*2

^ 0F c0j

to Q IMBS TECHNICAL NOTE 870

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

Sampling Techniques for Electric

Power Measurement
NATIONAL BUREAU OF STANDARDS

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OF STANDARDS

I-ihrary

8 -JUL 1975

r Sampling Techniques for Electric

Power Measurement

Dr. R. S. Turgel

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

*<

U.S. DEPARTMENT OF COMMERCE, Rogers C.B. Morton. Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued June 1975
Library of Congress Catalog Card Number: 75-600034

National Bureau of Standards Technical Note 870

Nat. Bur. Stand. (U.S.), Tech. Note 870, 35 pages (June 1975) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1975

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. 03.46:870). Price $1.05 (Add 25 percent additional for other than U.S. mailing.)
1. Introduction 1

2. Sampling Method 2

3. Description of Measuring System h

3.1 Analog Signal Conditioning k

3.2 Analog-to-Digital Converter k

3.3 Digital Computation k

3-h Timing and Control Circuits 5

k. Mathematical Model of A/D Converter 6

k.l Computer Program 6

5. Results of Simulation 8

5.1 Second Series of Tests 9

6. Conclusions 13

7. Appendix

Computer Printout "A" 18

Computer Printout "B" 23

Reprint from IEEE Transactions for Instrumentation and Measurement 27

in

SAMPLING TECHNIQUES FOR ELECTRIC POWER MEASUREMENT

R. S. Turgel

\ A system is described that determines average electric

power by periodically sampling current and voltage waveforms

and calculating the result from digitized values of measured

instantaneous currents and voltages. System performance is

modeled on a digital computer to investigate the effects of

noise, harmonics and sampling time errors on the result of a simulated power measurement* With 15 bit analog-to-digital

conversion and 512 measured sample points an accuracy of

0.01% can theoretically be obtained from dc up to 5 kHz.

Key Words: Analog-to-digital conversion; digital; electric

power; electricity; measurement; sampling; simulation; wattmeter .

1. Introduction

This report contains a study of the feasibility of a new method for

making precision measurements of audio frequency electric power. The

study examines the limitations imposed by the nature of the measurement

process itself as distinct from the additional uncertainties introduced

by the departure from ideal characteristics of the actual hardware.

None of the currently used methods to measure low frequency ac power is an absolute method. As in the measurement of ac voltage and

current some transfer device is used to relate the measurements to the

corresponding dc units. The use of transfer devices is based on some assumptions regarding the relative response of the device to ac and dc

quantities. Generally an upper limit of the ac/dc error can be deduced

from theoretical considerations.

The definition of (average) ac power is given by

W = ~- fl eidt, (1)

where e = instantaneous voltage,

i = instantaneous current

,

t = time, and

T = some definite time interval, usually a period (or

multiple thereof) of the fundamental frequency

component.

1
Integration of equation (l) is performed within the measuring device, making use of mechanical or thermal inertia, by the first three of

the following well-known methods of measuring electric power:

(a) Electrodynamometer

(b) Electrostatic

(c) Electrothermic (thermoelement, calorimeter)

(d) Hall effect

The Hall effect devices produce dc voltages proportional to the instantaneous product of e and i which are then usually averaged (integrated)

by the readout system.

In all the above methods a physical transducer is used to obtain a measure of the product of current and voltage. Their relative advantages and disadvantages derive mainly from the nature of that transducer.

The method to be described in this report follows directly from the

fundamental definition of ac power, equation (l). Instead of carrying

out the necessary integration with an analog system, instantaneous values

of current and voltage are measured and the integration is performed by

numerical (digital) methods. The integration is therby replaced by a summation:

N

''k'iV (2)

The problem is therefore twofold. The number of samples of instantaneous voltage and current and the method of summation must be chosen

so that the value of the integral can be approached within a given tolerance. Further, the measurement of current and voltage must be made

with sufficient accuracy in a given time interval.

