IEEE TRANSACTIONS ON INSTRUMENTATIION AN1) MEASUREMENT, VOL. IM-18, NO. 2. J tNE 1961The first method is limited by the availability of ac-

curate RF resistance values below 1 ohm. The secondmethod gave values of r that were higher than calculatedby 0.2 ohm, at 25 and 58 MHz. Some of this extra lossmay have been due to the capacitor.

DIRECT INDICATING Q METERThis method could be made the basis of a direct-read-

ing Q meter useful for rapid measurements on numbersof similar tuned circuits. The manual counting of thenumber of cycles between two peak amplitudes couldreadily be replaced by a high-speed integrated circuitcounter. Suitable choice of the ratio V1/V2 would give,

for example, a multiplying factor of 10. Similarly, twofeedback loops could be arranged 1) to vary I to set V1equal to a selected value at t1 and 2) to vary time t, tomake V2 again equal to the selected value.

REFERENCES[1] 0. W. Dopheide, "Method for the measurement of 'Q',"

IEEE Trans. Instrumentation and Measurement, vol. IM-15,pp. 109-112, September 1966.

[2] G. H. Kramer, "A new type of Q meter using variable widthpulse excitation," IEEE Trans. Instrumentation and Measure-ment, vol. IM-16, pp. 315-319, December 1967.[3] W. H. Hartwig, "Cryogenic resonant circuits," Electronics,vol. 36, pp. 43-47, February 1963.

[4] F. E. Terman, Radio Engineers' Handbook, lst ed. NewYork: McGraw-Hill, pp. 34-36, 1943.

Digital Instrumentation for AngularVelocity and Acceleration

ALAN DUNWORTH

Abstract-An instrument is described for the measurement ofangular velocity and angular acceleration of a rotating shaft, basedupon an optical angular transducer and an associated operationaldigital (pulse rate) system.The angular transducer produces a pulse rate directly proportional

to the instantaneous angular velocity and this quantity and its rate ofchange are processed by the pulse rate system and presented as par-allel binary-coded decimal representations driving in-line digital dis-plays.The operational digital system comprises essentially an electronic

register controlling a variable rate pulse generator that tracks theinput pulse rate in a frequency lock loop. Changes of loop frequencyare brought about by changing the register contents by positive ornegative correction pulses, and the rate at which these are suppliedprovides an accurate measure of acceleration or deceleration, re-spectively, provided that the loop is locked on and hence follows thechanges of input pulse rate accompanying changes of angular veloc-ity. The error correction pulse rate may be measured in a secondfrequency lock loop and the registers in the two loops used to controldigital in-line decimal displays of angular velocity and acceleration,of ranges up to 10 000 r/min (+1 r/min) and 1000 r/min/s (i 1 per-cent), respectively.

INTRODUCTIONA RECENT article [ 1 ] has summarized methods

for measuring angular acceleration based onanalog methods or hybrid digital-analog tech-

niques and describes in detail a purely digital methodfor obtaining acceleration/time, velocity/time, and ac-celeration/velocity information for rotating shafts. The

Manuscript received December 2, 1968.The author is with the University of New South Wales,

Sydney, Australia. He is now on leave at the University of BritishColumbia, Vancouver, Canada.

method is based on an angular transducer that gives muniformly spaced pulses per rotation of its rotor, andinvolves the counting, in two successive intervals ofduration T seconds, of the numbers q1 and q2 of trans-ducer pulses. From these values the estimated angularvelocity V and angular acceleration at the halfwaypoints are given by

(q1 + q2)2mT r/s (1)

and

-

_ (q1 -q2)ml" r/s2. (2)

This method gives good sensitivity and accuracy es-pecially when a small number of samples per second,and correspondingly large values of q1, q2, and T, can betolerated.

