ENGIEfrING RESEARCH INSTITUTE UNIVTERSITY OF MICHIGAN ATIN ARBOR THE FREQUENCY OF LINEAR OSCILLATORS Technical Report No. 27 Electronic Defense Group Department of Electrical Engineering By: J. L. Stewart Approved by: -. A. Boyd Assistant Supervisor Electronic Defense Group Project 1970 TASK ORDER NO. EDG-8 SIGNAL CORPS, DEPARTMENT OF TRE ARMY DEPARTMENT OF ARMY PROJECT NO. 3-99-04-042 SIGNAL CORPS PROJECT NO. 29-194B-0 March, 1954

TA[BLE OF CONTENTS Page ABSTRACT iv 1. THEORY 2. FUNCTIONS HAVING ZROS ONLY AT THE ORIGIN 4 3. APPLICABILITY 4 4. EXAMPLES 5 DISTRIBUTION LIST 11 LIST OF FIGURE Fig. 1 Black Diagram of Conventional oscillator 2 Fig. 2 Phase Shift Development of Several Conventional Oscillators 6 Fig. 3 Reactance Modulated Oscillator 7 Fig. 4 Extended Range Oscillator 9 Fig. 5 An Ineffective Extension 10 iii

ABSTRACT A general method is derived for determining the oscillation frequency of linear oscillators and application is made to a widerange reactance-modulated oscillator. In addition, it is demonstrated that all oscillators can be treated as ordinary phase-shift oscillators. The generalized equations derived here allow the relative advantages and values of certain types of circuits to be easily ascertained. The method actually amounts to Barkhausen's criteria aided with the concepts furnished by poles and zeros. iv

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN THE FREQUENCY OF LIhEAR OSCILLATORS 1. THEORY A conventional oscillator can be diagramed as shown in Fig. 1. Let some driving voltage en be applied at the input and let the feedback path be opened. Then, the ratio of output to input voltages eo/en is given by - G - K A/B where K is a real constant and where A and B are polynomials in the complex variable p having the form 2 m A = a0+ + ap a2p +.. + a p 2 n B = bp + bp +b +. +bp (1) The steady state prevails for p = j w. When the loop is closed e/en = - G/(1 + G) and oscillation will take place at the frequency where the phase shift of G is -180 degrees provided that I G i> 1 at this frequency. In obtaining the open loop transfer function, all impedances at the input to the network Pre referred to the output of the network such that the input impedance can be assumed to be infinite. More generally, when K may be either positive or negative, oscillation ill take place at a frequency where the phase shift of the open loop transfer ~unction is some multiple of 180 degrees. The tangent of such phase angles is zero.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TUBE AND PHASE SHIFT INPUT N TWORK OUTPUT NETWORK -G= -K__ B FEEDBACK FIG. I BLOCK DIAGRAM OF CONVENTIONAL OSCILLATOR. The open loop transfer function can be expressed in real and imaginary parts according to - K(A/B) = - K (AB/BB) = - (K/BB*) [ (Ev A Ev B* + Od A Od B*) + (Ev A Od B* + Od A Ev B*) (2) where the first term in the brackets is the real part and the second term is the inaginary part. It is assumed that the equation is evaluated at p = j. "Ev" and "Od" are the operators that take the even and odd parts of the 2 polynomials upon which they operate, respectively, i. e., Ev A = a + a2p + 4 alp +.... B* is the complex conjugate of B at p = j. Along the jc axis, Ev B* = Ev B and Od B* = -Od B thus allowing the phase angle G of (2) to be represented as Od A Ev B - Ev A Od B j tan = -A(3).(. Ev A Ev B - Od A Od B pj which has poles and zeros only along the jc axis. The numerator of (3) factors as Od A EvB - EvA Odi B = cp(p2 + 2 )(2 + 2).(4)

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where the constant c is the coefficient of the highest power of p and where the W i are the frequencies at which occur phase shifts 9 that are multiples of 180 degrees. At these frequencies, the numerator of (3) and hence tan 9 becomes zero. As a side interest, it can be observed that the denominator of (3) factors as Ev A Ev B - dA d B -= d(p2 + (2 + ) (5) ) )(p +Wb (5) where the frequencies are those at which occur phase shifts that are odd multiples of 90 degrees. The loop gain at one of the frequencies W~ is found by using the Condition that (4) be zero in Eq 2 to obtain Ev A Ev B - Od A Od B -K(A/B)p=j oi= -K BB*. (6) L J p=j co which must be equal to or greater than unity in order for oscillation to take place. If the phase shift is an odd multiple of 180 degrees at the oscillation -requency, (6) will be positive. Of particular interest is the case when the transfer function has no.eros. Then, A = ao and the left side of (4) reduces to - a Od B = cp(p2 +0)(p2 +o2)... (7) Lnd the loop gain at one of the frequencies Oi becomes K(A/B) -K a/(Ev B)p (8) ______________________ 3 _______

