Precise measurement of phase difference and the amplitude ratio of two coherent sinusoidal signals

George S. K. Wong Division of Physic& National Research Council of Canada, Ottawa, Ontario K1A OR 6, Canada

{Received 20 May 1983; accepted for publication 1 December 1983}

This paper describes the theory of the interchange reference method for the precise assessment of phase difference 0 and amplitude ratio R of two coherent sinusoidal signals. Systematic errors are analyzed in depth. For 0 equal to 10', the uncertainties in R and 0 are of the order of 2 ppm and less than 0.01', respectively. Experimental verification confirms the theory of the above method.

PACS numbers: 43.85.Mt, 43.88.Yn, 43.60. Qv

LIST OF SYMBOLS

A n harmonic component E n dc contribution due to odd harmonics el, e2 ac signals eA, es signals to null detector ei inphase component

j x( -1 N nulling sensitivity .R, .R m gain = e2/e 1 Y cos 0

O, 0 m phase angle between el and e 2 • phase shift 0R ratio-transformer phase shift

attenuator readings (where x is a numerical suffix)

e' amplitude perturbation /• • attenuator uncertainty due to amplitude

perturbation V• null detector resolution R h gain perturbation due to attenuation

uncertainties

Y h perturbation of cos 0 due to attenuation uncertainties

Unless otherwise stated, and with the exception of those shown above, parameters marked with a prime, such as a•, /• •, etc., are small perturbations of their respective param- eters.

INTRODUCTION

In scientific and engineering investigations, it is often necessary to perform precision ac measurements. The mea- sured quantity usually involves the phase and amplitudes of sinusoidal signals. The phase and the frequency responses of electronic and electroacoustical devices, and ac bridge mea- surements are common examples. The need for precise phase information has recently become more urgent in the rapidly expanding field of sound intensity measurements: The matching requirement of the phase responses of the two mi- crophone technique 1--4 is very stringent, and phase errors introduce large uncertainties in sound intensity measure- ments when the spacing of the microphones is small and particularly at low frequencies. A recent study 5 recommend- ed initiation of standardization activities on phase calibra- tion of transducers. The precision of conventional phase and amplitude ratio measuring methods is of the order of 0.1 ø in phase angle and 0.1% in amplitude ratio. For the measure- ment of very small phase differences, 6 the uncertainty is less than 10%. The aim of this paper is to describe a precision ac nulling method for laboratory calibration of phase and fre- quency response of microphones and preamplifier systems.

I. THEORY OF THE INTERCHANGE REFERENCE METHOD

The general arrangement of the interchange reference method 7 is shown in Fig. 1. Signal el from a signal source is

applied to the test device which has a gain of R, and the phase of the output signal e2 is at an angle 0 with respect to Switch S selects either el or e2 as the reference signal for the ac null detector. The amplitudes of el and e2 are modified by attenuators a and/g before reaching the differential inputs A and B of the null detector, which is a lock-in amplifier. The need for two attenuators is only to simplify the theoretical discussion here. In practice, one attenuator is sufficient for the calibration, although it must be shifted either tO the A or the B input, depending on the signal amplitude ratio R or phase difference 0. With switch S at position 1, el is the selected reference signal. Attenuators a and/• are adjusted until the inphase component is at null. The vector represen- tation of the inphase null condition for the two positions of switch S is shown in Fig. 2, and the following equations can be deduced

e2fl 1 COS 0 -- elo• 1 = 0, (1)

el/•2 cos 0 - e20• 2 = 0, (2)

from which the general equations for the amplitude ratio R = e2/el and cos 0 are found

a = [(al/•2) ( •/]•1)] 1/2, (3) cos 0 = [(a: 1 a:2)/( ,i• 1 •)]1/2. (4)

For the particular conditions shown in Fig. 2, where e 2 > e l, one may assume a l =/•2 = 1.

