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TRANSIENT STABILITY of the WIEN BRIDGE OSCILLATOR I

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TRANSIENT STABILITY of the WIEN BRIDGE OSCILLATOR I TRANSIENT STABILITY OF THE WIEN BRIDGE OSCILLATOR i TRANSIENT STABILITY OF THE WIEN BRIDGE OSCILLATOR by RICHARD PRESCOTT SKILLEN, B. Eng. A Thesis Submitted ··to the Faculty of Graduate Studies in Partial Fulfilment of the Requirements for the Degree Master of Engineering McMaster University May 1964 ii MASTER OF ENGINEERING McMASTER UNIVERSITY (Electrical Engineering) Hamilton, Ontario TITLE: Transient Stability of the Wien Bridge Oscillator AUTHOR: Richard Prescott Skillen, B. Eng., (McMaster University) NUMBER OF PAGES: 105 SCOPE AND CONTENTS: In many Resistance-Capacitance Oscillators the oscillation amplitude is controlled by the use of a temperature-dependent resistor incorporated in the negative feedback loop. The use of thermistors and tungsten lamps is discussed and an approximate analysis is presented for the behaviour of the tungsten lamp. The result is applied in an analysis of the familiar Wien Bridge Oscillator both for the presence of a linear circuit and a cubic nonlinearity. The linear analysis leads to a highly unstable transient response which is uncommon to most oscillators. The inclusion of the slight cubic nonlinearity, however, leads to a result which is in close agreement to the observed response. ACKNOWLEDGEMENTS: The author wishes to express his appreciation to his supervisor, Dr. A.S. Gladwin, Chairman of the Department of Electrical Engineering for his time and assistance during the research work and preparation of the thesis. The author would also like to thank Dr. Gladwin and all his other undergraduate professors for encouragement and instruction in the field of circuit theor.y. Acknowledgement is also made for the generous financial support of the thesis project by the Defence Research Board·under grant No. 5501-06. iii TL.::'"~.B OF :;c:\;'_-.£.~TS --·- ----~-·----- Page J. Introduction 1 II Simple Feedback Oscillator 5 (1) Conditions for Oscillation 5 (2) Selectivity 7 (3) Phase-Frequency Statility 7 Beta Networks for R-C OsGillators 9 (1) R-C Oscillator'" 9 (2) The R-C OsciTJ..z,tor li!ct-:.:ork 9 (3) The Phase S!t2_ft I~etv:or~c 12 (4) The 1:Jien l:;:oicit;t:o Hetl·.·o:ck 14 ( 5) The Tlvin-'l.' l\: et~·:ork 16 (6) Phase-Frequer:cy Characteristic of Bridge Net·,.;orks 18 (7) The Nonlinear Bl~idge Compensator 19 IV 'l'emoerature Deoendent Resisto::-s 22 (1/ Incandesc~nt Lamps 22 (2) Lamp ThJYlBJnic Behavio'J.r 23 (3) Lmv Frequ~mcy Re~3ponse 28 (4) Thermistors 31 v Hien Bridge Oscillato2' Vath Lr::.plitude Limiter 35 (1) Sensitivity of \·Jien B::.'idge Circuit 35 (2) Oscillator Sta't::"clity 38 (3) Transient Response of \.·.;,,:;n Bridge Oscillator 39 (4) Effect on 'I'ran:>i<:::nt Response for Nonlinear Amplifier 43 VI 'l'he ExperimEn:..:::.l Circuit 59 (1) The Basic Oscillator 59 (2) The Freque;ncy SelectiYe Net>vork 59 (3) The Then:..s.l Bridge .Arlll 60 (4) 'I'he Linea.r Amplifier 61 ( 5) The I:ifferential A.'Tlplifier ( I ) \,0 The Envelope Detector VII :S:x:perimental Outline 63 (l) General Out:Line 63 (2) Heasurement of Static Lamp Cnaracteristics 63 (3) Measurement of Them,al Tir:1e Constant 68 (I+) Low Frequenc:y L2.mp Response 71 ( 5) Transient Oscillator Response 76 (6) Hum Enhancenent 9::. Conclusion and S1.:::;gestions 98 }~efere::-~ces 101 Bibliography 103 iv LIST OF ILLUSTRATIONS Figure Page (1) Feedback Oscillator 5 (2) R-C Oscillator Beta Network 9 (3) Locus of Roots for R-C Oscillator 11 (4) Phase Shift Ladder Network 12 (5) Wien Bridge Network 14 (6) Twin-T Network 16 (7) Twin-T Oscillator 17 (8) Lamp Stabilized Oscillator 20 (9) Lamp Bridge Arm 24 (10) Lamp Bridge Circuit 28 (11) Thermistor Static Characteristics 32 (12) Thermistor Bridge Arm Characteristics 33 (13) Thermistor Bridge Arm 33 (14) Lamp Stabilized Wien Bridge Circuit 35 (15) Wien Bridge Oscillator 39 (16) Oscillatory Transient Envelope Response 56 (17) Experimental Wien Bridge Oscillator 59 (18) The Frequency Selective Network 59 (19) The Lamp Compensator 60 (20) The Envelope Detector 62 (21) Experimental Circuit for Thermal Time Constant 68 (22) Lamp Dynamic Response Curve 69 (23) Lamp Bridge for Low Frequency Response 72 (24) Unwanted Low Frequency Output 75 (25) Envelope Response to Switching Disturbance 78 v Figure (26) Oscillator,y Transient Response 78 (27) Diode Clamping Circuit 99 Table (1) Static Lamp Characteristics 67 (2) Thermal Time Constants 72 (3) Low Frequency Unwanted Output 74 (4) Experimental-Transient Envelope Results 81 ( 5) Calculated Results-Envelope for Linear Amplifier 87 (6) Calculated Results-Envelope for Nonlinear Amplifier 92 Graph (1) Current-Voltage Lamp Characteristic 65 (2) Resistance-Voltage Lamp Characteristic 65 (3) Loy Resistance-log Power Lamp Characteristic 66 (4) Thermal Time Constant Versus Lamp Voltage 69 (5) Unwanted Bridge Output Versus Input Voltage and 75 Frequency (6) Transient Peak Versus Square Root of Oscillator 83 