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 2 FUITCTIONS HAVING ZEROS ONLY AT TEE ORIGIN Many types of oscillators have open loop transfer functions whose zeros are all at the origin of the complex frequency plane. This has the effect of giving a large leading phase angle at low frequencies which must be partially overcome with the lagging phase angle contributed by the poles in order for oscillation to take place. If there are q zeros at the origin and q is even, Eq 4 gives the possible frequencies ci as -pq d B = cpq+l (p2 + W)(p2 + )... (9) which reduces to the equation used when the transfer function contains only poles. If the number of zeros r at the origin is odd, Eq 4 gives pr Ev B = cpr+l(p2 + c ) (p2 + 1). (10) which is quite similar to that used for transfer functions containing only poles. Certain types of R-C oscillators have open loop transfer functions that contain several zeros at the origin. The tuned plate, tuned grid and Hartley oscillators also belong in this catagory. 3. APPLICABILITY* Conventional oscillators can be treated as phase shift oscillators -- a point of view that is all to frequently misunderstood. In order to demonstrate * For a comprehensive discussion of oscillators, see "Vacuum Tube Oscillators", W. A. Edson, John Wiley and Sons, Inc., New York, 1953 _____._______ 4 ____________

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN this assertion, a few one tube circuits along with their phase shift developments are shown in Fig. 2. In obtaining these equivalents, the cathode is assumed to be grounded even though it may not be grounded in the physical circuit. Then, it can be observed that the feedback quantity is always applied to the grid of the tube. In the equivalent circuits of Fig. 2 the tube is represented as a current generator driven with an external driving voltage in series with the grid —this facilitates setting up the transfer functions. Evidently, the only essential difference between the various circuits is the magnitude of the transfer function at the oscillation frequency. The study of such circuits enables one to estimate the dependence of the frequency of oscillation upon such parameters as the tube plate resistance and interelectrode capacitances. Capacitances in Fig. 2 include tube and miscellaneous stray and other capacitance. The resistance terminating the network represents in part the grid conduction resistance if the grid draws current. The plate resistance of the tube is rp and the transconductance is gm. 4. EXAMPLES As an example, consider a network with characteristics similar to those of a three section R-C network with shunt C or a Pi section L-C network with shunt C loaded with a resistance. These networks have three poles and no zeros in their transfer functions. Then, - K A/B = Kao/(bo + b1p + b2p2 + b3p3) (11) Setting the odd part of the denominator equal to the corresponding factored expression b3p(p2 + co ) the oscillation frequency is immediately found ______,,5,5

b79-6-~ W 3r LLZ-1.-V O61 L L TO GRID C1 0 C2 R gme ) rp C2 R STANDARD PHASE SHIFT OSCILLATOR Rl. C, -''C^ ^ 2 t'W~' <! ___lv_ _ < TO GRID R;I c L n r 9 m en j {C3 ^R2 -= g rp L C3T2 a COLPITTS OSCILLATOR C3 C3 TO GRID CL — ^R2 1C2 IL2 rpg ) 2 2p 7R2 jC2 L2 1C1 R1 L1 LI; TC| <R,- f R g TUNED PLATE TUNED GRID OSCILLATOR C4 C4 IR > s-TO GRID T'~ TT " — Tc- gmen (x L 2 (LR rp C3 RCL c, I M G3 - HARTLEY OSCILLATOR R L / ACTO GRID Cp cP g men ( p'PS p;ix o:e R R Lx. LARGE CLAPP'S OSCILLATOR FIG. 2 PHASE SHIFT DEVELOPMENT OF SEVERAL CONVENTIONAL OSCILLATORS. 6

