PROCEEDINGS OF THE I.R.E.-Waves and Electrons Section

An Analysis of the Sensing Method of AutomaticFrequency Control for Microwave Oscillators *

EUGENE F. GRANTt, ASSOCIATE, IRE

Summary-An analysis is made of the type of automatic-fre-quency-control circuit that uses a simple cavity resonator for thestable element and either frequency modulation of the controlle4oscillator or modulation of the cavity resonance frequency to obtaineffective discriminator curves which give a null output for a fre-quency equal to the average cavity resonance frequency. An analysisis made of the complete automatic-frequency-control loop gain, in-cluding the transmission through the cavity as a function of thevarious parameters and the frequency-modulation swing. A discus-sion is presented of the best method of decreasing the pulling of theoscillator frequency by the cavity, and the pulling of the cavity fre-quency by a load which has a variable susceptance.

GLOSSARYA1= the power attenuation' between the oscil-

lator and the cavity.A2 = The power attenuation between the cavity

and the detectora, b, c, d =the constants characterizing the general

four-terminal networkC(W, V, n) =the fourier component coefficient of the

frequency modulation present in the de-tected output

6 = per unit frequency changeD(W, V, n) =the second fourier component coefficient

of the frequency-modulation frequencypresent in the detected output

E(t) =the detected output voltage as a functionof time

Ei= the fundamental component of the sensingfrequency in E,(t)

E2= the second-harmonic component of thesensing frequency in E,(t)

F1=the voltage across the input terminals ofa four-terminal network

E2=the voltage across the output terminalsof four-terminal network

f =the instantaneous frequency of the oscil-lator

fo = the resonant frequency of the cavityAF=the total frequency modulation or cavity

resonant frequency swingAf= the center frequency deviation of the

oscillator from fofm=the frequency-modulation sensing fre-

quencyG =the loop-gain, measured by breaking the

loop at any point

* Decimal classification: R355.6XR119.39. Original manuscriptreceived by the Institute, November 17, 1947; revised manuscriptre-ceived August 19, 1948.

t Electronic Research Laboratories, Air Materiel Command,Cambridge 39, Mass.

Ga=the ratio of the output of the phase dis-criminator to the input of the amplifier

Gf = the ratio of the frequency deviation of theoscillator to the control-element displace-ment. (This may be either a voltage or amechanical displacement depending onthe control method used.)

1=the current flowing into a four-terminalnetwork

12 the current flowing out of a four-terminalnetwork

Kn= detector constant; E, (t) - K P_nf2n =the detector-law exponent; n is equal to

unity for a linear detector, and to 2 for asquare-law detector

Pose= the power output of the oscillatorP,=power incident on crystal detectorQ= the input Q of the cavityQ2= the output Q of the cavityQo -the unloaded Q of the cavityQL= the loaded Q of the cavity

Q.,, =the loaded Q of the oscillatordC(O, V, n) aC(W, V, p)

= ~W = Oaw aw

T=power relative to incident power trans-mitted through a cavity

t1-27fmtV= QLA\F/fo, which is the frequency-modulat-

ing swing in terms of the cavity 3-dbbandwidth

W=2QLAf/fo, which is the frequency devia-tion in terms of one-half the cavity 3-dbbandwidth

Yin, Yout-admittance relative to the admittance ofmatched load.

INTRODUCTION7ff HERE ARE several methods of accomplishing

automatic frequency control of oscillators operat-ing in the microwave region. The five methods

which are commonly used at present are:1. The Pound dc discriminator.1,22. The Pound ac discriminator.2'31 R. V. Pound, "Frequency stabilization of microwave oscillators,"

PROC. I.R.E., vol. 35, pp. 1405-1415; December, 1947.2 C. G. Montgomery, "Technique of Microwave Measurements,

Radiation Laboratory Series No. 11," McGraw-Hill Book Co., Inc.,New York, N. Y., 1947.

3 W. G. Tuller, W. C. Galloway, and F. P. Zaffarano, "Recent de-velopments in frequency stabilization of microwave oscillators,"PROC. I.R.E., vol. 36, pp. 794-800; June, 1948.

