The suitability of operational transconductance amplifiers (OTAs) as the main active element to obtain basic building blocks for the design of programmable nonlinear continuous-time networks is examined. The main purpose is to show that the OTA, as the active element in basic building blocks, can be efficiently used for nonlinear continuous-time function synthesis. Two efficient nonlinear function synthesis approaches are presented. The first approach is a rational approximation, and the second is a piecewise-linear approach. Test circuits have been integrated using a 3-μm p-well CMOS process. The flexibility of the designed and tested circuits is confirmed

Linares Barranco, Bernabé

Ramírez Angulo, Jaime

Rodríguez Vázquez, Ángel Benito

Sánchez Sinencio, Edgar

English

idUS. Depósito de Investigación Universidad de Sevilla

OTA-B ased Non-linear Function Approximat ions Edgar Shchez-Sinencio’, Jaime Ramirez-Angulo’, Bernabe‘ Linares-Barranco’*’, Angel Rodriguez-Viizquez2 ‘Department of Electrical Engineering, Texas A&M University College Station, Texas 77843-3128, U S A .  2Departamento de Electr6nica y Electromagnetismo Universidad de Sevilla, 41012-Sevilla, Spain Abstract - The suitability of operat ional  transcon- ductance amplifiers (OTAs) as t h e  main active ele- ment to obtain basic building blocks for  the design of programmable non-linear continuous-time net- works is presented. T h e  ma in  purpose is to show t h a t  the OTA, as t h e  active element in basic build- ing blocks c a n  be efficiently used for  non-linear continuous-time functions synthesis. Two efficient non-linear function syntheses approaches are pre- sented. T h e  first approach is a rat ional  approxima- t ion a n d  t h e  second is a piecewise-linear approach. Test circuits have been integrated using a 3j1 p-well CMOS process. T h e  flexibility cif the designed a n d  tes ted circuits is confirmed. I. INTRODUCTION Lately, several authors [I] - [5] have been successfully using the Operational Transconductance Amplifier (OTA) as the main active element in continuous-time active fil- ters. The OTA’s programmability nature’ and the fact that OTAs have only a single high impedance node, in contrast to conventional op amps make the OTA an excel- lent device candidate for high frequency and voltage (or current) programmable analog basic building blocks. The applicability of OTAs as components to the design of lin- ear networks has been extensively discussed elsewhere [l], [6] and not repeated here. The objective of this paper is to examine the applicability of OTAs as the basic elements to the design of non-linear networks. There is not much reported in the literature on the use of OTA for designing non-linear block components [7] -[8]. There are reported excellent contributions 191 - [ l l ] ,  I161 of non-linear circuits dealing with particular important non-linear problems. In our proposed approach rather than try to tackle a specific problem, we focus our attention in a general approach deal- ing with non-linear basic building blocks using OTAs as the main active elements. At this point no emphasis was done to optimize the circuit performance but to explore the potential and applicability of the OTA-based non-linear system approach. ’ The output current Io of an OTA due to a differential input Vid is 10 = gmUid and gm is a voltage (current) controllable pa- rameterll], [6], 171. 96 ISCAS ’89 11. BASIC BUILDING BLOCKS Multiplier Block. A two input four quadrant multiplier has an output(current) given by where the multiplier constant KM has units of A/V2. If Vl and Vz can take any positive or negative sign, the mul- tiplier is called a four-quadrant multiplier. This multiplier is represented in Fig. l(a). The corresponding OTA-based implementations are shown in Fig. l(b). The block ‘an represents a signal attenuator, its function is such that the maximum voltage swing of Vl and V2 are equalized, and -Vbb is the usual bias control of the OTA. An active at- tenuator can be implemented in CMOS technology [15]. Although not indicated in Fig. 1, assume the power sup- plies of the OTAs are VDD and - V ~ S .  For the circuit of Fig: l (b)  we obtain and I , = - g  mZ v 1 - -K(VI, + VSST)Vl (2b) where K is a process- and geometry-dependent constant, V s s ~  = Vss -Vt, and Vt is a transistor threshold voltage2. The output current becomes IO = -aKVlV2 = -/KMIVIV~ where IKMI = aK. We have the flexibility of making the sign of KM positive or negative by injecting V2 to OTAl instead of to OTA2. Divider Block. A two input divider has an output which is the ratio of the two inputs, multiplied by a dimensional (in volts) constant K R ,  i.e., VO = K R ~ .  