Variable-Frequency Bridge-Type Frequency-

Stabilized Oscillators *W. G. SHEPHERDt, ASSOCIATE, I.R.E., AND R. 0. WISEt, NONMEMBER, I.R.E.

Summary-Results are given of a theoretical and experimentalinvestigation into two types of bridge-stabilized oscillators incor-porating a thermal device for amplitude control. One circuit em-ploys only resistances and capacitances in the frequency-determin-ing network and consequently is useful for low-frequency operation.The other circuit uses an inductance-capacitance network which iswell adapted to the higher-frequency network. Conditions foroptimum stability and the variation of the stability with frequencydetermined experimentally are found to be in general agreementwith theoretical results.

NCREASING precision of measurements hasnecessitated the development of variable-fre-quency oscillators of greatly improved frequency

stability and reduced harmonic content. Many of themeans employed in achieving these desirable character-istics for fixed-frequency oscillators cannot be adaptedto oscillators of wide frequency range in which re-quirements of constancy of output, simplicity of con-trol, portability, and ease of construction must be met.

In achieving frequency stability it is of course neces-sary that circuit elements be chosen which are affectedas little as possible by temperature and humidity. Evenassuming ideal circuit elements, however, variation infrequency with supply voltage and with unavoidablechanges in transconductance of tubes will remain inmost variable-frequency oscillators. It is this variationin frequency with gain which constitutes the circuitproblem in the design of wide-range frequency-stabi-lized oscillators and to which a solution is presented inthis paper.

In many oscillator investigations undertaken in thepast a complete solution of the circuit problem hasbeen found impossible because of the complication in-troduced by the nonlinearity of the vacuum tubes inthe oscillator. Recently, however, the separation of thefunctions of amplification and amplitude limitationhas made it possible to treat any oscillator so con-trolled as a linear-network problem and thereby toobtain complete expressions for the conditions andstability of oscillation. The present work deals withtwo oscillators of this type.

Early efforts in the direction of improved oscillatorcircuits attempted to refine the method of amplitudelimitation. One of the first of these was the use of agrid leak and condenser which provide an automaticbias for the oscillator tube. Such an arrangement tendsto reduce the nonlinear behavior required of the gridcircuit in comparison with that necessary for the case

* Decimal classification: R355.9XR133. Original manuscriptreceived by the Institute, March 2, 1942; revised manuscript re-ceived, January 30, 1943. Presented, Winter Convention, NewYork, N. Y., January 13, 1942.

t Bell Telephone Laboratories, Inc., New York, N. Y.

of a fixed bias. The grid leak and condenser do, how-ever, require that grid current be drawn which resultsin the production of harmonics. Furthermore, the tubeinput and output impedances depend to a considerableextent on the supply voltages which in turn affect theoscillating circuit. Some of the methods1 undertaken toeliminate these effects consisted principally in provid-ing sufficient loss between the tube and the oscillatingcircuit so that the effect of changes in the impedancesof the tube on this circuit were reduced more or less inthe ratio of the amount of loss allowable.

Llewellyn has indicated2 how it is possible to im-prove the stability of oscillators under certain condi-tions by including appropriate impedances in the gridor plate circuits or both. These impedances were suchas to reduce the phase shift in the oscillating circuit tozero at the operating point. The analysis however islimited in that it assumes that the oscillating circuitoffers a negligible reactance to any harmonics gener-ated. It is also necessary in most cases to change theseimpedances with frequency which complicates the net-works.

In recent years many investigators3"6 have recog-nized that harmonics produced by the nonlinear be-havior of the tubes affect the frequency of oscillation.This effect occurs since any changes in the supplyvoltages which produce relative changes in the ampli-tude of the harmonics will cause a change in the re-actance at the fundamental frequency and hence ashift in that frequency.

Moullin' proposed to eliminate the harmonic ef-fects by building up the oscillating circuit in such afashion that the harmonics were short-circuited. Thiswork was applied particularly to a dynatron oscillatorbut the same results should apply equally well to atriode or pentode oscillator. Moullin found that theresidual instability which remained could be blamedprincipally upon the changes in the interelectrode ca-pacitances with the supply voltages.Arguimbau3 recognized the effect of harmonics and

proposed to eliminate them by operating the vacuum1 J. W. Horton, "Vacuum tube oscillators-A graphical method

of analysis," Bell Sys. Tech. Jour., vol. 3, pp. 508-524; July, 1924.2 F. B. Llewellyn, "Constant frequency oscillators," PROC.

I.R.E., vol. 19, pp. 2063-2094; December, 1931.3 L. B. Arguimbau, "An oscillator having a linear operating

characteristic," PROC. I.R.E., vol. 21, pp. 14-28; January, 1933.4E. B. Moullin, "The effect of the curvature of the character-

stics on the frequency of the dynatron generator," Jour. I.E.E.(London), vol. 73, pp. 186-195; September, 1933.

5 J. Groszkowski, "Oscillators with automatic control of thethreshold of regeneration," PROC. I.R.E., vol. 22, pp. 145-150;February, 1934.

6 C.-J. Bakker and C. J. Boers, "On the influence of the non-linearity of the characteristics on the frequency of dynatron andtriode oscillators," Physica, vol. 3, pp. 649-665; July, 1936.

Proceedings of the I.R.E. June, 1943256

Shepherd and Wise: Frequency-Stabilized Oscillators

tube as a linear amplifier. This was done by rectifyinga portion of the output to furnish grid bias to thevacuum tube in order to limit the amplitude. This ac-complishes the effect of the grid leak and condenserwithout the disadvantage caused by grid current. Asomewhat similar idea was suggested later by Grosz-kowski5 who used an automatic volume control to main-tain a dynatron oscillator at the optimum operatingcondition on the threshold of oscillation. While bothof these are steps in the proper direction they are stillopen to the objection that in order to control the ampli-tude a tube parameter, namely the grid bias, must bevaried, leading to associated changes in the tubecapacitances.Meacham7 has recently proposed a fundamental im-

provement in oscillator design which has led to a re-markable betterment in performance. By utilizing athermal element as an amplitude control the non-linearity of the amplifier becomes an unnecessary evil,since the amplitude limitation is completely externalto the amplifier. When the thermal element is suitablychosen for the frequencies involved, it is a quasilinearcircuit component so that disturbing harmonics are notproduced.