1949 943

PROCEEDINGS OF THE I.R.E.-Waves and Electrons Section

3. The use of absorption lines.44. The method of frequency modulation.5. The method of resonant-circuit sensing.'Some of the design details of the methods (4) and (5)

are to be discussed in this paper. As in all control sys-tems, the general method of operation is as follows: Thefrequency of the oscillator is compared with that of thestable element (in this case a resonant circuit or a cavityresonator), with the result that a voltage (or a displace-ment) proportional to the amount of deviation is pro-duced. This voltage is then applied to a frequency-con-trolling element on the oscillator in such a direction asto return the oscillator frequency to the frequency of thestable element. In the microwave region, a transmissioncavity can be made to have the property that the powertransmission is a maximum at a particular frequency,but decreases very rapidly as the frequency is deviatedfrom that of maximum transmission. Such cavities canbe made to have very high Q's and a relatively stableresonant frequency. A cavity cannot be used directlyas the stable element, since the transmission is an evenfunction of the frequency deviation from resonance.Therefore, no information can be obtained from theoutput, whether the deviation is plus or minus. Onemethod of realizing a voltage which shall be positive ornegative dependent upon the direction of the deviationof the oscillator frequency is to apply a small amount offrequency modulation to the oscillator frequency. Itwill be observed in Fig. 3 that the output of the cavitywill be amplitude modulated by an amount which isdependent upon the relative amount of frequencymodulation and the average oscillator frequency. Inparticular, it will be observed that the phase of the re-sulting amplitude modulation reverses with respect tothe phase of the frequency modulation when the oscil-lator frequency passes from above the cavity resonancefrequency to below it. This fact is made use of by pro-viding an amplitude-modulation detector (usually asilicon crystal detector for the microwave region) at theoutput of the cavity, and a suitable amplifier to operate

USE/rUtF'R-SQUAICY OUTPO OTPUOSFC/uLrOR ALreencycviref~~~~~M cJela,odw/I/on, COt2PLING

E/eo,enfro//tnemen t

DISCIMINATUOR SoeEUECE Y T CAeo1rY

|A4P1 IFIER ENVLOP

Fig. 1-Schematic of the frequency-modulating methodof automatic frequency control.

4WVv. Smith, Jose L. Garcia de Quevedo, R. L. Carter, andW. S. Bennet, "Frequency stabilization of oscillators by spectrumlines," Jour. Appl. Phys., vol. 18, p. 112; January, 1947.

5 G. G. Gerlach, "A microwave relay communications system,"RCA Rev., vol. 7, pp. 560-600; December, 1946.

a phase detector. The output of the phase detector isthen not only proportional to the magnitude of thedeviation of the average oscillator frequency from theresonant frequency of the cavity, but agrees with thealgebraic sign of the deviation. Fig. 1 is a completeschematic of such a system of control. The sensing fre-quency is defined as the rate of the frequency modula-tion on the oscillator.From the point of view of the over-all system be-

havior, it makes little difference whether the oscillatoris frequency-modulated or the reasonance frequency ofthe cavity is caused to vary about a mean frequency atthe sensing frequency rate. The operation is essentiallythe same. This is the method of resonant-circuit sensing.See the schematic of this system in Fig. 2. Some over-all

(comroLIE C/S.FOSILAMER )U'u, ~~~~~~~~~COUPLI/NG

AMPLIFIER EVELOPE l

Fig. 2-Schematic of the resonant-circuit sensing methodof automatic frequency control.

design considerations will dictate which method is pref-erable. There are several methods available for causingthe resonant frequency of a resonator or cavity to vary.(Since, in general, this paper will be concerned with amicrowave system where cavities are generally employedas the resonant system, the word cavity will be used toindicate a resonator. However, the arguments and theanalysis may be generally applicable to a system whichoperates at a frequency where lumped components maybe used.)

1. The cavity may be equipped with a diaphragm ora plate which is displaced harmonically at the sensingfrequency, causing a variation of a suitable dimensionof the cavity.

2. The cavity could be constructed in such a mannerthat it could be resonant mechanically to the sensingfrequency and driven in a suitable manner; a convenientmeans would be a piezoelectric crystal, a magnetostric-tion bar, or even an electromechanical drive. Thechanging dimensions would effect a periodic change inthe resonant frequency of the cavity.

3. A variable reactance could be coupled to thecavity. A nonlinear element, such as a silicon or ger-manium crystal, may be used if placed in a suitablenetwork.

In order to simplify the ensuing discussion and argu-ments, only the method of frequency modulation willbe discussed. However, from the standpoint of analysis,

944 August

Grant: Loop Gain of Automatic-Frequency-Control Circuit

the systems are practically identical. It is only a ques-tion of the reference for the time scale. Hence, to applythis discussion to a resonant circuit sensing system, itis only necessary to realize that the frequency swing ofthe cavity from its resonant point is identical to thefrequency-modulation swing of the oscillator.

