IEEE JOURNAL OF SOLID-STATE CIRCUITS, JUNE 1976418CorrespondenceThe Differential Pair as a Triangle-Sine Wave Converterv-ROBERT G. MEYER, WILLY M. C. SANSEN, SIK LUI,AND STEFAN PEETERS1R~/R~Abstract–The performance of a differential pair with emitter degen-eration as a triangle-sine wave converter is analyzed. Equations describ-ing the circuit operation are derived and solved both analytically and bycomputer. This allows selection of operating conditions for optimumperformance such that total harmonic distortion as low as 0.2 percent“-has been measured.-vEE(a)I. INTRODUCTIONThe conversion of ttiangle waves to sine waves is a functionoften required in waveshaping circuits. For example, the oscil-lators used in function generators usually generate triangularoutput waveforms [ 1] because of the ease with which suchoscillators can operate over a wide frequency range includingvery low frequencies. This situation is also common in mono-lithic oscillators [2] . Sinusoidal outputs are commonly de-sired in such oscillators and can be achieved by use of a non-linear circuit which produces an output sine wave from aninput triangle wave.The above circuit function has been realized in the past bymeans of a piecewise linear approximation using diode shapingnetworks [ 1] . However, a simpler approach and one wellsuited to monolithic realization has been suggested by Grebene[3] . This is shown in Fig. 1 and consists simply of a differen-tial pair with an appropriate value of emitter resistance R. Inthis paper the operation of this circuit is analyzed and relation-ships for optimum performance are derived.II. CIRCUIT ANALYSISThe circuit to be analyzed is shown in Fig. 1(a). The sinusoi-dal output signal can be taken either across the resistorR orfrom the collectors of Q 1 and Q2. The current gain of the de-vices is assumed large so that the waveform is the same, in bothcases.The operation of the circuit can be understood by examiningthe transfer function from Vi to current i flowing in R.This isshown in Fig. 1(b) and has the well-known form for a differ-ential pair. The inclusion of emitter resistance R allows thecurvature to be adjusted for optimum output waveform, aswill be seen later.When a triangle wave input of appropriate amplitude is ap-plied as shown in Fig. l(b), the output waveform is flattenedManuscript received August 4, 1975; revised December 15, 1975.Research by R. G. Meyer and S. Lui was sponsored by the U.S. ArmyResearch Office, Durham, NC, under Grant DAHC04-74-GO151. Re-search by W. M. C. Sansen and S. Peeters was sponsored respectively bythe Belgian National Science Foundation (NFWO) and the Belgian Na-tional Fund for Scientific Research (IWONL).R. G. Meyer and S. Lui are with the Department of Electrical En-gineering and Computer Sciences and the Electronics Research Labo-ratory, University of California, Berkeley, CA 94720.W. M. C. Sansen and S. Peeters are with the Laboratorium Fysica enElektronica van de Halfgeleiders,Katholieke Universiteit, Leuven,Belgium.I ----J ‘“bI-v,rZr t— — — —-I-I -----M‘M VIw772TFig. 1. Triangle-sine wave converter. (a) Circuit schematic. (b) Trans-fer function.by the curvature of the characteristic and can be made to ap-proach a sine wave very closely. As with all such circuits, thedistortion in the output sine wave is dependent on the inputamplitude and this must be held within certain limits for ac-ceptable performance.In the following analysis, Q 1 and Q2 are assumed perfectlymatched, although in practice mismatches will occur and giverise to second-order distortion (typically less than 1 percent).However, introduction of an input dc offset voltage has beenfound to reduce second-order distortion terms to negligiblelevels and they will be neglected in this analysis. The presenceof such an offset does not affect the following analysis. FromFig. l(a)Vi= VBEI + iR - VBE2but(1)(2)CORRESPONDENCE419VBE2 = VT in ~(3)~,vi=:++(ly+:(l)+ . . . .(15)whereBy comparison of (12) and (15) it is apparent that in order torealize the desired transfer function, it is necessary (but notVT = ~.(4) sufficient) thatSubstitution of (2) and (3) in (1) givesvj=~-R+~Thh.IC2(5)If a = 1 for Q1 and Q2 thenIcl=I+i(6)Ie2= I-i.(7)Substitution of (6) and (7) in (5) gives()1+:Vi i IR—..— +ln —VT I VT1-;”(8)Equation (8) is expressed in normalized form and shows thatthe output signali/I normalized to 1 depends only on normal-ized input voltage Vi/ VT and factor lR/ VT. Because of thesmall number of parameters in (8), it is readily solved in nor-malized form by computer to yield a series of curves specify-ing the circuit performance. Before this is pursued hclwever, itis 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 series1+;In-=1-;2:+%)3+ $(+r+””” “)for+<1.(lo)Substitution of (9) in (8) for the circuit transfer function givesThis cad be expressed as1Vj “ 2 1()i3-—=~+— — —VTI3 IR I=+2VT—+2VT21is()+———+. . . .5IRI+2~(12)The desired transfer function for the circuit is [see Fig. 1(b)]i= K1sin K2Vi(13)whereK ~ and K2 are constants, and thusK2 Vi = arcsin ~.(14)Expansion of ( 14) in a power series givesKI=I(16)1 1K2=—..._—. —VT “-%2VT(17)Equation (16) shows that the peak value of the output currentshould equal the current source value 1. If the input trianglewave has peak value VM then (13) indicates that for a perfectsine wave output it is necessary thatK2 VM = ;.(18)Substitution of ( 17) in (18) givesv~—= 1.57 %+3.14.VT vT(19)Equation (19) gives the normalized input triangle wave ampli-tude for minimum output distortion.The circuit transfer function given by (12) is to be made asclose as possible to the arcsin expansion of ( 15). If we equatecoefficients of third- and fifth-order terms in (12) and (15) weobtainIR/ VT equal to 2 and 3.33, respectively. It is thus ex-pected that the best performance of the circuit will occur forthis range of values, and this is borne out by experiment andcomputer simulation.III. COMPUTER SIMULATION AND EXPERIMENTAL RESULTSThe solution of(8) was obtained by computer simulation forvarious value$ of VM/ VT (normalized triangle wave amplitude)and factor IR/ VT, and the output signal was analyzed into itsFourier comfionents. Third-harmonic distortion (HDs ) is de-fined as the ratio of the magnitude of the signal at the thirdharmonic frequency to the magnitude of the fundamental.Total
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