In addition to attaining the possibility of linearoperation of the vacuum tubes, Meacham used abridge network to determine the frequency of the oscil-lator. By incorporating the control as a component ofthe bridge he was then able to make use of the fullgain of the amplifier. The bridge network has theproperty that the Q of the coil is multiplied by thegain of the amplifier. Hence any changes in the gainor phase of the amplifier produce a minimum of fre-quency variation. Thermal control, moreover, elim-inates changes of the tube capacitances which are in-herent in automatic volume control involving vacuumtubes. With these improvements Meacham has con-structed fixed-frequency oscillators whose stabilitycompares with the best standards available.

Scope

This paper discusses variable-frequency oscillatorswhich function in a manner similar to those describedby Meacham but employing electrical-frequency-de-termining networks better adapted to the necessityof covering a range of frequencies. It should be realizedthat the requirement that an oscillator cover a rangeof frequencies imposes difficulties not present for fixed-frequency oscillators. Thus, at a fixed frequency it is arelatively simple matter to adjust the phase shift in theamplifier to an optimum value while over a wide rangeof frequencies elaborate circuits become necessary. Inaddition, physically larger elements are necessary be-cause of the variability requirement so that largerparasitic capacitances may be expected. It must beaccepted, therefore, that the stability of the oscillators

I L. A. Meacham, "The bridge-stabilized oscillator," PROC.I.R.E., vol. 26, pp. 1278-1294; October, 1938.

to be discussed will be considerably inferior to thestability which can be achieved with single-frequencynetworks. However, while the discussion which followsconsiders these circuits from a variable-frequencystandpoint it will be seen that they may be employedfor fixed-frequency applications and under such circum-stances are capable of a very high degree of stability.The circuits employed are two examples of a class of

bridged-T and parallel-T circuits whose advantageshave been discussed recently by Tuttle8 and Honnell.9One of the circuits employed is particularly applicableto low frequencies because only capacitances and re-sistances are employed. This is a major advantage atlow frequencies where it is usually necessary to uselarge coils which are much less stable than those usedat high frequencies. This results from the fact thatthey must either be very bulky or use iron cores. In thelatter case the inductance may be a function of thecurrent density. It is possible, however, to obtain re-sistances which are as stable as the best coils availablefor high frequencies so that this network offers thepossibility of much better performance at low fre-quencies. The other network has a capacitance-in-ductance frequency-determining network which maybe adapted to any frequency range. In the presentwork it has been applied particularly to frequenciesabove the audio range. The two types of oscillators to-gether cover a frequency scale from a few cycles persecond to several megacycles per second. In each cir-cuit the amplitude is controlled by means of a thermaldevice which is an element of the network. This ampli-tude control permits the analysis of these circuits tobe made by linear-circuit methods.

In the following, the general operating conditionsfor the oscillators will be considered. A theory for thefrequency stability of these oscillators for conditionsof large amplifier gain and for the low-frequency oscil-lator under actual operating conditions has been de-veloped. The results of a comparison of the theoreticalresults with those obtained with experimental oscil-lators will be shown. A description will be given ofoscillators for laboratory use which were adapted andmodified from the experimental models.

Oscillator Circuits

The circuit schematics for the two types of oscil-lators are shown in Figs. 1 and 2. Fig. 1 shows a re-sistance-capacitance parallel-T network which forbrevity will be referred to as the CR oscillator. Asimilar parallel-T network has been described byAugustadt'0 for use as a filter and by Scott"' for use

8 W. N. Tuttle, "Bridged-T and parallel-T null circuits formeasurements at radio frequencies," PROC. I.R.E., vol. 28, pp. 23-29; January, 1940.

9 P. M. Honnell, "Bridged-T measurement of high resistances atradio frequencies," PROC. I.R.E., vol. 28, pp. 88-90; February,1940.

10 U. S. Patent 2,106,785.11 H. H. Scott, "A new type of selective circuit and some ap-

plications," PROC. I.R.E., vol. 26, pp. 226-235; February 1938.

257

Proceedings of the I.R.E.

as a selective analyzer and, with the addition of anotherregenerative path, as an oscillator circuit. In the net-work as used in the present case the elements are soaltered as to make any additional path unnecessary,and further, one of the elements is made a function ofthe oscillating amplitude so as to provide automaticamplitude control. This control adjusts the feedback

AMPLIFIER

Fig. 1-Schematic circuit diagram of capacitance-resistanceparallel-T oscillator.

to the proper value so that the tube functions as a

linear amplifier, resulting in better frequency stabil-ity.

Fig. 2 shows a bridged-T type of network which willbe referred to as the LC oscillator. This circuit whichis well adapted for high-frequency oscillators has char-acteristics somewhat better but similar to those of theCR circuit. The schenmatic indicates that the shuntresistance is a thermally controlled resistance whichautomatically balances the feedback and controls theamplitude. Another circuit arrangement which has thesame type of loss and phase characteristic is obtainedby employing a fixed resistance in place in the ther-mally variable element R and using as an amplitudecontrol a thermal element indicated by RT and con-

nected as shown by the dashed lines. The amplitudecontrols RT and R must have opposite thermal char-

Fig. 2-Schematic circuit diagram of bridged-T oscillator.

acteristics. For reasons which will be mentioned laterthe circuit employing the control RT is the more usefulfor variable-frequency applications of the LC oscil-lator.The phase and attenuation characteristics of the

two types of networks are similar. Sample character-istics for purposes of illustration are given in Figs. 3

and 4 for the parallel-T network of Fig. 1. In this casefor a critical value of the shunt resistance the networkoffers an infinite loss to the frequency for which thephase shift is 180 degrees. For all values of this resist-ance less than the critical value the loss of the 180-degree point is finite and its value depends upon hownearly the critical condition is approached. The phaseshift passes through 180 degrees at the critical fre-quency for a shunt resistance less than the criticalvalue, but for larger resistances the phase shift passesthrough zero. In order to make the circuit automatic

300K=I

280

K-0.99'