Fig. 3-Cavity characteristics.

Fundamental to the design of a system which uses acavity in this manner for the stable element, is the rela-tionship between the frequency-modulation swing andthe sensitivity of the output of the phase detector interms of the various circuit parameters. An importantelement of the system is the amplitude-modulation de-tector. Usually, this detector will be a crystal rectifier,which has a law relating the input amplitude to theoutput amplitude not easily described. However, forlow-level operation the behavior will approach a square-law relationship, and for high-level operation, the be-havior will approach a linear relationship. The analysisis carried out for the square-law detector and the linear-law detector. For the case where the operation is notdescribed by either of these two laws, a heuristic ap-proximation will have to be made on the basis of experi-ence and the analyzed cases. It is more convenient todiscuss the behavior of a cavity in terms of its Q's andresonant frequency fo than in terms of its equivalentlumped-constant parameters. The Q's and fo are moreadapted to direct measurement in the physical system.An automatic frequency-control system using the

method of frequency modulation was developed by theauthor for use as the beacon-frequency automatic-fre-quency-control system for the AN/A PS-6 airborneradar.

GENERALThe various parameters of the control-system design

will be taken up in term. It is assumed that the fre-quency at which the system is to operate has been de-cided. It remains, then, to decide the various othercavity parameters, the loop gain of the system, and the

padding that must be inserted at various locations. Acavity must be chosen which has a resonant frequencyof the desired stability. The normal frequency drift ofthe oscillator is ascertained to determine the amount ofloop gain necessary to control it within the necessarylimits. Since the system is a normal negative feedbacksystem, the frequency drift of the controlled oscillator,relative to the resonant frequency of the cavity, is equalto the frequency drift of the oscillator uncontrolled di-vided by one plus the loop gain. As will be seen later,the loop gain may be increased by increasing the un-loaded Q of the chosen cavity, other factors remainingthe same. The unloaded Q of a copper cavity is a func-tion of the resonant frequency chosen and of its physicalsize. That is, a large E-mode cavity will have a muchhigher unloaded Q than a small re-entrant H-modecavity for the same frequency. The physical size of theequipment may be a deciding factor in the unloaded Qobtainable.6 For economy in the design of a controlsystem, each component should be used to its fullcapability. The loop gain of the automatic-frequency-control circuit is controlled to a very large extent by thecavity parameters, and it would be wise to adjust thoseparameters such that the effective gain is the maximumobtainable commensurate with the other factors whichmust be considered.The analysis of the loop gain in a frequency-control

system with frequency modulation will be based uponthree assumptions.

1. The sensing frequency and the frequency modula-tion swing is sufficiently low that the cavity may beconsidered to be in steady state for each instantaneousfrequency. This will be true if the sensing frequency issmall compared to the 3-db bandwidth of the cavityand if the maximum rate of change of the instantaneousfrequency is less than the square of the 3-db bandwidthof the cavity. (The bandwidth must be expressed incycles per second.)7

2. The relationship between the envelope amplitudeapplied to the detector and the output voltage can berepresented by either a square-law relationship or alinear relationship.

3. The phase discriminator output is linearly relatedto the amplitude of the fundamental component of thedetector output.

ANALYSIS OF EFFECTIVE GAIN THROUGH THE CAVITYThe cynosure for the analysis is the cavity behavior.

It is convenient to treat the high-frequency energy interms of the incident power rather than in terms of itsvoltage. The reason for this is that the incident power isa relatively easy quantity to measure. Further, for work

6 T. Moreno (editor), "Microwave Design Transmission Data,"Sperry Gyroscope Co., Manhattan Bridge Plaza, Brooklyn 1, N. Y.,1944; chap. X.

7Balth van der Pol, "The fundamentals of frequency modula-tion," Jour. IEE (London), vol. 93, pp. 153-158; May, 1946.

1949 945

PROCEEDINGS OF THE I.R.E.-Waves and Electrons Section

of this kind, it is not necessary to know absolute im- By Fourier's theorem, the fundamentpedance or admittance values, but only the relative the sensing frequency in the detected cvalues in the circuit. The Q's of the cavity are de- 1 -'fined or measured under the same conditions that it will FC1 =-J Ec(t) sin Odhave in the operating system. Some methods of measur- T ring the various Q's of a cavity have been described.28 Combining (5) and (6), it follows that

E01 = K.(2QL) [nPOAIA2]nh +f [1 + (W + V sin 0)2]-n/2 sin Ode.