The sign of the phase angle 0 is confirmed by noting the

967 J. Acoust. Soc. Am. 75 (3), March 1984 0001-4966/84/030967-06500.80 967

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TEST SIGNAL

DEVICE UNDER CALIBRATION

INPUT e I .• GAIN = R OUTPUT e 2 ß • PHASE=G

(2)

I

e2

S

(•)•

! e^ ^.C. DETECTOR

'(A - B)•.• (LOCK-IN -- • I• AMPLIFER) REFERENCE INPHASE QUADRATU SIGNAL '

FIG. 1. Schematic arrangement of the interchange reference method for precise measuremeats of R and 0. The input signal to the null detector can be (e.• -- es) as shown; or (eA + es) via summing arrangement (not shown) into input A.

signs of the quadrature components during the inphase null measurements represented by Eqs. (1)and {2}. In the case shown in Fig. 2, when el is the reference signal, the quadra- ture component is negative, and when e2 is the reference, the quadrature component is positive. These two conditions in- dicate that e2 lags e•.

The interchange reference method described above has several special conditions which are discussed here.

A. When the phase angle is rr/2

As 0 approaches rr/2, the nulling sensitivity decreases (see error analysis). When 0 - rr/2, the arrangement shown in Fig. 1 has to be modified as follows: (a) with el as the

(a)

•el

Quadrature e2 'j Component

(eA-e 8)

(b) Quadrature

Component _ e• (eA- es)• .-- ---/

e?•._ _ __•.__ [/'•O -- + e2•z 2

,-j

FIG. 2. Vector diagram showing the inphase null conditions with e I or e 2 as the reference signal. 0 is the phase angle of the signals, a and/g are attenua- tor readings.

(a) +i (b) / +i e 2 (eA-e s) /•'•. ,•

phase shift • • reference I• .reference e2a2 • /phase

• r • . •e 2 - ••a• phase - O= • hift

' FIG. 3. When the phase angle 0 = •/2, the phase of the reference sisal (shown in Fig. 1) is shift• by an angle • with respect to e• in (a) and ee in (b).

reference, see Fig. 3(a), signals from the attenuators are add- ed (e• + e•) with a summing amplifier or summing resistor network, either of which can be provided extelally before the sisal is applied to input A of the lock-in amplifier, which no longer operates in the differential mode, and (b) the phase shift provisio• i• the lock-i• amplifier is utilized to advance the phase of the reference signal by an angle • with respect to e• and e• as shown in Fig. 3.

With e• as the reference (switch at position 1, Fig. 1), the attenuators are adjusted for the inphase null condition. It can be seen that

= tan (5)

Similarly, with e• as the reference [the input signals are rc- tu•ed to the (ex - e• ) condition by removing the summing arrangement] one has

&) = ta, (6)

Dividing •. (6) by Eq. (5), and rearranging to give the ampli- tude ratio, one obtains

R = [(a, (7) Since R = e•/e•, substituting •. (7) into Eq. (6) and

rearranging, one also obtains

tan = &)]'/'. (8) If the phase shift provision were to be such that the reference phase lags e•, see Fig. 3(a), the value of the fight-hand side of •. (8) is negative, and •. (7) is unchanged. For the condi- tions shown in Fig. 3, ee>e•; one in practice can set • = • = 1. •uation (8)then becomes

tan = (a, a,)'/'. (9) It can be seen that either •. (8) or (9) provides a means

to verify the phase shift • introduced by the lock-i• amplifi- er. Thus, the phase shift controls of the lock-in ampUfier can be calibrated with two quadrature signals. A similar method to •ce•in • is given by •. (A7)in the Appendix.

B. When the phase angle approaches rr/2

As 0--•rr/2, the nulling sensitivity of the arrangement (shown in Fig. 1) reduces, and this is discussed in more detail in the error analysis section. Again with the phase shift pro- vision of the lock-in amplifier, an angle • can be introduced between the phase of the reference signal with respect to that of the input signals el and e2, such that the phase of the reference leads el and e2 (see Fig. A1 in the Appendix). Based on inphase and quadrature null measurements, the equa- tions for R and 0 are as follows:

968 J. Acoust. Sec. Am., Vol. 75, No. 3, March 1984 George S. K. Wong: Measurement of sinusoidal signals 968

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where •11• •12• •21• •22• P11• P12• P21• an• P22 are ratio-tr•n$- former readings obtained from inphase and quadrature null me•urements. If the phase shift provision were to be such that the reference lags e, in Fig. A 1 {a), similar equations for R and cos 0 are as shown in Eqs. {A8) and {A9), respectively, in 'the Appendix.

