LC Comparison System Based on a Two-Phase Generator

344

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-34, NO. 2, JUNE 1985

means of a digital double-sinewave generator," IEEE Trans. InACKNOWLEDGMENT strum. Meas., vol. IM-29, pp. 370-372, 1980. The author wishes to thank H. Bachmair for many stimulat- [21 , "Digitaler Generator speist Impedanzme1briicke," Elektronik, no. 20,pp.95-99, 1983. ing discussions and R. Vollmert and R. Harke for their assis[3] B. Fuhrmann, "Induktiver Spannugsteiler mit hoher Eingangsimtance in the experimental work. pedanz," PTB-Mitteilungen, no. 3, pp. 196-201, 1975.

[4] G. Ramm, "Hochauflosende Wechselstrom-Mei3brucke fur WiderREFERENCES stande," Elektronik, no. 13, pp. 101-104, 1983. [1] H. Bachmair, and R. Volimert, "Comparison of admittances by

LC Comparison System Based

on a

Two-Phase

Generator

FRANCO CABIATI

AND

Abstract-A solution to the problem of measuring a standard inductor in terms of the unit of capacitance using a stable ac two-phase generator is presented. Accurate separation of inductive and conductive components is obtained by means of a self-adjustment feature of the system, whose quadrature phase, in principle not essential for LC comparison, allows exclusion of any reference resistor. Standard deviations s < 1 X 10-6 and overall uncertainties near to 1 X 10-6 (la) are obtained with a test implementation of the system.

I. INTRODUCTION THE development of two-phase generators, favored by the use of digital techniques for sinewave synthesis [1], [2], gives new suggestions for designing ac measurement systems, and particularly for admittance comparisons [3], [4]. The present paper concerns itself with an application of a stable two-phase generator to the problem of measuring a standard inductor in terms of the unit of capacitance. Bridges using a

quadrature phase, obtained by passive phase-shifting networks,

are well known [5] -[8]. More recently, as in the above cited systems, the quadrature phase is supplied by a low-impedance source belonging to the generator. In a broad sense the system described below can be included among the quadrature bridges

for unlike admittance intercomparisons, and, with regard to the generator, to the latter group. The measurement circuit is based on inductive ratios, variable voltages and fixed reference standards deflned as two-terminal, or four-terminal N-pair devices. In fact the extension of coaxial techniques to inductance measurements has become more convenient since the introduction of shielded standard inductors with toroidal winding. The choice of a parallel res-

GIAN CARLO BOSCO

onant circuit, with a current summing point on the detector side, seemed to be the more suitable to the use of admittance-

defined standards. Of course, measuring a standard inductor as an admittance (that is referring to its parallel equivalent circuit) requires a current balance at the low-voltage terminal by means of two orthogonal current components, one of which in quadrature with the voltage. These current components can be obtained by means of either two unlike reference admittances with a single-phase generator, or two like admittances with a twophase generator. In the first case the measurement system may have the basic structure of a transformer bridge for like admittances, except that an accurate separation of the main component of the admittance is more critical than for capacitors or resistors, due to large impurity component of precision standard inductors. In particular a standard resistor is necessary whose time constant must be determined by other means and also its ac conductance must be known if the parallel to series conversion of the equivalent inductance is required. To this class of bridges recently presented solutions belong [9], [10] . In the second case, two reference capacitors can be used, at least one of which should be a gas dielectric capacitor with negligible loss angle. It will be shown that the apparent redundancy of a two-phase generator can be justified by some selfadjustment and self-calibration features of the system. II. PRINCIPLE OF THE METHOD

The schematic diagram of Fig. 1 shows the structure and the main components of the system. The circuit appears in its normal configuration; a second configuration is obtained by Manuscript received August 20, 1984. The authors are with the Istituto Elettrotecnico Nazionale "Galileo changing both the switches from the normal (N) to the reverse position (R). Ferraris," 1-10135 Torino, Italy.

0018-9456/85/0600-0344$01.00 © 1985 IEEE

CABIATI AND BOSCO: LC COMPARISON SYSTEM34

345

C,61

Fig. 1. Basic diagram showing the principle of the method. U2

U2

kIG Ic'

7te--

1.0

U'

Vi

Le.k

kIL

U1

'-C2

IL

(a)

(b)

Fig. 2. Vector diagram of the main voltages and currents involved in the (a) normal and (b) reverse balance.