2. Sampling Method

To reconstruct the waveform from sampled data the minimum sampling

rate must be at least twice the frequency of the highest sinusoidal frequency component contained in the signal according to the sampling theorem . Since instantaneous power varies at twice the frequency of the

current and voltage, the sampling theorem would require a minimum sam- pling rate of four times the highest sinusoidal frequency component con- tained in the current and voltage waveforms. However, we are interested

in "average power", and as shown below the minimum sampling rate is

thereby reduced. It should be remembered that conditions encountered in

practice differ from the ideal assumed for theoretical derivations, and

therefore the minimum sampling rate is seldom sufficient in actual

cases

Raised figures indicate literature references on page IT.
In the present applica1 ion the average integral

7?f

T f(x)dt

T o

must he approximated by summing a sufficient number of samples during one

period T so that

1 Z f(x.) 1 - T (

v In j- ?

/ f(x)dt

The trapezoidal rule has been shown to converge at least as fast as

7- The last term of the summation represents the error of the approxi- 1

2 N

mat ion

l\ f(x)dx = h[^Si+ f(a+h) +....+ f(a+(N-l)h)+^] - ^^

f"U)

where

b-a

a<£<b and h = —zr~

For sampling at random intervals the error term decreases at a rate

proportional to - . It appears therefore that random sampling is not

the right approach if high precision is desired.

For the special case of periodic functions, the trapezoidal rule

is exact for the 2N periodic functions: 3

1, sin (x), cos (x),..., sin (W-l)x, cos (N-l)x, sin (Nx)

Therefore, the integral representing the average power of sinusoidal

current and voltage with phase angle

-t— f sin (x) sin (x+0)dx = -—/

v [cos (0) - cos (2x+0)]dx £-" O ^TT o

can be evaluated exactly with only three terms, (N=3), of the summation

formula. This can be verified by direct substitution. Theoretically,

therefore, the number of samples per cycle need be only 2K+1, where K

is the order of the highest harmonic present in the input signal. If

harmonics up to the fourth are included at least 9 samples must be taken

in each period. If the input waveform is no longer a combination of har-
monically related pure sinusoids, but includes small random fluctuations

(noise), or if the sampling time is subject to small fluctuations (jitters), then a higher sampling rate is necessary. In practice, therefore,

the minimum sampling rate is never sufficient. To ease implementation

of binary arithmetic, the number of samples, N, has always been chosen

as an integral power of 2 in this study.

3. Description of Measuring System

The system is capable of measuring rms current, or voltage, and

average power with either a real load or phantom load connection. For

purposes of discussion it can be divided into four main functional

blocks as shown in fig. 3.1.

( 3.1 Analog Signal Conditioning

Signal conditioning preamplifiers translate the input voltages and

currents to the required 10 volt level at the input to the A/D converter.

Figure 3*2 shows one such possible arrangement.

To preserve the accuracy of the overall system the closed loop

gain- frequency characteristic of the amplifiers has to be constant within 50 ppm from dc to the upper frequency limit. This requires a loop

gain of at least 80 db. The preamplifiers can be built from commercially

available operational amplifiers that satisfy these requirements.

3.2 Analog to Digital Converter (A/D)

The A/D subsystem samples the input signals at predetermined intervals and converts them to two 15 bit binary numbers.

Two configurations are possible. (l) The signals are captured in

two sample-and-hold amplifiers and then successively applied to the

converter through an analog multiplexer, or (2) the sample-and-hold

amplifiers feed two parallel converters. The second scheme has the advantage of eliminating additional errors due to the multiplexer; on the

other hand, it requires equality of the A/D's or suitable corrections.

Specifications of commercially available units indicate that an overall

short term accuracy of 0.01% can be maintained under favorable circumstances .

3-3 Digital Computation

This subsystem has the following functions:

1. Temporarily stores pairs of successive numbers from the

output of the A/D converters.

2. Forms the product of these numbers.

3. Adds the product to a total.

k
h. Divides the total by the number of products added to

obtain an average.

5. Displays the average on an output device.

6. Applies numerical corrections where necessary.

There are two general approaches to the problem of handling the

digital data from the A/D converter (s)

.

1. A general purpose minicomputer can be used to store sets of

data during the measurement, and to perform the computation

after the measurement is completed.

2. Special arithmetic hardware can be built to perform the

necessary calculations in real time.