As long as the phase shift •b remains constant in both conditions: {a)with e, as the reference, Fig. A1, and {b) with e2 as the reference, Fig. A2, the magnitude of •b need not be known. It can be shown that the optimum phase shift •b is equal to 7r/4 if the nulling sensitivities were to be the same for both of the above conditions.

C. When the phase angle exceeds •r/2

When 0 exceeds ½r/2, the phase shift 0• is unnecessary; e, and e2 are added with the summing arrangement discussed above. From the inphase null conditions shown in Fig. 4, the following equations can be derived

e,a, - e2]?, cos0r - 0)-0, (12)

e2ct2 -- el B2 cos(•r -- 0) = 0. (13)

From Eqs. (12) and (13), it can be shown that the amplitude ratio R and cos 0 are given by

R [(a, p2)/(a2 pl)] 1/2, (14) cos 0 = -- [(a,a2)/(/3,/32)] '/2. (15)

Equations (14) and (15) are also valid for 0 = A summary of the signal nulling conditions and input

requirements for the null detector (Fig. 1) for various values of 0 is shown in Table I.

TABLE I. A summary of the nulling signal arrangement (shown in Fig. 1 ), and the corresponding phase shift requirements for various values of 0.

Signal nulling conditions and Phase input requirements for the Phase shift Vector angle 0 null detector requirements diagram

0,•r/2 Inphase null; (eA -- es) None

Inphase and quadrature null; 0--,•r/2 (eA + ez) and (e• -- ez) •b = d- •r/4

Inphase null; 0 = •r/2 (e• + ez ) and (e• -- ez) •b = d- •r/4

O> •r/2 Inphase null; 0 = rr (e,• + es) None

Fig. 2

Figs. A1 and A2

Fig. 3

Fig. 4

y,= 8Y al + 8Y 8Y 8Y where a[ , a;,/3 ;, and/3 ;• are small perturbations, and

(17)

Y = cos O. (18)

Note that the variation is obtained by summing all perturba- tions of the parameters regardless of sign. From Eqs. (3) and (4), it can be shown that the maximum possible errors due to the uncertainty of the attenuator readings are

RA max = OIl a2 -•-• + , (19)

120) It must be noted that ( ¾• )max is the maximum variation in cos O, and Eq. {20} enables the computation of the perturba- tion in 0. Some other sources of errors are discussed in the

following.

II. ERROR ANALYSIS

The accuracy of the above interchange reference meth- od depends on the phase difference and amplitude ratio of the signals, system nulling sensitivity, and uncertainty ofthe attenuator readings. For small perturbations of the above parameters--in this section denoted by primes--the vari- ation of R and cos 0, see Eqs. (3)and (4), are given by

R OR bR bR bR '=•a; + al+ B'+ /3' (16)

(o)

,•lal

(e^.

{b) e• •+_•• p• (e^ + %) el I•' x. • O\\

\ ß 2 - +

-i

FIG. 4. When the phase angle 0 > •r/2, phase shift is not required for the reference signal. (a) With el as the reference, (b) with e2 as the reference.

A. Nulling sensitivity

From the vector diagram shown in Fig. 2, the inphase component ei {not shown) is gradually reduced to zero by the attenuation of the signal amplitudes. At the inphase null condition, ei = 0. The nulling sensitivity N is defined as the change in ei with respect to the change in attenuator setting

.'.N = ei/( 1 - attenuator reading), (21 ) where N is a function of R and 0. In the condition shown in

Fig. 2(a), e2 > el, and in practice, only one attenuator is re- quired (al = 1). The inphase component before the attenu- ation e2 is

e i = (1 --/3,) e: cos 0. (22)

Since the change in attenuator setting is (1 -/3,), substitut- ing Eq. (22) into Eq. (21) one obtains

N = e2 cos 0. (23)

From Eq. (23), the nuiling sensitivity is reduced to zero when 0= rr/2.