This paper describes an alternative purely digital sys-tem in which the pulse rate produced by the angulartransducer is measured in an operational digital systemthat presents a parallel binary-coded decimal represen-tation of the angular velocity V. The pulse rate is alsodifferentiated to produce a measure of the rate of changeof angular velocity, i.e., angular acceleration pl.The operational digital system comprises essentially

an electronic register controlling a variable rate pulsegenerator that tracks the input pulse rate in a frequencylock loop. Changes of loop frequency are brought aboutby changing the register contents by positive or negative

132

Dl-NWORTI I)DIGITAL I NS-,TRUMENTFA\TION FOR ANGULAR VElLOCITY N)A \DACCEL,TrIoN

w

-- rotating shaft

( fo ) ( 1 )1000 8Fig. 1. Angular velocity measurement, basic system.

correction pulses and the rate at which these are sup-plied provides an accurate measure of acceleration ordeceleration, respectively, provided that the loop islocked on and hence follows the changes of input pulserate accompanying changes of angular velocity. The out-puts of the binary stages comprising the register con-stitute a parallel binary-coded representation of theangular velocity and are used for driving a digital in-line numerical display. The error correction pulse rate,corresponding to angular acceleration, may be measuredin a second frequency lock loop or by conventionalpulse rate to analog voltage convertors (diode pumpcircuits).A system has been built up using a low-resolution

angular transducer (m = 360) that permits the measure-ment of velocities up to 10 000 r/min in steps of 1 r/minwith an effective loop time constant of 1/6 second, andangular accelerations up to 1000 r/min/s with a resolu-tion of 1 percent and an effective loop time constant of1/10 second. Using a high-resolution angular transducer(mt = 3600) it should be possible to measure angularvelocity to 10 000 r/min wiith a resolution of 1 r/min anda much reduced loop time constant of 1/60 second, whileretaining the angular acceleration specification of 1000r/min/s with a resolution of 1 p)ercent and a loop timeconstant of 1/10 second.

THE BASIC SYSTEMThe basic system insofar as velocity measurement is

concerned is illustrated in Fig. 1. The angular transducerconsists of a metal disk mounted on the rotor shaft anddrilled with a series of holes at 3-degree spacings. A set

of three lamps and photodiodes are positioned on eitherside of the disk so that a pulse output is produced fromone of the three photodiode outputs for each angulardegree of rotation. The associated transducer logic pro-duces a composite pulse train of frequency frotor consist-ing of one pulse for each degree of rotation. The se-quence in which the photodiode outputs are turned onenables a SIGN-level control voltage to be generated thatindicates the direction of rotation of the rotor.

frotor and its associated SIGN are fed into a frequencylock loop and compared with the loop output frequencyfmonitor and its corresponding SIGN control level. fmonitor isthe output of a variable rate pulse generator that con-sists of a reversible 4-decade counter and sign binary,the states of which control the gating of a set of pulsetrains derived by frequency division from a single-crystal oscillator. A composite pulse rate is generatedthat is proportional to the product of the counter con-tents and a reference frequency divided by the maxi-mum coiunter capacity, and this pulse rate and the SIGNbinary ouitput are compared with the input fmonitor anditS SI(-GN in the unit labeled "Kirchoff digital adder" inFig. 1. Thiis latter unit is the digital equivalent, withregard to pulse rates, of the well-known Kirchloff resis-tive adder for analog voltages and currents. It producesatn error pulse rate fe and an associated SIGN level thatare used to correct the counter content. As will be men-tioned in more detail below the method used for gen-erating fmonitor suffers the limitation that there are ir-regular fluctuations in actual pulse positions about theideal pulse positions that would occur for a truly regularpulse train at the same average frequency. Thus even

1l,4EE TR.ANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, JUNE 1969

fo 3+ n2 +n + no1 =(Ofo)J N l1 6 10 100 1000 16 l10,00O0Fig. 2. Variable rate pulse generator.

Fig. 3. Production of noncoincident pulse trains.

13s4

DUNWORTH: DIGIT.UL I\STRt NIE.:N rATlION FORE ANGULAiR VELOCITY AND X,WCELERATION1

if the mean frequenicies of the inputs to the Kirchoffdigital adder are exactly equal there will be an outputpulse train f, together with a SIGN level that will changefrom positive to negative in such a way as to give anintegrated error output (long term) of zero. There ir-regular fluctuations are smoothed out before being ap-plied to the register, this function being carried out bythe backlash unit in the following manner. Following achange of error SIGN level, and hence a change in countdirection, the backlash unit absorbs the first eight pulsesbefore passing any further pulses to the counter. Thus forequal fmonitor and frotor pulse rates (and equal signs) thepeak to peak amplitude of the pulse count irregularitiesmay amount to eight pulses before correction to thecounter is applied. This is adequate to compensate for theirregularities inherenit in the method of generation offriionitor.