Method 1 is preferable during the development period, because it

provides flexibility of operation and permits the introduction of diagnostic routines. On the other hand, when the immediacy of results is

of prime importance (e.g. certain control applications), method 2 is

more appropriate. Minicomputer data manipulation is generally too slow

in this case, and hard-wired integrated circuits are required to perform

the arithmetic operations in between sampling periods.

The final result can be displayed on electro-optical readouts ,• or

,

if a minicomputer is used, on a printer.

3.^- Timing and Control Circuits

The timing and control section is of vital importance to the correct performance of the whole system. The two main functions of the

control circuit are:

1. to determine the proper sampling time,

2. to coordinate the various subsystems so that the signals and

digital information are processed in the correct sequence.

Of these two functions the first is the more critical. The degree

to which the result from the sampled data approximates the integral

desired is directly dependent on the timing of the sampling interval.

Although a certain amount of random fluctuation (jitter) of the precise

time of the sampling command can be tolerated, any systematic error in

the periodicity of the samples will produce a proportional error in the

final result. It is therefore very important to synchronize the timing

pulses with the fundamental frequency to be measured. At low frequencies all the necessary samples for one reading can be obtained during a single cycle of the fundamental of the singal frequency. For instance,

with an input frequency of 60 Hz 512 samples can be taken per cycle

(fig. 3-3). At higher frequencies, or when more samples are required,

the measurement time has to be spread over several cycles.

5
The timing of the sampling period is related to the input signal

frequency by means of a frequency divider and phase-lock circuitry. The

sample command pulse enables the "hold" mode of sample-and-hold amplifiers and thereby starts the conversion. Delays to allow for settling

time can be programmed independently or derived from the clock circuit

of the A/D converter.

The time required to complete the analog-to-digital conversion will

occupy the major portion of the period between samples. If calculations

are performed by a minicomputer, enough additional time must be provided

to transfer the results of the conversion into memory. A direct memory

access (DMA) channel is useful for this purpose. If special hardware

is used, computations can be carried out simultaneously with the con- version of the next set of sampled data. In fact, with a pipeline type

arithmetic operation computation can be spread over several sampling

periods if necessary, provided a sufficient number of registers for

storage of intermediate results are available. The additional delay in

obtaining the final result will still be only a small fraction of the

total measurement period.

h. Mathematical Model of A/D Converter

In outlining the proposed system for measuring low frequency ac power a number of assumptions were made regarding accuracy and the num- ber of samples required. Most of these assumptions were based on purely

theoretical considerations. In an actual system disturbances due to

noise, sampling time fluctuation, round off due to finite number of

digits, harmonics in the signal, etc., contribute to the overall error. Because some of these are interrelated, estimation of the overall un- certainty is difficult. It seemed appropriate, therefore-, to simulate

the system operation to determine its characteristics.

The design of a mathematical model of the A/D conversion process is

fairly simple. Into such a model the various error sources can be introduced by defining or specifying their bounds and using normalized random

numbers to simulate perturbations. Separate calculations have to be

performed for each set of conditions and at each sample point, but total

computation time is relatively short.

U.l Computer Program

The object of the simulation is the determination of the error limits of the measurement process. That is, the calculation of the difference between the "exact" integral (eq. 1.), which represents the

analog input, and the response of the sampled data system (eq. 2.) under

various conditions. For this purpose the voltage and current inputs

are represented by

Yl = sin (x) + A 12 sin (2x + 12 ) + A\ 3 Sin ( 3x + 13 ) +

A 14 sin (kx + Q lh ) (3)
Y 2 = sin (x + 21 ) + A22 sin (2x + 22 ) + A23 sin ( 3x + 23 ) +

A2i+ sin (hx + lt+ ) (1+) The amplitudes, A, of the 2nd, 3rd, and Uth harmonic and the respective

phase angles, 0, are assigned during execution of the program from input

data. The sampling time x is expressed as a fraction of 2ir depending on the number of samples, N, to be taken per cycle.

x =2L ( n _ I),

( n = 1....U). (5)

To each sampling time x, is added a "timing error" to simulate fluctuations occurring in a real system, and a term representing "noise" is

included in the calculation of the instantaneous amplitude Y. Beth of

these values are obtained using normalized random numbers (range -1 to

+l) multiplied by suitable parameters specified in' the computer program

input data. To obtain maximum resolution and to avoid cumulative rounding errors for the desired number of significant figures, the values of

analog quantities are computed with "double precision."