240

220I ~~~RR

200 1-

180

K=0.9RI c 8

40 C. C. . . . . . . . . . .

o I~~~~~~~~~f

120~~~~~~~~~~~~~~~-

I - K-I~~~~~~~~~~~~RI=RW 80

140

z 20

Fig. 3-Phase characteristic of parallel-T network.

the shunt-resistance arm is an element whose resist-ance increases with temperature. The cold resistancemust be sufficiently low so that the network loss is lessthan the amplifier gain at the 180-degree phase point.The amplifier phase shift is assumed to be nearly 180degrees. From Fig. 4Z it will be noticed that the net-work loss at frequency fo increases with increasingshunt resistance. Thus when the amplifier is turned on,oscillations build up, simultaneously increasing theshunt resistance until the network loss is just equal tothe amplifier gain.One of the most desirable features of circuits havring

these characteristics, as will be apparent from thesefigures, is that all harmonics are fed back degenera-tively; that is, in such a manner as to reduce their netvalue over that present with no feedback. Since the

258 June

Shepherd and Wise: Frequency-Stabilized Oscillators

generated harmonics are at a very low level as a resultof the linear operation of the amplifier, this further re-duction by feedback results in a very nearly sinusoidaloutput.

It is useful to consider the conditions which lead to anull for transmission through the networks. For areasonable amplifier gain these conditions will be ap-proximately satisfied when oscillation occurs. Also, theexpressions which result are useful in understandingthe operation of the circuits. Table I lists the importantconditions for an infinite transmission loss through thenetworks. The plus or minus signs of (1) indicate thatthe same behavior may be obtained with either in-ductances or capacitances. If one assumes the moreuseful case for a variable-frequency oscillator that allthe reactances are capacitive, (1) may be solved for

TABLE I

RC LCi4/il =0 is/i. =0

Positive CoefficientControl Positive Coefficient Negative Coefficient

Control Control

QpoL QopPoLX-±2RR, Ro_ (5) R< (5a)

2 2

Xi = 2 + Rfo2 1 (6) fo2- (6a)i2RR= 8sr2LC 8r2LC

fo 1(2) Z1inQpoL (7) RTO-

4R

(6b)2irR V2CiC 21<

RC1QLoPLR =K (3) QLOPoL

4cK = I for nullK < 1 for oscillation

R(1-jiV)

Zin= (4)

2C)

whereX and X1 are the react- Ro=value of R for a RTO =thermistor resist-ances of the series and null ance for a nullshunt capacitances at poL is the reactance of R =value of shunt resist-null frequency fo. each coil at the null ance which must be less

Zi,, is the impedance ap- frequency fo than Ro if RTO is to bepearing between the in- po =27rfo finiteput terminals at null R1 =resistance of eachfrequency coil Q, QCoQLo

Q pL QLo+QCoR1 Qco =RTOPOC

PoLQLO=s

Ri

the frequency of the transmission null fo, given byequation (2), which is for practical purposes the fre-quency of oscillation. The equations (1) may also becombined to give the condition for a null in terms ofthe circuit constants, as expressed by (3) when K = 1.From (2) it is seen that the frequency of oscillation

may be varied by changing R, C1, or C. The amplitudeof oscillation will be determined by the value of R, re-quired to make the loss through the bridge just equalto the gain of the amplifier. This gain should be con-stant over the frequency range. The value of R1 thendepends upon the circuit constants and the amplifiergain. Thus, if the frequency is varied by changing C,C1, or R individually, R1 will be a function of frequency.

This is true first because the value of R1 to produce agiven loss in the network will be changed and secondbecause the input impedance of the network as givenby (4) will be altered and produce a variation in theamplifier gain which must be balanced by the networkloss. If the frequency is varied by changing the capac-itances simultaneously with a fixed ratio between them

C0-i

N

7 6(,T - R R650-_ _v - l

25- -

C

-LC

K0 99x x

45 - - - --

15_ I _ . _ _ _

4RrC 14

35 -- -CI'_R I

30~~~~\ 0

25-

IC~~~~~~~~~~~~~~~~~~~~~~f1

1.4 1.5 1. 1.7 1.8 1.9 2.0

F=ffo

Fig. 4-Loss characteristic of parallel-T network.

and a constant value for R, then both the control re-sistance R1 and the input impedance Zi, will be inde-pendent of frequency. It may be demonstrated alsothat the attenuation-phase characteristics of the cir-cuit depend only on the ratio of these capacitances.As a result, the fundamental and harmonic levels andfrequency stability of the oscillator will be independentof the frequency, provided that the amplifier character-istics are independent of frequency. For some applica-tions it may not be advantageous to vary frequency bycapacitance variation but to change R instead. Inorder to do this a wide-range control resistance is re-quired which is very sensitive to current changes. Suchresistances will be discussed in a later section.A typical set of phase characteristics for the bridged-

T oscillator with a positive temperature-coefficientcontrol at R is given in Fig. 5. It may be noted thatwhen R is zero the circuit becomes a Hartley oscil-lator, the phase characteristics of which are alsoshown on Fig. 5. The rate of change of phase with fre-quency is much more rapid for the bridged-T oscil-lator than for the Hartley oscillator. As a result anychanges in the amplifier phase shift will have less effectthan in the Hartley circuit. A circuit with similarcharacteristics which degenerates into a Colpitts oscil-lator is obtained by exchanging inductance for ca-pacitance and vice versa.

For the LC oscillator the problem of maintaining aconstant amplitude over a range of frequencies is moredifficult than in the case of the CR circuit. The valueof the control resistance of the positive thermal-coeffi-cient type necessarily varies with frequency unless the

1943 259

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1i.3

Proceedings of the I.R.E.

Q of the coil varies inversely with the frequency. Thedifficulty is especially serious for this type of resist-ance since the most suitable of these which are readilyavailable require relatively large changes in amplitudeto produce the necessary resistance variations. For thisreason the circuit utilizing the negative temperature-coefficient resistance RT iS more advantageous. Theresistance R is then fixed at a value sufficiently belowthe null value so that the gain around the loop isgreater than zero. The parallel-T network with a

36020

32 00 9

280 Q=2

Zw200V

I' 140 v I90 7

0.98 0.99 1.00 1.01 1.02 1.03f = fe

Fig. 5-Phase characteristic of bridged-T oscillator. The phase plotfor a Hartley circuit is shown for comparison.

fixed shunt resistance R feeds back the fundamentalregeneratively while the resistance shunting the coilsupplies a degenerative feedback just sufficient to satis-fy the conditions for oscillation.

