INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS Int. J. Circ. ¹heor. Appl., 26, 215—218 (1998) LETTER TO THE EDITOR A CAUTIONARY NOTE ON STABILITY OF CURRENT CONVEYOR-BASED CIRCUITS R. CABEZA AND A. CARLOSENA Dipartmento de Ingenierı& a Ele& ctrica, y Electro& nica, ºniversidad Pu& blica de Navarra, 31006-Pamplona, Spain INTRODUCTION The invention of the Operational Amplifier dates back to the period of the Second World War but such amplifiers only became commercial as monolithic active devices 20 years later.1 It is therefore not strange that some circuits, including ideal (infinite gain) differential amplifiers, were proposed prior to the availability of the device. As soon as the opamp was accepted, its use extended among designers of those circuits, and other new ones were designed and tested with opamps.2 Simple and accurate models for the opamps were developed in order to predict and explain deviations from the ideally expected performance, giving rise to a tremendous research effort which established the now well known body of the theory on RC- and even R-Active Circuits.2,3 Current conveyors have followed to some extent a similar pattern, although they are far from producing the influence that opamps had at their time. First, they were proposed in the late 1960s4 and were followed by a number of publications suggesting a plethora of circuit applications that made use of the different versions of current conveyors (i.e. first, second and third5 generation, positive and negative) always as ideal devices. About 20 years later, the first commercial version of the second-generation positive current conveyor was available in the market. The list of applications for current conveyors is today endless,6 but despite the commercial availability of more than one monolithic version (and many others in discrete form) many papers report circuits that have not been tested in the lab and not even simulated. It goes without saying that theoretical analyses do not go in most cases, unlike it happened with opamps, beyond the calculation of ideal transfer functions and sensitivities, in spite of the availability of some models.7 In many cases experimental results are based on non-optimum implementations for current conveyors (e.g. unity feedback opamps with current output mirroring). Consequently, a comparison among the apparently similar topologies for filters, biquads, amplifiers, simulated impedances, etc., using current conveyors is not possible. Even a comprehensive classification would be very difficult to achieve. However, the most negative consequence is that some of the circuits proposed are conditionally stable when not completely unstable. The prediction of such unacceptable behaviour is not easy to achieve, but this does not justify, from our point of view, the proposal of unsound circuits. In this letter we will illustrate, through a very simple but significant example, three unquestionable facts: f When designing circuits containing current conveyors as active devices in feedback configurations, one should take into account basic and well-known feedback rules for optimum results. f Working with current conveyors requires in general a much more complicated model than the one we routinely use with opamps. f Even a very simple stage, consisting of a single CCII and resistive negative feedback, may be stable only within very narrow design space. CCC 0098—9886/98/020215—4$17.50 ( 1998 John Wiley & Sons, Ltd. Received 13 October 1996 Revised 12 March 1997 216 LETTER TO THE EDITOR EXAMPLE Consider the circuit shown in Figure 1 containing a single CCII-, where we are interested in the current transfer ratio I /I , which assuming ideal conveyor is o i I R o"! f (1) I R i g The circuit has been obtained as the adjoint of a typical inverting voltage amplifier and it is not, intentionally, an optimum design since its output impedance is not high and parallel feedback is applied to the high impedance node Z. Building blocks of this kind are contained in many so-called current mode filters.8 To analyse its stability, we use the model shown in Figure 2, which is well established in the literature,7 but for our purposes we do not consider the output impedance at Z terminal (and hence not included in Figure 2). This necessary simplification is based on our experimental results with the circuit of Figure 1, and does not affect the final conclusions. It can be demonstrated, although not shown here, that making use of such a model for the CCII- and supposing: (i) impedance at X terminal is a single resistor, and (ii) a(s) and b(s) containing one single pole, the new nonideal current transfer function results in an unconditionally stable circuit; The new denominator of (1) is a second-order polynomial with positive coefficients. Note that the model is not particularly simple. This result could move us to consider that the circuit under test is unconditionally stable. Nothing further from reality. Let us consider now the following model: (i) parasitic inductance, ¸ , and series resistance R at terminal X, x x (ii) a(s)"1 and (iii) two pole model for the current buffer, i.e. i b 0 b(s)" z" i b s2#b s#1 x 2 1 (2) Figure 1. Simple feedback circuit with single CCII Figure 2. Small signal CCII model Int. J. Circ. ¹heor. Appl., 26, 215—218 (1998) ( 1998 John Wiley & Sons, Ltd. 217 LETTER TO THE EDITOR According to the above conditions the new current transfer function contains a third-order characteristic equation which can be analysed making use of the Routh-Hurwith criterion to give a necessary and sufficient condition for stability: b b2 b R2 R( 1 ¸# 1 R# 1 x (3) g b b x b b x b ¸ 0 