Each Y value then represents the analog voltage or current applied

to the input of the A/D converter. The converter, however, has a finite

resolution of lU bits (plus 1 sign bit). The remainder of the calculations are therefore carried out in the integer mode by multiplying Y by

2

14 with appropriate rounding off by 1/2 bit. The number of bits re- tained in each subsequent step of the calculation closely parallels the

real time calculations that would be performed by special hardware in an actual system following the A/D conversion. For convenience the

final result was normalized and compared to the calculated "true value"

of the integral.

The program has provisions for specifying the following parameters:

1. The amplitudes and phases of the harmonics (up to Uth

order)

2. The phase angle between the fundamentals

3. The percentage "noise"

h. The timing error (in nanoseconds)

5. The frequency of the fundamental component

6. The number of samples per cycle for the initial run and

whether the number of samples in the succeeding run is

twice or ten times as large

7. The limit on the number of runs and/or the limit of error

7
of the summation approximation

8. Whether the error is to be expressed as percent of reading

or percent of full scale

9. The percentage difference between the fundamental frequency and the appropriate submultiple of the sampling

frequency (Sychronization error)

10. Ordered or random sample points.

5. Results of Simulation

The performance of the hypothetical system of power measurement is

shown in computer print-out "A" in the appendix.

For this calculation a basic set of parameters was chosen to cor- respond to a typical situation. There "Reference" Conditions are listed

in Table 1. Each parameter, one at a time, is then varied to test how

the system behaves. The other parameters remain the same, except for

the determination of timing error where the fundamental frequency is

increased to 2000 Hz from 60 Hz in order to make the resulting changes more apparent. The computation is terminated for each set of conditions

when the summation agrees with the integral within the specified tolerance (0.005$). The maximum number of sample points was also limited

when the tolerance could not be reached.

Table 1

Reference Conditions

Phase Angle between Fundamentals 60°

2nd, 3rd, ^th Harmonic content 1.0$

Superimposed Noise 0.01$

Timing Error 100 ns

Frequency 60 Hz

The first four sets of data show the variation with phase angle.

In each case no more than 32 sample points are required to reach an accuracy of 0.005 percent. For the 80 degree point and the 89-999 degree point the error was calculated as percent of full scale.

The next set demonstrates that when using random sample intervals

8192 samples are not enough to assure even 1 percent accuracy.
The following three sets of data show that increasing the harmonic

amplitudes from to 10 percent of the fundamental does not significantly

affect the number of samples required.

In the next three sections the influence of noise in the input signals is demonstrated. For a noise amplitude of 0.01 percent or less not

more than 32 samples are necessary. If the noise\ is increased by an order of magnitude 102U samples are required to obtain the desired accuracy. Although this is a considerable increase in the number of

samples, it does indicate that the method can produce valid results even when the noise level is high.

Similarly, the next nine sets of data show that the accuracy is

relatively insensitive to random timing errors (in the sampling time),

but very sensitive to systematic timing errors.

The last five sets of tests show the effect of signal frequency on the number of samples required. For a "clean" signal results can be obtained with 1021+ samples even at 10,000 Hz.

5.1 Second Series of Tests

The results of the first series of tests suggested that 512 sample

points would be a reasonable number. With this in mind, and considering

that in practice an A/D converter has a limited throughput rate, the

simulated mode of operation of the system was somewhat modified. To

make efficient use of the A/D converter (s ) , it was decided to operate

as close to the maximum throughput rate as practicable at all input frequencies. The number of sample points per cycle was therefore made dependent on the fundamental component of input frequency as shown in

Table 2 which assumes a throughput rate of about 50 kHz.

Table 2

Input Frequency

(Hz)

Number

points

of sample

per cycle

50 - 100 512

100 - 200 256

200 - Uoo 128

^00 - 800 6U

800 - 1600 32

1600 - 3200 16

3200 - 6U00 8

6U00 - 12800 k
The computer program was modified accordingly, and tests were run

to determine the attainable accuracy under the specified conditions.