STABILITY THEORY

An analysis for both oscillators has been carriedthrough which is valid for the limiting conditions ofvery large amplifier gain. In the case of the RC net-work by a more exact treatment this analysis has beenextended to cover the gain achieved experimentally.While the latter theory gave a more complete agree-ment with the experimental results the expressions ob-tained are cumbersome. They indicate, moreover, thelimiting correctness of the asymptotic expressions whichwill therefore be given since they are instructive fordesign purposes. The method of analysis is illustrated

in the Appendix. In the analysis it is assumed that theoscillator amplifier is a constant-current generator andall parasitic capacitances are neglected.

It is perhaps well to repeat at this point that we arenot considering in this paper the effects of temperatureon the circuit elements but are restricting ourselves tothe effects of changes in the supply voltages.

Resistance- Capacitance OscillatorThe asymptotic analysis for the RC oscillator shows

thatd bf 9 sin 0 dg

dE fo /2 Rg2 dE(8a)

where fo =the frequency at the null=f the operating departure from fo0 =the amplifier phase shiftg =the transconductance of the amplifierE =any supply voltage

Equation (8a) indicates that the frequency stabilityagainst supply-voltage shifts should be independent ofthe frequency of oscillation if R and the amplifier phaseshift are fixed and should be infinite if the phase shiftof the amplifier is zero.'2 It also predicts that thegreater the transconductance of the amplifier the morestable the oscillator becomes.

Experimentally the frequency stability againstchanges of transconductance varies more rapidly withtransconductance than (8a) predicts. There is also adiscrepancy in the optimum phase angle. These dis-crepancies may be ascribed to the failure experimen-tally to meet the theoretical conditions. These requirethat the departure of the control resistance underoperating conditions from the null value should besmall. To fulfill this condition required a considerablyhigher value of transconductance than that experi-mentally realized.A more rigorous analysis of the RC circuit leads to

the following equations:

K2R2/ 1 2R2 1 21- +- I-K

4 j2 +2F2 F

[R 2 +3) +R3(1-K) + 2

1g2R32

andtan 0 = - tan (0pi - 2).

w here

2(1 - KF2)tan 4, = V2K(F2- 1)

tan 42 =-R3(K+8)

V/2[R3(t - K)-R - 3)]

12 Zero amplifier phase shift with reference to the required 180degrees.

260 June

Shepherd and Wise: Frequency-Stabilized Oscillators

1x = -XO

FR,= KRjo.

Xo, Rio are the null values.

These expressions apply for the case of equal capaci-tances.A numerical solution of these equations for a given

set of circuit parameters leads to a series of curveswhich are given in Fig. 6. These show the frequencyof oscillation referred to the frequency for an exactnull for transmission through the network as a func-tion of the transconductance of the amplifier. Theamplifier phase shift is taken as parameter. The curvesindicate that for small values of transconductance theoptimum amplifier phase shift is not zero. They alsoshow that as the transconductance becomes verylarge the stability tends to become independent of

gain so that the optimum phase shift depends on theoperating region. The optimum crossover phasechanges very slowly with the gain as indicated by thecurves of Fig. 6.

In order to test the foregoing theory, an experi-

0.97 0.98 0.99 1.00 1.01 1.02 L.t

fo

Fig. 7-An expanded section of the phase characteristic forthe parallel-T network.

mental oscillator was built according to the schematicof Fig. 1. The control element was composed of 12Western Electric C-2 switchboard lamps in series.These lamps were found to have the most suitablecharacteristics of any available control of this type.The frequency was varied by changing the three ca-pacitances simultaneously with the resistances R fixed.

Fig. 6-Theoretical relation of the frequency of oscillation to thefrequency of zero transmission as a function of the amplifierphase shift and transconductance. These data were computedfor the circuit constants of the experimental model of the CRoscillator.

phase shift. Moreover since these curves depend onlyon the ratio of the shunt and series capacitances, thestability should be independent of frequency when fre-quency changes are produced with fixed ratio.

It is illuminating to consider the phase-shift-versus-frequency curves of Fig. 7. These curves display on anextended scale a section of the curves of the type previ-ously shown in Fig. 3. The curves are plotted for aseries of values of K. The significance of K, as can beseen by inspecting the attenuation curves of Fig. 4,is that the more nearly K approaches unity the largerthe gain must be in order that oscillations may besustained. Thus for any particular frequency at whichthe amplifier phase shift is fixed, any change in gainwill mean a shift from one of these curves to another.From this it can be seen that the optimum operatingphase shift will be that which results in the smallestchange in frequency for small changes in K about theoperating value. If the crossover were common for allvalues of K then the stability could be made infinitefor any gain by incorporating the proper phase shift.Actually the region of crossover is a function of the

0

a.tLz0

-.

za.4t

20

C------

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32FREQUENCY IN KILOCYCLES PER SECOND

Fig. 8-The variation of the frequency instability of the CR oscil-lator against line-voltage fluctuations as a function of the fre-quency of oscillation. These measurements are for an oscillatorin which frequency changes were produced by simultaneouslyvarying the three tuning condensers and maintaining a con-stant ratio between them.

The amplitude of oscillation for the model was notabsolutely constant with frequency as expected for thistype of control but varied about 2 decibels over a fre-quency range of 50 to 20,000 cycles. Fig. 8 gives thefrequency stability as a function of frequency. It willbe observed that the stability tends to become inde-pendent of frequency as the frequency increases. Thedeviations from the predicted performance may beascribed to phase shift in the amplifier. This will be-come evident in particular for the frequency stability

1943 261

Proceedings of the I.R.E.

at low frequencies where the phase shift caused by theplate-supply choke, cathode-bias network, and otherelements becomes pronounced.

Several experiments were carried out to test thetheory of stability. In one of these the oscillator wasoperated at an intermediate frequency sufficiently lowso that effects from the input capacitances of the tubes

75

70

0 EXPERIMENTAL85 _ EXTENDED THEORY

--- ASYMPTOTIC THEORY

60

550

o50

z

2 45WW 4

-J

40a. 0

1-35

30

25

20 003

5

10-

5 -

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

EFF.