Frequency (7) Modulation Frequency Versus Square Root of Oscillator 84 Frequency (8) Trans.tent 'l'ime Constant Versus Oscillator Frequency 85 (9) Hodu1ation F'requency Versus Square Root of Oscillator 89 Frequency (10) Transient Peak Versus Square Root of Oscillator Freguency 90 (11) !! " II II II " II II 94 (12) Transient Time Constant Versus Oscillator Frequency 95 vi LIST OF PRINCIPAL S1MBOLS Symbol Meaning s complex frequency~+jw~ of modulation j complex operator uu frequency in radians per second equal to 2~ f frequency in cycles per second t time in seconds e exponential operator v instantaneous voltage in volts peak voltage in volts instantaneous current in amps radiated pcwer in watts steady state radiated power in watts supplied power in watts to lamp Wo steady state supplied power in watts to lamp R lamp resistance in ohms Ro steady state lamp resistance in ohms Ri for i equal 1,2,3, ••• Resistances in ohms c capacitance in microfarads e temperature of filament in degrees Kelvin steady state temperature of filament in °K. time constant of transient response in seconds thermal heat capacity of lamp in watt-sec/degree differential thermal conductance in watts/degree lamp bridge thermal time constant in sec. constant - slope of log R versus log P for lamp temperature coefficient of resistance for tungsten rli Symbol Meaning A gain of operational amplifier S(w) feedback function of frequency-determining network A' closed loop gain of system L(s) Laplace transform of change in output voltage amplitude JEl change of output voltage amplitude in peak volts ~El peak value of 6E1 extBpolated back to time t=O &R small perturbation of lamp resistance = dRf(t) € ratio of dR to R0 or relative change in lamp resistance wo steady state oscillator frequency in radian/sec. fo steady state oscillator frequency in cycles/sec. Wm transient modulation frequency in radian/sec. fm transient modulation frequency in cycles/sec. D3 fractional third harmonic distortion of output voltage 1 I Introducri:,ion In recent years~ resistors which exhibit nonlinear electric properties by virtue of a change in resistance with applied power, have found many new uses. One application lies in exercising control over the amplitude of a sinusoidal wavefonn without introducing the customary nonlinearity associated with most simple nonlinear limiters. Although many devices exhibit this property to some extent, we will primarily concern ourselves with two; the incandescent lamp, and the solid state thermistor. Although the electrical characteristics are extremely different for the two, they have in common the key to their ability to control without distortion; their associated time constant due to thermal inertia. Although the thermal properties of the incandescent lamp had been quali.htively understood for a considerable time, Meacham(l) was probably the first to analyse and employ its dynamic characteristics to stabilize the amplitude of a Meachan Oscillator in 1938. His work aroused considerable interest at that time in the nonlinear aspects of the l&~ps and Glynne(2) and othersC3),(4),(5),(6) followed by studying the dynamic behaviour of the lamps. By the middle of the century, thermistors were also being analysed and employed for this control problem. Bollman and KreerC7) were two of the first to realize the potential value of these solid state devices and to set up the differential equation for their dynamic behaviour. PatchettC8) employed a directly heated type of therrrastor in a bridge circuit and showed that it had a figure of merit, which he defined as the change in output voltage for a change in 2 input voltage, of about fifty. Typically the figure of merit for the lamp bridge is less than unity. Thus the use of the thermistor would seem to indicate a more sensitive amplitude control for oscillator amplitude stabilization. Theoretical work concerning the amplitude stability of harmonic oscillators had attracted considerable interest in the years prior to the thermal amplitude control. With its introduction some re­ newed interest was brought to bear on the subject. Edson(9) examined the amplitude stability in a lamp stabilized oscillator by examining the changes in amplitude and phase of the modulation vector when a lightly modulated wave passed through the open loop feedback circuit. For the modulated wave to persist the open loop gain and phase shift of the input signal were required to be zero. An increase in gain would indicate instability or growth, while a decrease would indicate attenuation of the modulation. In his analysis the low frequency component is not considered and the feedback factor must be symmetrical about the centre frequency for a bandwidth of at least twice the highest modulating frequency used to insure that the output will be independent of phase ~odulation. Gladwin examined the problem of amplitude stability in valve oscillators(ll). For the class of oscillators known ~separable oscillators he examined the stability pf the system from.
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