The total number of sample points for each run was varied from 512 to

20U8.

The results are shown in the computer print-out "B" (Appendix),

and are also summarized in Tables 3 and k. Some rounding errors are evident (e.g. Table U, column h) which represent 1 in 2

1

. Disregarding

these, there appear to be no trends in the calculated errors as function

of frequency, phase angle, noise content, or random timing fluctuations.

This indicates that 512 samples per measurement are sufficient and

that the system can tolerate reasonable amounts of noise and timing

errors and remain within the 0.01% accuracy limit. Because of occasional rounding errors it is advisable to make more than one measurement.

10
Table 3

Percentage error of the simulated power measurement as a function of

phase angle, frequency, and the number of samples with a timing error of

50 ns and 0.01% noise amplitude.

Frequency

(Hz)

Phase angle

60

(degrees)

89.999

Total no. of

sample points

60 0.009 0.007

•)

0.005 512

0.009 0.007 0.005 102U

0.009 0.007 0.005 201+8

i+oo 0.003 0.007 0.005 512

0.003 0.007 0.005 102U

0.003 0.007 0.005 201+8

1000 0.009 0.007 0.005 512

0.003 0.007 0.005 102*+

0.003 0.007 0.005 201+8

2000 0.009 0.007 0.005 512

0.003 0.007 o.ooi 1021+

0.003 0.007 0.005 201+8

5000 0.012 0.005 o.oi 512

0.006 0.019 0-005 1021+

0.006 0.005 0.005 201+8

*) percent of full scale

11
Table k

Percentage error as a function of noise, timing error, and total

number of sample points for ^00 Hz and 2 kHz input with 60° phase angle

,

Timing error Maximum noise amplitude (%) Total number

(ns) 0.0 0.01 0.03 0.1 of samples

1*00 Hz

50

100

0.007 0.007 0.019 0.005 512

0. 007 0.007 0.007 0.005 102U

0.007 0.007 0.007 0.019 20^8

0.007 0.007 0-007 0.019 512

0.007 0.007 0.007 0.019 102U

0.007 0.007 0.007 0.007 20U8

2000 Hz

50 0.007 0.007 - 0.007 512

0.007 0.007 0.019 102U

0.007 0.007 0.007 20U8

100 0.019 0.007 - 0.007 512

0.007 0.019 0.007 102U

0.007 0.007 0.007 20U8

12
6. Conclusions

A system has "been described tnat will measure audio frequency power

by computation from the numberical values of simultaneous samples of the

instantaneous current and voltage. A mathematical model of the system

has been investigated and has shown that the method is capable of an accuracy of 0.01 percent. If the maximum sampling rate is 50 kHz, 512

sampling points for each determination are adequate in most cases. In

the calculation quantizing errors arising from finite resolution of the

A/D converters were taken into account, but non-linearities and offsets

were not considered. The results therefore correspond to the best that

can be obtained by this sampling method with the assumption of a 15 bit

conversion resolution under conditions where the hardware does not con- tribute additional errors.

A reprint of a paper (previously published) 1* describing the experimental verification of this method is included in the appendix. Measurement results show agreement with the NBS wattmeter generally within a ±0.02% error band for frequencies up to 5 kHz. Contributing to the ex- permintal uncertainty is the deviation from linearity of the A/D con- verters, the frequency response and voltage dependent offsets of the

sample and hold circuitry. Improvements in the operating characteristics of the hardware could reduce the overall uncertainty by perhaps

50 ppm.

13
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References

Douglas T. Ross, Chapter II "Sampling and Quantizing" Notes on Analog-Digital Conversion Techniques", A. K. Susskind, Editor,

pp 2-2 to 2-7, Technology Press, M.I.T., 1957-

Philip J. Davis, Philip Ratdnowitz, "Numerical Integration", p. 17,

Blaisdel Publishing Company, Waltham, Mass., 19^7.

« 3 ibid, p. 57.

4 R. S. Turgel, "Digital Wattmeter Using a Sampling Method", IEEE

Trans. Instrum. and Meas., Vol. IM-23 . No. k, pp. 337-3^1, Dec. 197^

17
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.uuu .600041 .7^0125 -44.01*954 1. .0100 100. 60.