Fig. 9-CR oscillator. A comparison of the theoretical and experi-mental variation of the frequency instability against line-voltagefluctuations as a function of the amplifier transconductance.The curve marked asymptotic theory was fitted to the experi-mental results for go=geff.

could be neglected and high enough so that the low-fre-quency phase shift was negligible. The effective trans-conductance of the amplifier was then varied by usinga potentiometer grid leak. The frequency change of theoscillator for changes in the supply voltage with refer-ence to the oscillating frequency f~fo will be accordingto the asymptotic theory,

d (6f) 9 sin 0 dgeffdE (fo) 2 Xog2eff dE

on the assumption that 0 is independent of E. For theCR oscillator this assumption is justified since anychanges in the tube capacitances are negligible in com-parison with the circuit capacitances. For this methodof varying the transconductance, geff =lgo where I isdetermined by the potentiometer tap. Hence

1 dgeff 1 Idgo IV__d - dE (11)g2eff dE 12go2 dE I

where N is a constant. The expression indicates thatthe stability should be proportional to 1/1. Fig. 9shows the results of this test in which the frequencystability is plotted against 1/1. For comparison thetheoretical result predicted by (11) is indicated by thecurved marked asymptotic theory. The experimentaland theoretical curves were made to coincide for 1= 1.For large values of the transconductance the experi-mental results approach a linear variation as (11) pre-dicts but deviate for small values. The data for thecurve marked extended theory were calculated fromthe curves of Fig. 6 which were computed for the cir-cuit constants of the experimental model. For therange of data available from the curves of Fig. 6 theagreement between experimental and theoretical re-sults is good. In this case the data are not fitted, theagreement being numerical. This indicates how wellthe oscillation amplitude control is removed from thetube since in the theoretical analysis the tube wastreated as a linear element.As a second test, it was desired to investigate the

stability for changes in the supply voltage as a func-tion of the amplifier phase shift. Unfortunately nomeans were readily available which permitted phaseshift to be produced without simultaneously changingthe gain. The following experiment was carried outwhich does not constitute an adequate test but forwhich the results are thought to be significant.The test was made at such a frequency that the im-

pedance of the plate-supply choke was very high com-pared to the input impedance of the network so thatthe phase shift from the choke was negligible. Thiswas also true of phase shifts from other sources suchas the cathode bias and the grid by-pass condenser andleak. The amplifier phase shift was assumed to be zeroin the normal operating state. Phase shift was thenproduced by shunting the input of the network withreactance.

Curve C of Fig. 10 shows the experimental frequencystability against changes in the supply voltages plottedagainst the phase shift. Curve A gives the theoreticalresults calculated from the curves of Fig. 6 upon theassumption that the phase shift in the amplifier wasproduced without changing the gain. Curve B wascalculated on the same basis as curve A but assumedthe gain as half the gain for curve A. At the frequencyof oscillation a capacitive reactance given approxi-mately by (4) appears between the input terminals ofthe network. Thus a shunting inductance tends to in-crease the amplifier gain and a capacitance to reducethe gain. This explains the unsymmetrical characterof curve C. It is also significant that the optimumcondition, even for a reduced gain, occurred when alagging phase shift was produced, agreeing in signwith the theoretical result.The addition of a shunt capacitance to produce the

optimum phase shift offers a convenient and simplemethod of obtaining high-frequency stability even

262 June

Shepherd and Wise: Frequency-Stabilized Oscillators

with a low-gain amplifier. It is useful chiefly for fixed-frequency oscillators since it would be necessary tovary the shunting capacitance inversely with fre-quency for a variable oscillator. However, for a vari-able-frequency oscillator the desirable characteristicof an improving frequency stability as the frequencyincreases may be obtained by shunting the networkwith a fixed capacitance which will produce the opti-mum condition at the maximum frequency.

Inductance-Capacitance OscillatorIn the case of the LC oscillator using a control with

a negative thermal characteristic, the asymptoticanalysis shows that

poL sin 0 +PO

dd_(61\ 2R' dg_ ,\-- ~~~~~~~~(9a)dE Kfo 4g2R'2 dE

where bf and fo are defined as for (8)

Xo= poL =the impedance of each of the coils atthe null frequency

R' =R+ (R1/2) where R is the value of the shuntresistance

0=the amplifier phase shiftg =the amplifier transconductance

The expression obtained from an asymptotic analysisof the LC oscillator in which the amplitude control is apositive temperature-coefficient resistance is

d ~~poL sin (0Ro dg

dEf0 4g2RO2 dE

Now R' must always be less than Ro, the value of shuntresistance required to produce a transmission null inthe absence of RT. Hence a sacrifice in stability is en-tailed in the use of a negative temperature-coefficientthermistor, but this is outweighed by the gainsachieved in other directions. The stability expressionsare similar in other respects except for the sign of theoptimum phase shift.

It is assumed in this that the angle 0 is independentof changes in E. For the higher-frequency ranges ofthis oscillator where the tube capacitances become im-portant this will not be strictly justified and one mayexpect deviations in performance from the theoreticalpredictions.

Equation (9a) indicates the optimum stability oc-curs for a small phase shift 0 = - (poL/2R'). For anamplifier for which 0 is very small

d Qf po2L2 dgdE fo, 8g2R3 dE

At first sight (9b) appears to indicate that for anygiven frequency the sta.biJity will be greater the smaller

the inductance. However the conditions for oscillationrequire that R should vary approximately proportion-ally with L on the assumption of a fixed value of QL.Hence the inductance should be chosen as large aspossible consistent with other circuit requirements.For an amplitude control in which a negative thermal-coefficient resistance shunts the coil, R is fixed. Thus

THEORETICAL

~ ~ ~ ~ ~ ~ ~ ~ -B A

550

- - - - - - e1- -7-i25 - - /

IL~~~~~~~~~~~~~~~zI

5 __t- -__ ___

-2-2 -1 -1 -5 - S 0152-2-0 S

PHASE CD

I<- / /zo %~ ~ I

-25 -20 -15 -10 -5 0 5 10 IS 20 25 30 35PHASE SHIFT IN DEGREES

Fig. 10-CR oscillator. Frequency instability against line-voltagefluctuations as a function of the amplifier phase shift.