It can be shown the instantaneous power transmittedthrough a cavity for a unit incident power is (see Ap-pendix I)

T =4QL2 1Q1Q2 1+ [2QL(f -fo)1f

fo

(1)

tal component of)utput isl. (6)

(7)

1 rTC(W, V, n) - I [I + (W + V sin 0)2]-n/2 sin OdO. (8)

7r JT

C(W, V, 1) and C(W, V, 2) are evaluated in AppendixII. Figs. 4 and 5 show the plots of these functions with

The instantaneous frequency of the oscillator may beexpressed in terms of the sensing frequency fin, the totalfrequency swing AF, and the deviation of the center fre-quency from the cavity resonant frequency and time.It is seen that

AiFf= fo+- Af-+-sin 2rffmt.f=Jo~~~ (2)

Since the high frequency is most easily described interms of incident power, it is convenient to express thedetector law in terms of the relationship between itsincident power and the voltage output. Since it isexpressed this way, it is very easy to obtain the coeffi-cient in the expression for the limiting cases experi-mentally. Let the power from the cavity be fed throughan attenuator to the crystal detector, which then will beconsidered to have an instantaneous output voltage re-lated to the input power as follows:

Ec(t) = KnPc71/2* (3)The combination of the previous expression and theeffect of an attenuator between the cavity and thecrystal detector and the effect of the decoupling betweenthe oscillator and the cavity results in

E(t) =Kn(2QL[ 1 ]

+QA QL/~F 2iTfmt)]* 1+ (-+ - sin 2,rfmt (4)

The above expression can be much simplified by the useof some dimensionless constants. Substituting W, V, and0 for the expressions 2QLAfl/fo, QLAF/JfO, and 27rft, re-spectively, results in the following:

E,(t) KK(2QL) [ Q1Q2[1 + (W + V sin O)2]-n/2. (5)

8 W. Altar, "Q-circles-a means of analysis of resonant microwavesystems," PROC. I.R.E., vol. 35, pp. 355-361; April, 1947.

2.0w

Fig. 4-Effective discriminator curves for a cavity and linear de-tector, W being proportional to the oscillator frequency deviation,while -C(W, V, 1) is the relative amplitude of the detectoroutput at the sensing frequency.

0.4 as 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4Dw

Fig. 5-Effective discriminator curves for a cavity and a square-lawdetector, W being proportional to the oscillator frequency, while-C( W, V, 2) is the relative amplitude of the detector output atthe sensing frequency.

various values of V as a parameter, C(W, V, n) is anodd function of W, so it is only necessary to plot thefunction for positive values of W.

Since, in general, the operating point of the controlsystem is near the W origin, the function C(W, V, n)may be approximated by the first term in its Taylorexpansion

dC(O, V, n)C(W,V,n)_ aw W

Grant: Loop Gain of A utomatic-Frequency- Control Circuit

The response of the output to a frequency variation onthe input is approximately linear, and the systemmay be treated as a linear system. The expressiondC(O, V, n)/dW may be computed as a function of V.(See Appendix III.) These results are plotted in Fig. 6.

QLn+l(Q1Q2) n/ (12)

partially with respect to Q, and Q2, equating to zero, andsolving, it is seen that for maximum EC, as a function ofQ, and Q2,

Fig. 6-Slope of the effective discriminatorcurves at resonance.

These curves indicate that the function 9C(0, V, n)/o Wgoes through a maximum as a function of V. For nequal to unity, the maximum occurs at V equal to 0.88,and for n equal to 2 the maximum occurs at V equal to0.707. This indicates that in the system under design,the amount of frequency modulation of the oscillator(or the amount of frequency swing of the resonator)can be adjusted to provide a maximum gain throughthe cavity system.The relationship between a small frequency deviation

in the frequency of the power incident on the cavity andthe fundamental component of the sensing frequencycan now be completely stated and interpreted. From(7), (8), and (9), by substitution

