PROCEEDINGS OF THE I.R.E.

The Response of Frequency Discriminators to Pulses*EUGENE F. GRANTt, ASSOCIATE, IRE

Summary-An analysis is made of the time response of a simpleshunt resonant circuit and the results applied to the behavior of theRound-Travis and Foster-Seeley frequency discriminators. The rela-tionship between the parameters of the circuit are derived such thatthe discriminator will have only one crossover frequency in the de-sired frequency band for the application of pulsed signals.

INTRODUCTIONTEHERE ARE MANY applications where a fre-

quency discriminator of any of the popular typesRound-Travis or Foster-Seeley-is used for the

frequency discrimination of pulses of short duration.An application of this use is the automatic frequencycontrol of pulsed oscillators; in particular, the control ofmicrowave oscillators with respect to maintaining a

constant difference frequency with a pulse oscillator.As might be suspected, when the frequency spectrumof the pulse to be passed through the discriminator hasa width which is of the same order as the pass-band ofthe discriminator, an anomalous behavior is presented.Specifically, if one observes the response of the dis-criminator to pulses of short duration as a function ofthe center frequency of the pulse, one may find thatthere are several crossover points. This type of behavioris confusing to an automatic-frequency-control circuit.By making a time analysis of the frequency-sensitive

elements of a discriminator, design formula may bederived such that the response of the discriminators willhave only the required crossover for a particular pulselength within a specified band of frequencies.There are two types of discriminators in common use:

the Round-Travis, and the Foster-Seeley (Figs. 1 and 2).

CURRENTiNPUr

VOLTJAGEoUrTr

Fig. 1-Round-Travis discriminator circuit.

In the Round-Travis discriminator, use is made ofthe envelopes of the output of two simple resonant cir-cuits displaced in frequency. The rectified outputs are

* Decimal classification: R361.217. Original manuscript receivedby the Institute, September 25, 1947; revised manuscript received,April 1, 1948.

t Electronic Research Laboratories, Air Mat6riel Command,Cambridge 39, Mass.

added in opposition, so that, at a frequency equal tothe mean of the individual resonant frequencies, theoutput is zero. On the application of an input on eitherside of this frequency, one side dominates, and thereis an output. Since, then, the output is a superpositionof two identical circuits displaced slightly in frequency,it will be sufficient to make a time analysis of only oneof them.

NVPTrO U TPU TVOLTAGEVOAE

Fig. 2-Foster-Seeley discriminator circuit.

In the Foster-Seeley discriminator, use is made ofthe variation of the phase shift of the response of aresonant circuit as a function of frequency. The resonantcircuit has a coil which is loosely coupled to a sourceand is center-tapped and fed from the same source. Atthe resonant frequency, the voltage across the coil is 900out of phase with the voltage injected at the tap. (Thisis true since there are losses in the circuit.) The differ-ence in the rectified voltages from the ends of the coilwith respect to the return circuit is zero at the resonantfrequency. For other frequencies, the phase shift from900, causing a small component of voltage to be addedto one side and to be subtracted from the other, resultsin an output.

GLOSSARYp =jwEl=value of voltage independent of time (or fre-

quency)E(t) =output voltage as a function of timeE,(t) the envelope of E(t)E(p) =output voltage in steady state as a function

of frequencyI=value of current independent of time

I(t) =applied current as a function of time1(p) = applied current in steady state as a function of

frequencyc = radian frequency = 2irfWa= crossover frequency of the discriminatorw)2 = resonant radian frequency of the high-fre-

quency circuit of the Round-Travis discrim-inator

3871949

PROCEEDINGS OF THE I.R.E.

w3 = resonant radian frequency of the low-fre-quency circuit of the Round-Travis dis-criminator

wo=radian frequency of the applied currentA =the peak amplitude of the envelope of E(t)H=the peak amplitude of the envelope of E(t) at

resonanceBr=the Bromwich contour of integration. It is a

path in the p plane extending from -j oo to+j oo in such a manner as to pass all the polesof the integrand on the positive real side

a= the time duration of the applied pulsey, =roots of the equation p2+W1p/Q+w12=0V= the total rectified output of the discriminatorL =the inductance of the generic circuitC=the capcitance of the generic circuitwj=resonant radian frequency of the generic cir-

cuit = 1/LCQ = "Q" of the generic circuit = R/(w,L) = RC/L

C(p) =the frequency spectrum of the input pulse.,Aw=(wo-w,I-1/4Q2x = 2QALcwy = aw1/2Q

AWb =the minimum frequency deviation from wo forwhich a departure from monotoneity exists

Aw = the minimum frequency deviation for whichan additional crossover exists.

