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Frequency Stability of L-C Oscillators

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Frequency Stability of L-C Oscillators Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1951 Frequency stability of L-C oscillators Strelow, Reuben E. Annapolis, Maryland; Naval Postgraduate School http://hdl.handle.net/10945/24729 if ^UilUl. DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA 93943-5002 FREQUENCY STABILITY OF L-C OSCILLATORS R. E. Strelow FREQUENCY STABILITY OF L-C OSCILLATORS by Reuben E. Strelow Lieutenant Commander, United States Navy Submitted in partial fullillment of the requirements for the degree of MASTER OF SCIENCE in ENGINEERING ELECTRONICS United States Naval Postgraduate School Annapolis, Maryland 1951 5 8-3 This work is accepted as fulfilling the thesis requirements for the degree of IvIASTEH OF SCIENCE in ENGINEERING ELECTRONICS from the United States Naval Postgraduate School. PKEFACE This thesis, a requirement for a master's degree, v/as prepared and written during the second and fourth terms of the school year 1950-51 while at the U.S. Naval Postgraduate School, Annapolis, Md. , and during the third term which was spent at RCA laboratories in Camden, N.J. The \vork was in- spired by a basic interest in oscillator theory and a desire to gain a better understanding of the subject than can be obtained from standard text books. The material available on L-C Oscillator Stability is spread tnrough the literature of the last quarter century or more. Accordingly, an effort was made to collect and review the pertinent articles on the subject hoping that future students of electronics might benefit thereby. In conclusion, I wish to acknowledge the assistance and encouragement given me by the various professors at the U.S. Naval Postgraduate School, particularly to professors A. Sheingold and G. H. Giet. R. E. Strelow (ii) • t ( TABLE OF CONTENTS Page Certificate of Approval i Preface - il List of Illustrations V Introduction 1 Chapter 1 Fundamentals of Oscillators 3 1, Elements of an Oscillator 3 2. A Simple Electrical Oscillator 3 3« Resonance in a Simple Parallel Circuit 5 4o A Non-linear llaintaining Element 7 Chapter II Feedback Amplifier Stability Criteria 11 1. Barkhausen Criterion 11 2. Nyquist's Stability Criterion 13 3. Llewellyn* s Criterion for Oscillation 20 Chapter III The Vacuum Tube as a Ivlaintaining Element 22 Chapter IV Llewellyns Method of Obtaining Improved Frequency Stability 29 1. First Method 30 2. Another Type of Stabilizii]g Technique 38 3. Design procedure 41 4. Another Type of Stabilization 42 Chapter V Relationship of Circuit Q, to Frequency Stability 46 Chapter VI The Clapp High Stability Oscillator 52 Chapter VII The Electron Coupled Oscillator 59 ill I Page Chapter VIII The Cathode-Coupled Oscillator 64 Chapter IX Other Reference on Oscillator Stability 72 Bibliography 77 iv I I • LIST OF ILLUSTiiATIONS Figure P&ge 1. Simple Electrical Oscillator 4 2. Oscillator Equivalent-Circuits 4 3. Parallel Circuit 5 4* Circle Diagrams 6 5« Circle Diagram of a Kon-Linear Circuit 8 6« A Feedback Amplifier 11 7. Map in S-plane 17 8. Map in K- plane 17 9« A Feedback Oscillator 18 IC, Kyquist Plot 19 11. Equivalent Circuit of a Tube 22 12. General Oscillator Circuits 25 13. Modified Colpitts Circuit 29 14. Modified Tuned-Grid, Tuned-Plate Oscillator 29 15. Llewellyn's Ec^uivalent Circuit 30 16. Hartley Oscillator 34 17. Tuned Grid-Tuned Plate Oscillator 41 18. Equivalent Circuit 42 19. Unity Coupling Type Oscillator 45 20. Oscillator Circuit 47 21. Affect of ;^ on Stability 48 22. Reactance versus Frequency 50 23. Reactance versus Frequency 50 24 Tapped down Tank Circuit 51 i Figure Page 25. Affect of a Shunting Element. 52 26. Clapp Oscillator 53 27. Equivalent Circuit 53 28. Electron Coupled Oscillator 59 29. Frequency versus Screen Grid Voltage 60 30. Frequency versus Plate Supply Voltage 61 31. Cathode Coupled Oscillator Using Tv/in Triode 64 32. Equivalent Circuit 65 33 Analysis on liodal Basis 65 34. Simplified Oscillator Circuit 67 35. Improved Oscillator 73 360 Resistance Stabilization 75 37o Phase Stabilization 75 vi INTRODUCTION This paper is concerned primarily with the principles to be employed when designing L-C oscillators with a high degree of frequency stability. The material presented, essentially a review of articles published in periodicals, is intended to give the reader adequate knowledge of tiB factors affecting oscillator stability to be able to design circuits with high frequency stability. The L-C oscillator is first broken up into two parts, a frequency sensitive element and a maintaining element* The contribution of each part to frequency instability is indicated and it is demonstrated hov/ non-linearity of the maintaining element causes changes in frequency. Because of the difficulty in obtaining solutions of non-linear differential equations, analyses are usually carried out using small