IEEE JOURNAL OF SOLID STATE CIRCUITS JUNE 1976 418 Correspondence The Differential Pair as a Triangle Sine v Wave Converter ROBERT G MEYER WILLY M C SANSEN AND STEFAN PEETERS SIK LUI R R 1 Abstract The performance of a differential pair with emitter degeneration as a triangle sine wave converter is analyzed Equations describing the circuit operation are derived and solved both analytically and by computer This allows selection of operating conditions for optimum performance such that total harmonic distortion as low as 0 2 percent has been measured vEE a I INTRODUCTION The conversion of ttiangle waves to sine waves is a function often required in waveshaping circuits For example the oscillators used in function generators usually generate triangular I output waveforms 1 because of the ease with which such oscillators can operate over a wide frequency range including very low frequencies This situation is also common in monolithic oscillators 2 Sinusoidal outputs are commonly desired in such oscillators and can be achieved by use of a nonlinear circuit which produces an output sine wave from an input triangle wave The above circuit function has been realized in the past by means of a piecewise linear approximation using diode shaping networks 1 However a simpler approach and one well suited to monolithic realization has been suggested by Grebene 3 This is shown in Fig 1 and consists simply of a differential pair with an appropriate value of emitter resistance R In this paper the operation of this circuit is analyzed and relationships for optimum performance are derived from the collectors of Q 1 and Q2 The current gain of the devices is assumed large so that the waveform is the same in both cases The operation of the circuit can be understood by examining the transfer function from Vi to current i flowing in R This is shown in Fig 1 b and has the well known form for a differential pair The inclusion of emitter resistance R allows the curvature to be adjusted for optimum output waveform as will be seen later When a triangle wave input of appropriate amplitude is applied as shown in Fig l b the output waveform is flattened Manuscript received August 4 1975 revised December 15 1975 Research by R G Meyer and S Lui was sponsored by the U S Army Research Office Durham NC under Grant DAHC04 74 GO151 Research by W M C Sansen and S Peeters was sponsored respectively by the Belgian National Science Foundation NFWO and the Belgian National Fund for Scientific Research IWONL R G Meyer and S Lui are with the Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory University of California Berkeley CA 94720 W M C Sansen and S Peeters are with the Laboratorium Fysica en Elektronica van de Halfgeleiders Katholieke Universiteit Leuven Belgium I r v I b w J M I M Zr t VI 77 2T II CIRCUIT ANALYSIS The circuit to be analyzed is shown in Fig 1 a The sinusoidal output signal can be taken either across the resistor R or Fig 1 Triangle sine wave converter a Circuit schematic fer function b Trans by the curvature of the characteristic and can be made to approach a sine wave very closely As with all such circuits the distortion in the output sine wave is dependent on the input amplitude and this must be held within certain limits for acceptable performance In the following analysis Q 1 and Q2 are assumed perfectly matched although in practice mismatches will occur and give rise to second order distortion typically less than 1 percent However introduction of an input dc offset voltage has been found to reduce second order distortion terms to negligible levels and they will be neglected in this analysis The presence of such an offset does not affect the following analysis From Fig l a Vi VBEI iR VBE2 1 but 2 CORRESPONDENCE 419 VT in VBE2 3 where VT 4 Substitution vi ly l 16 K2 1 vj R Thh IC2 15 By comparison of 12 and 15 it is apparent that in order to realize the desired transfer function it is necessary but not sufficient that KI I of 2 and 3 in 1 gives 5 2 VT 1 VT 17 If a 1 for Q1 and Q2 then Icl I i Ie2 6 I i Substitution Vi i VT I 7 of 6 and 7 in 5 gives IR VT ln 8 1 2 3 1 r for 1 lo of 9 in 8 for the circuit transfer function gives This cad be expressed as 2 VT Vj VT I 21 5 IR 2 3 i 1 3 IR 2 VT I is 2 I 12 The desired transfer function for the circuit is see Fig 1 b i K1sin K2Vi where K and K2 are constants 13 and thus K2 Vi arcsin Expansion v VT of 17 in 18 gives 1 57 3 14 vT 19 Equation 19 gives the normalized input triangle wave amplitude for minimum output distortion The circuit transfer function given by 12 is to be made as close as possible to the arcsin expansion of 15 If we equate coefficients of third and fifth order terms in 12 and 15 we It is thus exobtain IR VT equal to 2 and 3 33 respectively pected that the best performance of the circuit will occur for this range of values and this is borne out by experiment and computer simulation III COMPUTER SIMULATION AND EXPERIMENTAL RESULTS In 1 Substitution 1 18 K2 VM 1 Equation 8 is expressed in normalized form and shows that the output signal i I normalized to 1 depends only on normalized input voltage Vi VT and factor lR VT Because of the small number of parameters in 8 it is readily solved in normalized form by computer to yield a series of curves specifying the circuit performance Before this is pursued hclwever it is useful to consider an approximate e analytical solution of 8 which gives some insight into the circuit operation The log term in 8 can be expanded as a power series Substitution Equation 16 shows that the peak value of the output current should equal the current source value 1 If the input triangle wave has peak value VM then 13 indicates that for a perfect sine wave output it is necessary that of 14 in a power series gives 14 The solution of 8 was obtained by computer simulation for various value of VM VT normalized triangle wave amplitude and factor IR VT and the output signal was analyzed into its Fourier comfionents Third harmonic distortion HDs is defined as the ratio of the magnitude of the signal at the third harmonic frequency to the magnitude of the fundamental A Total harmonic distortion THD is HD HD typical plot of HD3 and THD is shown in Fig 2 for IR VT 2 5 It can be seen that the THD null and the HD3 null occur at about the same value of VM VT and this is true for any value of IR VT The measured points in Fig …
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