ANALYSIS

It will be assumed in the following analysis that theeffect of the loading of the rectifiers on the resonantcircuits will be that of a shunt resistance, and that theoutput of the diodes is a voltage which is the envelopeof the ac applied to them. In general, this will be ap-proximately true. At least for an ideal linear diode, theoutput is proportional to the input. Specifically, then,the analysis will amount to finding the envelope of thevoltage across a shunt resonant circuit resulting fromthe application of a pulse of ac.

In terms of the steady-state behavior of a shunt-reso-nant circuit, the rejationship between the driving cur-rent and the voltage across the circuit in terms of itsresonant radian frequency and Q is

RI(i)E(p) = + +i (1)

1 +Q(PICO,+@1P

By the method of the Laplace transform, the instan-taneous voltage resulting from the application of a

pulse of current of duration a may be derived. (See Ap-pendix I.) In general, the phase of the alternating wavewithin the pulse will affect the shape of the resultingvoltage. However, the parameters of discriminatorsgenerally are such that this effect is small enough to beneglected. (The magnitude of this discrepancy is notedin Appendix I.) With this approximation, then, the peak

amplitude of the envelope of the response reduces to atractable expression. The peak of the response is chosenfor discussion for two reasons. For automatic-frequency-control work, the peak response of the discriminator isusually chosen to operate the control mechanism. Forany other use, since the behavior is assumed linear, anyother important quantity will be proportional to thepeak amplitude. (As, for example, the integral of theresponse with respect to time.)Then, subject to the restriction that 1/(4Q) is much

smaller than unity and the region of frequency deviationis such that Awcowl is smaller than unity, the followingexpression for the peak voltage out of the rectifier isvalid:

1 - 2e-awi/2Q cos aACo + e-a"'IQEv(a) = / ( 2)

. (2)

~+ (2zXo)

This expression has some rather interesting ramifica-tions. As one would expect, if the pulse length were verylong compared to the proper circuit parameter, then theresponse of the system should be that of a simple reso-nant circuit. This is easily seen from the above expres-sion. As the quantity aw/2Q is made very large, the,exponentials disappear, leaving the expression

1

1+(2Q,Aw)2(3)

which is the approximation to a resonance curve insteady state for frequencies near resonance.

If, on the other hand, the circuit Q were made verylarge, one might expect that the resulting peak responseas a function of frequency would be proportional toamplitude of the frequency spectrum of the pulse. Thisis seen to be the case by taking the limit of the expressionfor the peak voltage as Q gets indefinitely large, yielding

(awl\ sin (aAow/2)kY2Q (aAw/2)

(4)

There is a factor in the expression which vanishes witha large Q. It is seen, then, that as the circuit Q gets verylow, the frequency behavior of the circuit approachesthat of the steady-state case. On the other hand, if thecircuit Q is very high, the resulting output depends onthe time behavior within the pulse itself, and the outputis of the form of its frequency spectrum. This lattercharacteristic is not a desirable one to have for the ele-ments of the frequency discriminator, since the form ofthe responses is independent of the resonant frequency.The fundamental problem is to arrive at the appropri-ate circuit Q such that the peak response of the circuit

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Grant: Frequency-Discriminator Response to Pulses

is nearly that of the steady-state resonant response andshall be monotonically decreasing from the peak output(at resonance) over the desired band of operation. Ifthere should be a response which has a succession ofmaxima and minima, then it could be expected thatthere will be several crossover points as a function offrequency when two such circuits are used in the dis-criminator system. (Fig. 3 portrays a discriminator thathas this difficulty.)