2 0 2 0 x The above condition indicates that the load resistor cannot exceed a given value. For a practical CCIIimplementation,9 equation (2) is particularized to a low-pass response with 0)99 DC gain, 315 MHz cut-off frequency and 0)82 for quality factor. Values for R and ¸ are 29)6 ) and 48)2 nH, respectively. If we x x introduce in the above expression (3) such experimental values, the upper limit for R value can be estimated g to be about 172 ). Also, the condition indicates that stability is independent of feedback resistor. We can more precisely estimate the value obtained above by employing a slightly more complicated model for the CCII-. Assuming now: (i) a purely resistive impedance for Z ; (ii) two-pole model for the current buffer x as shown in (2) and (iii) two-pole model for the voltage buffer, i.e. v a 0 a(s)" x" (4) v a s2#a s#1 y 2 1 similar, though more complicated calculations with a fourth-order characteristic equation, give now the following necessary and sufficient condition for stability: a b a2#a a b #a b2#a2b !2a b #a b b #b2 1 2 1 2 1 1 2 2 2 1 1 2 2R (5) R( 1 1 2 x g a b (a b #a b )2 2 1 1 2 0 0 The numerical estimation for the CCII- by hand gives a more restrictive value of 102 ). For this calculation we have used a low pass response in (4), with 0)99 DC gain, 1)14 GHz cut-off frequency and 0)86 for its quality factor. For this model we have not taken into account inductive effects at terminal X because on doing so, the denominator in equation (1) becomes a fifth-order polynomial, whose stability conditions have not a closed Figure 3. Step response of the circuit shown in Figure 1 ( 1998 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 26, 215—218 (1998) 218 LETTER TO THE EDITOR form expression like (5) or (3). On the other hand, it could be demonstrated that a relevant point in the stability of this circuit is the phase lag in the high-frequency response, i.e. high-order model, of current and voltage buffers. In fact, using one single-pole model for both buffers and an inductance in series with a resistance at terminal X, the upper bound for R is estimated around 630 ), a value that lies too far away g from simulated results shown in the next section. SIMULATION RESULTS We have verified the conclusions in the above section, resulting in a circuit stable for R values lower than g 96 ), very close to the theoretical value. In Figure 3 we show the step response obtained when R and R are f g equal to 90 ). An oscillation with a slow decreasing exponential envelope is clearly observed, typical of a system in the verge of instability. FINAL DISCUSSION AND CONCLUSIONS The circuit example analysed in this letter has many resemblances to similar ones built-up with CFOAs. In the latter, the feedback resistor has a lower bound to assure stability, while in the former it is the load resistor which is limited by an upper bound. In both cases we have to resort to second-order effects in the active device.10 It is convenient to remark that simple models for CCII7 are unable to predict high frequency phase lag, mainly due to the inductive effect at terminal X, and of course to higher order poles in the voltage and current buffers. All these effects play an important role particularly in high performance CCII implementations. Their effect is more noticeable and negative when the circuit is not designed to optimize the own characteristics of the CCII, as is the case of the example circuit. A similar analysis, with the corresponding negative conclusions could be carried out over many current conveyor applications reported in the recent literature. So, the main conclusion of this letter is to point out the high sensitivity of the theoretical predictions with respect to the CCII model used, specially with high performance ones, and on the other hand to warn about the necessity to perform an in-depth stability study of CCII-based circuits. ACKNOWLEDGEMENTS Gobierno de Navarra (OF 441/92, OF 497/93) and Ministerio de Educación y Ciencia (TIC 94/0544) are acknowledged for financial support. REFERENCES 1. D. Bowers, ‘The impact of new architectures in the ubiquitous opamp’, in Analog Circuit Design, Kluwer Academic Publishers, Dordrecht, 1993. 2. L. P. Huelsman, (Ed.), ‘Active RC-filters: theory and applications’, Benchmark papers in Electrical Engineering and Computer Science, Hutchinson and Ross, 1976. 3. R. Schaumann, M. A. Soderstrand, and K. R. Laker (Eds.), Modern Filter Design, IEEE Press Selected Reprint Series, 1981. 4. A. S. Sedra and K. C. Smith, ‘A second-generation of current conveyor and its applications’, IEEE ¹rans. Circ ¹heory, CT-17, 132—134 (1970). 5. A. Fabre, ‘Third-generation current conveyor: a new helpful active element’, Electron. ¸ett. 31, 338—339 (1995). 6. C. Toumazou, F. J. Lidgey and D. G. Haigh (Ed.), Analogue IC Design: the Current-mode Approach, Peter Peregrinus, London, 1990. 7. A. Fabre, O. Saaid and H. Barthelemy, ‘On the frequency limitations of the circuits based on second generation current conveyors’, Analog Integrated Circ. Signal Process. 7, 113—129 (1995). 8. S.I. Liu and H. W. Tsao, ‘New configurations for single CCII biquads’, Int. J. Electron. 70, 609—622 (1991). 9. R. Cabeza and A. Carlosena, ‘Analog universal active device: theory, design and applications’, Analog Integrated Circ. Signal Process, 12, 153—168 (1997) 10. J. Mahattanakul and C. Toumazou, ‘A theoretical study of the stability of high frequency current feedback op-amp integrators’, IEEE ¹rans. Circ. Systems 43, 2—12 (1996). Int. J. Circ. ¹heor. Appl., 26, 215—218 (1998) ( 1998 John Wiley & Sons, Ltd.

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