The variable rate pulse generator is shown in greaterlogical detail in Fig. 2. From a basic crystal-controlledfrequency fo are derived sixteen pulse rates of values:

(4)' 8 ) 16

01o

NJ =- f" lo oo (5)

where N is equal to the total integrated counter contentand

f k 16/The pulse rate f is generally subject to nonuniform

spacing as illustrated in Fig. 4 for the case of a singledecade, i.e., n2 = n, = no = 0. The actual number ofpulses in a given time is irregularly distributed aboutthe ideal inumber produced by a pulse train at the sameaverage rate. The resulting pulse count errors are notcumulative but repeat over one complete cycle of theinput pulses.The irregularities may be expressed as a positive or

negative (leviation between the number of pulses ac-tually generated in a given time and the number thatwould theoretically occur for a uniform pulse rate at thesame average frequency. A measure of the irregularitycan then be expressed by calculating the maximum peakto peak positive and negative deviation. For the case ofa pulse rate multiplier controlled by a pure binary coun-ter of p stages the maximum deviation has been shownto be [21If()()I I( ()I)

100 2 ' 100 4 '100 \8' 100O16Ofo I{ fo I(8 f 81 fo 81\1000 \2' 1000 4 i'\O(\8} 100\lO 16^

These pulse trains are generated in such a way there areno coincident pulses, and hence any number of the pulsetrains may be adlded to produce a composite pulse trainwithout the loss of any pulse members. The sixteen pulsetrains are generated as shown in Fig. 3 where the systemdepicted is based upon the principle that for any numberof cascaded couniters connected to count in the forwarddlirection there will not be more than one counter in-creasing its content at any given time. Thus the fourmodulo-10 counters can produce noncoincident pulseoutputs at frequencies (fo), (fo/10), (fo/100), and(fo/lOOO) by detecting, for example, the transitions fromcounter states 4 to 5. Similarly the four modulo-2 coun-ters ( . 2 stages) into which to is fed can produce non-coincident frequencies of (fo/2), (fo/4), (fo/8), and(A/16) by detecting the 0 to 1 transitions.Referring again to Fig. 2, the selection of constituent

puilse trains in the composite pulse train is controlled bythe states n3, n., un, and n0 of the four decade counters,Cand the mean frequeney of the output f is given by

= fo n-3 + on2 + fo )( ( 3)16 10 16 100 16 R1000 16/or

__0f_1OOOn+ lOOn9 + iOn, + nJl(16 10 000 (4)

maximum deviation =- + + ( 239 9.21 (6)Thus for an equivalent pure binary system with p13 or 14 the maximum deviation is between five and sixpulses. For the case of a binary-coded decimal pulse ratemultiplier as shown in Fig. 2 it is difficult to express thedeviation in closed form but a recent investigation [31has showni that the error is somewhat larger than for thepure binary case but not significantly so. Accordingly,the capacity of the backlash counter in Fig. 1 waschosen to be eight pulses.

VELOCITY LooP EQUATIONSThe basic equations of the velocity frequency lock

loop will be developed from the simplified block diagramshown in Fig. 5. The following relations may be writtendown

frotor = mVrotor (7)where Vrotor angular velocity of the shaft in revolu-tions per second, m = number of pulses per revolutionof the slhaft, and frotor = output pulse rate from theangular transducer in pulses per second.We also have

-f-=N (8)where fg = loop output pulse rate, ft = reference clockfrequencyv, JNT maximum capacity of the loop register,and N = actual counter content.

1.3.5

136

_ one cycle

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, JUNE 1969

of input pulses W

I I I I1I111 1 1 1

Illll

I I I II III I

lI

Fig 4. Pl s tri oupt fo a inledc e rat m ltiplier.

