Performance Analysis and Compensation of MT-Type Digital Tachometers-pAK

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 4, AUGUST 2001 965

Performance Analysis and Compensation ofM = T -Type Digital Tachometers

Richard C. Kavanagh, Senior Member, IEEE

AbstractNew analyses are presented of the widely used-type digital tachometer. This device incorporates accurate

time measurement with pulse counting. It is shown that theaccuracy of velocity measurement in real implementations is sig-nificantly poorer than indicated previously. Encoder nonidealitiesare found to introduce a bias in the measured velocity output. Asignificant rms output error can also result. The insensitivity ofthe error to the form of the encoder noise model is demonstrated,and two analyses of differing complexity are presented. Thesepermit calculation of the tachometer error measures, based ona knowledge of the magnitude of the error characteristics of theencoder. This facilitates the implementation of a compensationfunction that will reduce the influence of the bias. Experimentalresults are presented which demonstrate the existence of the bias,

verify the validity of the derived formulae, and show the benefit tobe gained by compensation.

Index TermsDigital measurements, error analysis, errorcompensation, optical transducers, optical velocity measurement,probability, quantization, tachometers.

I. INTRODUCTION

ANUMBER of digital tachometers which have been de-

scribed intheliterature[1][4]arebased onthe -type

digital tachometer (MTDT) developed by Ohmae, et al. [5]. The

application of this technique is motivated by the fact that the

velocity value obtained from a digital position transducer via

simple pulse counting (or digital differentiation of the output)will result in poor resolution due to spatial quantization. How-

ever, the addition of an auxiliary counter in the MTDT allows

compensation for the fact that encoder edges occur asynchro-

nously with the sampling instants of a microprocessor- or DSP-

based control or instrumentation system operating with a fixed

sample interval.

The principle of the tachometer is illustrated in Fig. 1; a high-

frequency counter can be used to measure accurately. This

counter is reset at an encoder edge. Its use facilitates the calcu-

lation of an accurate velocity estimate (in units of code changes

per second)

(1)

(It is worth noting that this concept, which was independently

developed for use in high-accuracy motion control applications,

has similaritiesto the work of Nutt [6]and, more recently, Kalisz

et al. [7], [8] on time-interval measurement).

Manuscript received May 4, 2000; revised April 6, 2001.The author is with the Department of Electrical and Electronic Engineering,

University College Cork, Cork, Ireland (e-mail: [email protected]).Publisher Item Identifier S 0018-9456(01)07353-3.

Fig. 1. Operation of the auxiliary counter in an M = T tachometer.

Henceforth, the nominal angular position change between en-

coder output transitions and the number of such transitions oc-

curring per sample-time are used as the units of position and ve-

locity (rate), respectively. Using these units, the estimated rate

is defined by

(2)

in code changes (transitions) per sample-time (this unit is ab-

breviated to Transitions in the figures below). At very low

speed, when no transition has occurred during a sample-period,

the velocity calculation is adapted to

(3)

where is the number of samples since the previous encoder

edge.

II. INFLUENCE OF ENCODER ERRORS ON PERFORMANCE

Analyses have been presented by Prokin [3] and Ohmae et

al. [5] that illustrate the very high potential accuracy of the ve-

locity measurement obtained using an MTDT. These analyses

have concentrated on the error that results from the digital na-

ture of the auxiliary counter. The relative error due to this ef-

fect isusually approximatedat most rates by , where

is the auxiliary counter frequency (for example, an un-

certainty of the order of 0.01% is expected if ms andMHz). However, the actual performance is usually

substantially poorer than indicated in such analyses due to the

nonideal nature of the incremental encoders which provide the

digital pulse-train inputs.

These nonidealities have been well described and catego-

rized by Yien [9]. The variation in encoder-edge locations from

the ideal equidistant locations leads to both differential- and

integral-type errors [10]. The former appear as high-frequency

errors when viewed over the circumference of the encoder disk,

while integral-type errors display a low-frequency (usually

00189456/01$10.00 2001 IEEE

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966 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 4, AUGUST 2001

once-per revolution) characteristic. The differential-type tran-

sition-noise will typically be the dominant error source in an

MTDT.

