IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-23, NO. 1, MARCH 1974

A Wide Range Time Constant Measuring Technique

HELENIO ARQUEOSAD.MJR,kN LUIS A. ESTEVANOT

Abstract-A method is presented and analyzed that allows theimplementation of a technique for the direct measurement of thetime constant of single and/or multiple event exponentially decayingsignals of the form: Y = K exp (-at). The decaying exponentialsignal is fed into monolithic comparators which detect two pre-selected voltage levels of the signal. By means of TTL logic circuitrythe output of the comparators are converted into a pulse. The dura-tion of the pulse is directly proportional to the time constant of thesignal and can thus be measured by different means. The techniquedescribed is capable of a dynamic range with a lower limit of 50 nsand a measured higher limit of several milliseconds, however, thereis no theoretical upper limit with this method. The accuracy of themeasurements remains within 1 percent, utilizing standard inexpen-sive components. Both lower limit and accuracy can be improved bythe use of selected components. Inaccuracies of the measurementsdue to temperature variations from 0 to 80°C could not be detectedby standard laboratory methods.

f,

V(t)o -> J

VR10 ft t2

t2 G

VR2 - --IC2

v(t) t

VR1

VR2

INTRODUCTION

SEVERAL methods for obtaining the time constantfrom a voltage signal of the form K exp (- at) are

described in the literature [1], [2]. Typically, the signalis displaved on an oscilloscope, a photograph is taken,and the curve is plotted on semilogarithmic paper. Thetime. constant is obtained from the slope of the line.

In a variation of this method, the signal is passedthrough a logarithmic amplifier, the result is displayedon an oscilloscope, a photograph is taken, and then ameasure of the slope of the straight line is obtained. Thesemethods are tedious, require adjustments, and are de-pendent on operator judgment. Other methods cancel thesignal displayed on an oscilloscope with exponentials gen-erated with precision passive components. A major dis-advantage, other than those previously mentioned, is thenecessity of a repetitive signal or the use of multiplephotographs in order to achieve a proper null.We present an accurate, direct, simple, and inexpensive

method for the measurement of time constants with atested dynamic range from 10-7 seconds up to severalseconds, although there is no theoretical upper limit. Themethod has the intrinsic capability of measuring singleevent signals.

GENERAL

Considering a signal V (t) = K exp (-t/Tr), where r =I/a, we can particularize for two time values: t = t1 andt = t2. Thus V(t1) = V1 = Kexp (-t1/r) and V(t2) =V2 = K exp (-t22/) . The ratio of the values is

VI t2-tlV-=exp . (1)1V2T

Manuscript received May 16, 1973.The authors are with the Departmento de Ingenieria, Instituto

Venezolano de Investigaciones Cientificas, Caracas, Venezuela.

©

---I--

t,tt

,2 t

t, t2 1

Fig. 1. Simplified block diagram for measurement of time constantT of decaying exponentials. V(t) is the signal, reference voltagesVR1/VR2 = e, and t2 - tl = r-

Taking the- logarithm (e = base of natural logarithm)

V1 _t2-t1ln-v -

then

t2 - tl

ln(Vl/V2)If we make V1/V2 = e, then ln V1/V2=1 and

7 = (t2 - tl)- (2)From (1) and (2) we conclude that the time required foran exponential to decay through two arbitrary values V,and V2 such that V1/V2 = e is equal to one time constant.Based on this simple relation we designed and con-

structed an electronic cirouit in which the input is a decay-ing exponential and the output a rectangular pulse witha duration equal to one time constant. Fig. 1 shows asimplified block diagram of the electronic circuit and theassociated time diagram.As indicated in Fig. 1, a signal V (t) = K exp (-r/T) is

fed into two voltage comparators (C1,C2). When the signaldecays to the reference voltage (VRI) connected tu7 thenoninverting input of the C1 comparator, the output risesfrom an initial value of 0 volts (logical "0") to a logical"1,. Similarly when V(t) decays to the reference voltageVR2 connected to the inverting input of C2 comparator,the output of C2 falls from a logical "1" to a logical "0".

49

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, MARCH 1974

Output

10

Fig. 2. Final diagram of circuit where signal V(t) is fed into samepolarity inputs of two voltage comparators. Delay introduced byinverter G-1 is compensated by gate G-2 and flip-flop (FF) isintroduced to block output during initial transition of signal.

The AND gate G conforms the pulse fronm t1 to t2, andthen, by measuring the pulse duration (t2 -ti), we obtainthe value of the decaying exponential time constant, pro-

vided VR1/VR2 - e.