Fig. 5. Velocity loop basic block diagram.

differential equation to be valid:

fe = frotor fg (9)where fe = output from the Kirchoff digital adder.

Again,

AN

T f (10)where AN = change of counter content occuring in atime AT where AT is an integral multiple of the meanperiod of fe.The quantity N does not vary continuously but in

steps of 1 count, i.e., in steps of I/N, of the maximumvalue Nc. We will assume that Nc is sufficiently large,and the steps AN sufficiently small, for the following

dN=o (11)dt

This assumption yields an acceptable approximationfor the transient response of the loop but does notpredict the oscillations about the true value of N thatoccur if the steady-state value of the latter, correspond-ing to the input frotor, is nonintegral [4].Thus from (8), (9), and (11) we may readily deduce

dN NdN + Jc N = frotor. (12)

This equation may be solved by elementary methodsfor the case of particular inputs frotor.

fo

fo/8fo/16

n = 1

n = 2

n = 3

f = fo(n )n =416

n = 5n = 6

n = 7

n = 8

n = 9K

Further,

DUTNWORTH: DIGITAL INSTIll\ILNT\TIN UOR \NGI L.LR VELOCITY AND \(CELRFAI\TI(ONStep Change of Velocity

Suppose the shaft velocity changes at time t = 0 fromV1 to V2, and the angular transducer pulse rate changesfrom frotor 1 to f 1oto,r.- where

frotorl - 7V1m

frotor2 = V2m.

(13)(14)

Nve1 (fc- A)2 (f- l c)I (25)

From (23), the ideal loop correction pulse rate faeewould be given by

dN a mO3faae e Je - dt (f,/N,) ((f,/N,) (26)The solution to (12) for time t > 0 is given byN(t) = N2'I1 - exp [-(fc/Nc)t]} + N1 exp [-(fC/Nc)t]

(15)where]

and

N2 _- frotor 2(f./Nc)

V _frotorl

(16)

(17)Thus the response is an exponential type of time con-stant:

Te= (Ne/fe). (18)It should also be noted that f, is the frequency corre-sponding to Vmax and hence

The actual loop correction pulse rate f',c obtained bydifferentiation of (24) yields

fac = (fa/N-) {1 - exp [-(f,/N,)t] (27)or

fLce = fac{ 1 - exp [-(fI/Nc)t]}. (28)Thus the pulse rate f',, as a measure of the angularacceleration f,, tends towards the value facc (mO)/(fe/N,)along an exponential curve of time constant T. = (Nl/fI).

ACCELERATION FREQUENCY LOCK LOOPSuppose fc is fed into a second frequency lock loop

for which the parameters are Nd and fd (maximum countercontent and clock frequency, respectively) such that theloop output frequency f is given by

f= fdN (29)fc = mlVmax I (19)

Step Change of AccelerationWe here suppose that the shaft velocity is V1 initially

and at time t = 0 there is a step change of accelerationfrom 0 to a value a3 r/s/s. Thus V (t), the shaft velocityat time t > 0, is given by

V(t) = V1 + At.The transducer output frotor(t) is given by

frotor(t) = frotor 1 + mot

(20)

For a step change of acceleration at time t = 0 from0 to A r/s2 we have a change of fle from 0 to a value face(m3)/(fc/Nc). This change may be regarded as a stepchange if the value of the velocity loop time constant TC 0, would be ideally

N(t) N1 + t(23)but the solution of (12) actually yields

N(t) = N, + - t1 - exp [-(fc/N0)t]}.There is thus an indicated velocity error in the steadystate given by

The final value stored in the loop register is givenfrom (29) and (30) by

Nace facc{e } (32)and hence the acceleration loop counter contents aregiven as a function of time by

N fa{dI}{1 - exp [-N t]} (33)or

N = mf Qc)(t){d1 - exp [ (fd) t (4

137

(34)

Td =fd

138

The steady-state value of N is given by

Nac (M Nf,)(N) (35)and if /8 = /max corresponds to a counter setting NNmax = Nd then we can derive an expression for therequired loop reference frequency fd:

fd = mmax(Nc/fc). (36)SYSTEM PERFORMANCE

GeneralA velocity measuring system with provision for ac-

celeration monitoring has been constructed according tothe above principles and has been used extensively inthe development of a precision digital control systemfor a variable speed motor [5]. The system has beenimplemented using the Fairchild range of industrialmedium-power silicon-integrated digital circuits (/,tL900,,uL914, ,uL923 series) and the in-line digital display ofvelocity is provided by Philips (Miniwatt) numericaldisplay tubes derived by the velocity loop decimal coun-ter stages.