Assuming that the null-detector resolution is V•, the corresponding uncertainty in attenuation is

13 • = VS/N. (24)

969 J. Acoust. Soc. Am., Vol, 75, No. 3, March 1984 George S. K. Wong: Measurement of sinusoidal signals 969

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B. Amplitude stability

The amplitude perturbation e', contributed by el and e2, increases the uncertainty in the measurement of R and 0. The effects of e' on R and 0 may be expressed in uncertainty of attenuation

13; = e'/N. (25)

When the amplitude perturbation is due to the signal source, both el and e2 are affected, depending on the com- mon mode rejection ability and the selected time constant of the null detector, an error may be introduced.

C. Phase orthogonality

With inphase and quadrature measurements, deviation from true orthogonality increases the uncertainty of the measurement. The deviation is of the order of 0.1 ø in most

lock-in amplifiers, and it can be ascertained by connecting a sinusoidal signal simultaneously to signal input •1 and the reference input {the lock-in amplifier is in the single input mode); the phase shift control is adjusted for maximum and minumum output for inphase and quadrature measure- ments, respectively. The departures of the phase shift from 0 ø and 90 ø phase are the corresponding orthogonal errors, which can be subtracted from experimental phase readings.

D. Signal harmonics

The general principle of operation of the lock-in ampli- fier is based on the phase sensitive detector {PSD) which provides a de output proportional to the amplitude of the input signal, and varies with the cosine of the phase 0 between the reference and the input signal. If the reference signal were to contain harmonics, the frequency response of the PSD can be contaminated with the harmonic compo- nents of the input signal which are coincided with the odd harmonics of the fundamental reference signal. The de con- tribution of the odd harmonics is

E n = (2• n/1'1,•') cos(nO ), (26)

where n is the odd harmonic number, and '•n is the ampli- tude of the corresponding harmonic component of the input signal.

For modem lock-in amplifier systems which are based on heterodyne techniques, the third harmonic is attenuated by more than 60 dB. Nonetheless, for signals with 1% har- monic distortion, assuming that the reference signal con- tains a third harmonic component, the maximum uncertain- ty, which may be expressed in terms of attenuation E ;, ?N, is of the order of 2 ppm.

E. Attenuator phase shift

The phase shift 0R of the ratio-transformer depends on the attenuator setting and the frequency. For a particular seven decade ratio-transformer, the maximum phase shift is approximately 0.003 ø per kHz for settings above 0.1. For acoustical applications, the maximum phase shift is of the order of 0.06 ø .

With the above interchange reference method, the ef- fects of 0R on the amplitude ratio R in Eq. {3) is canceled by the interchange procedure, and on the cosine phase function

cos 0, Eq. (4), the effect is to multiply the fight-hand side of Eq. (4} with cos 0R. For 0R = 0.06 ø, the uncertainty in 0 is less than 0.002 ø when 0 is above 1 ø. However, when 0 is 0.1 ø, the uncertainty is 0.017 ø.

F. Maximum possible error

Based on the above analysis, and from Eqs. (16)-(20) the maximum possible errors for R and cos 0 are

max = T I• 1 a' 2

+Z-, + 2 ' e' ) N/?(F'd+ +E;) ,

{z 1 I• 2

/31 /3;. 2 e' ) In 'practice, the unce•nty of the attenuator readings

(a' and fl '} and the null-detector resolution V• expressed in terns of signal amplitude are less than 1 ppm in each case. The contribution from amplitude pe•urbation e' depends on the system configuration, including the common mode re- jection ability and selected time constant of the null detector. For the pu•ose of this analysis, e'/e is assumed to be 10 ppm, and the effects of si•al hamonics E; is taken to be 2 ppm. From •. {27}, the maximum possible e•ors for the assessment of R is approximately 130, 20, and 14 ppm, corre- sponding to 0 equals to 85', 45', and 10', res•ctively. If the common mode rejection of the a•angement were to be veu effective, such that the effects of amplitude pe•urbation were to be negli•ble, then the above maximum possible er- rors for R are reduced to approximately 23, 2.8, and 2 ppm, r•p•tively.

During inphase assessment, •d with the phase shift provision {4 = • •/4} discussed above, the phase angle between the signal vectors (e• or e2}, and the reference axis need not exceed •/4. Based on conditions similar to the as-

sessment of R above, for phase •gles between 45* and 5', the maximum possible error for 0 is less than 0.01'. However, when 0 decreases, the finite errors, which multiply with cos 0, become significant and the unce•ainty in 0 increases. For example, the unce•ainties in 0 are 0.085', 0.1', and 0.13', corresponding to 0 equals to 0.5', 0.4', and 0.3', respectively.