the frequency stability in this case is inversely propor-tional to the square of the frequency.The small phase shift for optimum operation cor-

responds to the region of crossover for the phase curvessimilar to that discussed for the CR oscillator. In thecase of the LC oscillator the crossover region is muchmore restricted as is evident from the inset of Fig. 5and less dependent upon the amplifier gain than forthe CR oscillator. This, as would be expected, resultsin better stability against gain changes. The LC circuitgives results an order of magnitude better than the CRoscillator with the same amplifier gain.No complete experimental investigation of the sta-

bility of the LC oscillator to check the stability theorywas made. The arrangement in which a positive tem-perature-coefficient resistor in the shunt arm serves asa control element is impracticable because of largevariations of amplitude which result in covering areasonable frequency range for any existing thermalelements. Hence all the experimental investigations ofthe high-frequency oscillator were made with the con-trol provided by a negative temperature-coefficientthermistor shunting the coil as a "Q control." It hasbeen found that the experimental stability curves canbe well fitted in form by curves of the type of (9b).This is indicated by the curves of Fig. 11 which apply

2631943

Proceedings of the I.R.E.

a.z0

-i

tru

n

cr

z

-i

*nz

0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 23 4 5FREQUENCY IN MEGACYCLES PER SECOND

Fig. 11-LC oscillator. Frequency instability against line-voltagefluctuations as a function of the frequency of oscillation for anexperimental model.

to an experimental model utilizing a negative coeffi-cient thermistor control.

LABORATORY MODELSGeneral Requirements

Figs. 12 and 13 show the circuit details of the twotypes of oscillators discussed above.

In each case the oscillator proper is followeXd by a

broad-band feedback amplifier to obtain a stablesource of amplification and to prevent reactions be-tween the oscillator and the load circuits. The prob-lem of singing in such a single-stage feedback amplifieris usually not a serious one. In the present case, how-ever, the difficulties are increased by the fact that thecoupling network has a high loss at the frequency of

desired oscillation and a small loss at other frequencies.If, therefore, a source of spurious phase shift of theorder of 180 degrees combined with low coupling lossexists between the grid and plate, oscillation will occur

at the frequency of this phase shift rather than at thedesired frequency of oscillation. In general, becausethe frequency of this spurious oscillation is not gov-

erned by the control provided, it is undesirable for it

Fig. 12-Circuit diagram for a laboratory model of the CR oscillatorincluding an output stage incorporating negative feedback. Fre-quency changes are accomplished by simultaneous variation ofthe three condenser arms.

will produce nonlinear behavior in the tubes. An in-elegant but very effective cure for this trouble wasfound in the use of a small resistance (10 to 100 ohms)at the grid and plate-socket prongs of the tube.

Other general design considerations consist chieflyin producing an amplifier section for the oscillator

Fig. 13-Circuit diagram for a laboratory model of the LC oscillator. The output amplifier consists of aninverter stage and a push-pull stage. The oscillator amplifier in this application was operated with the platesat alternating potential ground in order to reduce the effects of the parasitic capacitances of the tuningelements.

2.0 -0 EXPERIMENTAL- THEORETICAL

1.8 ---

0

i ~~~~~~COILS / 0 D

1.0 --~~~~~~~~~~

0.6 6~~~~~~~~~~~~

0.4 ---

0 _-

264 June

Shepherd and Wise: Frequency-Stabilized Oscillators

having reasonably flat phase and gain characteristicsover the wanted range of the oscillator.

A mplitude- Control CharacteristicsThe use of the tungsten-filament lamps for a control

places some restrictions upon the oscillator circuits.The most suitable lamps commercially obtainable re-quire about 2 milliamperes of current flowing throughthem in order to operate on a steep portion of theirresistance-current characteristic. For a permissibleplate swing of the oscillator this limits both the shunt-and series-resistance arms of the network if the con-densers are to be equal. An inspection of (8a) showsthat this restricts the stability of the oscillator. Physi-cally it means that an upper limit is imposed by thepossible amplifier gain. If the shunt and series con-densers are not made equal then more amplifier gainmay be attained but this requires more capacitance tocover a given frequency range if frequency changes areproduced by simultaneous variation of the condensers.The control resistance, as may be seen, is an impor-

tant factor for the successful operation of both the CRand LC oscillator circuits so that it is appropriate tomention some of the characteristics required of it. It isnecessary that this resistance should be linear in thesense that it should not vary over a cycle of the oscil-lation.A satisfactory solution to the problem of a highly

sensitive control is obtained by fixing R1 at the largestvalue possible for the range desired and connecting aresistance having a very large negative temperaturecoefficient between the grid and plate as a separatefeedback path. These negative resistance-coefficientthermistors are obtainable in a wide range of imped-ance and current ratings and have a very high rate ofchange of resistance with current."3 Their simplicityand compactness also makes them highly useful forcircuit applications. A typical characteristic is shownin Fig. 14.The method of applying these thermistors may be

exemplified by reference to the LC oscillator. As anapproximation to the impedance facing the amplifierone may take the value QLoPoL. One then chooses avalue for this as large as is consistent with the require-ments of the broad-band amplifier considering straycapacitances, etc. The value of the thermistor for anull is given by (6b) as

4RRTO= 2

1QLOPOL

The departure from this value at the operating point is

ART RTO(6c)

RTO 4gR213 G. L. Pearson, "Thermistors, their characteristics and uses,"

Phys. Rev., vol. 57, p. 1065; June 1, 1940. Presented, AmericanPhysical Society, Washington, D. C., April 10, 1940.

Combining (6b) and (6c) we obtain

ART 1

RT (gR (i ,2R)(6d)

Now, referring to the stability condition (9b), it willbe seen that it is desirable to make R as large as pos-sible. However the whole analysis is dependent uponthe assumption that 5RT/RT0<K1 and we thereforechoose a value so that this is satisfied. We can make8RT/RTO small by making the amplifier transconduct-ance large and this is more desirable than manipulat-ing R which cannot in any event be made greater than

300,000 ._-

200,000."

