IEEE T'RANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-26, NO. 2, JUNE 1977

IV. CONCLUSIONS

Two new methods for measuring the complex permit-tivity of dielectric yarn and textile has been presented. Themeasured results show that the methods are sufficientlyaccurate for most applications. These methods are easy touse and could be extended to measure the moisture contentof textiles and dielectric properties of ultra-thin dielectricfilms. The first method may be used to measure the per-mittivity of liquids after some modifications.

REFERENCES

[1] F. Horner et at., "Resonant methods of dielectric measurements atcentimeter wavelengths," J. Inst Flec. Eng., vol. 93, pt. III, p. 53,

[21

[31

[41

[51[61

[71

[8]

1946.M. Sucher and ,J. Fox, Handbood of Microwave Measurenments, 3rded. Brooklyn, NY: Polytechnic Press, 1963, vol. II.A. Kumar and D. G. Smith, "Measurement of the complex permit-tivity of sheet materials using resonant window in the rectangularcavity," IEE Conf. Pub., (London, England), no. 129. p. 151, 1974.R. F. Harrington, Time-Harmonic Electromagnetic Fields. NewYork: McGraw-Hill, 1961.A. Kumar, unpublished work.A. Kumar and D. G. Smith, "The measurement of the permittivityof sheet materials at microwave frequencies using an evanescentwaveguide technique," IEEE Trans. Instrum. Meas., vol. IM-25, p.190, Sept. 1976.A. R. Von Hippel, Dielectric Materials and Applications. New York:Wiley, 1954.C. P. Aron, "Effect of degenerate EilL mode in Ho,, mode cavity onthe measurement of complex permittivity," Proc. Inst. Elec. Eng.,vol. 114, no. 8, p. 1030, 1967.

Phase and Amplitude Modulation Effects in a Phase Detector

Using an Incorrectly Balanced Mixer

REMI BRENDEL, GILLES MARIANNEAU, AND JEAN UEBERSFELD

Abstract-A very simple theoretical model of a phase detectorusing a double- or single-balanced mixer is described. The maincharacteristics of this device are depicted and the effect of thebalance in the mixer is analyzed. Experiments show that the modeldescribes the behavior of the unbalanced phase detector in a sat-isfactory way.

I. INTRODUCTIONACTUALLY ONE of the most interesting setup

for the measurement of frequency stabilities ofhigh-quality signal sources is based on the phase or fre-quency comparison between two oscillators. Fig. 1 showssuch a simplified system [1], [2].The phase fluctuations of the oscillator under test are

measured with respect to a reference oscillator. Both os-cillators are connected to the inputs of a double balancedmixer used as a phase detector, and which is operated atits quadrature point by means of a phase shifter. In thiscase, the dc output signal of the mixer is proportional tothe phase difference between the two oscillators.For a better utilization of such a measurement system,

it is necessary to know its peculiar noise. To this end, oneconnects the same oscillator to both the mixer inputs as

Manuscript received December 16, 1976.R. Brendel is with the Laboratoire de Chronometrie, Ecole Nationale

Superieure de Chronom6trie et de Microm6canique (ENSCM), 25000Besan~on, France.

G. Marianneau and J. Uebersfeld are with the Laboratoire de Physiqueet M6trologie des Oscillateurs du C.N.R.S., 25000 BesanVon, France.

ReferenceOscillator

OutputMixerSi nal

Oscillator Phase -ShifterUnder Test

Fig. 1. Block diagram for relative stability measurement.

shown in Fig. 2. In this case it can be easily shown that thephase noise of the local reference oscillator is removed andthe analysis of the output signal of the mixer gives the noiseof the measurement system which might be independentof the local oscillator.

However, experiments have shown that the output sig-nal spectrum depends on the local oscillator used for thismeasurement. This means that the mixer is not perfect andit is necessary to evaluate its improper balance in order toknow the consequences on the measurements performedwith this system.The problem can now be summarized in the following

way (see Fig. 1), since e1, e2 are two harmonic or quasi-harmonic signals which do not necessarily have the samelevel:

1) What is the influence of a balance defect on theoutput signal V. ? How is this phenomenon related withthe phase shift 0 between e1 and e2 and with their ampli-tudes?

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BRENDEI et al.: MODULATION EFFECTS IN A PHASE DETECTOR

VI

Fig. 2. Block diagram for the determination of the measurement systemnoise.