El

and E2

exactly equal

voltages

the

are

complex

whose

to the

are,

-El

U1

the

EMF's of the

ac

ratio is

supposed

imaginary

unit.

can

two-phase generator,

capacitors C1

stable, although

for

At the

not

switching terminals

respectively,

U2=

be varied

by

iUl1(+ a +

means

phase

a

and

C2, respectively.

inversion

on

IC2

in the

Also evident is the need

reverse

consideration for stray parameters, but

balance. With

neglecting

more

second-order

terms, balance conditions for the normal configuration result

3)

where the fractional differences tion

to be

a

of the

in

--E2+ AE and

from the ideal situa-

complex injected

increment

I-1

ksLN

cu2LC1

15

Q

Q

AE. The

use

voltage

of

a

current

comparator in addition

to

be

C1

is the

applied

to the detector

capacitance compared

diagram

in

Fig.

2 shows the main

currents involved in the normal balance

balance

kIL

with opposite

with the inductance L

measured, while C.2 balances the conductance G.

The vector

values

(b).

All currents

affecting

and

whose

SGNCSC2

the variable

are

the detector.

is

,k,

can

and in the

intended in terms of It is evident

kIG of the equivalent

phase angle

(a)

voltages and

current

as

through

equivalent

the

+

,)+i-

(1)

where

Q

=

I~-

&iLG

SG

OC2

reverse

the components

inductor,

be balanced by the currents

Q5

ksLNG

dividers extends the range of measurable inductance

and allows currents to be

phases.

to

through

is the storage factor of the inductor-

C1, 2 ;

of the relevant capacitors to ratio and

phase

errors

a1, 2

'51,2

of the inductive

of the current comparator.

are

are sums

the loss

angles

of terms related

voltage dividers and

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-34, NO. 2, JUNE 1985

346 Ul

=

The value of Q is given by SG and the ratio C1/C2, as it was pointed out. This ratio can be measured with the same system, as an additional self-calibrating procedure. In fact both capacitors can be connected to the same voltage and to opposite sides of the current comparator, while a smaller capacitor applied to the other phase can balance the loss-angle difference.

1 1 1G + Ek + - 7L + 7k Q Q Q 1

L-

-

U2 = EL eG + Ek Q'L Qr7k being SL(l + T?L +IEL), SG(1 + ?G + 6G), k(l + TIk +jEk)

III. BALANCING PROCEDURE As each of the two balances is dependent on four variable

the actual value of readings SL, SG, and of ratio k. From conditions (1) one obtains

L(N)- k2Cs 1G(

SGN W

Q

(62 + a2) +IQ

P[ Q(61 + 1) + a]. ksLNI

(N

parameters (SL, SG, a, and 1) convergence of the double balance is not easy to achieve, unless a defined procedure is followed. In order to plan it, the following expressions of unbalance deviations (as indicated by a phase-sensitive detector) are useful:

(2)

-

Similarly, for the reverse configuration the balance conditions

result in

kSLR

-

co'LCI

I

1

-

Q

(52 + a2) -

1

9 -

Q

ksCNGCC _ 1 - Q(61 + a1)

° (3)

a

G(R)

[I

kSLR

1

SGR JC2 [1 ksLR

(62 +u)

1131

(4)

-Q(61 + 1) -a]

As and : are assumed to be small, switching from the riormal to the reverse balance requires little changes in SL and SG, so that their errors (essentially depending on the most sigrnificant figure) can be considered equal in both configurati ions and then also a1 and 02. It is interesting to note that a an appear with the same coefficient in (2) and (4), but with op site signs. This can be utilized to eliminate such quanti from the measure; for example, by assuming a

L-2

(L(N)

+

L(R))

G

(G(N)

+

G(R))

A similar procedure is followed with conventional quadrat bridges [5], [8] and also with more recent systems [3], sentially intended for RC comparisons. Furthermore, in the present system one of the genera voltages, namely, U2, can be adjusted, by means of the co plex increment AE, in order to achieve both the normal the reverse balance with the same values of SL and SG. As iit is evident, by inspection of (1) and (3), this is possible onl Y if adjuistments lead to the condition = = 0. Once such a condition is achieved, and as long as it can be supposed to hold, the normal balance only is sufficient to de termine L and G on the basis of (2), which becomes a

L-

G

kSL

2C

L1Q

SGWC2 [1 kSL

-

(62U+ 2)J

Q(61

+

a1)].