In the above analysis the biggest source of error is due to mplitude pe•urbation {10 ppm} of the ac signals. In some applications, such as frequency response of passive circuits, the common mode rejection ability, plus the averaging mode of the lock-in raplifter can reduce amplitude pe•urbation to a second order of magnitude. Thus, for 0 equals to 10', the unce•ainties in R and 0 are of the order of 2 ppm and less th• 0.01', respectively.

III. EXPERIMENTAL VERIFICATION

The theory of the above interchange reference method was confirmed experimentally by comparing readings of R and 0 obtained,by the above method with those obtained

970 J. Acoust. Soc. Am., Vol. 75, No. 3, March 1984 George S. K. Wong: Measurement of sinusoidal signals 970

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with a phase meter {model 351, Wiltron Co.), and a measur- ing amplifier {model 2606, Bruel and Kjaer, Denmark}. The device under test was a R-C circuit, coupled with an oper- ational amplifier to ensure low output impedance from the circuit. The experimental arrangement is similar to that shown in Fig. 1. The setting uncertainty of the attenuator (model DT 72A, Electro Scientific Industries Inc.} was less than 1 ppm. The null detector was a lock-in amplifier (model 186A, Princeton Applied Research Corp.). Depending on the amplitude ratio of the input {1.2 V rms} and the output signals, the attenuator served in both positions as a and • as shown in Fig. 1. To ensure that the amplitude of the refer- ence signal is at the optimum value of 1 V rms (as recom- mended by the manufacturer of the lock-in amplifier), a 10 k •2 multiturn potentiometer and a digital rms voltmeter were used to adjust and to monitor the reference. During the ex- periment the reading uncertainties of the phase meter and the measuring amplifier were estimated to be of the order of 4- 0.1 ø and _ 0.1 dB, respectively.

It can be seen from the experimental results {Table II), that over a frequency range from 100-1000 Hz, the maxi- mum differences in readings obtained with both methods were 0.13 ø and 0.184 dB for 0 and R, respectively. In view of the relatively high inherent accuracy of the interchange ref- erence method, the above experiment can only confirm the theory of operation of the method, and show that it is at least as accurate as this phase meter method. The numerical val- ues of R and 0 shown in Table II are rounded to three and two decimal places, respectively.

In practice, most ratio-transformers have provision for an output ratio which approaches 1.1 times the input signal; and for applications when the signal amplitude ratio R lies within the range 0.9<R < 1.1, only one attenuator at a fixed position is needed.

The attenuator used for the experimental verification described above is a seven decade ratio-transformer, and the assessment of amplitude ratio and phase difference of the two sinusoidal signals can be very precise.

TABLE II. The theory of the interchange reference method for measuring R and 0 is confirmed by the relatively small differences between readings obtained with the above method and those obtained with a phase meter and a measuring amplifier.

With phase meter With interchange and measuring reference method amplifier Differences

Frequency R 0 R m 0 m R -- Rm 0- 0 m Hz (dB) (degree) (dB) (degree) (dB) (degree)

100 2.583 -- 3.62 2.6 125 2.585 --4.52 2.5 160 2.585 -- 5.76 2.6 200 2.580 -- 7.19 2.5

250 2.565 -- 8.97 2.5

315 2.533 -- 11.25 2.4

400 2.470 -- 14.21 2.3 500 2.365 -- 17.63 2.2 630 2.184 -- 21.95 2.0 800 1.874 -- 27.37 1.8

1000 1.413 -- 33.37 1.4

-- 3.7 --0.017 0.08 -- 4.6 0.085 0.08 --5.6 --0.015 --0.16 --7.0 0.08 --0.19

-- 8.9 0.065 -- 0.07

-- 11.3 0.133 -- 0.05 -- 14.3 0.17 0.09 -- 17.6 0.165 --0.03 -- 22.0 0.184 0.05 -- 27.3 0.074 -- 0.07 -- 33.5 0.013 0.13

IV. CONCLUSION

The interchange reference method is a very precise method for the assessment of phase and frequency response. The above technique is useful for comparing the perfor- mances of electronic and acoustical devices when the phase difference is less than •r/4. For some applications, such as the reciprocity calibration of condenser microphones, the uncertainty of the amplitude ratio is of the order of 1 ppm, and the phase uncertainty is less than 0.01 ø.