.000 .60u041 .bal794 -30.348276 1. .0100 100. 60.

.UUU .600041 .535919 -7.1/5113 1. .0100 100. 60. .uuu .600041 .665162 -10.623318 1. .0100 100. 60.

.UOU .50U041 .440918 11.823586 1. .0100 100. 60. .uou .600041 •4b84bb 8.514349 1. .0100 100. 60.

.UOU .60U041 .613672 -2.726012 1. .0100 100. 60.

.000 .b0l)041 .b0897l -7.7b5414 1. .0100 1U0. 60.

.uou .600041 .4*5331 .941902 lr .0100 100. 60. .uou .50U041 .494537 1.1UU580 1. .0100 100. 60. .uou .500041 .496918 .624545 1. .0100 luu. 60.

.000 .600041 .6u827u -l.b45778 1. .0100 100. 60.

60. UOU .<i5u0UU .260031 -.0122U7 0. .0100 100. 60.

6U.U0U .250000 .2499b9 .U122U7 o. .0100 loo. 60.

6U.UUU .^5U0U0 .260U00 -.U0U0U0 0. .0100 100. bO.

Timing tKROK an nanoseconds

* = LKROK CALCULATE mS PtKOElJl OK KULL SCALE

COMPUTER PRINTOUT V

18
DIGITAL PuwEK IviEaSUREMENT page 2 a

NO. 0>- PHASg ilMTtGRAL SOi PekClnt PCNT. PCNT. TIMING ERequ

pOXNTb ANGLE EhROh HARM. NOISE ErkOr HZ

4 bU.uOu .<ib004l •2b5035 -1.997553 1. .0100 100. bO.

8 60.000 .*bo04l .2t>0061 -.UU8132 1. .0100 100. 60.

16 60.000 .250041 .2b0031 •UU4074 1. .0100 100. 60.

4 bU.OUU .<i54071 .301117

| -16.516986 10. .0100 100. 60.

8 6U.U0U .254071 .2b7935 -1.52U784 10. .0100 100. 60.

16 6U.U0U .254071 .<ib4059 .UU4670 10. ..0100 100. 60.

4 bU.UUU .<ib0041 .^34974 -1.973143 1. .0001 100. 60.

8 60.UOU .2b0 041 .2b003l .004074 1. .UU01 100. 60.

4 6U.U0U .2b004l .iibbOOb -1.965349 1. .0010 100. 60.

8 bU.UOO .2b0041 .2bU061 -.0U8132 1. .0010 loo. 60.

lb 60.000 .2b004l .230031 .UU4U74 1. .0010 loo. 60.

4 oU.UOO .^bUUHl .ab4974 -1.973143 1. .OlUO 100. bO.

8 bU.000 .^bliOHl .2bOUbl -.UU6132 1. .0100 loo. bO.

16 bU.UOO .^bUOHl ,2dOU31 .004074 1. .0100 100. 60.

4 bU.UUU .<;bO04l .254791 -1.699913 1. .1000 100. bO.

8 hU.UUil .^bUU41 .^jU«il4 -.0b91bb 1. .1000 loo. bO.

16 bU.UUU .«ibO04l .«d49939 .U4U669 1. .10U0 100. oO.

32 bU . uUu .2bu04l .2dOUo1 -.UU6132 1. .10U0 luO. 60.

64 bU.UUU .250041 .^b009<i -,U2U33b 1. .1000 100. 60.

A 26 bU.UUU .250041 .2b00bl -.008lo2 1. .1U00 10 0. 60.

«:56 bU.UUU .c:bUU4l • eJ^OUOO ,0lb2t>0 1. .1000 100. 60.

ol2 bU • uUO .«ibu04l .«ib0U0O .Ulb260 1. .1000 100. 60.

1024 bU.UUU ,c:bU041 .250031 .004074 1. .1000 100. bO.

4 hu.UOO .dbu041 • <i^)bu3b -1.997553 1. .0100 0. 2000.

8 bU.UUU .^5u0h-1 ..ioOUbl -.UU61.22 1. .0100 0. 2000.

16 bU.UUU .251)041 .^oOUUU .Ulb2b0 1. .0100 0. 2000.