I00,000 - - - - -

80,000 - --e - _- _80000 _ - _- - _ - -

50,000

040,000-- - - / -z/30,000 -w IIz<20,00 01 - - -In~~~~~~~~~~~~~~

U)

8000--- ----

6000 ------ -5000-----1---------4000---4--

3000--01- --2000 OHMS

2000--- ---\-- ~~~~INSERIES1000 OHMSIN SERIES

00oo0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 680 6.5 70

POTENTIAL IN VOLTS

Fig. 14-An example of a resistance-voltage characteristic for abead thermistor. The effect of adding series resistances on thecharacteristic is also illustrated.

QLOpoL/2. For any given value of g a maximuLm for (6d)will be obtained for R = QLOPOL/4. How much greaterthan this value R is made will be determined by thetransconductance. When R is thus fixed a value ofRTO is determined. The foregoing calculations shouldbe made for the low-frequency end of the band for anygiven coil since this will be the worst case. A thermistorfor which a typical characteristic is shown in Fig. 14 isthen selected such that the operating range of resist-ance required will lie in the region where the resistancechanges most rapidly with the voltage in order tominimize variations of amplitude. The resistance char-acteristic may be further improved by the addition ofa fixed resistance in series with RT. The effect of this isshown in Fig. 14.The negative temperature-coefficient thermistor

may also be applied to the RC oscillator. This applica-tion makes possible alternative methods of frequencycontrol without the wide variations of amplitude which

1943 265

0

Proceedings of the I.R.E.

would result with these methods if lamps are employed.The negative coefficient thermistors are bridged acrossthe frequency-determining network as in the LC oscil-lator.There is one undesirable feature involved in a ther-

mal amplitude control which for certain applicationsmay make its use impractical. When sudden variationsoccur in the supply voltages or any other disturbanceswhich suddenly upset the gain-loss balance of the cir-cuit a fluctuation of the amplitude of the oscillationwill occur. This fluctuation is the result of the thermal

I0

40 50 100 200 500 100 200 50FREQUENCY IN CYCLES PER SECOND

Fig. 15-Frequency instability against changes in the supply volt-age as a function of the oscillating frequency for the laboratorymodel of the CR oscillator illustrated in Fig. 12. Frequencychanges are accomplished by gang variation of the condenserarms. The thermal control consisted of tungsten lamps.

lag of the thermistor and it damps out with a decre-ment determined by the thermal and electrical con-stants of the thermistor in combination with the con-stants of the rest of the circuit. The effect is minor forthe LC circuit but is most severe as might be expectedfor the CR circuit at very low frequencies where thetime constants are long. Such fluctuations for manypurposes, while annoying, may not be so serious as tomilitate against the desirable quasi-linear character-istic of a thermal element. In other cases it has beenfound that a successful substitute for the thermistorsmay be obtained by using such elements as diodes,copper-oxide varistors, or gas tubes. The suggestedelements are all for the case of the negative tempera-ture-coefficient thermistors.

Condenser-Resistance OscillatorIn one low-frequency oscillator, illustrated in Fig.

12, the three condensers in the network were madeequal and varied simultaneously to change the fre-quency. As pointed out previously this means that thepositive coefficient thermistor R, should have a valueindependent of frequency and hence may be adjustedto remain at the steepest point of its current-resistancecharacteristic resulting in good amplitude control. Theamplitude variation of the oscillator proper was foundto be less than + 0.5 decibel over a frequency range of40 cycles to 50 kilocycles with most of this variationcaused by phase shift at the extremes of frequency.

Fig. 15 illustrates frequency stability as a functionof frequency. Decreasing stability at low frequenciesresulted from phase shift introduced by the plate-supply choke.The harmonic content of the oscillator proper was

found to be principally second harmonic and was about50 decibels below the fundamental over the entire fre-quency range. The output feedback amplifier whensupplied from a pure sinusoidal source had a harmoniccontent 60 decibels below the fundamental at fullpower output of 20 decibels above 1 milliwatt. A switchis provided to remove the feedback for use where har-monic content is of lesser importance, in which casean output of over 1 watt is available.Another type of low-frequency oscillator made prac-

ticable by the use of a negative coefficient thermistoris one in which the frequency is varied by changing theresistance arms simultaneously. It has been pointedout that under these circumstances the network inputimpedance and hence the amplifier gain will change.However, the wide range of resistance change for asmall change of voltage across the negative coefficientthermistor makes it possible to cover a 10-to-i fre-quency range without undue amplitude variation.

Fig. 16 illustrates a model of this type where threeresistances are ganged together and are continuouslyvariable over a 10-to-1 range. The position of therange is changed by varying the condensers together in

Fig. 16-Circuit diagram for a laboratory model of a CR oscillatorwith a two-stage output amplifier. Frequency variation in thiscase is accomplished by a continuous variation of the three re-sistance arms over a limited range. Four separate ranges arecovered by switching the three condenser arms. The resistancearms permit a 10-to-1 frequency range and the over-all range ofthe oscillator is from 14 to 50,000 cycles.

decade steps. Here a range of 12 to 50,000 cycles wasobtained in four steps. The negative coefficient ampli-tude control made it possible to increase the gain andhence reduce the harmonic content of the oscillatorcircuit to a value of 60 decibels below the fundamentalat the high-frequency end of each range, and about 80decibels at the low-frequency end.

June266

1-

Shepherd and Wise: Frequency-Stabilized Oscillators

Inductance-Capacitance OscillatorIn the high-frequency bridged-T oscillator, shown

in Fig. 13, continuous frequency control over a re-stricted range is obtained by varying the capacitance.A range control using four coils covered in steps thefrequency range from 12 kilocycles to 6 megacycles.

In the laboratory models of the LC circuit the largeparasitic capacitances associated with a wide-rangevariable condenser make it preferable to place theplate end of the condenser array at ground with thetube cathode off ground. It will be observed that underthese conditions the tube acts as a triode since the sup-pressor and screen grids are at the same alternatingpotential as the plate.

Reference to the stability criterion (9b) shows thatfor greatest stability the L/C ratio and QL should bekept as large as possible. The desirability of the largeL/C ratio, which is contrary to conventional oscillatorpractice, arises from the fact that the impedance ofthe network and hence the gain and permissible feed-back increase with L. In practice, where the frequencyof operation was such that the plate-cathode and grid-cathode capacitance limited the gain, a low L/C ratiowas found to be preferable in order to minimize phaseshifts.

