Peter and Strandberg: Phase Stabilization of Microwave Oscillators

The relations (8a) and (8b) show that infinite direc-tivity can be obtained when the primary line (line Ain our case) is mismatched as well as when it is matched.(We call line A "matched" when Pai = 0.) Having foundthe general condition for infinite directivity [(8a) and(8b) ] let us consider some particular cases of practicalinterest.

CASE OF SMALL COUPLINGLet us define "small coupling" by the condition:

K2 << 1. (10)

In this case, the conditions of infinite directivitygiven by (8a) and (8b) become:

lim Z22ZS3 = 1

lim ZllZ44 = 1for K2- 0.

Comparing (11) (which has been rigorously derived)with the conditions Z22Z33 =1 and Z1Z44 = 1 given byFirestone for infinite directivity, we see that these lastconditions hold only in the case of weak coupling. Al-though the case of weak coupling is of considerablepractical importance [e.g. when a transmission-line di-rectional coupler is used for the measurement of swr, inthe way suggested in (H) ], cases of strong coupling mayalso be important; in such cases, the conditions (11) forinfinite directivity are not valid any more, and (8a) and(8b) have to be used instead.

CASE OF MATCHED LINESLet us determine under which conditions the reflec-

tion coefficient on line A is zero and the directivity ofthe system is infinite.From (9) we find the condition for ra = O to be:

1

1-K2 (12)

From (8) we find the condition of infinite directivity(D1= oo) whenr., =O to be:

Z33 = 1. (13)

One notices in (12) how the coupling K affects thematching condition of line A.As a conclusion of this analysis we may summarize

our results as follows:1. Infinite directivity may be obtained with trans-

mission-line directional couplers for mismatched as wellas for matched lines. The condition of infinite directivityin its general form is given by the relations (8a) and(8b).

2. In the case of weak coupling, the conditions ofinfinite directivity reduce to the conditions Z22Z33=1and Z11Z44=1

3. In the case of matched lines and infinite directivity,(Pat=0 and D1= oc) the matching impedance of theprimary line is a function of the coupling coefficient[cf. (13a)].

Phase Stabilization of Microwave Oscillators *M. PETERt AND M. W. P. STRANDBERGt, SENIOR MEMBER, IRE

Summary-A circuit has been developed with which microwaveoscillators may be phase-locked to weak but stable reference signals.The circuit was operated with S-band oscillators (707B klystron;2C37 triode oscillator) and a 2K50 K-band klystron. It is possible tolock a microwave oscillator directly or through a cascade of suchcircuits to a quartz-stabilized oscillator. The statistical theory ofrandom noise is used to obtain an analysis of the stabilizing effect ofthe circuit, and the power spectrum of the stabilized microwavesource is calculated. The scheme can also be applied in divider opera-tion. Modifications are discussed. A modified circuit that uses carrier-suppressed modulation of the reference signal has also been realized.In another circuit, the oscillator frequency is converted by means of astable reference, and compared with a second reference that can beof low frequency and tunable. These latter circuits allow eliminationof the excess noise introduced by crystal diodes. In the originalstraight dc circuit this noise cannot be eliminated, but calculationshows that its influence on the output power spectrum is very small.

* Original manuscript received by the IRE, February 25, 1954;revised manuscript received, May 6, 1955. This work was supportedin part by the Signal Corps; the Office of Scientific Research, AirResearch and Development Command; and the Office of Naval Re-search.

t Dept. of Physics and Res. Lab. Elec., Mass. Inst. Tech., Cam-bridge, Mass.

INTRODUCTIONT HIS PAPER will discuss the phase stabilization

of microwave oscillators. It should be clearlyunderstood at the outset that phase stabilization

is quite distinct from frequency stabilization in the con-ventional form. A frequency discriminator with a(fo-h-1 control circuit would give essentially a phase-stabilization type of control, but such a (fo f)-f controlis neither realizable physically nor defined analyticallyfor the operating region, i.e., for f=fo. If instead of afrequency discriminator a phase discriminator is usedat the outset, all necessary components are realizable.

Note also that frequency stabilization allows one toestablish a frequency to an accuracy which is constantwith time. Phase stabilization establishes a mean fre-quency with an accuracy directly proportional to thelocking time. The interest then in phase stabilizingmicrowave oscillators is to realize the transference offrequency stability from one frequency region to anotherwith any desired precision.