Fig. 3-Response of a discriminator to pulses; Round-Travistype with resonant-circuit spacing of 2QAcw/wa=c=.

There are at least two approaches to the design of a

proper circuit given the conditions under which it mustoperate. One is to plot the value of (1) for various valuesof the parameters involved and then select the appropri-ate curve that would provide the proper operation. Thisprocess can be simplified with the substitution of

2QAw aw,x- - and y= (5)-

2Q

x then has the character of a frequency-deviation param-

a20AWWI

eter, and y is a dimensionless quantity involving thebandwidth of the circuit and the pulse length.

Rewriting (1), it follows that

/1a-2e-Y cos xy + e-281 + x2

(6)

A discussion of this equation in terms of the x's and y'sthen will apply to all circuits with these parameters.Figs. 4, 5, and 6 plot E,(a) over a convenient range of xwith several well-chosen values of y as a parameter. Fig.7 is the curve of a discriminator designed by thismethod.

8

nLa(,

Fig. 5-Normalized resonant-circuit response to pulses.

The other method is perhaps more erudite: That is,differentiate the function E,(a) with respect to Awo anddeduce the circuit Q for the condition that the deriva-tive shall be nonzero in the desired range of operation(other than at the peak of the response at resonance).A nonzero derivative insures that the function shallbe monotonic.

Let, then, Awb be the point at which it is permissiblethat the slope of the curve be zero. Under the condition

w

a

49

WIFig. 6-Normalized resonant circuit response to pulses.

.9,

.8

y- 1.0

.7 y.I0

.8

.4

.2 _

C 4 8a 12 16 20 24 28 32

1949 389

X _sswFig. 4-Resonant-circuit response to pulses.

390PROCEEDINGS OF THE I.R.E.

that this point is removed from the 3-db point of theresonance curve, an approximation may be made whichleads to a rather simple solution for the appropriate Q.If the conditions are such that this approximation maynot be made, the equation to be solved remains tran-scendental with its inherent difficulties. Then, given thepulse lengths, center frequency, and the frequency over

which monotoneity is desired, the following formulagives the circuit Q which is necessary. Equation (7) issubject, of course, to the restriction that the quantityx corresponding to Awb must be greater than unity. Thisfact must be determined after the Q is calculated.

awliaACb .

2 arc cosh - sin aL\ b + COS aAcob2J

(7)

Although, for discriminator action, it is not essentialthat the individual resonant curves be monotonically de-creasing from the resonant point, it is usually more con-

venient to make them that way, for the slope of theover-all discriminator curve may be maximized or madeto meet other conditions arbitrarily.The slope of the output versus frequency in the

Round-Travis discriminator may be maximized rathereasily at the crossover point in steady state for a fixedQ and a variable CW2 and C03. For pulsed operation, the Qhas been established subject to other requirements.While the rectifiers are connected so that the voltageoutputs subtract, the rates of change of the voltage inthe two circuits add in the region near the crossover

point. For maximum slope, the spacing between theresonant frequencies should be such that the points ofmaximum slope coincide. It is easily shown by differ-entiation of the resonance curve that the frequency spac-ing should be 2/Q multiplied by the crossover frequency.

The previous results may be heuristically applied tothe Foster-Seeley frequency discriminator as well. Inthis type of discriminator there is only one resonant

circuit, and the frequency-discriminator action resultsfrom the addition of a voltage which is 900 out of phasewith the voltage of the circuit at resonance. The over-allbehavior is nearly that of the Round-Travis discrimina-tor. (See Appendix II for the mathematical discussion.)In this case, it is easier to treat the additional crossover

points directly, yielding the following formula for theQ of the coil subject to the approximation that the re-

sulting xc shall be large compared to unity:

acoac=-2 log, cos aAw,

(8)