IN 'Q)

INQ= R

from which

L(R)

INI

kG

iNI1 U1oC1

=

IRI

U1CoC1

WoC,

C2 C1

SL -SGu+61

k

l -,2LC SL +

k

1

co2LC, SL

SG C2 C1

C2 62 aSG

a

SGC-2 1C

C2 au ksLG 0

82 C1

WC

kG kSLG C2 SG = 'RQ IR col5WC, C+ISc+ c a U,

(6)

Again subscripts N and R denote the normal and reverse configurations, while I and Q refer to components in-phase and in quadrature with respect to U1. Deviations i are intended as fractional values of equivalent current components I, and referred to the +1 terminal of the comparator and to the current through capacitance C1 for the amplitudes. From (6) the sensitivities of the balances to the variable parameters are easily evaluated as partial derivatives. The dependence of small deviations near the balance condition on increments of the variable parameters is summarized by the following matrix equation: AiNI

AiNQ AiRI

AiRQ

1

-1

-Q

0

ASL

0-Q

ASG

SLQ SG Q -lI 62 -1 Q SL-1 SG Q 1 -l1 -62 0-

SL

_ SLQd

SG Q -1 1

SG Q

(7)

Q

0

For immediate comparison of the coefficients, Q has been introduced in place of less evident expressions. By inspections of (7) one can observe that: (a) all deviations are either sensitive to a and insensitive to ,B, or vice versa; and (b) deviations iNI and iRQ are inversely conjugated with respect to SL or SG and to a or : (in the sense that they are equally dependent, but with opposite sign, on each of those pair of variables), and the same is for iNQ and iRI Of course, optimal convergence is obtained if, once one balance is achieved, the other is also performed without affecting the first one. This can be done by the following procedure.

CABIATI AND BOSCO: LC COMPARISON SYSTEM

347 C,

Fig. 3. Simplified diagram of the measurement circuit set up as an experimental implementation of the system.

Normal balance is performed first; and then, after having switched to the reverse configuration, unbalance deviations iRI and IRQ appear. From (6) they result in -r=R SG&)C2 +kSLG

C, WC,

iR ao_SG)C2O + kSLG a IRQ~ WC, ~

-2

Q

a

2-

Q'

As a and ,B are to be adjusted to zero, increments AP = -,B and Aca = -a should be applied. These yield, as it is evident from

(7), AlPJ-Q-

2

AiR

n-Qa

R20

Thus halving deviations " and i"Q leads closer to the ideal adjustment of U2. The remaining halves of the same deviations can be reduced to zero by adjusting SL and SG. In fact, for remark (b) above, when the effects of ASL or ASG are equal and concurrent with respect to those of Aa or AP in perFig. 4. The experimental setup for testing the method. forming the reverse balance, they are equal and opposite for the normal balance. Thus when switched back to the normal analog type was used. It was phase-locked to a quartz oscillaconfiguration, the previous balance will be found almost un- tor and followed by two power amplifiers in order to lower changed-just as is required for optimal convergence. the output impedance. An additional control was applied for locking the output voltages to a fixed complex ratio. This IV. EXPERIMENTAL IMPLEMENTATION OF THE SYSTEM locking could be essential with a two-phase generator, as it In order to test the method, experimental implementations takes place of inductive couplings in bridges with a single-phase of the system outlined in Fig. 1 have been set up. All of them generator. can be represented by the simplified diagram of Fig. 3, the difFor this purpose a resistance and capacitance network with ferences concerning essentially the generators and the solutions good short-term stability was applied to the generators to form adopted for the correct definition of the parameter to be mea- a balanced bridge, whose unbalance current was used in a feedsured. Also a general view of the setup is given in Fig. 4. As back loop affecting El. As a result E1 was tracking both ama first solution for the generators, a two-phase oscillator of plitude and phase variations of E2. Reductions of instability