E,1 _ Kn(2QL) n Po[ AFP08 nA122C(O, V, n)-but

W = 2QLAf/fohence

Ec -- Kn(2QL)n+l [P 1A2 1n12 aC(0, V n) (11)It has been pointed out in a preceding paragraph thatC(0, V, n)/5Wmay be maximized by the proper choice

of V for the detector law in effect. In addition, the ex-pression for E,1 may also be maximized with the properchoice of Q relationships. As was pointed out in generalsection, the unloaded Q is usually determined by theother physical factors in the system; so it remains to dis-cover if there is a proper choice of the ratios of the win-dow or loaded Q such that the ratio of E,1 to Af is a maxi-mum. Differentiating the expression

orQ1 = Q2 = 2QoQo = 2QL for n= 1, (13)

and

orQ1 = Q2 = QOQo - 3QL J

for n-=2. (14)While it is very interesting that such a maximum oc-

curs, it frequently happens in the design of a system thatis not desirable to take advantage of it for reasons to beconsidered.DIscusSION OF OSCILLATOR STABILITY AND FREQUENCY

PULLINGIn coupling resonant cavities to microwave oscillators

a certain amount of care must be exercised to preventtwo undesirable effects:

1. If the cavity is coupled too strongly to the oscilla-tor, the system may exhibit frequency discontinuities orjumps and refuse to oscillate at the cavity resonance fre-quency.

2. If the system is the type that uses the cavity-resonance frequency sensing, the varying susceptancepresented to the oscillator through the coupling maycause a slight frequency and/or amplitude modulationof the oscillator at the sensing rate.An excellent discussion of the first point appears in

the literature." 9 Ford and Korman derived a formulawhich relates the frequency pulling of an oscillator tothe change in load susceptance

(susceptance relative to matched admittance)2QO8C . (15)

The load susceptance must, of course, be referred to theoscillator terminals. In the case of a microwave oscilla-tor, the terminals may be defined as the point in thetransmission system where the frequency of the oscilla-tor is independent of the load conductance. The sus-ceptance at the cavity terminals may be calculated fromthe formula (see Appendix I):

Yin Ql Yout+Ql + j 2QlfQ2 QO fo (16)It is observed that there is a pure susceptance changewith frequency (either resonance or applied), but witha reflection through a transmission system, this sus-ceptance variation with frequency may manifest itselfas conductance variation as well as a susceptance varia-tion at the oscillator terminals. As may be seen with the

I J. R. Ford and N. I. Korman, "Stability and frequency pullingof loaded unstabilized oscillators," PROC. I.R.E., vol. 34, pp. 794-799; October, 1946.

1949 947

PROCEEDINGS OF THE I.R.E.-Waves and Electrons Section

aid of a Smith chart,'0 the proper length of transmissionline may favor either conductance variation or a sus-ceptance variation. If the system under considerationuses a frequency varied resonator, the reflected suscept-ance variation will cause a frequency modulation of theoscillator, but the reflected conductance variation willcause an amplitude modulation of the oscillator at thesensing frequency and possibly at its harmonics. An en-gineering decision will have to be made as to the amountof these modulations that can be tolerated if this systemis to be satisfactory. In the- event that the coupling tothe cavity must be reduced to bring these effects to atolerable minimum, it may be necessary to sacrifice loopgain by the insertion of attenuation or other means toachieve the desired coupling. This point will be consid-ered again in the discussion of the loop-gain equation.

In a similar manner to the pulling of the oscillator fre-quency by a susceptance presented to its terminals, thecenter frequency of the cavity may be pulled by a sus-ceptance connected to its terminals. (This is one methodof varying the resonance frequency.) Since, however, thecavity is to be used as the stable element of the controlsystem, its frequency pulling must be considered, formany of the crystal detectors for microwaves have verybroad tolerance on input admittance. Equation (19)may be used to calculate the pulling of the cavity by asusceptance of the detector if Q.G, is replaced by Q2.

DERIVATION OF LOOP-GAIN LIMITATIONSA loop gain may be assigned to the complete loop as

would be done in the case of an ordinary feedback am-plifier. This is possible since Ea, is linearly related to theoscillator frequency deviation (11). All that is necessaryto determine the loop gain is to multiply all of the indi-vidual gains together, including the various constantsthat relate the controlling sensitivity of the oscillator.(This last quantity may be influenced by a frequency-sensitive load upon the oscillator.1"9

Hence, from Fig. 1 and (11) it follows thatGf F POSCAIA2P/~2 aC(o, V, n)G = TG2Kn(2QL)n+l L Q2 aV (17)

The following equation results for the gain optimuizedwith respect to the cavity Q's and the amount of fre-quency modulation.