1955 869

PROCEEDINGS OF THE IRE

Recent developments in the techniques of molecularbeam measurements and microwave spectroscopy1 makeit possible to observe substances in a state where theyabsorb electromagnetic energy at one or several ex-tremely sharply defined microwave frequencies. A sub-stance in such a state is therefore analogous to a cavityof very high Q (107 or better) with a persistently accu-rate resonance frequency. In order to measure the centerfrequency of one of these resonances it is desirable tohave microwave oscillators whose output power is asmonochromatic as possible.At lower frequencies, an oscillator controlled by a

quartz crystal may be used to generate a signal withvery high stability for a period of hours or days. Thislow frequency can be multiplied by means of vacuumtube or silicon diode multipliers. Conventional multi-plication usually yields a high-frequency spectrum thatis not monochromatic but has sidebands, arising fromlower frequency modulations, that remain because ofthe finite selectivity of the circuits. Furthermore, sincemultipliers with a gain of less than one (silicon diodes,for example) introduce additional noise into the spec-trum, it is not desirable to multiply a frequency by morethan a factor of 10 in these diodes.

In this paper we describe a stabilization circuit thatallows a microwave oscillator to be locked to a harmonicof a stable reference oscillator. Through iteration of thisprocess, the stability of a quartz-controlled oscillatorcan, essentially, be transferred to a K-band oscillator(23,040 mc). Description of the experiment is followedby an analysis of the stability and performance of aphase-locking circuit.

PHASE STABILIZATION OF A K-BAND OSCILLATOR

The circuit that has been successfully used to stabilizethe frequency of a 2K50 klystron is shown in Fig. 1.The circuit consists, essentially, of a single, absolutelystable feedback loop. Any phase modulation in the klys-tron is detected in the phase discriminator: the result-ing signal is amplified in the differential dc amplifierand applied to the repeller of the klystron to producean opposite and stabilizing phase modulation.

|AUDIO AMPLIFIER OSCILLOSCOPE DIFFERENTIALAND HEADPHONES MONITOR AMPLIFIER

ICORRECTION SIGNAL

Fig. 1-Block diagram of phase-stabilizing circuitfor K-band oscillator.

Phase Discriminator

The reference signal and oscillator are introducedthrough the noncoupling arms of a waveguide hybridjunction, or "magic tee." The signals that arrive at thedetector crystals on the two remaining arms, 1 and 2,

l M. W. P. Strandberg and H. Dreicer, "Doppler line-width re-duction," Phys. Rev., vol. 94, pp. 1393-1394; June, 1954.

ESi sin (wct+771) and ES2 sin (Wct±+'2), are shown by thevector diagram in Fig. 2.These fields are the sum of a signal coming from the

reference source, ERi sin coCt and ER2 sin coet, and a siginalcoming from the oscillator, eol(sin wct+01) and E02(sin cowt

ARM I -K

ARM )

E02

ARM 2

Fig. 2 Vector diagram of signals in hybrid junction.

+¢)2). From symmetry properties of the magic tee it isseen that 1 = 02 +7. Since detected power PD in a silicondiode as a function of input power Pi is given by2

PD = SP22 (S = conversion gain per watt ; 104 (watts)-')it follows that the detected output signal is:

Vj = -2VRSPojPRj cos q j + VNRS(Poj + PRj), j = 1, 2.

Here, Pod, PR2 are the power of oscillator and referencesignal in each arm; V1 is the phase discriminator de-tected output of each of the crystals. It can be seen thatif Po and PR are divided equally between arms 1 and 2,then VI - V2 is independent of variations of Po and PRfor Ck;-ow/2. Amplitude modulation of the two signalsis therefore proportional to the balanced-out control sig-nal, and hence second order, being negligible in the limitof small control signal. Sinice the insensitivity to ampli-tude variation allows discriminator to be operated ata high power level in spite of the small PR, crystalscan be operated in a region of good conversion gain. ForP0 = 200 ,ut watts, R = 100 ohms, PR = 8 ,u watts, we ex-pect a differential output of gp = 0.04 volt per radian.As indicated in Fig. 1, this output is amplified in a

differential dc amplifier. A cross-coupled circuit3 wasused for this purpose; the actual circuit is shown inFig. 3. The circuit uses all readily available techniquesto achieve stable, hum-free operation. The heaters ofthe 12AX7 tubes are fed in series from the negativepower supply. A K-band spectrum analyzer, an oscil-loscope, and headphones are used to tune the oscillatorto the reference signal. Once a klystron is within about1,000 cps of the reference, it will phase-lock itself auto-matically. Since the 2K50 klystron is microphonic, goodsound isolation is essential. The experiment was, there-fore, carried out in an anechoic chamber. But, any goodacoustic isolation for the klystron should be sufficient.The reference signal was supplied from a very stable,

cavity-tuned, planar triode S-band oscillator. This oscil-lator, in turn, was locked by an analogous circuit to thetenth harmonic of the output of the M.I.T. frequencystandard.4 The correction signal was applied to the plateof the oscillator triode.

