IS THE QUADRATURE OSCILLATOR A MULTIVIBRATOR

IS THE QUADRATURE OSCILLATOR A MULTIVIBRATOR?Erik LindbergØrsted•DTU Department,348Technical University of DenmarkØrsteds Plads,Kgs.Lyngby,DK2800,Denmarke-mail:el@oersted.dtu.dkWWW:http://www.es.oersted.dtu.dk/∼el/Abstract—The aim of this paper is to give insight in the behaviour of the quadrature oscilla-tor by means of a study of the eigenvalues of the time varying linearized Jacobian of the differential equations describing the oscillator.The mecha-nism behind the oscillations is compared with the behaviour of the multivibrator.I.IntroductionO SCILLATORS are nonlinear circuits.By means of the Barkhausen criteria a sinu-soidal oscillator is normally designed as a linear amplifier with a frequency determining feed-back circuit so that the poles of the whole circuit are on the imaginary axis of the complex frequency plane.In order to startup the oscillations some parameters of the circuit are adjusted so that the poles of the linearized circuit are in the right-half plane,RHP,i.e.in the dc bias point the circuit is unstable and signals will start to increase.Due to the nonlinearities,the physical nature of the circuit,the signals will always be limited in some way.The crucial point is whether there will be oscillations or just a transition to a new stable dc bias point.Very little is reported about how far out in RHP the poles should be placed initially.Also many authors assume that due to the nonlinearities the poles arefixed on the imaginary axis at a certain constant amplitude of the signals.It is an open question if this assumption is true.At a certain instant you may investigate the linearized small signal model of the oscillator.It is obvious that the signals will increase in amplitude if the poles are in RHP and decrease if the poles are in the left-half plane,LHP,of the complex frequency plane.If you assume that the poles are moving back and forth between RHP and LHP then the mechanism behind the behaviour of an oscillator may be described as an act of balance between the energy you gain from the dc power source when the poles are in RHP and the energy you loose when the poles are in LHP.In the following a study of the eigenvalues of the linearized Jacobian of the common multivibrator and the quadrature oscillator is made.II.OscillatorsAn oscillator is a circuit which for constant input signal(DC battery)produces an oscillat-ing output signal(a steady state time varying signal).An old rule of thumb says that if you want to design an oscillator try to design an amplifier instead and if you want to design an amplifier try to design an oscillator(Murphy’s Law).Figure1shows a perfect linearamplifier Fig.1.An amplifier with positive and negative feed-back.with positive and negative feed-back.The input impedance of the amplifier is assumed infinite and the gain of the amplifier is assumed constant,i.e. V3=A(V1−V2).If we introduce memory ele-ments-capacitors,coils,hysteresis-in the four impedances various types of oscillators may be obtained[7].If we observe time varying signals for zero input signal V in=0we have an oscillator.If the poles of the circuit are in the left half plane(LHP)of the complex frequency plane the signals are damped.If the poles are in RHP the signals are undamped. Only if the poles are on the imaginary axis the signals are steady state signals.This is of course impossible in a real world circuit.The ideal har-monic oscillator may be started with any initial condition and keep its amplitude and frequency constant.If we want to build an oscillator we must intro-duce an amplifier with nonlinear gain so that for small signals the poles of the linearized circuit are in RHP and for large signals the poles are in LHP. In this case we may obtain steady state behavior based on balance between energy we obtain from the battery when the poles are in RHP and energy we loose when the poles are in LHP.In other words an oscillator is a feed-back am-plifier with an unstable DC bias point.Due to the nonlinear components the linearized small signal model corresponding to the instant bias point will vary with time.The dominating behavior of the circuit is based on the instant placement of the poles of the linearized model.If the poles are in RHP the signals will increase in amplitude.If the poles are in LHP the signals will diminish in amplitude.mon multivibratorIf we replace Z D with a capacitor C D=0.5µF and the resistors with R A=1kΩ,R B=1kΩand R C=2kΩwe have a common multi-vibrator with real poles moving back and forth between RHP and LHP.The hysteresis is the second“memory component”needed for oscilla-tions.Figure2shows the voltages as functions of time using a RC4136operational amplifier.Fig-ure3shows the resistance R p=V2/(−I(C D)) as function of time.The time constant becomes τ=RC=−((R A R C)/R B)C D=−1ms for A very large andτ=R C C D=1ms for A=0. This is in agreement with Fig.2.Assuming a perfect op.amp.with gain A the pole of the circuit is+1k for A very large and −1k for A very little.For A=2the pole pass from+infinite to−infinite as shown on Fig.4. Please note that this is not a jump but a smooth transition.IV.Quadrature oscillators Quadrature oscillators are based on a feed-back loop with at least one almost ideal integrator for which input and output are two sinusoids with90degree phase difference.In[4]an active RC in-tegrator and a passive RC integrator is combined with a negative resistance.The mechanism with a complex pole pair moving