GfG = (0.23)-G2Kn(A,Po80)n12Q0. (18)

foThe ratio of the frequency stabilities before and after

the loop is closed can now be determined. The deviationin the oscillator frequency from the resonance frequencyof the cavity after closing the loop is equal to the devia-tion before closing the loop divided by the quantity(1+G).

10 P. H. Smith, "A transmission line calculator," Electronics, vol.12, pp. 29-33; January, 1939.

In general, it may be stated that the relation betweena small output-admittance variation and input-admit-tance variation is as follows for a matched coupling sys-tem.

Ayin AYout 1Yin Yout (Power attenuation) (19)

where Yin and Yout are the input and output admit-tances respectively. This expression applies to the casewhere the cavity is decoupled from the oscillator outputby means of a directional coupler or a "magic tee" aswell as by a conventional attenuator. For proof of thisexpression, see Appendix V. It is seen, then, that the in-put admittance variation varies inversely with the powerattenuation. In deciding the admittance variation as afunction of frequency due to the cavity and its effectupon the oscillator, it is observed from (19) that the ad-mittance at the input terminals of the cavity will bevery large at resonance and infinite at frequencies farfrom resonance. This is due to the fact that Y2 is nearlyunity and the ratio of the input Q to the unloaded Q willbe chosen to be greater than unity. Hence, when used,the cavity will be placed an odd multiple of quarter-wavelengths from the oscillator terminals (if used in aseries system, it will be placed an integral number ofhalf-wavelengths from the oscillator terminals). There-fore, as seen at the oscillator terminals, a small variationof admittance will be observed as the frequency is variedthrough the cavity resonance frequency. The magnitudeof this variation is seen to vary inversely with the inputtermination Q, since the input admittance to a quarter-wave section of transmission line is the reciprocal of theoutput admittance. (In the series case, impedance is thereciprocal of admittance.) Therefore, from the point ofview of decreasing the oscillator pulling, is it better toincrease the attenuation due to the coupling between thecavity or to increase the input termination Q of the cav-ity? The resonance frequency of the cavity is also sensi-tive to admittance variation presented to its terminalsand ordinary mismatches that occur in an operatingsystem may be responsible for some frequency variation;hence, from that point of view the termination Q of thecavity should be as high as possible. In looking at (17),it is noticed that the loop gain is influenced in one termby the ratio of attenuation to Qj; hence, this term is in-dependent of the decoupling for a constant amount ofpulling on the oscillator. However, since the gain is pro-portional to a power of QL and QL varies in the same di-rection as Q' there is benefit to be reaped by increasingQ, over that of increasing the attenuation. These samearguments will apply to the decoupling of the cavityfrom the admittance variations of the crystal detectorwhich might occur due to temperature variations orother effects. It follows, therefore, that for the reasonsconsidered, it is preferable to decouple the cavity by theuse of large input and output Q's with no additional at-tenuators.

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Grant: Loop Gain of Automatic-Frequency-Control Circuit

DERIVATION OF THE SECOND-HARMONICOUTPUT OF THE DETECTOR

It may be important for design consideration to knowthe magnitude of the second harmonic of the sensing fre-quency contained in the detector output. In some cases,this component may have to be reduced to the properamplitude by suitabte filtering if the amplifier and asso-ciated circuits are to operate properly. In operation, themagnitude of the second-harmonic component may bemany times that of the fundamental component result-ing in the saturation of the amplifier. By Fourier'stheorem,

EC2 fE(t) cos 20dM (20)KPoAAlA 2 - nV1Kn(2QL) L- Ql j D(W, V, n) (21)

APPENDIX I

If it is assumed that the cavity has only a single modewithin the desired range of operation and has a band-width (to the 3-db points) which is small compared toits center frequency, then a number of simplifying as-sumptions can be made to expedite the analysis of itsbehavior. For the purposes of this paper, the cavity maybe described sufficiently by the constants Ql, Q2, Qo, andfo. The restrictions and assumptions of the lumped-cir-cuit model which will be assumed are described.9 Thispaper is generally concerned with cavities that have onlyone window or loop, but the arguments are applicable totransmission cavities as well. Fig. 8 is a schematic of themodel cavity. Use is made of a similar model in describ-ing the behavior of cavities used as wavemeters;11 theexpressions used have a slightly different notation. Thefollowing are the definitions of the Q's:

(Maximum energy stored in the cavity)(Power dissipated in the walls of the cavity)

(Maximim energy stored in the cavity)Q1 = 2 rfo ---24o(Power transmitted back through the input window)

Q2 =22rfo (Maximum energy stored in the cavity)(Power transmitted through output window)

o { (Maximum energy stored in the cavity)VL - 2rJ0 (Total power lost to the cavity)1 1 1 1QL Qo Qi Q2

where

D(W, V, n) = cos 20d6[1+ (W + V sinO)2]n/2 . (22)D(O, V, n) is evaluated in Appendix IV. Fig. 7 shows itas a function of V for n equal to 1 and 2. The values of

v

Fig. 7-Quantity of second harmonic of the sensingfrequency in the detected output.