A first-order analysis, showing that this error can cause a sig-

nificant rms error in speed measurement was briefly outlined

in [11], in which it was also shown that substantial additional

hardware is required if this error is to be reduced. More accurate

analysis, to be described in this paper, shows that the transitionnoise introduces a bias into the velocity measurement, which

is not removed by linear filtering techniques. This bias occurs

because of the nonlinear nature of the MTDT velocity calcula-

tion. While usually small in magnitude, this error has potentially

important consequences for high-accuracy measurement appli-

cations in which the MTDT is likely to be employed.

III. TRANSITION NOISE MODEL

It was illustrated in [12] that uniformly distributed noise can

provide a reasonable model for the transition noise. It was also

shown that the parameter describing the width of the transition

region (i.e., the range of actual positions at which a change indigital position code occurs) can be obtained, for a particular

sensor, by analyzing the error spectrum that results if a simple

digital differentiator is used for shaft velocity measurement.

However, the actual noise distribution will inevitably differ

somewhat from the simple uniform distribution model. To

investigate the influence of the noise model chosen, the errors

caused by uniformly distributed noise are compared with those

due to a truncated Gaussian distribution with the same standard

deviation. Such a choice is motivated by the fact that one might

expect a high probability that the code change will occur near

the nominal position, but a simple Gaussian distribution is not

physically realistic because of the ordered sequence of code

changes. When the truncated distribution is limited to the range, where represents the standard deviation that

would accrue without truncation, this variable can be shown

to be related to the actual standard deviation with truncation

by

(4)

where

(5)

By equating the standard deviation of the uniform and truncated

Gaussian distributions, based on the assumption that the uni-

formly distributed position error will be in the range ,

the sensitivity of the MTDT error to the error characteristic can

be investigated. In Fig. 2, computer simulations are used to il-

lustrate the effects of transition noise error. It is clear from this

figure that both models predict very similar velocity error char-

acteristics. Therefore, the simpler uniform distribution can be

employed for tachometer analysis.

The encoder model used to predict tachometer performance

has assumed a simple random distribution of encoder transition

location errors. This assumption is not valid in a quadrature de-

Fig. 2. Average andrms errors e and e associated with MTDT, assumingeither uniformly distributed noise ( = 0 : 2 ) , or truncated Gaussian noise ( k =2 ) (portions of the plots are shown magnified by 360% to aid discrimination ofthe very similar curves).

coded system if the errors are mainly due to phasing errors be-

tween positive and negative transitions of a particular channeland/or between channels (state width errors). The model is still

found to give a good approximation to the system behavior, but

with reduced accuracy, as demonstrated below.

IV. ANALYSIS OF THE MTDT

The tachometer is analyzed using the encoder model shown

in Fig. 3. A complete and exact analysis is deceptively difficult.

Probability of overlap or, otherwise, between the sampling in-

stants and the transition regions, is central to the analysis. When

samples of the pulse count and auxiliary timer values are taken

at a position that is in the transition region, the values sampled

will depend on whether or not the position change has actuallyalready occurred. This is illustrated in Fig. 4. A large number

of distinct conditions are possible, particularly when the rate is

close to an integer number of transitions per sample time, so

that a double overlap condition becomes possible (i.e., when

adjacent samples may occur in transition regions). Similar anal-

ysis has been undertaken when considering the intercept time

problem in radar warning receivers [13].

While possible, a complete analysis yields very cumbersome

expressions for both average and rms errors. Two simplified

analyses are presented in this paper. The first is a simple first-

order analysis that yields an intuitive insight into the system op-

eration. The second is more accurate and involved and produces



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KAVANAGH: PERFORMANCE ANALYSIS AND COMPENSATION OF -TYPE DIGITAL TACHOMETERS 967

Fig. 3. Encoder model for MTDT analysis.

Fig. 4. Relationship between sample interval T and actual encoder transitionlocations [in (a) the first sample occurs prior to the transition, in (b) it occursafter].

accurate error prediction formulae at most input rates. How-

ever, a simplified linear interpolation is used to yield an approx-

imate expression for sensor bias at those rates for which double

overlap is possible.