between RHP and LHP may be observed.If two amplifiers are coupled as shown in Fig. 5you may have a quadrature oscillator if the impedances are chosen as resistors according to [6].In this case you make use of theopera-Fig. mon multvibrator,PSpice analysis, V(1,2)and V(3)as functions oftime.mon multvibrator,PSpice analysis,re-sistance R p=V(2)/(−I(C D),as function of time.tional amplifier poles as frequency determining memory(“integrating real poles”)and the non-linear saturation characteristic of the operational amplifier as the amplitude limiting mechanism. The quadrature oscillator is made from three cir-cuits.For thefirst circuit(Z1,Z2and A1)we have:V3V1=1+R2R11+1A11+R2R1For the second circuit(Z3,Z4,Z5,Z6and A2)we have:V6V3=11+R3R4R5R5+R6−1A2−R3R4Fig.4.Multivibrator pole as function of amplifier gain A.Fig.5.Quadrature Oscillator.The third circuit is the feed-back circuit(Z7and Z8)for which:V1 V6=R7R7+R8With reference to[6]we may now design the os-cillator.With R1=R4=R7=10kΩwe get R2=33.5kΩ,R3=R8=3kΩ,R5=0and R6=infinite i.e.V(4)=V(0).Fig.6shows the result of a PSpice analysis with two uA741 OA’s.The frequency is530kHz and the output amplitude of any of the two OA’s is155mV.Fig. 7shows a simple macro model for the OA with two negative real poles.Thefigures8and9show the result of the optimization of the simple model with only one nonlinear element:the piece wise linear voltage controlled current source GI1.Fig. 8shows the close agreement between the output voltage V(5)of the PSpice library macromodel and the output voltage V(15)of the macromodel infig.7in the fraquency range1mHz to1THz. Fig.9shows that the hysteresis of the PSpice library macromodel is not modelled infig.7. There are only two nonlinear components GI1A1 and GI1A2in the quadrature oscillator modelfig. 5when the simple op amp model offig.7isused.Fig.6.Quadrature Oscillator.V(3)as function ofV(6).Fig.7.Op Amp macromodel.Eout=A(s)Vac These nonlinearities are piece wise linear.There are four combinations of large gain and zero gain so we may calculate the poles as follows:For small signals i.e.when V(1,2)and V(0,5)both are in the range from−47µV to+47µV,there are two complex pole pairs,one in RHP and one in LHP: real±jimag real±j imag+401k4.50M−11.3M4.50M When V(1,2)is outside the range from−47µV to +47µV and V(0,5)is in the range from−47µV to+47µV,there are four negative real poles: real real real real−31.5−1.71M−9.17M−10.9MWith V(1,2)in the range from−47µV to parison of the open loop ac-transfer char-acteristics of the Op Amp macro model of Fig.7 and the PSpice library macro model.Logarithmic frequency scale.parison of the dc-transfer characteristicsof the Op Amp macro model of Fig.7and the PSpice library macro model.+47µV and V(0,5)outside the range from −47µV to +47µV,there are four negative real poles:real real real real −31.5−1.71M −9.17M −10.9MWhen V(1,2)and V(0,5)both are outside the range from −47µVto +47µV,there are two real double poles in LHP:real real real real −10.9M −10.9M −31.5M −31.5MWe see that only in the case where the input volt-ages of A1and A2both are small signals,a com-plex pole pair in RHP is present corresponding to the unstable DC bias point.In the other cases the poles are real and negative and the circuit is damped.This is in agreement with fig.10and fig.11.It is seen how the two op amps synchronize so that only one op amp is active at a time.A certain amount of energy is moving between the memory elements corresponding to the input level of 47µV above which the gain is zero.Fig.10.Input voltages of op amps (V(1,2),V(4,5))and nonlinear currents of op amps (GI1,GI2)as functions of time (initial transient).Op Amp macro model of Fig.7used.Fig.11.Input voltages of op amps (V(1,2),V(4,5))and nonlinear currents of op amps (GI1,GI2)as functions of time (steady state).Op Amp macro model of Fig.7used.V.ConclusionsWe must rewrite our text books and replace statements concerning ”nonlinearities pulling the poles back to the imaginary axis”with proper statements concerning the mechanisms behind the steady state oscillations as e.g.sinusoidal oscillation where a complex pole pair is moving between RHP and LHP or relaxation oscillation where real poles are moving between RHP and LHP.A mechanism where only in the unstable DC bias point a complex pole pair occur in RHP is reported in this paper.In the steady state all poles are real and negative and relaxation oscil-lations similar to the multvibrator performance occur.References[1] E.Lindberg,“Oscillators and eigenvalues”,in Proceed-ings ECCTD’97-The 1997European Conference on Cir-cuit Theory and Design ,pp.171-176,Budapest,Septem-ber 1997.[2] E.Lindberg,“Colpitts,eigenvalues and chaos”,in Pro-ceedings NDES’97-the 5’th International Specialist Workshop on Nonlinear Dynamics of Electronic Sys-tems ,pp.262-267,Moscow,June 1997.[3] E.Lindberg,K.Murali and A.Tamasevicius,“De-sign of chaotic oscillators”,ECCTD’01-the 15th Eu-ropean Conference on Circuit Theory and Design ,Tu-torial nr.T5,http://www.it.dtu.dk/ecs/tutor-01.htm,Helsinki,August 2001.[4] A.S.Sedra and K.C.Smith,Microelectronic Circuits 4th Ed ,Oxford University Press,1998.[5]L.Strauss,Wave Generation and Shaping ,pp.1-520,McGraw-Hill,1960.[6] E.Vidal,A.Poveda and M.Ismail,“Describing Func-tions and Oscillators”,IEEE Circuits and Devices ,Vol.17,No.6,pp.7-11,November 2001.[7]J.R.Westra,C.J.M.Verhoeven and A.H.M.van Roermund,Oscillators and Oscillator Systems -Classi-fication,Analysis and Synthesis ,pp.1-282,Kluwer 1999.。