348

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-34, NO. 2, JUNE 1985

to almost two order of magnitudes were easily obtained, so that i. was possible to operate at the ppm level. Then a digitally synthesized two-phase sine wave source was up

applied for the generators, utilizing 12 bit, deglitched, digitalto-analog converters (DAC), followed by the same power amplifiers of the analog system. In spite of the low-pass filters, interposed between the DAC's and the output amplifiers, the overall short-term stability allowed balance with a resolution of the order of i07. No tracking control between the two phases was found necessary. The injected increments Aht and AO3 were derived from E2 and E1, respectively, and were controlled manually. Seven-decade inductive voltage dividers were used for the variable ratios SL and SG. Additional variable admittances, not shown in figure, have been provided in parallel with these dividers, in order to equalize the loads on the generators. Their use did not prove necessary in most cases, due to the low-output impedance of the power amplifiers and also to the tracking control affecting E1 and E2 (when applied). However an adjusting procedure was stated, which uses the auxiliary RC bridge (cited with regard to the tracking control) as a short-term reference for the complex ratio of U1 and U2. C1 and C2 are typically gas-dielectric reference capacitors of 1 nF. Larger capacitances can be convenient for low values of inductance or frequency, but their loss angle should be known, being possibly not negligible. For this purpose the procedure for intercomparison of different reference capacitors, cited above as a self-calibrating feature of the same system, could be applied. For the standard inductor a connection technique typical of four terminal-pair standards [8] has been provided in order to exclude the series impedances of the cables and the windings of the voltage and current inductive ratios, thus avoiding the additional measurement in short-circuit condition. As is evident from the figure, the equivalent current generator at the voltage side of YL is adjusted for a zero on the current detector sensitive to the output current of the voltage divider, while at the low-voltage terminal the equivalent voltage generator is adjusted for a zero on the relevant voltage detector. In practice the outlined operations can be performed with different means. In the present test system the variable current and voltage generators have been implemented by means of variable resistances and capacitances connected to both ends of the voltage divider. More conveniently these auxiliary balances could be achieved automatically. For this purpose an updated application of a previously developed technique, based on synchronous amplification [11 ] would be suitable to prevent dynamic instability. In fact this proved difficult to avoid, with a feedback control based on direCt amplification, due to transformer couplings necessary for reasons of isolation and shielding. A shielded transformer for bridge purposes was used as a current comparator. In this, besides terminals +1 and -1, other taps with ratios of at least 0.1 and 0.01 are useful in order to operate with a sufficient number of decades on the

voltage dividers, even with low-value inductances. The detector winding of the comparator is short-circuited during switching operation, as indicated by the shorting coaxial plug. Otherwise the winding is closed on the low input impedance of the pre-

amplifier (PA). This tends to maintain the voltage at the low terminal of the admittances near to zero also out of balance. As the bridge balance is strongly dependent on frequency, any residue of harmonic content of voltages U1 and U2 must be carefully filtered out. Therefore, multipole low-pass and bandpass active filters (F) have been interposed between the preamplifier and the phase-sensitive detector (PSD). This last was of the multiplying type that is intrinsically less sensitive to harmonic frequencies. Additional care was required with the digitally synthesized sinewave in order to prevent intermodulation of high-order harmonics in the input stage of the preamplifier. The overall harmonic rejection of the system was tested by injecting voltages at harmonic frequencies in series with the generator voltages and comparing the deviation at PSD output with that of a known unbalance. Harmonic voltages of 0.1-percent level produced output deviations corresponding to few parts in 107 for both even and odd orders. As a figure of precision the result obtained with a 1 H standard inductor, with Q t 15, fed by a current of 0.5 mA at the angular frequency X = 10 krad/s, can be reported. On a series of ten measurements, carried out on different days with interposed warm-up cycles of the system and correcting for tempperature variations within ±0.1 K in the inductor, a standard deviation for a single measure of 0.7 X 10-6 is achieved. In the same conditions the accuracy is limited essentially by that of the voltage dividers, 62/Q being negligible. Thus the uncertainty can be estimated to be about 1 X 10-6 (1 a value) assuming the inductance defined with zero readings on the auxiliary detectors. The accuracy decreases, depending on the inductive ratio performances, for lower values of inductance or frequency, because k