CIRCUIT DESCRIPTIONAs shown in Fig. 2, several components were added in

order to improve the performance of the basic circuitpreviously described. The signal is fed into the same

polarity inputs of two uA710 high-speed voltage com-

parators (Cl,U2). This is done in order to minimize theerror in pulsewidth due to the difference in transitiontime between logic level changes at the output of thecomparators [3]. Therefore the pulse is initiated andterminated by a low to a high logic level transition.We need now to invert the output of the comparator

(02) triggered by the lower level of the signal. This intro-duces an undesirable delay on the pulsewidth, however,this delay is easier to compensate for than the error intro-duced by utilizing the high to low logic level transition ofthe comparator. The compensation of this delay due tothe high to low logic level transition of the SN7400 NAND

gate G-1 is accomplished by introducing the delay of thelow to high logic level transition on an SN74H11 AND

gate (G-2) into the pulse starting path [4]. Note thatthe modifications introduce additional delays but thesehave no overall effect on the pulsewidth.The outputs of gates G-1 and G-2 are fed into a pulse

shaping SN74H11 AND gate G-3 which produces an out-put pulse proportional to the time constant of the expo-

nential signal. As in practical circuits it is impossible toreach the initial value K of the exponential without pass-

ing through the values VR1 and VR2. A flip-flop (FF) isintroduced with the purpose of blocking the output gateduring the initial transition of the input signal from 0volts (t = 0) to K volts (t = 0+).The bistable used is an SN7476 in which inputs J, K,

and the clock are grounded. Preset and clear are accom-

plished with the outputs of gates G-1 and G-2.Since we want to increase the noise immunity of the

comparators, we introduce hysteresis in the transfer char-acteristics by using external positive feedback [.5)]. Be-cause we use only the transition from low to high level,

f - 0 999459TM = 52.35 nsr, - 52.50%e - +029

10 20 30 40 50 ,uu

t (nonosec)

Fig. 3. Correlated curve for circuit-measured time constant To of52.5 ns and correlation factor of f = 0.99946. V(t) is input ex-ponential and slope is time constant value Tm - %E is To -

Tm/Tm X 100.

the increase. in hysteresis does not effect our operatingpoint. However, we should notice (Fig. 2) that the refer-ence voltages VR1 and VR2 are obtained from the samereference voltage supply VR and that they are dc coupledto the noninverting inputs of both comparators. Withthis positive feedback they also become coupled to theoutputs of the comparators. Therefore, when the higherlevel of discrimination is achieved (VR1), the output ofthe corresponding comparator (Cl) goes high and it isreflected to VR2 via the feedback resistor. To minimizethis effect we keep the output impedance of the referencevoltage supply as low as possible.The ratio of the reference voltages VR1/VR2 must be

made equal to e (base of natural logarithm). This isaccomplished by means of a simple voltage divider usingtwo precision resistors and a small trimmer for fine adjust-ment (Fig. 2).

RESULTSWe tested the circuit by generating decaying exponen-

tials using a passive RC network and exciting it with afast rise time pulse.The loading effect of the two comparators, as well as

the effect of the pulse generator, was taken into accountin computing the RC network time constant. The com-bined input equivalent of the two comparators was as-sumed to be another RC parallel nietwork. As for thepulse generator, a single series resistor was assumed.We wanted to be completely positive that the input

signhl was truly an exponential, so we displayed the signalat the input of the comparators on an oscilloscope screen,theni took a photograph and measured several points onit. Fig. 3 shows the best fitting curve obtained with thesemeasured values by a linear regression and plotted by acomputer/calculator HP9100B for a 52.5-ns time constant.

This procedure was repeated for different time con-stants up to several seconds. Table I shows the time con-stant values obtained and the computed correlation fac-tors.

L .I- 1- - ^ rn Inn rr)

50

50()1000

100[

13)U

ARQUE et al.: TIME CONSTANT MEASURING TECHNIQUES

rT.IOOnSec

50nS#c./Dlv.

rA.lOLssec.

TABLE I

Tm To Units f %E

52.35 52.5 nSec 0.99946 +0.29

99.70 100.0 nSec 0.99985 +0.30

0.852 0.85 p,Sec 0.99969 -0.28

10.87 10.97 pSec 0.99997 +1.0

101.0 100.0 pSec 0.99989 -1.0

322.0 320.0 pSec 0.99986 -0.63

1.71 1.71 mSec 0.99993 0.0

13.72 13.5 mSec 0.99963 -1.62

2.98 3.85 Sec 0.99873* 25.0

T = Time constant obtained from correlated curve.

To = Time constant measured at the circuit output.

f = Correlation factor.

XE = Error percentage.

Notice that the generated exponential is not perfectfor high time constants. The error is not due to the

circuit but to the decaying exponential generator.

O°C

Z3k

5mSec.lDiv.Fig. 4. Photographs showing input exponentials and output

pulses for time constants ranging from 52.5 ns to 13.5 ms. Lengthof output pulses is equal to time constants (t2 - tl = T). Verticalaxis represents voltage amplitude and horizontal axis representsreal time.