2) What is the influence of the balance defect on theoutput signal when the same signal is connected to bothinputs of the mixer, one of them being out of phase withrespect to other one? (see Fig. 2.)

3) What is the simplest theoretical model of an unbal-anced mixer depicting the observed behavior?

II. DESCRIPTION OF THE MODEL

The phase detector used for frequency stability mea-surements is generally a double-balanced mixer or ringmodulator [3] shown in Fig. 3.

Another phase detector shown in Fig. 4, can be also used[4]. It is known as the single-balanced mixer or half-ringmodulator.When dealing with dc or slowly variable output Vs, there

is no fundamental difference between the two previoustypes of mixer, the first only has a better efficiency becauseof the double rectification. For these reasons, the theo-retical analysis can be performed with either the double-balanced or the single-balanced mixer model for a startingpoint. The latter has been chosen for the theoretical cal-culations for its simplicity.

III. PHASE DETECTOR RESPONSE

A. Equation Derivation

Let el and e2 be the signals with the same frequencyapplied at the mixer inputs (Fig. 1)

el = E cos wt

e2= E'cos (wt+). (1)

From Fig. 4 it is obvious that

e= e2+ el = (E+ E' cos () cos wt - E' sin sin wt

Ve = e9-e1 = (E' cos E- B) cos cot - E' sin ¢ sin wt.

(2)The two RC networks of Fig. 4 are equivalent to the

low-pass filter of Fig. 3 which is used in order to obtain theenvelope of the function defined by (2). For an ideal phasedetector, we should have for the output signal (Fig. 4)

Vs = \u -

where

u2 = E2 + E2 + 2EE' cos

s 2 + F'2-2EE' cos 0.

Fig. 3. Double-balanced mixer.

Fig. 4. Single-balanced mixer.

The balance defect can be introduced now by affectingat each input a coefficient of efficiency iq which is not thesame for each one. Hence, the output signal becomes

Vs = nv- ?'Xv (5)

Then, the phase detector response can be set in theform

V8= E 1+R

2R - ^ 2R cos )IV +R2V1+R2J

(6)

where R = E'/E is the input rms voltage ratio and 6 = q'lqdefines the balance defect between both sides. For an idealdevice 6 must be unity. It will be shown later how this de-fect can be evaluated.

Fig. 5(a) shows the response of an ideal phase detector,that is the output signal V, versus the phase shift 0 fordifferent values of the input signals ratio R.

Fig. 5(b) shows the calculated effect of a balance defecton the phase detector response.To use the mixer as a phase detector, the two input sig-

nals must be operated in quadrature. Let's introduce asmall difference a(a << 1 rad) defined by the followingrelation

( ==r/2 + a. (7)

Then (6) becomes

ii Ra _/ Ra\Vs=~E l+R(1l-l +R2) I+ R2 )

(8)(3) This function, plotted in Fig. 5(c), shows that the sensi-

tivity (which is the slope of the curve) changes when theratio of the input signals is modified and the zero outputsignal is no longer obtained for a ir/2 phase shift as in aperfect detector but for a phase shift /0 which is a function

(4) of the input signals ratio R.

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, JUNE 1977

.5 =4

a6 R= <-1

-1 R=-l-

0 45 90Input Signals Phase-Shift

(a)

135 180(Degrees)

76

zO 45 90 1

nptSgas Phs-hf (egres

N R - :(E R=1.5t_._-. I

0 45 90 135 180Input Signals Phase-Shift (Degrees)

(b)

Input Signals Phase-Shift (Deg. near 900)(c)

Fig. 5. Phase detector response: (a) for an ideal phase detector; (b) fora detector with balance defect; (c) for a detector with balance defectnear 900. R is the input signals ratio.

B. Phase Detector SensitivityThe sensitivity k of the phase detector can be defined

by the slope of the response curve in the vicinity of the zerooutput signal, hence,

k= 3Vs ER(1 + i)k =

=7 1+R2 (9)

where k is expressed in volts per radian or per degree.Fig. 6 shows the sensitivity k as a function of the input

signals ratio. It can be noticed that the balance defect donot strongly affects the sensitivity of the phase detector.