signal theory and assuming linear elements© Consequently, a discussion on Kyquist*s stability Criterion is included o Nyquist»s Criterion furnishes information as to whether a circuit is oscillatory and hence can be used to determine the minimum coupling, between output and in- put circuit, necessary to induce oscillation* The vacuum tube as a maintaining element is then presented as thg source of circuit non-linearity and it is shown how changes in electrode voltages may influence stability© Using techniques suggested by Llewellyn, the frequency of oscillator circuits may be made independent of the //^ ^^ and iC, of the tube so that the change of frequency with supply voltage can be reduced to zero* Tlie desired effect is brought about through proper selection and adjustment of circuit elements. The work of lieising confirms Llewellyn's principles and adds that a high c^ in the tank circuit is not, in general, a factor which ixaproves frequency stabili- ty* He points out that a low -±^ ratio decreases the range of frequency stability and that a high (^ if accompanied by loose coupling between input and output circuit may do like- wise, but to reduce the variation of frequency with plate voltage to zero careful circuit design following the princi- ples suggested by Llewellyn is necessaryo Several circuits from which high frequency stability may be obtained are presented* The technique for making the electron coupled oscillator immune to supply voltage changes is discussed. Two circuits of recent origin, tne Clapp oscillator and the Cathode coupled oscillator, are analysed and methods of obtaining frequency stability are pointed outo Finally, a brief summary of a number of papers on L-G oscillator theory is included. The theory of oscillation is so vast that this seems to be the only possible way to call attention to some of the related information available in the literature. CKAPTER I FUNDAMENTALS OF OSCILLATORS 1. Elements of an Oscillator. It is worth while before starting an analysis of oscilla- tion to examine briefly some of tiie fundamental principles which apply in general to all oscillators. Any network capa- ble of sustaining continuous oscillation at a fixed frequency must fundamentally consist of a frequency controlling ele- ment and a maintaining element. The maintaining element, under steady state oscillatory conditions, supplies an a- mount of energy to the system equivalent to the energy lost in the frequency controlling element. If a system is os- cillatory the amplitude of o sc ilia t ion will increase until limited by the non-linearity of the maintaining element. A steady state is reached in which the energy relations prescribed above are fulfilled. Neglecting the possibility of compensating errors, it appears evident that the fre- quency stability of an oscillatory system can never exceed that of its frequency controlling element. Thus, when analyzing any oscillatory system, it seems expedient to assume in each case that the frequency-controlling ele- ment is intrinsically stable, and the analysis of the sys- tem then reduces to an analysis of the maintaining element. 2. A Simple Electrical Oscillator. The simplest electrical frequency controlling element is a tuned gircuit consisting of a pure inductance and a pure capacitance. In practice such a circuit will have i I I losses, tile ciiief losses occurring in the inductive element. In order to replace the energy dissipated a source of energy must be inserted in the circuit. In theory this can be accomplished by inserting a negative resistance in series with the loss resistance. Such an element would be a perfect main- taining element since it accomplishes its purpose without affecting the loop resonant frequency at which oscillation would be maintained. The theoretical circuit is shovm in Fig.l. An alternative method of representing the same basic circuit is to consider the parallel circuit equivalent. In Fig. 2(a) the maintaining element - ^ is connected across tlie parallel combination of C and the "lossy** inductance. In. Fig. 2(b) the L and H of Fig, 2(a) are replaced a o r* Oioii I + by their parallel equivalents© Simple mathematics shows that the relationship between the I I -R t=r<j 2.CG.) F.^. ^Cb) i £^ a I €> O s c 1 1 a + o r q vj \v n V ^i r c u I'fs series and parallel components are given by: R "^ _R^+_uK^ ^, ^ ^ L -^ _Jii These equations show that the value of L^ and hence the fre- quency of oscillation is no longer independent of the value of R« Another significant difference is , for a high Q cir- cuit, the value of jR is small, wnereas the value of R^ is very large » This property of the two circuits becomes especially important when the negative resistance is shunted by a capacity as it is in the practical case» It will be shov/n later that the manner in which the negative resistance is connected will determine to some extent the degree of frequency stability.
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