(27)

Since the actual values of the admittance of the load andthe source to which the cavity is connected are notknown, the effective shunt resistance cannot be known.However, for the purposes of this paper, it is not neces-sary to know this; it is only necessary to know the ad-mittance rela4ive to the value for which the proper Qwas determined.

i 2PEArFEcr

rTAMASORSAERPEjeFcr

J/2ANSfORMER

Fig. 8-Equivalent circuit of a cavity.

Since the cavity will, in general, be rather looselycoupled to the oscillator, the most useful way of describ-ing the power transmission will be in terms of the ratioof incident power to the output power. Taking accountof the reflected power at the input terminals of the cav-ity, by conventional circuit analysis12 there results1s

D(W, V, n) are not important as a function of W sincethe operating point is near W equal zero. The value ofthe second-harmonic component is maximum at Wequal zero.

11 See chap. 5 of footnote reference 2.12 E. A. Guillemin, "Communications Networks," John Wiley and

Sons, New York, N. Y., 1931 and 1935.13 These expressions may be obtained directly from page 65 and

page 291 of footnote reference 10 with the corresponding change innotation.

(23)

(24)

(25)

(26)

04

0.4.~~~~~~~~~~~SUR LAW DETECTOR

0.2

A Aa2 1.6 2.0 2.4 2.6 3.2 3.6 4qC

9491949

PROCEEDINGS OF THE I.R.E.-Waves and Electrons Section

Y1 =Qi Y2+ Qi + f2Q,j-Q2 Qo fo (28) For n equal to 2, the following integral nmust-be eval-(2)uated( r (Wsin OdOC(W, V, 2) = I

7r , 1+(W+V sin0) IT (2QL)' 1

Q1QL (2QLAf)21 +fo2

APPENDIX IIFor n equal to unity, the following integral is to be

evaluated1 r~7r sin OdO

C(W, V, 1) = j V--WVio2.(30)C(WF 1) X J xX + (W + v sin 0)2From a practical point of view, it is most expedient toevaluate this by a series expansion about V equal tozero. Expanding the denominator by a Taylor series,then, it is seen that

W 1 7 O dm Vmnsinm+lOdoT J r n=o dWm nl + Wy )

Exchanging the order of integration and summation andintegrating,"4

(29) 13y expanding the integrand in partial fraction, it is seenthat'5

C 7rjYW + jC(W, V, 2)=27j r- W + i + V sin 6

_W-j 1d0.W-j + V sin oi

1 ~ ~ ~~1C(Wj VJ 2) = [ 1 1 )

1 ( 1.

By a few algebraic simplifications, it is founld that

(35)

(36)

C(W, V, 2) = - 2VW2+1VV(W2 + 1)2 - 2V2(W2 - 1) + V4

2V2Wsin 1 arctan + - 1 (37)

(W2 + 1)2- V2(W2 -1)

(32)

Performing the differentiating and factoring gives therequired series

C(w, V, 1) --WV 3(2W2-3)C=(1+W2)3/2L 8(1 W2)2

5(8W4-4W2 + 15)+ 64(t + W2)435(16W6-i68W4 + 210W2 - 35)

+ - -V6. .. (33)81,920(t + W2)6

aC(O, V, 1) - 4V r/2aW r(1 + V2)3/2

This series is convergent for V2 less than 1+ W2 since theradius of the circle of convergence of the series for1/1+(W+z)2 in the Z-plane is V/1+W2, when expandedabout Z=0.

14 Mathematical Tables from "Handbook of Chemistry andPhysics," Chemical Rubber Publishing Co., Cleveland, Ohio, 5thedition, 1936; no. 337, p. 271.

