) can be obtained from (6) after making b,( A) = hk( A) and b2=f10/Q. Equation (5) is thus valid for designing either quadrature or bandpass-based oscillators including a limiting type scheme, while (7) applies either for quadrature or bandpass AGC oscillators. There is a particularly simple implementation of the limiting-type bandpass oscillator concept where the nonlinearity is of the ON-OFF type. The corresponding block diagram is readily obtained from Fig. 3(a) by substituting the cascade of the amplifier and nonlinear block by a comparator. The existence of oscillations for this system is a consequence of the very high gain of the comparator and thus does not require any other design conditions to be fulfilled [16]. Besides, the local stability of the resulting limit cycle is guaranteed because dN(A)/dA < 0 for an ON-OFF nonlinearity. The oscillation frequency is also given by (5). However, since now it is b,= hk >> 1, the amplitude is not conveniently expressed using this equation but as A, = 4E,hQ/mQo [16], where E, denotes the comparator saturation levels. This concludes this review section on oscillator models. Next sections of the paper address the problem of implementing previous models by using TAC structures. 111. GENERATION OF TAC OSCILLATOR STRUCTURES Until otherwise stated, we will use the ideal OTA model of Fig. 4. The transconductance gain of the OTA is assumed to be a positive number. Interchanging the OTA input terminals is equivalent to substitute g, by - g,. The systematic application of the OTA for different useful linear functions can be read elsewhere [18]-[20]. AS can be seen from Fig. 1 the basic building blocks for Authorized licensed use limited to: Universidad de Sevilla. Downloaded on April 06,2020 at 13:49:55 UTC from IEEE Xplore. Restrictions apply. RODRIGUEZ-VAZQUEZ et d. : SINUSOIDAL OSCILLATORS 201 ,... ............................................... ................................................. (4 ................................................ I ......................... ........................................................................., Fig. 5. TAC quadrature oscillator structures. ........................................................... ,_........ viq$J---+vo g'm3 - - - - (4 L____________ ...................................................... vJ-Fqz - - - (b) Fig. 6. TAC bandpass oscillator structures. TABLE I FOR THE DIFFERENT OSCILLATOR STRUCTURES o,, k, b,, b2 AND b VALUES AS FUNCTIONS OF THE OTA TRANCONDUCTANCE GAINS I I I I I I I I quadrature oscillators are integrators. In a similar way, state-variable realizations of second-order bandpass filters also use TAC integrators as the basic building blocks [19]. A. TAC Oscillator Structures Different TACO structures are given in this subsection and analyzed using linear models. The control of the amplitude in these structures will be addressed in Section IV. Fig. 5 shows two quadrature TACOS. They have been obtained from Fig. 1 by using TAC lossless and lossy integrators and OTA-simulated linear resistors [20]. Both circuits in Fig. 5 can be described by a second-order characteristic equation: (s2- bs+Q~)Vo(s)=O. (9) The first two rows of the first column in Table I give the expression of fi0 as a function of the transconductance gains for these quadrature TACOS. The corresponding rows of the sixth column give the expression for b. We shall postpone any further comment on quadrature TACO design until the next section. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on April 06,2020 at 13:49:55 UTC from IEEE Xplore. Restrictions apply. 202 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS. VOL. 37, NO. 2, FEBRUARY 1990 Fig. 7. Basic single feedback oscillator model and characteristic equation. Fig. 6 shows two bandpass-TACO alternatives including the bandpass filter (enclosed by broken lines) and the voltage amplifier. The bandpass filter structures in Fig. 6 have been selected because they 1) exhibit high-input impedances, 2) allow us to independently tune the filter gain, the quality factor an the resonant frequency. These features give high flexibility in oscillator design [19], Looking at the first and sixth columns of Table I the thrd and fourth row give the expressions for Qo and b of these bandpass-TACOS. B. A Unified Feedback Model for TAC Quadrature and Bandpass Oscillators Most of the existing theory for sinusoidal oscillator analysis (see [l], [15], [22] as some significant references) is based on modeling the oscillator structure by the basic feedback loop in Fig. 7, where the characteristic equation of the model has been included. The bandpass TACO'S of Fig. 6 are directly modeled by Fig. 7, T(s) being the non-inverting transfer function of the bandpass filter and k = gm3/gk3. Analysis of Fig. 6(a) gives whle, for Fig. 6(b), In order to obtain a similar feedback-loop model for quadrature TACO'S, it is first convenient to redraw either Fig. 5(a) or (b) as illustrated in Fig. Sa, where the block labeled Z(s) corresponds to the portion enclosed by broken lines in those figures. The corresponding feedback loop is then shown in Fig. S(b), where, for convenience, the feedback signal has been normalized by g,, (g,, is defined in Fig. 8). Using Fig. S(a), the following transfer function results for the forward path of the feedback model corresponding to the quadrature oscillator of Fig. 5(a): 4The OTA g,',, of Fig. 6(a) is used to equalize the maximum voltage swings at <,i and yJ2 [19]. It could be eliminated in case equalization is not needed. In this case, Fig. 6(a) should have to be modified by either interchanging the input terminals of the OTA's g,, and gmZ or interchanging the input leads of the OTA gm3. Fig. 8. Modeling quadrature TACOs by the interconnection of oneports (g,, & g,,, for Fig. 5(a), g,,, & s,,, for Fig. 5(b). being, for ths structure, k = gm3/gm2. In the case of Fig. 5(b), the following T(s) results: with k = g,,Jgm4. In summary, the different TACO structures presented in the paper can be modeled by the feedback loop in Fig. 7, consisting of a forward path with a transfer function of the bandpass type, sh T(s) = s2 + s( Q,/Q) + and a feedback path described by a transconductance gain ratio k. This feedback model hence provides a unified framework for comparison among the different TACOs. Table I shows the expressions for the different relevant parameters associated to the feedback model as functions of the transconductance gains for Figs. 5 and 6. C. Some Observations on the Structures Design Considerations: Parameter b controls both the oscillation condition and the amplitude of the oscillations. Similarly, the level of distortion can be also shown to depend on b. Note from Table I that the OTA transconductance gains appearing in the expressions for Qo of Figs. 5(b) and 6(b) are different to that appearing in the corresponding expressions for b. It means that we can tune the oscillation frequency by changing any of the transconductance gains that control wo and without affecting neither the oscillation condition nor the amplitude of the oscillation. This feature is very appropriate for VCO operation. On the contrary, for Figs. 5(a) and 6(a) the oscillation frequency and the oscillation condition are not independent, what makes these latter structures to be disadvantageous from the tuning point of view. Observe from (lob) that the T(s) gain at the center frequency is fixed to 1 for Fig. 6(b), whle it is not constrained for Fig. 6(a). Since regeneration in the feedback loop must be guaranteed at the design stage, it means that the gain, k, of the feedback path is constrained to be greater than 1 for Fig. 6(b), while it can be set to 1 for Fig. 6(a). Hence, there is the possibility of substituting the two OTA's in the feedback path of Fig. 6(a) by a single wire. Note, on the other hand, that only 3 OTA's are required for Fig. 5(a) while 4 OTA's are required for Fig. 5(b). Summarizing, the structures of Figs. 5(a) and 6(a) are simpler than their counterparts in the sense that they Authorized licensed use limited to: Universidad de Sevilla. Downloaded on April 06,2020 at 13:49:55 UTC from IEEE Xplore. Restrictions apply. RODRIGUEZ-VAZQUEZ et al. : SINUSOIDAL OSCILLATORS 203 require less active components, what makes them interesting for those applications where tuning is not a critical task. Signal Scaling and Node Voltage Equalization: There is a limitation on the maximum OTA differential input voltage that ensures linear operation of the device. Besides, since in the proposed oscillator structures all the ungrounded OTA input leads are connected to an OTA output, voltage limitation is referred to both the input and the output of the OTA. Analysis of the maximum voltage swing at the different points inside the circuits is hence required for proper design of TACO’S. Table I1 shows a set of expressions giving the relevant voltage amplitude ratios as measured at the frequency of oscillations for the different TACO structures. For convenience the node voltages V,, and V,, as well as the OTA differential input voltages V,, are referred to Vo2 (Vu, is the input to the OTA of transconductance g,,); observe that by construction it is lVull = lVo21 for all the structures. Different choices are possible when specifying the voltage amplitude ratios for a particular TACO design. The implementation of any particular election requires consideration of Tables I and 11. Consider, for example, the TACO of Fig. 6(b) and note from Table I that the inequality g,, > g;, must hold