Fig. 4 shows different photographs where the generatedexponential curve and the output pulse are displayed. Thevalues measured with the pulses and the percentage devi-ation from the correlated values are also given in Table I.

Variations of the output pulse due to temperature were

also studied. No significant variations were detected, withthe means at our disposal, on the width of the output

pulse. Fig. 5 shows photographs taken at different tem-

peratures with a superposed calibration signal. No vari-ations are observed.

CONCLUSIONS

The method and technique described represent an easy

and effective way to measure time constants.The circuit can be implemented with standard, inexpen-

sive components without sacrifice of accuracy, stability,and dynamic range. Preliminary laboratory tests indicated

r0.aonSec.

20n Sec/Div.

Q

0~ S.=t ec.

2oonS ./Div.

.-961/Sec.

Fig. 5. Photographs showing input signals and output pulses fordiffere'nt time constants at 0 and 80°C. No measurable variationon pulse length can be observed. Vertical axis represents voltageamplitude and horizontal axis represents real time.

T.-52.snSec.

rO.8.5L Sec.

N3SZ0

a 0C5 fv.

Q

0

200I Sec /Dlv

T-13.5.Sec.

N

C

80'C

51

50nSecoDlv

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-23, NO. 1, MARCH 1974

the possibility of extending the lower limit to a few nano-seconds by utilizing faster comparators and logic.The method can also be used to ascertain whether or

not the input signal is a true exponential by taking areading, shifting the sampling points on the curve bychanging VR, and taking a second reading. If both read-ings agree within certain limits, we can assume that wehave measured a truly single time constant exponentialsignal.The possibility of obtaining different time constants in

signals of the type

y(t) -exp (-alt) + exp (-a2t) + + exp (ant)

is also present by automatinig the shifting of VR and re-cording the corresponding results, provided the signal isrepetitive.

REFERENCES[1] J. Le Page, A. Bernalte, and D. A. Lindholm, "Analysis of

resistivity measurements by the eddy current decay method,"Rev. Sci. Instr., vol. 39, Jul. 1968.

[2] P. Cotti, "A new size effect," Phys. Lett., vol. 4, Mar. 15, 1963.[3] M. Elphick, "Focus on IC comparators," Electron. Des., vol. 22,

Oct. 26, 1972.[41 The Integrated Circuits Catalog for Design Engineers, Texas

Instruments.[.5] J. N. Giles, Fairchild Semiconductor Linear Integrated Circuits

Applications Handbook, Fairchild Corp.

A New Device for Measurements of Pulses

or High-Frequency Currents

SEICHI OKAMURA, MEMBER, IEEE, AND TAKAHIRO OKABE, MEMBER, IEEE

Abstract-Resistors are frequently used for measuring wave-forms and magnitudes of pulse currents, assuming that the effect ofresidual inductance of the resistors is very small. However, when thevery high-frequency components of the current must be taken intoaccount, the effect of the residual inductance cannot be neglected.

In this paper, a new current measuring device with a resistor of aspecial type is described. It is so constructed that the effect ofresidual inductance does not appear in the observing circuit of thedevice. Consequently, the voltage observed becomes exactly theproduct of the resistance of the device and the current to be meas-ured, flowing through the device.As the effect of the residual inductance of the device does not

appear in the observing circuit, the resistance of the device can bemade very low, consequently, the circuit condition will be practicallyundisturbed by the connection of the device.

INTRODUCTION

IN THE measurements of both waveform and magni-tude of pulse currents or high-frequency currents by

the use of a conventional resistor, the effect of the residualinductance of the resistor must be taken into account inorder to obtain accurate results. Some methods have beeninvented to decrease the residual inductance, but it can-not be made completely zero.

This paper shows a new current measuring device,' usinga resistor of a special type, which is so constructed thatthe effect of the residual inductance completely disappears

Manuscript received August 23, 1973; revised November 10, 1973.The authors are with the Department of Electronics, Shizuoka

University, Hamamatsu, Japan.I S. Okamura and S. Abe, Japan Patent 576625.

E

Fig. 1. Conventional connection for measuring waveforms andmagnitudes of electric currents.

in the observing circuit of the device. Consequently, theresistance of the device can be made very small so thatthe circuit condition is not disturbed by the connectionof the new device.

Fig. 1 shows a conventional connection for measuringthe waveform and magnitude of electric currents. In thefigure, A is a conventional resistor, represented by a rod-type resistor, D and E are the lead-wires carrying thecircuit current i to be measured, and F and G are otherlead-wires which are necessary to measure the voltagedrop across the resistor. W1 and W2 are the terminals towhich an oscilloscope is connected in order to observe thevoltage drop.When current flows through the resistor, the voltage

drop is

V = Ri+Ldi-Ri+dd (1)dt dt

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