Lindberg, Erik

Crossref

Oscillators and operational amplifiers

A generalized approach to the design of oscillators using operational amplifiers as active elements is presented. A piecewise-linear model of the amplifier is used so that it make sense to investigate the eigenvalues of the Jacobian of the differential equations. The characteristic equation of the general circuit is derived. The dynamic nonlinear transfer characteristic of the amplifier is investigated. Examples of negative resistance oscillators are discussed

Online Research Database In Technology

                       General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal   If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.     Downloaded from orbit.dtu.dk on: Dec 17, 2017Oscillators and operational amplifiersLindberg, ErikPublished in:Proceedings of the 2005 European Conference on Circuit Theory and DesignLink to article, DOI:10.1109/ECCTD.2005.1522982Publication date:2005Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Lindberg, E. (2005). Oscillators and operational amplifiers. In Proceedings of the 2005 European Conference onCircuit Theory and Design (Vol. 2, pp. II19-II22). IEEE. DOI: 10.1109/ECCTD.2005.1522982Oscillators and Operational AmpliÞersErik Lindberg, IEEE Lifemember Abstract  A generalized approach to the design ofoscillators using operational ampliers as active ele-ments. A piecewise-linear model of the amplier isused so that it make sense to investigate the eigen-values of the Jacobian of the dierential equations.The characteristic equation of the general circuit isderived. The dynamic nonlinear transfer characteristicof the ampliÞer is investigated. Examples of negativeresistance oscillators are discussed.1 INTRODUCTIONWhen you want to design an oscillator theBarkhausen Criteria is normally used as a start-ing point i.e. a linear circuit with poles on theimaginary axis is designed [1, 2]. In order to start-up oscillations some parameters are changed so thatthe poles are in the right half of the complex fre-quency plane. The linear circuit becomes unstableand the signals will grow until inﬁnity i.e. we mustintroduce nonlinearity in order to limit the signalamplitude.• Oscillators are nonlinear circuits .In the following an oscillator design procedure is de-scribed. It is based on the idea of ”frozen eigenval-ues” [3]. By means of piecewise linear modelling ofthe nonlinearities the circuit is linear in time slots sothat it make sense to study the eigenvalues (poles)of the Jacobian of the diﬀerential equations andsee whether the signals are increasing or decreas-ing in amplitude in a certain time slot. If you ob-serve that the poles are moving back and forth be-tween the right half (RHP) and the left half (LHP)of the complex frequency plane the mechanism be-hind the behavior of an oscillator may be describedas an act of balance between the energy you gainfrom the dc power supply when the poles are inRHP and the energy you loose when the poles arein LHP. The characteristic equation for a generallinear circuit with a perfect ampliﬁer with a piece-wise linear transfer characteristic is derived. Thedynamic transfer characteristic of a speciﬁc oper-ational ampliﬁer (op amp) RC4136 is studied bymeans of PSpice analysis.∗Ørsted  DTU Department, 348 Technical University ofDenmark, DK2800 Kongens Lyngby, DENMARK, e-mail:el@oersted.dtu.dk, tel.: +45 4525 3650, fax: +45 4588 0117.2 THE CHARACTERISTIC EQUATIONFigure 1 shows the general circuit which is anop amp with positive and negative feed-back. Ifwe introduce memory elements - capacitors, coils,hysteresis - in the four admittances various typesof oscillators may be obtained. The nonlinearityneeded for oscillations may be introduced as non-linear losses or hysteresis in connection with thememory elements. If we introduce linear mem-ory elements with no hysteresis a nonlinear transfercharacteristic of the ampliﬁer must be introduced.Let us assume that the ampliﬁer is a perfect am-Figure 1: An ampliﬁer with positive and negativefeed-back, V (3) = A × V (1) − A × V (2)).pliﬁer with inﬁnite input impedance, zero outputimpedance and piecewise linear gain A. The gain isvery large for small signals and zero for large sig-nals. Now a network-function may be calculatede.g. the transfer function V (3)/V IN . The rela-tion between the output voltage V (3) and the inputvoltage V IN becomes:V (3)×Y BY A + Y B −Y CY C + D −1A(1)= V IN ×Y DY C + Y DIf we observe that V (3) is diﬀerent from zero whenV IN is zero then the coeﬃcient of V (3) must bezero i.e.Y BY A + Y B −Y CY C + Y D −1A= 0 (2)Equation (2) is called the characteristic equa-tion . The characteristic polynomial of the lin-earized diﬀerential equations describing the circuitAuthorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on December 2, 2009 at 04:46 from IEEE Xplore.  