CONCLUSIONSA discussion has been given and curves portrayed of

the pulsed behavior of a simple resonant circuit with a

view to finding the limitations of the parameters so thatthe discriminator would approximate steady-state be-havior upon the application of pulsed signals. It was

shown that, when the Q of the circuit was very low, thecircuit had an output which varied in the same manner

as if it were in steady state to pulses. When, on the otherhand, the circuit Q was very high, the simple resonantcircuit had an output as a function of frequency whichwas proportional to the spectrum of the signal pulse.The entire question then resolved itself to a determina-tion of parameters such that the resonant circuit doesnot exhibit any behavior as a function of frequencywhich would cause additional crossover points whenassembled into a discriminator. The simplest approachto this is to plot the behavior of the circuit as a functionof frequency with a parameter and then select the curve

.6 .,\STEADY STATE RESPONSE

4 RESP_~ONSE TO '/2 wSEd PULSES

w

.2

.4

.2 _ ___l__/ 1 l

A ~A

.6.4 tI d

F-MC

Fig. 7-Response of a discriminator to pulses; Round-Travis type with resonant-circuit frequency spacing of 1.75 Mc and a Q of 60.

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Grant: Frequency-Discriminator Response to Pulses

which would give the most satisfactory results. For mostpurposes, this is a sufficiently accurate method. If thismethod is not accurate, a more accurate formulation ofthe slope of the response can be represented analytically,and under certain approximations the circuit Q can besolved for which will give a monotonic-decreasing re-sponse in the desired range of operation.

APPENDIX IThe response of a simple resonant circuit to a single

current pulse of frequency coo will now be derived. Itwill be assumed that all of the dissipative loading on thecircuit can be considered as an equivalent resistance R.

z c 1 E(t

Fig. 8-The generic circuit.

It is easily seen from Fig. 8 that the relationship be-tween the input current I(p) and the output voltageE(p) in steady state is

E(p) 1I(p) 1/R + Cp + 1/Lp (9)

However, it is more convenient to discuss the behaviorin terms of R, Q, and w1, instead of R, L, and C; there-fore, by substitution, it follows that

E(p) R

I(p) 1 + Q(p/coi + wi/p) (10)

(14)CO1 p [l -ea(p-po) ]eP tdp

E(t) = ', p -eQ27rj (p - Po) (p - y) (P - ()

where

and

(3= 2 + jwl\Il - 1/(2Q)22Q= ___ - jwl-Il - 1/(2Q).

2Q

(15)

This integral may be evaluated by Cauchy's integraltheorem, or it may be found in the literature.1

Wl [I1- ea(PO-0)]e t=

Q( - A)ePo-

[1 - ea(Po-OJeYt}

Po - Jy(16)

Equation (16) is valid for t>.a. Simplifying further,letting Aw= woo-iV-11 1/4Q2.

-C~2QF1+ i-

1 - ecwlaj2QeiaAwcEC(t) = e-Xl t/2Q I +/Q _ L2 e+witQe L %VQ2 - 1_2QAcWi

v/4Q2 1 - j 1 - ew1a/2Qeja2wo-aAc,e7w1t

/402 - 1+ j j2Q(2wo - Awl .

1+ }(17)

This expression is difficult to interpret as it stands. Ifapproximations are made that Q is large and thatAwil is small, it reduces to a more tractable expression.

1 - eclaI2aeiaAwE(t) -- eiwcote-wl tl2Q.

1+].2QAcoCOI

(18)

The methods of the Laplace transform may be appliedto derive the response of the network to a pulse of cur-rent. For a current pulse of amplitude 1/R, durationa, and center frequency wo, the frequency spectrumG(p) is

G(p) =- epofe-P dt.Ro

By integration, it is seen that1 1 -ea(p-PO)

G(p) =-R p- po

Since the rectifiers may be thought of as providingan output equal to the time envelope (of course, thepass band of the output filters must be sufficientlygreat), the direct-voltage output will then be approxi-mately equal to

(11)

(12)

The response of the network is, then,1 r 1 1-ea(ppo)

E=2rj J 1+Q(P/w+w/P) P-PO ePdp. (13)It is to be noted that the real part of E(t) will be theresponse to a cosine wave of current input, and theimaginary part is the response to a sine wave of currentinput. Simplifying (5) further,

1 - eolaI2QeiaAwE,(t)--- e-co t /2Q.