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN to be l = (bl/b3) /2 at which the loop gain is -Kao/(bo - blb2/b3). As a second example, consider the transfer function 2 6 - K A/B = Kao/(bo + blp + bp2 +... + b6 (12) The odd part of the denominator of Eq 12 factors as p(b + b p2 + bp4)= bp (p2+ )(p2 + C2 (13) Equating coefficients, the frequencies W0 and W2 are found to be | 2 = b53/2b + [(b/2b) - blA] 1/2 (4) where the negative sign applies to the frequency where the phase shift is -180 degrees and the positive sign to the frequency where the phase shift is -360 degrees. The results of Eq 14 can be used to find the resonant frequency and range of frequencies of the reactance modulated oscillator described by Dennis and Felch. The circuit is shown in Fig. 3 in which the plate conductances of the tubes are neglected. Frequency variation is achieved by changing gm2, the transconductance of the modulator tube. gmz 2e2 el. e2 _e3 gi, L 1 I I L L __ \ 1X | ofG | gmeQ I S2 I f, I r_.i FIG. 3 REACTANCE MODULATED OSCILLATOR. F. R. Dennis and E. P. Felch, "Reactance Tube Modulation of Phase Shift Oscillators", BSTJ, vol 28, No. 4, pp 601-7, October,1949. | _~~~~ BSTJ. o 8 o4. ~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - Using a nodal analysis, the transfer function is found to be e3/en -g /[ p(2L2C5) +p(2L2C2G) + p5(6LC2) + p2(4LCG + LCg) + p(4C + LGgm) + (G + g) (15) Applying Eq 14 for the lower of the two possible frequencies, i (3/2LC) I l2-2L/ 1 (16) The mninimum frequency is given when g = 0 as 2.in /LC (17) and the maximum when the radical in (16) is zero as otax 3/2LC (18) which requires (m) = C/2LG (19) The tuning ratio is found to be max min Ratio W n) = 0.203 (20) (Co ax Cmin) It should be observed that there is nothing of basic importance about the relative values of L and C in Fig. 5. It would appear that the frequency range could be materially increased by choosing different values for L and C rather than those shown in Fig. 5. To illustrate the improvement that can be obtained, assume that all the C's in Fig. 5 are equal. Then, the transfer function is _______________________________ 8 _______________________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN e3/e = -g/ p5 (L2C3) p (L2C2G) + p3 (4LC2) + p (3LCG + LCg2) + p(5C + LGgm2) + (G + n2)] (21) in which case, 2 = (2/LC) L 1 - 2 j/ (22) 2.... and W~p~ Wmin Ratio= (0x in)/ / o.348 (23) (^max min A method for further extending the tuning range can be seen by examining (16); that is, gm2 can be made negative. To accomplish this, two reactance tubes can be used in the oscillator such that one or the other (but not both) has a finite transconductance at any given frequency. A circuit embodying such an arrangement is shown in Fig. 4. I I' FIG. 4 EXTENDED RANGE OSCILLATOR. It might be thought that the tuning range of the oscillator of Fig. 3 could be extended by adding a second reactance tube as shown in Fig. 5. However, an investigation of the transfer function quickly shows that only the ---------------------- 9 _____________

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - even part of the denominator is affected; hence, no change in the oscillation frequency will result with the addition of the second reactance tube. 1 —---- - I --- 11 ----,,,, I II FIG. 5 AN INEFFECTIVE EXTENSION.. -_____________________ 10 _________LO

DISTRIBUTION LIST 1 copy Director, Electronic Research Laboratory Stanford University Stanford, California Attn: Dean Fred Terman 1 copy Commanding Officer Signal Corps Electronic Warfare Center Fort Monmouth, New Jersey 1 copy Chief, Engineering and Technical Division Office of the Chief Signal Officer Department of the Army Washington 25, D. C. Attn: SIGGE-C 1 copy Chief, Plans and Operations Division Office of the Chief Signal Officer Washington 25, D. C. Attn: SIGOP-5 1 copy Countermeasures Laboratory Gilfillan Brothers, Inc. 1815 Venice Blvd. Los Angeles 6, California I copy Commanding Officer White Sands Signal Corps Agency White Sands Proving Ground Las Cruces, New Mexico Attn: SIGWS-CM 1 copy Commanding Officer Signal Corps Electronics Research Unit 9560th TSU Mountain View, California 75 copies Transportation Officer, SCEL Evans Signal Laboratory Building No. 42, Belmar, New Jersey For - Signal Property Officer Inspect at Destination File No. 25052-PH-51-91(1443) 11

UNIVERSITY OF MICHIGAN 11111I.11iIIYI i lllll lllll lllllllll 3 9015 03526 6967 1 copy H. W. Welch, Jr. Engineering Research Institute University of Michigan Ann Arbor, Michigan 1 copy Document Room Willow Run Research Center University of Michigan Willow Run, Michigan 1 copy Engineering Research Institute Project File University of Michigan Ann Arbor, Michigan 11 copies Electronic Defense Group Project File University of Michigan Ann Arbor, Michigan 12