Restrictions apply. may be derived from this equation. For innite gain(A = ∞) the equation becomes:(Y A× Y C) − (Y B × Y D) = 0 (3)For zero gain (A = 0) the equation becomes:(Y A + Y B)× (Y D + Y C) = 0 (4)Table 1 is derived from the characteristic equation.It shows that it is possible to place the frequencydetermining circuit in any of the four admittances.If one of the admittances of Fig. 1 is chosen as aCase Gain: A = 0 A =∞A Y A +RB −RB × RD/RCB Y B +RA −RA × RC/RDC Y C +RD −RD × RB/RAD Y D +RC −RC × RA/RBTable 1: The ”load” of an admittance ”YX” forgain zero and gain ∞ when the other admittancesare conductors (GX = 1/RX ).simple capacitor C or a simple coil L and the otherthree admittances are chosen as conductors (resis-tors) multi-vibrator oscillators might be designed.If one of the admittances of Fig. 1 is chosen as a sim-ple LC parallel or series circuit and the other threeadmittances are chosen as conductors (resistors)negative resistance sinusoidal oscillators might bedesigned.Case: D ARA 180Ω RLCRB 180Ω 180kΩRC 180kΩ 180ΩRD RLC 180ΩTable 2: Resistor values for two cases D and A.RLC is a two-terminal with a capacitor C = 1nFin parallel with a coil L = 256mH with series resis-tance RL = 14.8Ω.3 NEGATIVE RESISTANCE OSCILLA-TORSIn [4] a 10kHz negative resistance oscillator is de-signed where Y D is chosen as a coil L D = 256mH(with a series loss resistor RD = 14.8Ω) in parallelwith a capacitor CD = 1nF. The op amp RC4136 isused. The components Y A, Y B and Y C are chosenaccording to case D of Table 2.Figure 2 case D shows the dynamic transfer func-tion of the op amp. The gain A is varying almostpiecewise linear between zero and +2.3305 corre-sponding to the poles:− 2.801k±j × 62.44k (Q = +11.0, f = 9.94kHz)and+ 36.37k±j × 50.78k (Q = −0.86, f = 8.08kHz)respectively in the complex frequency plane.The quality factor Q of a pole speciﬁed above isdeﬁned asQ =( 2 +  2)/ (−2×  ) (5)It is a measure for the distance of the pole from theimaginary axis. It is seen that Q becomes ∞ forpoles on the imaginary axis. Q is negative for polesin the right-half-plane RHP and positive for polesin the left-half-plane LHP. The real part of the polemay be calculated from = /4×Q2 − 1 (6)or 2 = /Q for large Q.Now the sign of the gain of the op amp is changedFigure 2: Op amp dynamic transfer-characteristicV (3) as function of input voltage V (1, 2) for thetwo cases D and A.corresponding to inverting the input terminals orto interchange Y A, Y B and Y D, Y C. This corre-sponds to case A of Table 2.Figure 2 case A shows the dynamic transferfunction of the op amp. The gain A is varyingalmost piecewise linear between zero, −12.7 and−565 corresponding to the poles:−2.801k±j × 62.44k (Q = +11, f = 9.94kHz),+1.993k±j × 62.46k (Q = −16, f = 9.94kHz) and+2.729k±j × 62.43k (Q = −11, f = 9.94kHz)From Fig. 2 it is seen that both in case D and incase A the poles follows the whole trajectory i.e. acomplex pole pair goes to the real axis and split upinto two real roots. One real root goes against zeroand the other real root goes to ∞. The two realroots meet again and a complex pole pair leavesthe real axis and crosses the imaginary axis [4].Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on December 2, 2009 at 04:46 from IEEE Xplore.  Restrictions apply. In case D the fast transition from positive gain tozero gain via inﬁnite gain happens at large valuesof the op amp input voltage V (1, 2). In case A thefast transition from negative gain to zero gain viainﬁnite gain happens at small values of the op ampinput voltage V (1, 2).Please note that it is not a jump when the realFigure 3: Gain A = dV (3)/dV (1, 2) and op ampinverting input V (2) as functions of time, ”positivegain”, case D.Figure 4: Gain A = dV (3)/dV (1, 2) and op ampnon-inverting input V (1) as functions of time, ”neg-ative gain”, case A.pole goes from +∞ to −∞. The Figures 3 and 4show the gain as function of time for positive gainand negative gain respectively.An old rule says that if your oscillator do notwork then change the input terminals of the opamp. Here is given an example where the oscillatorworks for both positive and negative gain.4 A DESIGN PROCEDUREWe want to study a negative resistance oscillatorwhere Y C is a RLC series circuit i.e. case C.ZC = 1/Y C = sLc + Rc +1sCc(7)with Lc = 256mH, Rc = 14.8Ω and Cc = 1nF.From equation (2) we get:Y C +(1RD)×(NumDenom)= 0 (8)where Num = RB −RA× (A− 1)and Denom = RA + RB × (A + 1)The characteristic polynomial becomes:s2 + 2 α s + ω20 = 0 , (9)where2α = (10)(1Lc)×(Rc ×Num + RD ×Denom(RB −RA× (A− 1)))andω20 =1Lc × Cc (11)α = 0 forA =(RA + RB)× (Rc + RD)(Rc ×RA−RD ×RB) (12)Table 3 shows corresponding values of A and α.A α0 (Rc + RD)/(2× Lc)∞ (Rc ×RA−RB ×RD)/(2× Lc ×RA)Table 3: Corresponding values of A and αThe poles or the natural frequencies of the circuit- the eigenvalues of the Jacobian of the diﬀeren-tial equations - are the roots of the characteristicpolynomial.p1,2 = −α ± j√ω20 − α2 = −α ± j ω (13)In order to place the poles close to the imagi-nary axis α must be much smaller than ω0. ForQ = −10 and ω = 1/√256m× 1n = 62.5k, fromequation (6) α becomes 3.128913589k. From Ta-ble 3 (A = 0) the resistor RD is calculated asRD = 2α × Lc − Rc = 1.587203758kΩ and fromTable 1: choose RA = RB = 190Ω. From equation(12) the gain is calculated as A = −2.037649363which is negative. This indicates that we have toswitch the input nodes of the op amp correspond-ing to interchange Y A, Y B and Y D, Y C i.e. Y Bbecomes a RLC series circuit and we have a case Boscillator instead of a case C oscillator.The complex pole pair on the imaginary axisis found as: For A = +2.037649364, pole:+0.70µ ± j × 62.5k, Q = −0.5e + 12 andfor A = +2.037649363, pole: −64n ± j × 62.5k,Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on December 2, 2009 at 04:46 from IEEE Xplore.  Restrictions apply. Figure 5: Op amp dynamic transfer-characteristicV (3) as function of input voltage V (2, 1).Figure 6: Gain A = dV(3)/dV (1, 2) as functionof time.Figure 7: Gain A = dV(3)/dV (1, 2) as functionof time.Poles Real ImaginaryGain A  ±j  Q+1e+12 +3.071k 62.42k -10.0+38.83 +2.767k 62.44k -11.0+1e-12 -3.129k 62.42k +10.0-1e-12 -3.129k 62.42k +10.0-0.2136 -3.870k 62.38k +8.1-8.64 +4.939k 62.30k -6.3-12.26 +4.280k 62.35k -7.3-360 +3.106k 62.42k -10.0-1917 +3.078k 62.42k -10.0-1e+12 +3.071k 62.42k -10.0Table 4: The poles as function of the gain.Q = +0.5e+ 12. Figure 5 shows the op amp dy-namic transfer-characteristic: V (3) as function ofinput voltage V (2, 1). By zooming the character-istic the gain A = d(V (3))/d (V (1, 2)) is found atsome places. The Figures 6 and 7 show the gain asfunction of time. It is interesting to observe thatfor rising values of V (3) the inﬁnite gain point∞ ispassed 4 times whereas for falling values of V (3) theinﬁnite gain point ∞ is passed only twice. Table 4shows the poles for the selected values of the gain.When the gain A is in the interval from −2.20865to −1.81090 the poles are real. For A = −2 thereare no poles. The dc-gain is found to +38.83 whichcorresponds to a complex pole-pair in the RHP.Here the resistors RD and RC are chosen arbi-trarily equal to 190Ω. If we choose RD = 1900Ωand RC = 19Ω so that RD and RA = 1587Ω areof the same size and so that RC is of the same sizeas the resonance impedance of the series resonancecircuit Rb = 14.8Ω an investigation of the spectrumby means of PSpice FFT indicates that the contentof higher harmonics disappear i.e. it seems that itis possible to obtain a very clean sinusoidal signalby means of an optimization of the dynamic trans-fer characteristic. Also the impact of the choice ofQ should be further investigated.The results depends on the power supply voltageand the type of op amp. Besides RC4136 the opamps LF411, LM324 and uA741 have been investi-gated with similar results.5 CONCLUSIONSInsight in the mechanisms behind the behavior ofoscillators based on operational ampliﬁers is ob-tained by means of the frozen eigenvalues ap-proach [3]. The dynamic transfer characteristic ofa speciﬁc operational ampliﬁer (RC4136) is studiedin connection with negative resistance oscillators.A design procedure is presented.References[1] H. Barkhausen, Lehrbuch der Elektronen-Rohre, 3.Band, “Ru¨ckkopplung” , Verlag S.Hirzel, 1935.[2] A.S. Sedra and K.C. Smith, MicroelectronicCircuits 4th Ed , Oxford University Press, 1998.[3] Erik Lindberg, “Is the Quadrature Oscillatora Multivibrator ?”, IEEE Circuits & DevicesMagazine, November/December 2004, vol. 20,no. 6, pp. 23–28, 2004.[4] Erik Lindberg, “Oscillators - an approach fora better understanding”, (tutorial presented atECCTD’03 - http://ecctd03.zet.agh.edu.pl/)Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on December 2, 2009 at 04:46 from IEEE Xplore.  Restrictions apply. 

http://orbit.dtu.dk/ws/files/3984026/Lindberg.pdf

Oscillators and operational amplifiers

Abstract

Similar works

Full text

Available Versions

Crossref

Online Research Database In Technology