For the S-band oscillator, a klystron also could be2 M. W. P. Strandberg, "Microwave Spectroscopy," Methuen

and Co., London, Eng.; 1954.3J. N. van Scoyoc and G. F. Warnke, "A d-c amplifier with cross-

coupled input," Electronics, vol. 23, pp. 104-107; February, 1950.4C. G. Montgomery, "Technique of Microwave Measurements,"

M.I.T. Radiation Lab. Ser., McGraw Hill Book Co., Inc., New York,N. Y., vol. 11, pp. 347-375; 1947.

870 JUIlY

Peter and Strandberg: Phase Stabilization of Microwave Oscillators

TO'MAGIC TEE K "

INPUT 2lo PRECISION

. RESISTOR 68K SI

Fig. 3-Cross-cou

used; the 707B klystron was phase-locked with thissame equipment to the M.I.T. frequency standard.

ANALYSIS OF THE PHASE-STABILIZINGFEEDBACK LooP

The power spectrum of the output of a klystron, orany conventional oscillator, is not a single sharp line.There are three reasons. First, the klystron puts out anoise band as broad as the pass band of the loadedplate cavity. This noise may be thought of as simplediode noise.2 Second, variations of the supply voltageson the klystron electrodes impress a frequency modula-tion on the carrier. Third, microphonic pickup alsocauses frequency modulation in the klystron throughthe relative physical motion of the frequency-determin-ing elements of the oscillator.The frequency-modulated output is written

S =So sin {ct+f co(t)dt}, (1)

where c (t) is the frequency modulatioop, and (t) =fw(t)dtis the phase modulation. The frequency change pro-duced by voltage variations on an oscillator electrodemay be written as

-- radians sec-1/volt, (2a)

where m(t) is the random part of electrode voltage. Theacoustical pickup might be caused by a variation of thedistance of the grids in the gap of the plate cavity orby the vibration of the repeller perpendicular to thetube axis. The effect of this motion may, in general, beexpressed as

-M- e radians sec-1/cm, (2b)

ipled dc amplifier.

where d(t) is a characteristic distance in the physicalfrequency determining circuit. Both m(t) and d(t) areassumed to represent random noise having a normalamplitude probability distribution. Hence, co(t) can berepresented as5

N

@(t) = E C. COS (Q.t - Zn)n-1

I N r

,2(t) == -2 n cn2 = W(Q)dQ,2n==lb

(3a)

(3b)

where W(Q) is the frequency modulation power inradians sec'1, and b and r, the lower and upper cut-offfrequencies, will be discussed later [see (8a) and (12)].We would like to know the power spectrum of S withthe modulation (3a). The Fourier spectrum of a carrier,frequency-modulated by several independent sinewaves, has been calculated by Crosby.6 He found side-bands displaced by Q. from the carrier with amplitude1/2(c,,1/Q,), and cross-modulation bands of higher orderin c,,/Q,,. The energy of the modulated carrier is con-centrated within either twice maximum frequency devia-tion or twice modulating frequency, whichever is greater.Thus we have two different cases to consider,

(C,/Q,2,) >1 and (c,,1/0) <<«. In the first case we expectto find the power distributed within a band of width[W2(t) ]I/2 around w,. This assumption was verified in anexperiment in which a noise voltage of 10 cps band-width and of known rms voltage was applied to therepeller of a klystron whose output was observed in a

5 S. 0. Rice, "Mathematic Analysis of Random Noise,-Noise andStochastic Processes," Dover Publications, Inc., N. Y., N. Y.; 1954.

6 M. G. Crosby, "Carrier and side frequency relations with multi-tone frequency or phase modulation," RCA Rev., vol. 3, July, 1938.