to ensure a positive quality factor. Assume we specify lVo31 to be greater than lVo21. From Table I1 it means g,, > gL3; this resulting in a negative value of b from Table I, the circuit would not oscillate. Different situations regarding this problem may arise from the proposed oscillator structures, and for the type of limiter or gain control used in each case. A very interesting design specification concerning amplitude scaling is to equalize all node voltage swings at the frequency oscillation. Together with improving the dynamic range, equalization means the possibility of having multiple outputs with the same voltage level; in particular, two equal-amplitude 90” phase-difference signals would be obtained for all the structures, corresponding to the V,, and Vo2 node voltages. As seen from Table 11, the condition allowing us to equalize V,, and V,, for each one of the proposed TACOs is Clgm2 = C2gm,. Observe from Table I that the quality factor of Fig. 5(a) is fixed to a value Q = 1 as a consequence of imposing the equalization condition. It is a drawback that can result in an increased distortion level for this structure. On the contrary, the TACOs of Figs. 5(b), 6(a) and 6(b) can be designed for amplitude equalization and, at the same time, allow us to separately control the b, SI,, and Q values, which is a very appearing feature. Iv. AMPLITUDE STABILIZATION AND CONTROL Using the nonlinear saturation characteristic of the OTA is the simplest form of amplitude stabilization for TACO’s. The effect of this is equivalent, in a first order approximaf=GNL(v’ 14 slope:G, I Fig. 9. Nonlinear resistor for amplitude limitation and control of TACO’s. TABLE I1 RELEVANT VOLTAGE AMPLITUDE RATIOS FOR THE PROPOSED TACO STRUCTURES t I I I I Figure Sa Figure Sb Figure 6a Figure 6b IV.11= lVorl W.d= IVOZl I I I I I before the natural limitation of the OTA is a more sophisticated scheme providing better controllability. Finally, exploiting the programmability of the transconductance gain of the OTA is a natural and yet simple way to implement the AGC concept in TACO structures. A. Amplitude Control by External Limiters Limitations by Nonlinear Resistors: Three basic guidelines have to be followed when looking for limiting circuits for the proposed TACO’s: 1) the operation of the limiter must be compatible with the OTA operation principle (voltage to current conversion), 2) the equations describing the global operation of the oscillator (TAC structure plus limiter) must be in accordance with the oscillator models in Section 11, and 3) the implementation of the limiter must be possible for either the bipolar case or the CMOS case.A way to cope with all previous objectives is to use a nonlinear resistor described as in Fig. 9.5 Fig. lO(a) shows how to connect the nonlinear resistor to ensure proper amplitude control of quadrature TACOs, the block labeled Z(s) being the port impedance of the circuit enclosed by broken lines in Fig. 5. Fig. 10(b) illustrates the corresponding connection for bandpass TACO’s, T(s) being the transfer function for the circuit enclosed by broken lines in Fig. 6. tion, to decreasing the transconductance gain of the saturated OTA and thus Of b. We shall postpone the consideration of this possibility until Section v. Adding a nonlinear circuit which acts us to the ’In the discrete bipolar case this resistor can be implemented by using a linear resistor and a couple of diodes [23]. In the monolithic case the diodes can be substituted by MOS transistors [24] and the linear resistor simulated by OTA’s. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on April 06,2020 at 13:49:55 UTC from IEEE Xplore. Restrictions apply. 204 n L IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 2, FEBRUARY 1990 (4 (b) Fig. 10. Connection of the resistor for amplitude control of (a) quadrature TACOs and (b) bandpass TACOs. I slupe 42 = (b) (c) Fig. 11. (a) Unified nonlinear feedback loop model for TACOs. (b) Characteristics of the nonlinear block for quadrature TACO’s. (c) Corresponding characteristics for bandpass TACO’s. We can now compare (12) with (l), keeping in mind the conditions given in (2). Note that (dm,/dx)l,,, = (dmbp/dx)l,=, = 1. Thus for amplitude-limited TACOs the following inequalities must hold for quasi-sinusoidal oscillations to exist and be unique: where q2 is the slope of the outer segments in Fig. ll(b) and (c). The frequency and the amplitude of the oscillations can be calculated using the describing-function approach: where D denotes the value of the breakpoints of the characteristics of the nonlinear block of Fig. 11; being D = E for quadrature TACOs (Fig. ll(b)) and D = E/k for bandpass-based TACO’s (Fig. ll(c)). On the other hand, Nm(. ) is given by Nm(z) = (1- 42)N,(z) + q2 (154 