A. Simple Approximation

The actual digital position change, as indicated by the main

counter, is in error by due to the transition noise,

where and are two manifestations of the uniformly dis-

tributed transition noise process. Therefore, the error associated

with the digital differentiation of the main counter output is rep-

resented by a triangular distribution,

(6)

If the input rate is transitions per sample time, the average

value of , the position change per sample will equal the ac-tual speed (rate) of the encoder shaft in code changes per sample

time (the actual digital position change is usually , or

) where represents the largest integer number that is less

than, or equal to, the argument. The time duration corresponding

to the values of the auxiliary counter at successive samples will

differ from that expected of a noise-free system by an amount

represented by the variable (assuming ), where

is the probability density function of . The corresponding ve-

locity measurement error is

(7)

so that the average error is approximated by

(8)

and the mean-squared error by

(9)

When the error is defined instead by ,

the corresponding error measures being

(10)and

(11)

In Fig. 5, these error estimates (Approximation I) are compared

with those obtained by a simulation in which the errors are av-

eraged over 60 000 samples for each input rate (the rms error is

close to for , corresponding to the approximation

, which was utilized in [11]).

B. More Accurate Approximation

It is clear that the above analysis is not accurate in its estimateof average error. This is because it does not reflect a number of

facts.

There is a higher probability that a sample instant will

occur during an interval in which the transition noise re-

sults in a longer than average position variation between

transitions. The probability that the digital position change

is , rather than , is higher when the transitions

corresponding to and have a wider

than average separation. This can be related to the random

variable .

The nature of the probability density function (p.d.f.) gov-

erning the timer values varies when the sample in-stant occurs somewhere in the transition region associated

with a particular digital position. The sample instant may

occur before the actual transition location, in which case

the transition of relevance will have occurred in the pre-

vious transition region, approximately one bit previously,

as illustrated earlier in Fig. 4(a). This is found to result in

the standard triangular p.d.f. However, when the sampling

instant has already occurred [as illustrated in Fig. 4(b)],

the p.d.f. of the time interval used in the velocity calcula-

tion is found to change.

Taking these effects into account, it is shown in the Appendix

that a more accurate estimate of average measured speed pre-



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Fig. 5. Comparison between error measures (rms and average errors) obtainedusing computer simulation and estimated by Approximations I and II (the latteremployed to estimate e only).

dicted by the MTDT can be derived. This derivation produces

the following rate estimate when the actual rate is

(12)

when , and , where represents the

fractional value of . This latter condition guarantees that twoadjacent sample instants will not both be in transition regions.

To avoid a more cumbersome analysis when is close to zero

or one, use is made of the experimental observation that (12) is

accurate when . A linear interpolation for

the average velocity error is postulated for other rates

(13)

where is the integer closest to . It is shown

in Fig. 5 that this approximation (Approximation II) yields esti-

Fig. 6. Comparison between uncompensated and compensated experimentalbias measurements, simulation predication, and theoretical approximation (aportion of the plot is shown magnified by 360% to aid discrimination of thevery similar curves).

mates of the average error that are very close to those generated

by time-domain simulation, particularly when .

V. EXPERIMENTAL VERIFICATION

A low-cost, 500-cycle per revolution encoder was tested

using a low-velocity, high-inertia rig in quadrature-decoder

mode (given 2000 transitions per revolution and ms,

a rate of corresponds to 1.5 rpm). Examination of the

error spectrum of a simple digital differentiator applied to the

digital position derived from the encoder pulses can be used

to estimate the magnitude of the encoder transition noise [12].

This analysis yielded .

The average error is measured experimentally by rotating the

encoder at close to constant speed1 . In Fig. 6, the measured error

is compared with the estimate of derived using Approxi-mation II, and with that predicted by a time-domain computer

simulation of an encoder with the error characteristics assumed

in this paper. The curves match very well, except at those rates

that are close to multiples of four, and at very low rates. The

former rates equate to an integer number of encoder cycles per

sample-time (and compensation can be turned off at these rates).

The deficiency of the model at these rates can be explained by

the deterministic variation in state widths that exists in a quadra-

ture decoded system. However, the approximation is good at

most rates and can be used to reduce the steady-state bias in

systems where such bias would be significant.

Based on the velocity estimated using the method at

each sample, and a knowledge of the transition noise figure ofthe sensor, compensation is achieved by simply subtracting the

error measure obtained using an appropriate formula. (12),

which is used to achieve the compensated output displayed in

Fig. 6, is accurate when is greater than approximately ,

while (8) and (10) can be used at lower rates. When the rate is

reasonably large , it is found heuristically that

approximates very closely to (12) and is therefore suit-

able for simplified real-time implementation.