1 + j 2QAco

W

(19)

It may be shown that this expression is roughly within1/4Q of the actual envelope of (19) for frequencies nearthe resonance of the circuit.

It is interesting to note that by this process theresult is now independent of the use of a cosine wave ora sine wave for the pulse. The difference in the responsesto the cosine and sine wave was contained in the termwhich was dropped.

l G. A. Campbell and R. M. Foster, 'Fourier integrals for practi-cal applications," Bell System Monograph, pairs 453 and 207; 1932.

3911949

PROCEEDINGS OF THE I.R.E.

Equation (19) is valid for t.a, but it is a simple ex-ponential function of time, so it may as well be discussedat its maximum amplitude of t =a. Usually, it is thisamplitude which is the desired one. Simplifying further,

In the event that the approximation cannot be made,the original equation would have to be solved in com-plete form.

APPENDIX IIIE(-)2e-a1l- Q cos aAc,w + e-awllQE, (a)1= + (2QAW 2

( @1)+E,(a) may also Ifunctions.

Ev(a) =

be expressed in terms of the hyr

e/wl2 awl2e a co/2QIcosh - - cos aAw

\ 2Q /

1 + 2QA )2\ C1

The Foster-Seeley circuit may be treated in a some-what similar manner to that of the Round-Travis dis-criminator. The resonant circuit may be considered to

perbolic be fed from a loosely coupled, transformer (in order toapply the preceding theory) with the injection from afrequency-insensitive circuit. Due to the nature of thecoupling, the resonant circuit will be 900 out of phasewith the injection (in steady state at resonance). Let P

(21) be the amplitude of the injection compared to one-halfthe resonant circuit voltage. Then, since the rectifierssupply the difference of the absolute magnitudes,

APPENDIX IIThe problem is to find the Q which will cause the fre-

quency response to be monotonically decreasing out toAc\b. To do this it is expedient to find the first zero ofthe derivative of the resonance curve with respect tofrequency. Of course, there will be a zero at the centerfrequency and at infinite frequency, but these will besuppressed. Differentiating the response with respect toAco by conventional methods results in

dE Q y sinxy 2xd= E c(a) -dco WO cosh y -cos xy l+x2

Setting dE/dw= 0 results in

y sin xy 2x

cosh y - cos xy 1+x2

(22)

(23)

For most applications, the band of operation will belarge compared to the 3-db bandwidth of the resonantcircuit. (The 3-db bandwidth corresponds to x equalunity.) Hence, the approximation may be made thatx2 is much greater than unity. This simplification greatlysimplifies (23). Ignoring unity as compared to x2, it isseen that

sin xy 2cosh y cosxy xy (24)

cosh y -cos xy xy

or

Hence,

aCw aAwObcosh - sin aAwOb + cos aAWb.

2Q 2

awl

(25)

(26)2 arc cosh sin aAwb + COS aAwb]_2_

e- - eixY e-8-eixy

ou =1 +jx j 1 +jx

(27)

Eout will have a zero at x= 0. It will have many otherzeros. It will be desirable to adjust the parameter Qsuch that these zeros will fall outside the usable range.

E0,,ut = (e-Y -cos xy - Px)2 + (P - sin xy)2Eout=6 l~~~+ x2

(e-Y -cos xy + Px)2 + (P + sin xy)2-V 1 +x2 (28)

Equating to zero, transposing and squaring,

(eY - cos xy - PX)2 + (P - sin xy)21 ± X2

(eY - cos xy + Px)2 + (P + sin xy)21 + X2

(29)

Remove parenthesis, multiply by (1 +x2), transpose andfactor:

/sin xy0=4Px t- + e-0-cos xv . (30)

It is seen that one zero occurs at x equal zero. The loca-tion of the zeros is independent of P since it enters onlyas a factor. It remains to find the value of Q given a,Wa and Aw0.

sin aAco1O = 2Q-- + e-aal2Q - cos aAwj .

2Qcoc_- Wa

(31)

If the quantity x, = (2QAwc/Wa) is large compared tounity, then the first term may be ignored, and

aw,Q= 2 l c (32)2 log. cos aAw,co

392 A pril

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