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PROCEEDINGS OF THE IRE

spectrum analyzer. In the second case, the cross modula-tion was neglected, and we find

N C

S=0 jsin &,t + E 2- (sin [(W,. + 2In)t - Zn]

- sin [(.c Qn)t + Zn)} (4)

with the power spectrum

1 XW(Q)p(Q)dQ S02 =d Q. (5)

We now wish to calculate the effect of the stabilizingfeedback loop on the power spectrum. In order to obtainthe open loop gain, y(Qn), we consider the nth term in(4). This term is the result of a phase modulation, (X).This modulation, present in the output of the oscillator,gives rise to a voltage output, Gp P7Q) from the phasediscriminator. This output in turn is amplified by a fac-tor of AD in the dc amplifier and converted according to(2a) into frequency modulation. The latter process cailalso be described as phase modulation, with a gain of(/3/iQ) radians/volt. Hence,

1 r(Q) = GPLDI3 i=- with r GPID(. (6)

iQ iQ2If the feedback loop is now closed, the nth term will bereduced by a factor [1/(1 -,(Q))], and the power spec-trum of the stabilized oscillator, if r is assumed to bereal, is given by

1 W(Q)dS2PS(Q)dQ -2S02 -. (7)=2 r±Q

where p,(Q) is the stabilized noise power in watts/radiansec-'. With an estimate of W(Q) and r, the order of mag-nitude of the residual noise left in the stabilized systemmay be calculated. From (6) and typical numbersGP-0.04 volts/radian; ,UD = 2; , = 107 radians sec-'/volt,r may be computed as r = 8 105 radians sec- .This leads to a stabilization cut-off frequency 9,/2ir,where |I (Q.)I =1, of

QC= Vc= 130 kc. (8a)

27r

We can estimate W(Q) from (3b). If we assume it in afirst approximation to be independent of Q and use anempirical value of Q2(t) = 108 (radian sec-')2 for the un-stabilized klystron, and assume that these deviationsare the result of noise up to Q, we compute

W(Q) - 100 radians sec-1, for Q < Q,.

HenceP8(Q)- 8 10-11 PC in the pass band. (8b)

We estimate2Pdiode -- 10-pc. (8c)

By integrating (7) we findr 0dQlPo ,

P -= 100Pe J _- arctan 1 2 10-4P,, (8d).Jr+ 2 r

where Pc is the carrier power in watts; Pdiode iS the powerdensity resulting from diode noise in the klystron irnwatts/radian sec1-; and Pt is the total noise output ofthe stabilized circuit, up to vP, in watts.The phase angle v of the stabilized signal still has a

Gaussian amplitude probability distribution.

1P(~) =

( 7r_)1/e___ ~ /22

- Pa et2=P= 10-4

Pc(9)

where (W2)1I2 is the rms value of the phase variation,computed for the stabilized system.

Following conclusions are suggested by our results:1. Eqs. (8b) and (8d) show that the noise power in

the whole spectrum of the stabilized oscillator is farbelow the carrier. This justifies the assumption ot thevalidity of (4).

2. Eqs. (8b) and (8c) show that between coi - 27rv andcoi+2wrvp the noise power produced by the klystron diodenoise is negligible, compared with the frequency-modu-lation noise of the stabilized output. Although an ordi-nary broadband amplitude-modulation detector is in-sensitive to frequency modulation, so the diode noise isdominant, the high-Q experiments described in the in-troduction will detect the frequency modulation noise.

3. Eq. (8a) shows that the stabilization loop has apass band of only 130 kc. The dc amplifier will show nophase shift in this band; therefore, the loop will have aphase shift of 7r/2 and will be absolutely stable.

4. Eq. (9) shows that the rms phase shift is roughly1 degree and that a shift bigger than or/4 is quite im-possible. Hence, the phase discriminator operates in thelinear region of its discriminator characteristic.

5. The stabilization changes the klystron power spec-trum that was originally spread over a finite width (ofapproximately 100 cps) into an impulse function and avery low and broad noise band. This result rests on theassumption that the reference signal is monochromatic.In practice, the spectrtum width of the stabilized oscil-lator will be reduced to the width of the reference signal.

MODIFICATIONS OF THE STABILIZING CIRCUITHowever obvious it may seem, it does appear worth-

while to point out that this phase-locking circuitry mayalso be used to make a divider of particular use in themicrowave region where no other kind exists. The opera-tion of the circuit has been discussed from the point ofview of locking a microwave oscillator to the harmonicsof a lower frequency oscillator. However, the correctionsignal may also be applied to the lower frequency oscil-lator to transfer to it the stability of the high-frequencyoscillator. This divider type of operation would be quiteuseful, i.e., for the general utilization of output ofmolecular microwave oscillator frequency standards.78

7 H. Lyons, "Spectral lines as frequency standards," Ann. N. Y.Acad. Sci., vol. 55, art. 5; 1952.

8 J. P. Gordon, H. J. Zeiger, and C. H. Townes, "Molecular micro-wave oscillator and new hyperfine structure in the microwave spec-trum of NH3," Phys. Rev., vol. 95, pp. 282-284; 1954.