1Theuse ofa high-inertiatest-rig ensures that thespeedvariation is very smallduring each test. The unavoidable variation will cause a slight integral error thatwill alter the error measure at close to integer rates [10]. However, its effect isusually much less than that of differential error.



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KAVANAGH: PERFORMANCE ANALYSIS AND COMPENSATION OF -TYPE DIGITAL TACHOMETERS 969

TABLE IPOSSIBLE OPERATING CONDITIONS AND THEIR RESPECTIVE PROBABILITIES WHEN 2 "


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The average velocity is obtained by adding the integrals re-

lated to the various cases above, each integral being multiplied

by the probability of occurrence of that case

(15)

This equation can be solved using Mathematica to yield (12),

after some manual reformulations.

REFERENCES

[1] G. Bucci and C. Landi, Metrological characterization of a contactlesssmart thrust and speed sensor for linear induction motor testing, IEEETrans. Instrum. Meas., vol. 45, pp. 493498, Apr. 1996.

[2] R. C. Kavanagh, J. M. D. Murphy, and M. G. Egan, A high-perfor-mance positioning system using improved digital speed sensing, inProc. 3rd. Euro. Conf. Power Electron. Applicat. (EPE89), Aachen,Germany, Oct. 1989, pp. 9971002.

[3] M. Prokin, Double buffered wide-range frequency measurementmethod for digital tachometers, IEEE Trans. Instrum. Meas., vol. 40,pp. 606610, June 1991.

[4] , Extremely wide-range speed measurement using adouble-buffered method, IEEE Trans. Ind. Electron., vol. 41,pp. 550559, Oct. 1994.

[5] T. Ohmae, T. Matsuda, K. Kamiyama, and M. Tachikawa, A micropro-cessor-controlled high-accuracy wide-range speed regulator for motordrives,IEEE Trans. Ind. Electron., vol. IE-29, pp. 207211, Aug. 1982.

[6] R. Nutt, Digital time intervals meter, Rev. Sci. Instrum., vol. 39, pp.13421345, 1968.

[7] J. Kalisz, M. Pawlowski, and R. Pelka, Error analysis and design of thenutt time-interval digitiser with picosecond resolution, J. Phys. E, Sci.

Instrum., vol. 20, no. 11, pp. 13301341, 1987.[8] J. Kalisz, R. Szplet, J. Pasierbinski, and A. Poniecki, Field-pro-

grammable-gate-array-based time-to-digital converter with 200-psresolution, IEEE Trans. Instrum. Meas., vol. 46, pp. 851856, Aug.1997.

[9] C. Yien, Incremental encoder errors: Causes and ways to reduce them,in Proc. Intl. Incremental Motion Conf. (PCIM 92), Nrnberg, Ger-many, Apr. 1992, pp. 110121.

[10] R. C. Kavanagh, Shaft encoder characterization via theoretical modelof differentiator with both differential and integral nonlinearities, IEEETrans. Instrum. Meas., vol. 49, pp. 795801, Aug. 2000.

[11] , Improved digital tachometer with reduced sensitivity to sensornonideality, IEEE Trans. Ind. Electron., vol. 47, pp. 890897, Aug.2000.

[12] R. C. Kavanagh and J. M. D. Murphy, The effects of quantization noiseand sensor nonideality on digital-differentiator-based velocity measure-ment, IEEE Trans. Instrum. Meas., vol. 47, pp. 14571463, Dec. 1998.

[13] I. V. Clarkson, J. E. Perkins, and I. M. Mareels, Number/theoretic so-lutions to intercept time problems, IEEE Trans. Inform. Theory, vol.IT-42, pp. 959971, May 1996.

Richard C. Kavanagh (M95-SM01) was born in Cork, Ireland, in 1961. Hereceived the B.E., M.Eng.Sc., and Ph.D. degrees in electrical engineering fromthe National University of Ireland, Cork, in 1984, 1985, and 1998, respectively.

Heis a College Lecturer at theNationalUniversityof Ireland,wherehe directsthe Mechatronics Research Laboratory. He has previously worked as a SeniorResearch Scientist with PEI Technologies and as a Senior Project Engineer withSPS Laboratories, Ltd. His current research interests include improved sensordesign through advanced signal processing, the analysis of quantization effectsin data acquisitionsystems, and the design of hardware-in-the-loop test systems.

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Performance Analysis and Compensation of MT-Type Digital Tachometers-pAK