z being an auxiliary variable and NI(.) being otherwise. Fig. ll(a) shows a unified nonlinear feedback model valid either for quadrature or bandpass TACOs. The linear portion of ths model is defined in Table I for each particular TACO. For quadrature TACOs the signal y(t) in Fig. ll(a) is defined as Zf/gm3 where Zf is the port current for the block Z(s) as shown in Fig. lO(a). Since If = gm3Vo- GNL( V,), and taking, into account the definition of GNL( .) in Fig. 9, the characteristic of Fig. ll(b) results in the nonlinear block in the case of quadrature TACOs (G, in this figure is the slope of the external segment in Fig. 9). For bandpass TACO’s, y(t) = y(t)gA3/gm3, K(t) being defined in Fig. lob, and the characteristic for the nonlinear block is the one shown in Fig. ll(c). Analysis yields the following scalar differential equation for the unified nonlinear feedback TACO model: that applies to quadrature TACO’s (see (1)) by making m(-) = m,(.) and to bandpass TACO’s (see (8)) for m(.) = mpb(*)* Using (14b) together with Table I allows us to set the different OTA transconductance gains and capacitances for a given frequency and amplitude. For a given setting to correspond to a practical oscillator, we must also ensure that (14a) is fulfilled. Note that this latter equation constraints the sign of parameters b and b’ but does not impose any restriction on their values. Further insight on how to set these parameters can be gained by considering the distortion level at the output of the oscillators. Distortion Analysis: Assume the distortion level for x(t) in the model of Fig. ll(a) is small enough so that y(t) can be expressed as the Fourier expansion of the output of the odd/symmetric nonlinear block when a pure sinusoidal signal, A,sin(w,t), is applied to its input: 00 y(t) = B,sin(nw,t) n =1 Bn=~~2n/wm{A,sin(wot)}sin(nwot)dt. no (16b) The different harmonic components of x(t) can be calculated [ 11 by applying the superposition principle to the forward path in Fig. ll(a), the input signal being that Authorized licensed use limited to: Universidad de Sevilla. Downloaded on April 06,2020 at 13:49:55 UTC from IEEE Xplore. Restrictions apply. RODRIGUEZ-VAZQUEZ et al. : SINUSOIDAL OSCILLATORS 205 in (16a), x(t)= C,,sin(no,t++,,)= H,,B,,sin(nw,t++n) (174 m m n =1 n =1 hk n a,, = -tan-’ { - 1n!:2} + 3. Thus the following expression for the total harmonic distortion (THD) results: where b, A hk, b, A Q,/Q, and C, is the amplitude for the fundamental harmonic, it is, C, A A,, as calculated by (14b) using the describing function approach. On the other hand, B,, can be easily calculated from (16b) to give THDXlO0 Ih>h =U 995 25 11111111111 Q=l 20 - 15 - q=2 \-- lo - Q=4 i hlhl =O 999 8kL,l ,,,, I ,,,, I ,,,,~,,,,,,,, .i 091 080 055 0 -1 23 -395 -10 -23 53 Fig. 12. (a), (b) Illustrating the influence of the design parameters on the THD. (c) Normalized oscillator amplitude versus q2. (19) sin[(n-l)O,] - sin[(n+l)O,] n+l 0, n even where we define 0, A sin-’( D/A,), D representing the value of the breakpoint in Fig. ll(b) and (c). Observe from (19) that the actual value of the THD is a function of b,/b,, Q and q,. Fig. 12(a), (b) illustrate the dependence of THD on these parameters. Fig. 12(a) shows THD X 100 versus q2 (note from Fig. ll(b), (c) that q2 is always smaller than 1) for different values of Q and assuming b2/b, = 0.995; similar families of curves result from different b,/b, values. Fig. 12(b) shows THDX 100 versus q2 for different values of b,/b, assuming Q = 5. We note the THD decreases as Q increases and b2/b, increases. On the other hand, the THD seems not to be much influenced by the values of q2 with the exception of a sudden THD decrease when q2 is near to 1. We can exploit this for obtaining low values of the THD by using values of q2 near the unity. However, a drawback appears in that it corresponds to large oscillator amplitudes and thus can drive the OTA’s outside their linear input ranges. This can be seen from Fig. 12(c) where we represent D/Ao as a function of q2 for different b,/b, values. As it can be seen, decreasing q2 results in an increase of the ratio among the oscillation amplitude and the breakpoint D of the characteristic of the nonlinear block. The observations included in Section 111-C on the different structures are confirmed by the above distortion analysis. Those structures allowing separate control of b,/b,, Q,,, and Q, as is the case for Figs. 5(b) and 6(b), are potentially advantageous as compared to the others. An additional observation concerns the different way parameter q2 varies from either quadrature or bandpass-based TACO’s. As it can be seen from the definition of q2 in Fig. ll(b) and (c) this parameter can be made to change inside the interval (- m,1] for