APPENDIX IIIThe functions representing the slopes of the effective

discriminator curves at the origin are to be evaluated.The slope of the curves for the linear detector may beevaluated rather easily in terms of complete elliptic inte-grals from the original integral.

ac(0, V, 1) v r sin2 OdO(I-+ V2 0)112 * (38)aw irJ- (1+V2sin2')312Substituting 1 -cos2 0 for sin2 0 and 0 =7r/2

-0, it fol-lows that

(1 -sin2 4)d4_ V2 - 3/2I -

1+ V2sin2

(39)

Consulting de Haan,"6 and making a few algebraic sim-plifications, it is seen that

15 See no. 349, p. 272, of footnote reference 14.16 D. Bierens de Haan, "Nouvelles Tables D'integrales Defines,"

G. E. Strechert and Co., New York, N. Y.; 1939; p. 91.

and (34)

00 1.3. . .(2m + 1) d2m+l 1 V2m+lC(W, V, 1) = 2_ -mn0 2.4 ... (2m + 2) dW2m+l 1 + W2 (2m +1)!

950 A ugust

Grant: Loop Gain of Automatic-Frequency-Control Circuit

aC(O, V, 1) 4 V_ =^ ~K

a)"14Vr\/1 + V2 _ \/1 + V 2

-E( )] (40)

where K and E are elliptic integrals."7The slopes of the effective discriminator curves for the

square-law detector may be found most easily by dif-ferentiating the function directly, resulting in

aC(o, V, 2) -2V- (1 + V2)5'2 (41)d9W (I + V2)312

APPENDIX IVThe integral representing the amount of second har-

monic of the sensing frequency may be worked out at Wequal to 0. For the linear detector,

1 r7r cos 26d0D(0, V, 1) = 8X/I + V2 sn2 0 (42)

sin

Substituting 1 -cos2 0 for sin?2 0, n/2 -4) for 0, 1 -2 sin2for cos 26, and shifting the integration intervals, it isseen that

D(0, V, 1) = -4 r r2 (1-2 sin2' )dd7r'/1 + V2Jo /1 V2

sin2

1 + V2

It is found that'8

(43)

-4 _/V \D(O, V, 1) = + K v

XVl11 + V2L \V+/ V2

-2D V) (44)

For the square-law detector,1 r cos 20d6D(O, V, 2) 452 sin2

Substituting '(I -cos 20) for sin2 0, 0= 6/2, and shiftingthe interval of integration, it is seen that

2 r2T Cos 4d4)D(O, V, 2) = -- dq). (46)

1- cos2 + V2

Making some algebraic simplifications,"9 it is found that

D(0, V, 2) = - 1] . (47)

17 E. Jahnke and F. Emde, 'Tables of Functions with Formulaeand Curves," Dover Publications, New York, N. Y., 1943.

18 See p. 73 of footnote reference 17.See p. 272, no. 349, of footnote reference 14.

APPENDIX VThe expression for the relation between the variation

in input admittance and the variation in output admit-tance for a matched four-terminal network may bederived easily using matrix methods.12 Consider thegeneral four-terminal network characterized by the fourconstants a, b, c, and d. Not all are independent, for a bi-lateral system with a consistent set of units ad -bc mustequal unity. Consider the network operating into andadmittance Yout with an associated input admittanceYin. Let E, and 11 be the input voltage and current tothe network, and let E2 and I2 be the output voltage andcurrent. It follows, then, that

El[ a b 1 0 E2- ~xI1 c d Yout 1 0

Hence,E= (a + bY2)E2

and_l= (c + dY2)E2.

By dividing I, by El, it follows thatc + dYout

Yin =A + BYout

By differentiating with respect to Yout, it follows thatAout

AV.n _'__(a + bYout)2

The power attenutaion for the network can be derivedfor the relationship between E, and E2 if Yin and Yout beconsidered as pure conductances. If they are not, thenthe power input and output expression must containonly the real parts of Yin and Yut. To include this re-finement generalizes the final expression, but it does notincrease its usefulness. Hence,

(Power attenuation) 2out=El2 Yin

bYYt2outIt then follows thatAXYin AY|outjYin Yo,Yut (Power attenuation)

Even though the admittances Yin and Yout must be real,the variations do not have to be real and, in general,they will not be real. If it were possible to find the val-ues of a, and b and Y0,ut for the particular network, itwould be possible to calculate the relationship betweenthe real and imaginary components of the admittancevariation. However, if only the power attenuation isknown, only the relation between the absolute magni-tudes of A Yi,,/ Yi and A Yout/ Yo,ut can be found.

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