Usually, the sensitivity P is expressed in terms of thepeak to peak voltage V, of the output signal V, It is easyto show that

VDcR2= .+R2v/l R

for R > 1. (13)

C. Zero Output SignalFor an ideal phase detector, it has been seen that the

zero output signal occurs when the two input signals phasedifference is exactly lr/2 whatever their relative levels maybe. On the other hand, for an unbalanced phase detector,the zero output level occurs when the two input signalshave a phase difference 00 X wr/2 with q5o = r/2 + ao,equation (8) shows that V, = 0 for

1-b 1+R2ao1+ X I-I +6 R (14)

Vcc = 7E(1+ b)(1l + RI - 11 - RI). (10)

Experimentally V,,/2 can be easily obtained as a func-tion of R and then the efficiency is calculated as theratio

vcc

7= 4E'forR >; 1. (11)

Fig. 7 shows a typical plot of VCC/2 obtained from acommercial double balanced mixer and the correspondingtheoretical curve.

By comparing (9) and (10) we can now express the phasedetector sensitivity as a function of the output signal peakto peak value (Fig. 6)

vcck =

2 /1 +R2'Nfor R -< 1(

In Fig. 8 is shown the variation of oa0 as a function of theinput signals ratio R, experimental results have also beenplotted in this figure. This figure also shows that the moresimilar the input levels, the closer are their phase shift tor/2.

IV. MODULATED INPUT SIGNALSA. Phase Detector Output Signal

In the same way as before, we can study the behavior ofthe unbalanced phase detector when the input signals aremodulated both in amplitude and phase. In this case therelation (1) becomes

el - E(1 + m cos Qt) cos wt

e2= E' cos (wt + AO1 cos Qt + 1). (15)

100

(12)

BRENDEL et al.: MODULATION EFFECTS IN A PHASE DETECTOR

cr

.2

1 2 3Input Signals Ratio

4 5

D ,

1-

00

'-20c0-0-

,5-40'O

l-8 -4 0

Input Signals Phase-Shift4 8

(Deg. near 90°)

Fig. 6. Phase detector sensitivity versus input signals ratio.

I

AE

-J

0

200a

'150

100

0 .5 1 1.5 2 2.!Input Signals Ratio

Fig. 7. Theoretical and experimental output level versus input signalsratio: input level E = 250-mV rms; output level VCC/2 = 190-mV rms;phase detector efficiency: q = 0.37.

.L .1a

Input Signals Ratio

Fig. 9. AM modulation level versus input signals phase-shift. Levelsare expressed in dB (with arbitrary reference level) by considering AMand PM indexes of same order of magnitude.

quency is Q, the dc component being the same one as in(6).With the approximations (16), for X = ir/2 + a, the am-

plitude of the modulated output signal takes the form

Vsm lR{ ((1 _ Ra) _R (1 + 1+R2)

-Rl(1 I + Ra ++6 1 Ra,I 1 +R2

(17)

B. Discussion

Relation (17) is constituted of two terms, amplitudemodulation (AM) and phase modulation (PM). It is ob-served that the balance defect does not strongly affect thePM level. Fig. 9 shows how the AM level varies in terms ofthe input signals phase shift 0 when these signals have thesame level (R = 1). The corresponding experimental resultscan be used to evaluate the balance defect. In fact, theminimum of AM modulation is obtained when:

a' I -6

1.5

Fig. 8. Theoretical and experimental zero output signal position versusinput signals ratio: Reference input level E = 250-mV rms

The first is amplitude modulated, the second is phasemodulated. The same result could be obtained if only one

of them was modulated both in atnplitude and phase. Thesame frequency modulation is used for amplitude andphase because only one component of the Fourier fre-quency is analyzed at a time at the output of the phasedetector but this fact does not affect the generality of thismodel.

In practice, the input signals are modulated by ampli-tude and phase noise. The modulation indexes m and AO4are very small

m <<1, AO«<< 1, and Q«<<w. (16)

Then, in the same way as in Section III-A, the phasedetector output signal V, can be obtained. Now we are onlyinterested by the slowly varying component whose fre-

(a is expressed in radians). (18)

For the phase detector under test we have measured a

phase shift of about -2°30' for this minimum which cor-

responds to a balance defect 6 = 1.045. Fig. 9 also showsthat if the input signal phase difference is exactly 7r/2 theAM level is about 30 dB below the PM level.