A-symbol of the divider is shown in Fig. 2(a), where the notation n and d stands for numerator and denominator, respectively. The corresponding OTA-based circuit implementation us- ing the multiplier symbol is shown in Fig. 2(b). Squaring and High Powers (Ezponentiation) Blocks. A one input squarer has an output proportional to the square of We have Msumed equal K’s and threshold voltages vt’s for the OTAs. CH2692-2/89/0000-00% $1.00 0 1989 IEEE Ucpartmnt of Elccrncal and Computer EngLneenng and The coordinated Science Laboratory the input, 10 = KMV:. The implementation of the squarer is obtained by simply using a multiplier with equal inputs. To obtain an ezponentiation (raising to a power) block op- erator with an input V; and an output to be proportional to Yp where p > 2 ,  it is required ( p  + 1 ) / 2  multipliers for p odd and p / 2  multipliers for p even. Furthermore, since the proposed multipliers are of the transconductance type, the outputs must be converted into voltages to be able to use them as the inputs of following multipliers. This can be easily obtained by connecting an equivalent resistor at the output. An equivalent resistor using an OTA [5] - [6] is implemented by connecting the output to the negative OTA input and grounding the positive OTA input. Square-Rooter Block. A one input square-rooter has an output with the negative or positive square root of an input voltage multiplied by a constant of a proper polarity, e.g., v o = * l G l  , V , > O o r V o = + I J q I  , v i <  0. Fig. 3(a) shows the implementation of the square- rooter, where the output VO is given by VO = K R ~  which yields V, = 1-1. A more detailed description of the implementation is shown in Fig. 3(b). Piecewise-Linear Function Generators. Diodes intercon- nected with OTAs can simulate ideal diodes, thus allow- ing a piecewise-linear approximation to any desired non- linear function. The accuracy, naturally, improves with the number of line segments involved. The ideal basic building block for piecewise-linear function approximation is shown in Fig. 4. Note that I D  = 0 until the breaking point (voltage reference V,) is reached. .The slopes of the lin- ear segments are proportional to the gm’s. The diodes can be implemented with MOS transistors with their gate and drain tied together. If a step type input-output character- istic is needed to implement discontinuities in the function approximation, the linear OTA can be substituted by an OTA comparator which ideally simulates a large gm and a saturation (output) current of *trbiaS. 111. NONLINEAR FUNCTION SYNTHESES A rational approzimation that has the general form of a polynomial function or of a ratio of polynomials,i.e., where i is a positive integer number. In fact, the exponent i can be a fractional exponent of the form p/q, where p and q are negative or positive integers. The exponentation blocks are of the type of Fig. 5. If a negative - p / q  is needed, an additional divider has to be used. A piecewise linear approzimation can be obtained by using the basic building block of Fig. 4. Changing the PO- larity of diodes and input terminals of OTA’s allow the o b  tention of negative and positive slopes. Arbitrary functions with variable positive and negative slopes can be approxi- mated. Furthermore, the slopes are voltage programmable3 which gives an additional flexibility in the function approx- imation design problem. One example of an arbitrary func- tion approximation containing negative and positive slopes is discussed in the next section. Details on the practical considerations of the OTA-based piecewise-linear circuits are under consideration. Iv. EXPERIMENTAL RESULTS Several test-circuits containing OTAs and transistors connected as diodes were fabricated using a 3pm p-we11 CMOS process by MOSIS. The linearized OTA used to synthesize the different non-linear analog functions is re- ported in (31. The die area of each OTA is 220x700prnz  and it consumes 10 mW of power with f 5 V  supply volt- ages. A chip photomicrograph showing two complete OTAs of the test circuit is depicted in Fig 6. A. Pansconductance Multiplier. The structure used is shown in Fig. 6. The measured value of KM is 2.4pA/V2. In all measurements described here a LOOKS? load resistor was used. The large-signal character- istics of the multiplier are shown in Fig 7 .  Vi was held constant (at O.OV, &0.33V, f0.66V, fl.OOV), while the input Vz varied between k l V .  The non- linearity error is shown in Fig 8.  For Vz, a triangular 2 volts peak-to-peak signal was applied, while keeping Vi = 1V. The output current produced a triangular voltage signal of 660 mV peak-to-peak. Substracting this signal from an ideal triangular wave, the resulting peak-to-peak error signal was 17 mV which yields a non-linearity error of nearly 2%. Repeating the mea- surement but interchanging VI and Vz and being Vi a triangular signal of 2 volts peak-twpeak. A peak-to- peak error signal with an amplitude of 23 mV corre- sponding to a 3.5% non-linearity error was measured. Fig. 9 shows the multiplier being used as a modulator for the case where both input signals are sinusoidal. B. Piecewise Linear Approzimation. The intended trans- fer characteristic to be considered is shown in Fig. lO(a) and consists of three linear