quadrature TACO’s, and inside (0,1] for bandpass-based TACOS. Since the oscillator amplitude increases as q2 increases above zero (see Fig. 12(c)), quadrature TACO’s can be designed to exhibit smaller amplitude values than bandpass-based TACO’s. A potential advantage of these former structures can be observed by talung into account that the amplitude must be maintained below an appropriate safety level to avoid OTA nonlinearities. Amplitude Control by Voltage Comparators: Resorting to the unified feedback TACO model presented in Section I11 allows us to apply this technique to either bandpass or quadrature TACO’s. Fig. 13(a) shows the block diagram for bandpass and Fig. 13(b) shows the corresponding Authorized licensed use limited to: Universidad de Sevilla. Downloaded on April 06,2020 at 13:49:55 UTC from IEEE Xplore. Restrictions apply. 206 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 2, FEBRUARY 1990 - I (a) (b) Fig. 13. Amplitude control using comparators. (a) Bandpass TACO. (b) Quadrature TACO. diagram for quadrature (see (10) and (11) for T(s) and Z(s), respectively). Assume the saturation levels for the comparator for k E,. The oscillation frequency and the fundamental harmonic amplitude of V, are, respectively [16]: where E,’= E, for bandpass TACO’s and E,’= kE, for quadrature TACO’s (see Table I for the expressions of Q,, h and Q as functions of the transconductance gains). In steady state the feedback signal in Fig. 10 (Ir in Fig. 10(a) and V, in Fig. 10(b) is a square wave. By first expanding it into a Fourier series and then computing the response of the linear portion of Fig. 13, we obtain the following relation for the nth harmonic distortion at the bandpass filter output, 1 n odd (21) C” 1 _- - - where C, is the n th harmonic’s amplitude and C, A A,, A, being as calculated in (20). The very low input voltage range for linear operation of discrete bipolar OTA’s discourages the use of this technique for discrete TACO’s. However it can be very appealing in the monolithic case where the comparator can be implemented by using a simple CMOS inverter. B. Amplitude Control by AGC Exploiting the programmability of the transconductance gain of the OTA is a natural and yet simple way to implement the AGC concept in TACOs. Let us generically use z to denote the programming variable of the OTA, z being a bias current for discrete bipolar OTAs and a bias voltage for CMOS ones. Assume t is allowed to vary Az around a quiescent value to. The following inequality holds for both types of OTA’s: which means that he transconductance increases with z. For any of the proposed TACOs, the AGC concept can be implemented by making the transconductance gain g,, REMINDER OF THE TACO detector U Fig. 14. AGC loop for amplitude control of TACOs. Fig. 15. Second-order OTA macromodel to depend on the difference between the peak amplitude of the oscillator output and an external reference E,, as is illustrated in Fig. 14. Let us consider we want the oscillation amplitude to coincide wit the external reference level E,. Note from Table I that g,, controls parameter k for all the proposed TACO’s and that k controls the value of parameter b. For the circuit oscillating with A,= E,, we require b to be zero for such a value of the amplitude, i.e., the transconductance gains must be trimmed to fulfill the following equation: Since dk/dg,, > 0 and (dgm3/dA)Izo < 0 for Fig. 14, then (dk/dA)I,, < 0, which means that the roots of the characteristic equation return to the left half plane for the amplitude eventually increasing above A,, and hence that oscillatory behavior is stable. Different alternatives for AGC-TACO can be used involving control for one or more OTA’s. For the bipolar case, where z is a current, the required amplifier and the summing node can be implemented by an OTA and using Kirchoff s current law, respectively. Several possibilities for the monolithic case have been proposed by the authors and validated at the simulation level [21]. V. OTA PARASITIC EFFECTS Second-order effects are specially important in oscillators, where even small parasitics can produce pole displacements large enough to cause the oscillation to vanish or to increase distortion severely. There are three main OTA parasitics affecting the performance of TACOs: 1) Finite input and output impedances, 2) frequency-depen- dent transconductance gain, and 3) output current saturation. Fig. 15 shows a macromodel including all of these parasitics. Let us focus on the reactive behavior of the OTA modeled by Z,(s) in Fig. 15. A dominant nonideal effect in the frequency response of the OTA is the phase error Authorized licensed use limited to: Universidad de Sevilla. Downloaded on April 06,2020 at 13:49:55 UTC from IEEE Xplore. Restrictions apply.