Figs. 17 and 18 show the amplitude- and plate-voltage-stability characteristics of this oscillator. Itwill be noted that the stability curve for the highestfrequency coil shows a point of very high stability atabout 4.6 megacycles. This result was accomplishedby correcting the phase shift of the amplifier-networkcharacteristic to an optimum value by inserting a 10-microhenry coil in series with R1. This coil tends tocompensate for the phase shift introduced by the uppercutoff of the amplifier caused by parasitic capacitances.

bJ tI

3 _ii3ZL

)

~-0

I S

- ---:14

5N

COILS

-IJ,3 = -- _

II-l -I -I - .- - -_-20 30 50 100 200 300 500 1000 2000

FREQUENCY IN KILOCYCLEFS PER SECOND

difficulties of high-gain, high-feedback amplifier designand construction however discourage any great excur-sion in this direction. An oscillator employing a two-stage amplifier was built for the same frequency rangeas the one discussed above and showed a stability im-provement of two to three times. The harmonic con-tent was correspondingly improved.

It may be well to repeat that the oscillators as builtwere for variable-frequency work. The electrical and

ccUa

r2X

>wZ_ _,

a:-1<2'I'

10 20 30 50 10 200 300 500 1000 200 5000 10,000FREQUENCY IN KILOCYCLES PER SECOND

Fig. 18-Frequency instability against changes in the supply volt-age as a function of the oscillating frequency of the LC labora-tory oscillator of Fig. 13.

mechanical considerations necessary to obtain thewide frequency range make some sacrifice in stabilitynecessary. Fixed-frequency or narrow-band oscillatorsof the same basic design would show still better stabili-ties since the lead inductance of the capacitance andthe amplifier phase shift could be reduced or corrected.

ACKNOWLEDGMENT

The authors wish to express their indebtedness toDr. Eugene Peterson of these laboratories under whoseguidance this work was carried on. In particular weare indebted to him for the basic method of theasymptotic analysis. We also wish to express our ap-preciation to Mr. G. L. Pearson for his co-operation insupplying the negative temperature-coefficient ther-mistors and to Dr. H. W. Bode for many helpful dis-cussions on the theory of the feedback amplifier.

APPENDIX

It is assumed that the amplifier is a constant-currentgenerator. Then the mesh equations for the circuitofFig. 4 may be written

5000 10,000

Fig. 17-Amplitude-versus-frequency characteristics of the LClaboratory oscillator illustrated in Fig. 13.

The mean value of the amplitude characteristics ofthe four coils could have been adjusted to equality byan adjustment of RJ. A more uniform response can beobtained by using five or six coils instead of four. Theharmonic content of this oscillator and its associatedfeedback amplifier is about 60 decibels below thefundamental for power outputs up to 13 decibels above1 milliwatt.

Multistage amplifiers may be used for the oscillatorproper when very high stabilities are needed. The

(R1 + jXL) = [2R1+ j(2XL -XC)il

14- (R1 + jXL) ±+ jXC,

1i i1i2 1-4

0 = jXc - + 0 + (RT- jXc)-1 1i

14

13(R+RS+RI±jXL)-+ O-

1.2 - _

1.c __ _._ _ -

_- _I

0.4 - --_COILSI 2 3 4t0.2 - X __ -

O

R = -(Rl +jXL)

(1)

1943 267

10

Proceedings of the I.R.E.

These may be solved to give

Ai3- = RjRT(R, + 2R) + XLXC(2R1 + 2R) - RTXL'1

+ jlRTXL(2R, + 2R) - R,XC(R1 + 2R)- RTRXC + XL2XCI.

Let

RI + 2,R = 2,R', Qc-PCRT, n pLQL - -p

R,

Q, QCQ,;QQQQC + QL

R,XLXC is neglected in comparison with RTXL2 and2R,R'Xc in comparison with RTRXC. Then the condi-tions for Ai4/i = 0 are

R t- QO'XL02't

At the operating point

p = Po + 6P,

11-

1 QO'QLOfO2 = (3)8H12LC 11 +

QO.21+t

Q = Qo' + SQ.

An inspection of the equation for A shows that Achanges slowly near the null and hence we are justifiedin assuming A A0 at the operating point.

Further conditions for oscillation are

i3R3=V, i, = gV

2) where g may be complex. Combining,

gR3AIi3 Ai3-1= and 6+tan-'- = 0,

IA (i i, i(7)

where 0 is the phase shift produced by the amplifier.In order to write (7) as above it is necessary that Ashall be real which to a very good approximation willbe true near the null. From (7) one may derive the twoequations

Q= - ,p 12 + f1 tan 6a2 + al tan 0

A0 cos 0(a2 + a, tan 0)gR3(al3, - a,/3,)

(8a)

(8b)

When the values of a,, a2, p1, 12, and A0 are inserted

poPo

Then upon utilizing the null conditions, (2) may bewritten, if products of small quantities are neglected,as

2 sin 0 + tan-'-)Qo' I

rLQ2[- R 1Q+ QO 12QO1o2 RI I

- ~~~~QOQLO_

* (9)

Ai3 2po0RRTL 2R_RTL-=AQ' Q ,3 - Qo j + ap L

il QO/O

Equation (6) is of the form

8Q'[a, + ja21 + 6p[01 + j121. (5)

Equation (1), when solved for A at the null conditiondenoted by A0, yields, if R and RI are neglected in com-parison with R3, and R, in comparison with RT and ifwe assume Xc,- 2XLO,

Ao= 2R3R,RT [1+ -L:+QLXOQCO 2R3

+jRTR3 [2XLXC,(1--)CXL R 6RLLO X ~~RTR3 /1 QO12.(6

4R'RTL + 2R1R'L+ [ +Q"2] ]

-z O'rQ O

1 -

' ~~~~~~~~QoQLO -_

(4)

Now

R 1 1R- 1 -«1, and (,<<1.R'- QO'f2 QOVQLO

Hence

poL sin (6+ tan-' I)Po 4gR'2

(10)

which is the equation given in the text since 1/Qo' issmall.

Equation (10) may be combined with (8a) to deter-mine 3RT.

268