segments. The in- dividual slopes due to each OTA are indicated in the lower part of Fig. lO(a), and the composed resulting transfer characteristics are shown in the upper part of Fig. lO(a). The actual OTA circuit implementation is shown in Fig. 10(b) where an optional diode and volt- age sources have been added at the OTA (2 and 3) and shown with broken lines to improve high frequency performance of the circuit. Note that the slopes of the transfer characteristics can be easily modified by changing the OTA voltage-dependent transconduc- tances. The experimental results are shown in Fig. lO(c). V. CONCLUSIONS The suitability of OTAs as the main active element to obtain basic building blocks for the design of non-linear networks was established. Methods to implement prac- Additionally, if a resistive load simulated with an OTA is used, the slopes become ratios of transconductances which provida a very good temperature compensation and accuracy improvement. 97 tical non-linear circuits in a systematic design approach were developed. Two practical synthesis approaches were introduced. The programmability and flexibility of the OTA provides the potential to design adaptive non-linear circuits. Implementations of other non-linear synthesis approaches are feasible using the basic blocks here intro- duced. The proposed OTA-based building blocks can be incorporated in a CAD software [12] to fully exploit their. functionality and versatility. The test-integrated circuit experimental results verified the theoretical predictions. REFERENCES [I] R. L. Geiger and E. Sbnches-Sinencio, ‘Active Fdtm D d g n  U& ing Operational Panacondactance Amplifier: A TutmizJ*, IEEE Circuh and Dewice Magazine, voL 1, pp 20-32, March 1985. [2] C. Plett, M. A. Copeland and R. A. Hadaway, ‘Tontinuow-Time Filters Using Open-Loop h a b l e  Pannconductanca A m p l i 6 ~ * ,  Proc. ISCAS/IEEE. vol. 3, pp 1172-1176, San J a e ,  CA., May 1986. [3] A. P. Nedungadi and R. L. Geiger, ‘High-Frequency Voltage- Controlled Continuous-Time Lowpsas Filter Using Linearised CMOS Integrators”, EZectronicr Letters, voL 22, pp 729-731, June 1986. [4] F. Krummenacher and N. Joehl, ‘A 4-MHZ CMOS Continuous- Time Filter With On-Chip Automatic Tuning, IEEE J.  Solid- State Circuits, vol. 23, pp 750-758, June 1988. [5] E. Shcher-Sinencio, R. L. Geiger and H. Nevarer-Losano, ‘Gen- eration of Continuonu-Time Two Integrator Loop OTA Filter Structures”, IEEE Zhtcms. on Circuits and System, voL 35, pp 936-946, August 1988. [6] M. Bialko and R. W. Newcomb, “Generation of All Finita Linear Circuits Using The Integrated DVCCS’, IEEE lhu. on Circuit Theory, vol. CT-18, pp 733-736, Nov. 1971. [7] National Semiconductor Corporation, Lincar Dotobook, 1982. (81 B. Linares-Barranco, A. Rodrignes-Vasquer, J. L. Huetta, E. Shcher-Sinencio and J. J. Hoyle, ‘Generation and Deaign of Si- nusoidal Oscillators Using OTAs”, Proc. ISCAS/IEEE, vol. 3, pp 2863-2866, Helsinki, June 1988. 191 M. A. Chari, K. Nagaraj and T. R. V i a n a t h a n ,  ‘Broad-Band Precision Rectifier“, Proc. ISCAS/IEEE, vol. 3, pp 824-826, Philadelphia, May 1987. [lo] J. W. Fattamso and R. G. Meyer, ‘MOS Analog Function Syn- thesis”, IEEE J. Solid-State Circuits, vol. SC-22, pp 1056-1063, December 1987. 1111 B. S. Song, ‘CMOS RF Circuits for Data Communication Appli- cations”, IEEE J.  Solid-State Circuita, voL SG21, pp 310-317, April 1986. [12] L. R. Carley and R. A. Rutenbar, ‘How to Automate Analog IC Deigns”, IEEE Spcctrum, voL 25, pp 26-30, August 1988. (131 A. S. Sedra and K. C. Smith, Microelectronics Circuits, Second Edition, Holt, Rinehart and Winston, New York, 1987. 1141 C. G. Sodini, P. K. KO, and J. L. Moll, ‘The Effect of High Fields on MOS Device and Circuit Performance,’ IEEE Trans. on Electron Devices, vol. ED-31, pp 1386-1393, October 1984. (151 S. Qin and R. L. Geiger, ‘A f 5 V  CMOS Analog Multiplier”, IEEE J.  Solid-State Circuits, vol. SC-22, pp 1143-1146, Dec. 1987. 1161 E. Seevinck and R. F. Wassenear, ‘A Versatile CMOS Linear Thnsconductor/Square-Law Function Circuit’, IEEE J. Solid- State Cinuits, vol. SC-22, pp 3W377, June 1987. I. Fig. 1 Multiplier (a)  Symbol, (b) OTA Implementation 1. v. A Fig. 2 Divider (a) Symbol, (b) OTA Implementation. “Dr, I I T ’  Fig. 3 Square-Rooter, (a) Implementation, (b) OTA Implementation. Fig. 4 Piecewise-Linear (PL) Function Generator Building Block . I (0 )  v, **.: (b) Fig. 5 Exponentiation ( C )  (Raising to a power) Operation, (a) Squarer, (b) Cubit, (c) p-th. 98 Department ot ELecmcal and Computer Enpeenng and The coordinated Science Laboratory ~ Fig. 6 A Chip Photo Micrograph of Two Complete OTAs. Fig. 7 Large-Signal Characteristics of Multiplier. VI = *{1.00, 0.66, 0.33, 0.O)V. (a) Transfer Characteristic Variable Triangular Wave for Vz. (e) Experimental Results. 99 

OTA-based non-linear function approximations

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OTA-based non-linear function approximations

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