872 July

Peter and Strandberg: Phase Stabilization of Microwave Oscillators

STABILIZINGSIGNAL

Fig. 4-Stabilizing circuit using carrier-suppressed modulation.

Figs. 4 and 5 show two of the many modificatiorthe feedback loop. These two forms allow a discusthat is sufficiently general that it can be applievariations of the basic circuit.

In Fig. 4 the hybrid junction is used to apply carsuppressed modulation to the reference signalcrystal A) and to combine it with part of the oscilloutput. Detection of the resultant signal in crystalfollowed, after amplification, by a second detectionphase detector whose output is the stabilizing sigmThe whole stabilization problem may be transfor

to any convenient frequency, of course, by converthe oscillator signal with a stable reference. This maaccomplished in the manner shown in Fig. 5. Here,oscillator is converted by the reference to a frequiof f cps. The converted signal is amplified and compwith a second reference in a phase detector whoseput serves again as the stabilizing signal. The relastability of the second reference can be worse thanof the first reference by the ratio of their frequenThus the second reference may be obtained from aable source so that the frequency of the stable oscillcan be varied even if the first reference is fixed.

In the circuits of both Fig. 4 and Fig. 5 use is macIF power coming from a detector crystal; in the origcircuit dc power coming from two detector crystalsused. This is of interest in connection with the profof detector noise. If a crystal diode (1N26) rectifismall signal Pc, a noise power density Pk(Q) will apin excess of the thermal noise :2

106kTP,2PA = watts/radian sec-'.

If an IF carrier is generated in the diode, a similar rpower spectrum is found, distributed around the caas it was before around the dc carrier, in accordzwith (10).1"0 Whether this power results from bothquency modulation and amplitude modulation or famplitude modulation alone is not, at present, deciIt would seem that the latter is more likely. If this iswe have a means of minimizing the effect of the crznoise on the stabilizing signal. This AM crystal r

9 M. W. P. Strandberg, H. R. Johnson, and J. R. Eshbach,paratus for microwave spectroscopy," Rev. Sci. Instr., vol. 2576-792; August, 1954.

10 M. Strieby, 'Transistors: Circuit Noise Problems," QProg. Rep., M.I.T. Res. Lab. Elec., pp. 110-113; July 15, 1953.

I STABILIZINGOSCILLATOR t OUPTSIGNAL

l I-F AMPLIFIERMIXER CRYSTAL DEECO

[m <~~T~REFERENCE REFERENCE 2

Fig. 5-Stabilizing circuit using second reference.

may be cancelled by using a phase detector that is in-sensitive to amplitude modulation; for example, onethat is adjusted to work with zero output at equilibrium.If the oscillator is stabilized with the dc circuit the in-fluence of the crystal noise on the output power spec-trum cannot be balanced out because it arises from twoindependent sources-the two detecting crystals. Sincethis appears offhand to be a serious fault of the dcstabilization, we conclude with a calculation of influenceof crystal noise on stabilized oscillator spectrum.A noise voltage sN(Q)dQ introduced at the discrimina-

tor (see Fig. 1) is reduced by 1/(1 -,(Q)) by feedback,and appears as phase modulation JADOSN(Q)dl/i9(l-I)at the output. The noise power

pN(U)dQ = RpkPc dQQ2+ r2 (11)

is calculated in a fashion similar to p8(Q) in (7). We findthat PN(9) exceeds p8(R) only when

2R1016kTpC2/yD2:29< -= 350 radians sec-W(A)

if we set PN(Q) =p8(Q) and assume Pc= 10-4 watts. Thismeans that only in a band of 400 cps around the stabil-ized signal the noise introduced by the crystal exceedsp8(Q). Furthermore, even 1 cps from the carrier, PN(Q)< 10-'Pc. The contribution of the total crystal noise tophase modulation noise PNt is f'PN(Q)d&. We get

PNt =- 10'kTRPc3,.D2f2 -In [-2( + -2 r2 L2 b2/

(12)

For b = 10-4, the lowest frequency observable within anhour, we obtain PNt=6- 10-7P,. Comparing (12) with(8d) it may be seen that PNt is far less than P8j. Eq. (9)indicates that the probability of saturating the dis-criminator has not increased. This shows that the stabil-ity of the output signal is not significantly affected bycrystal noise. The circuit of Fig. 4 has therefore no spe-cial advantage. Its realization has been tried with somesuccess, but it is difficult to set all the adjustments prop-erly. The circuit of Fig. 5 makes the oscillator moreflexible frequency-wise but it does require a broad bandIF amplifier. The original dc circuit is found to be simpleand quite effective.

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