C. Influence of the Input Signals Ratio

Equation (17) shows that the position of the AM mini-mum depends on the input signals ratio, in fact, the AMterm of (17) is cancelled when

I1-6 1+R2a1ao = RX

1+6 R3

This curve plotted in Fig. 10(a) indicates that the largerthe input signal ratio, the closer to r/2 is the AM minimum.Experimentally, this was not-observed, indeed: when Rbecomes larger, the phase shift of the AM minimum goesto a limit (see Fig. 10(b)). If it is assumed that a small partof theAM is converted into PM, this limit can be explainedas a remainding part of phase modulation (theoreticalcurve in Fig. 10(b)). In this case, the limit corresponds to

5:u1:11n .

PM. Levet z + 6 d B,'O'

.I-J,-

-11IUnbal Ba(anced Mixeranc F.. I/

Mix IrT --. I

F

_rC ( I *IA I

I

101

I LIU4 I I l-

.5

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, JUNE 1977

8 a\ 1c 0100.

E(b)

I ~~ ~~I.

-o 2 4Input Signals Ratio

6 8 10

Fig. 10. AM minimum position versus input signals ratio: (a) with AMonly; (b) with AM and PM such as Aq/m 0.06

a ratio between PM and AM of about 6 percent, that isX/m 0.06.

V. INFLUENCE OF THE LOCAL OSCILLATOR FOR THEMEASUREMENT OF THE PHASE DETECTOR NOISE

In the introduction, it was described how one can mea-

sure the noise of a frequency stability measurement system(Fig. 2) and it was found that for an ideal phase detectorthe output signal is independent of the local oscillatorfluctuations.

In order to analyze the behavior of the unbalanced phasedetector, the same signal is applied to each of its inputs.Then, input signals in relation (1) take the form

el = E(1 + m cos Qt) sin (wt + zA4 cos Qt)

e2=E(l+mcosQt)sin(wt +0+ AcosQt). (20)

With the same approximations as in Section IV-A, theamplitude of the modulated output signal when 7X= r/2+ a and a << 1 rad can be put in the form

Vsm = V\Em ((i - 2) -6 ( + a)) (21)

This expression presents the same form as the AMcomponent in equation (17) with an input signals ratio R= 1. At the same time there is no contribution of the phasemodulation. But if the phase detector is unbalanced (b1), there remains an amplitude modulation term. Thisterm disappears only when the phase shift 0 takes thevalue O' = 7r/2 + a' with a' = 1 - 6 as in Section IV-B.

This fact explains why the noise spectrum of the mea-

surement setup (Fig. 2) depends on the local oscillator: the

output signal spectrum is a mixing between the peculiarnoise of the phase detector and the amplitude noise of thelocal oscillator. The latter noise component can be mini-mized by choosing a convenient value for the input signalphase shift (that is ir/2 + a'0).

VI. CONCLUSION

To explain with better precision the behavior of thephase detector it would be necessary to perform a moreelaborated model taking into account the characteristicsof the diodes. But the equations giving the output signalare in fact much more complicated and practically difficultto solve [5].With a very simple model it has been possible to describe

most of the effects of the unbalance in the phase detec-tor.

Especially it has been shown that the peculiar noise ofa frequency fluctuation measurement system dependsalmost always on the reference oscillator.

Hence, the model chosen is sufficient to understand theeffects of a balance defect of the phase detector and en-ables one to improve the possibilities of the frequencyfluctuations measurement systems.

ACKNOWLEDGMENT

The authors wish to thank J. J. Gagnepain for his aidand suggestions in the preparation of this manuscript andJ. Groslambert for his contribution in the experimentalaspects of this work.

REFERENCES

[1] R. E. Barber, "Short term frequency stability of precision oscillatorsand frequency generators," Bell Syst. Tech. J., vol. 50, no. 3, pp.881-915, Mar. 1971.

[21 L. S. Cutler and C. L. Searle, "Some aspects of the theory and mea-surement of frequency fluctuations in frequency standards," Proc.IEEE, vol. 54, pp. 136-154, Feb. 1966.

[3] J. V. Hauson and K. G. Sholtzhauer, "A closer look at ring diodemodulator," IEEE Solid-State Circuits, vol. SC-7, pp. 253-258, June1972.

[4] J. G. Gardiner and A. M. Yousif, "Distorsion performance of a sin-gle-balanced diode modulators," Proc. Inst. Elec. Eng., vol. 117, no.8, pp. 1609-1614, Aug. 1970.

[5] J. E. Colin and J. Salzmann, "De diverses considerations sur lesmodulateurs en anneau," Cdbles et Transmissions, vol. 22, no. 1, pp.19-37, 1968.

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