Velocity MeasurementMaximum velocity Vmax - 10000 r/min = 166.7 r/s,Velocity resolution = 0.01 percent = 1 r/min,Angular transducer resolution m = 360/r,Maximum loop input frequency = 60 000/s,Loop counter capacity NC = 10 000,Loop clock reference frequency f, = 60 000/s,Loop time constant T7 = (N,) = 1/6 second.

Acceleration MeasurementThe use of a low-resolution angular transducer (m-

360) enables steady-state accelerations up to 1000r/min/s be monitored to 1 percent using the frequencycorrection input to the velocity loop counter. A muchimproved performance could be obtained with a high-resolution angular transducer (m = 3600) as describedin the next section.

AN IMPROVED SYSTEM

GeneralHigh-precision high-resolution optical transducers are

available for which the number of pulses per revolutionmay be as high as m = 10 000. The frequency responseof such devices is limited, however, and would be in-adequate for high rates of angular velocity, but theavailability of high-speed semiconductor light detectorswould enable the following systems for acceleration andvelocity measurement to be realized.

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, JUNE 1969

Velocity MeasurementMaximum velocity Vmax = 10000 r/min = 166.7 r/s,Velocity resolution = 0.01 percent = 1 r/min,Angular transducer resolution n = 3600/r,Maximum loop input frequency = 600 000/s [from

(7)1]Loop counter capacity NA = 10 000,Loop clock reference frequency f =600 000/s,Loop time constant, T. = (NVj/(f,) = 1/60 second,Velocity error for an acceleration step /3max of 1000

r/min/s = 16.7 r/min [from (25)]= 0.167 percent of maximum velocity.

Acceleration MeasurementMaximum acceleration 3masx = 1000 r/min/s

-16.67 r/s2,Acceleration resolution = 1 percent = 10 r/min/s,Maximum loop input frequency 1000 pulses/s,Loop counter capacity Nd = 100,Loop clock reference frequency f, = 1000/s [from

(36) ],Loop time constant Td = (Nd)/(fd) = 1/10 second.

CONCLUSIONAn instrument has been described for the measure-

ment of the angular velocity of a shaft using a low-resolution angular transducer and an associated all-digital frequency lock loop. The system provides anin-line digital display of angular velocity up to 10000r/min in steps of 1 r/min, and the readings are definedto the accuracy of a single-crystal oscillator.The loop operation furnishes a signal suitable for

angular acceleration measurement for acceleration ratesup to 1000 r/min/s to a resolution of 1 percent. Thebasic equations governing the operation of the loop forvelocity and acceleration measurement have been pre-sented together with an outline design and expectedperformance characteristics of an improved system usinga high-resolution high-speed angular transducer.

REFERENCES[1] G. Hoffman de Visme, "Digital processing unit for evaluating

angular acceleration," Electron. Engrg., pp. 183-188, April1968.

[2] A. Dunworth and J. I. Roche, "The error characteristics ofthe binary rate multiplier," submitted to IEEE Trans. Com-puters.

[3] J. I. Roche, "An investigation of an operational digital fre-quency-locked-loop in three digit binary-coded decimalform," M. Eng. Sc. thesis, University of New South Wales,Australia, February 1967.[4] Y. Lundh, "Digital techniques for small computations," J.Brit. IRE, vol. 9, pp. 439449, January 1959.

[5] A. Dunworth, "A precise digital control system for a com-mutatorless electric motor," submitted to IEEE Trans. In-dustrial Electronics and Control Instrumentation.