ACKNOWLEDGMENT

The author would like to thank Dr. T. F. W. Embleton for his constructive comments.

APPENDIX

When the phase angle 0 approaches •r/2, the nulling sensitivity N reduces; and when 0 = •r/2, N = 0. To avoid the increase of measurement uncertainty due to low nulling sensitivities, the phase shift provision of the lock-in amplifier is used to introduce an angle • between the phase of the reference signal with respect to that of the input signals el and e2, such that the phase of the reference leads el and (1) With el as the reference signal (Fig. A 1 ):

(a) Inphase null: eA + es (modify Fig. 1 to the summing arrangement)

elall cos 0 -- e2 ]•11 cos[T/' -- (0 -I- • )] = 0.

(b) Quadrature null: e.• - es (as shown in Fig. 1)

jellsg 12 sin •b -je2 ]•12 sin[rr- {0 + •b )] = 0.

{2) With e2 as the reference signal (Fig. A2): (a) Inphase null: e• - es

(A1)

(A2)

el •1 COS(0 -- • ) -- e26921 cos • = 0.

{b) Quadrature null: e• + es

(A3)

jel ]•22 sin (0 -- 4 ) --je26922 sin 4 = 0. (A4)

By simple and appropriate manipulation of Eqs. (A1)-(A4), one obtains the following equations for amplitude ratio R, cos 0 and tan •b'

R = [(•11 _j_ •12'•/(Eg21 _j._ •22'•] 1/2 t\]g11 /91211 \]g21 /92211 '

(a)

FIG. A1. With e I as the reference signal. (a) Inphase null: eA + es, (b) qua- drature null: eA -- es.

971 J. Acoust. Soc. Am., Vol. 75, No. 3, March 1984 George S. K. Wong: Measurement of sinusoidal signals 971

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(a)

e2ct 21 • •

*i

•e A_ e B (b) N el

rence

•2

-J -i

FIG. A2. With e 2 as the reference signal. (a) Inphase null: eA -- ee, (b) qua- drature null: eA + es.

((R(•1•21/•21 -•- (•11/•11 R )•1,2 (A7) tan •b: \,•-•,2-•-•2.• • •••72i/ ' It must be pointed out that as long as • remains con-

stant, its magnitude need not be known. If the phase shift provision were to be such that the reference phase lags e• [see Fig. Al(a)], the equations for R and cos • are

= z [(ll + _ + .

It can be seen from Figs. A 1 and A2 that the value of 0 can be varied to ensure a high nulling sensitivity during the assessment of•b. In this respect, Eq. (A7) is more useful than Eq. (8) for the calibration of phase shift controls of lock-in amplifiers.

•J. Y. Chung, "Fundamental aspects of the cross-spectral method of mea- suring acoustic intensity," in International Congress on Recent Develop- ments in •4coustic Intensity Measurement (Senlis, France, 1981), pp. 1-10.

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3F. J. Fahy and $. J. Elliott, "Acoustic intensity measurements of transient noise sources," Noise Control Eng. 17, 120-123 ( 1981).

4G. Krishnappa, "Cross-spectral method of measuring intensity by correct- ing phase and gain mismatch errors by microphone calibration," J. Acoust. Soc. Am. 69, 307-310 (1981).

sV. Nedzelnitsky, "Calibration of phase response of transducers for mea- suring sound and vibration: Needs, feasibility, and possible standards ac- tivity for microphones," Report of a One-man Study Group to the Chair- man of ANSI $1 (November, 1982).

6F. Schauer, "Measurement of very small phase differences," Rev. $ci. In- strum. 52, 1776-1777 (1981).

?G. $. K. Wong, "Precise measurement of phase difference and amplitude ratio with an interchange reference method," J. Acoust. Soc. Am. $uppl. 1 73, S23 (1983).

972 J. Acoust. Soc. Am., Vol. 75, No. 3, March 1984 George S. K. Wong: Measurement of sinusoidal signals 972

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