52 bU . UUu .<=!b0041 .2 j0u31 •UU4U74 1. .oluu 0. 2000.

TiMlNb EKKOk IN NAwVbhCONu:>

19
uiGI'lAL PuaEk MEASUREMENT PAGE 3 A

No. OF PnAbE iNIEGRhL SUM PERCENT PCNT. PCNT. TIMING freomenc

PoINTb ANGLE ERROK HARM. NOISE ERKOR HZ

4 6u.u0tJ .2b004l .2o5005 -1.9tib349 1. .0100 50. 2000.

8 bO.uOO '.2.50041 .2b0l)00 .01b2b0 1. .oioo 50. 2000*

16 feo.ouu .^bOOHl •2D0031 .004074 1. .0100 50. 2000.

4 6 U . .<;5o04i .,ib491j -1.9467J4 1. .0100 100. 2000.

8 bu.000 .<i5004l .<;49847 .077304 1. .0100 100. 2000.

16 60.000 .^buom .249817 .089510 1. .0100 100. 2000.

32 60.000 .^bU0<4l .^DOObl -.008132 1. .0100 100. 2000.

o4 bU. 000 .^buom .^49939 .040889 1. .0100 100. 2000.

128 bU.OUO .<i5G04l .^bOUbi -.0U8132 1. .0100 100. 2000.

tt>b bU.UGu .250041 .<±b003l .004074 1. .0100 100. 2000.

•4 60.000 .^50041 .2b54b3 -2.1o84«i4 1. .0100 200. 2000.

8 60.000 .<db004l .^0456 -,lbt>79b 1. .0100 200. 2000.

it. 60 .oOO • *:5O04l .^b0^44 -.0rtl3b2 1. .0100 200. zooo.

yd bu.uUU .<:bU04l .^_>U12^ -.032541 1. .0100 200. 2000.

04 60.000 .^bOO'+l •2b0061 -.0081^2 1. .0100 200. 2000.

i^rt bU.UUU .^50041 .^4990rt .052895 1. .0100 2U0. 2000.

t_atj 6u.O0U .tibOOHl .<iS0o31 .U04U 74 1. .0100 200. 2000.

4 flU.UUU .^50041 •2b5005 -1.985349 1. .0100 100. 60.

8 60 .oOU .<d5u04l .^oOoCil .004074 1. .0100 100. 60.

4 bU.UUU .^50041 .C--.049 74 -1. 97^»143 1. .0100 100.$ 60.

8 bO.000 .,ibo04l .^D0tJ31 .004074 1. .0100 loo.s 60.

S-TIMiNG Pui_St_S bMbt-0 ON HEKlOiJ .010 PEKCENl LONGER

MAN PLKlUlJ OF t-uuUMiCNIAL. SloNAi- FkEUUENCY

.^5u04i

.*:bo04l

.^bu04l

.^bu04i

.<i5004l

• <d5o04l

.250041

.^5U041

.^bu04.l

S- TIMING i^JLSlS bAbt_L) ON Pt-KlUu

fHAN PLKlub Or FUNDAMENTAL. SioNAL FKEUUEl.'CY

4 6u • UOU

8 bu.uUU

lb bU . oOU

32 6 U . o4 bO.oOU

128 bO .OOU

<cbo bU .000

old bu • OOU

lu24 b U .

,^j4547 • -1.802273 1. .0100 100.* 60

.^49847 .0 77304 1. .0100 100.* 60

,,i'*978b .1U1714 1. .OluO loo.s 60

,24978b .101714 1. .0100 loo.s 60

.id49 78b .101714 1. .0100 100.

s

60

.«^4978b .1U1714 1. .0100 loo.s 60

.d-+978t> .101714 1. .0100 loo.s 60

.^+9 78b .101714 1. .0100 loo.s 60

.24978b .101714 1. .0100 loo.s 60

Uu .100 PEKCENl LONGER

Ti. i'i I No ERKOk IN NANOSECONDS

20
uibllAL POWEk MEASUREMENT PAGt 4 A

NO. Op

POiNTs

4

8

lb

32

t>4 x28

Cd6

bl2

lu24

$='( IMiNG

PhAbL

AImGlE

INT^gHmL bUi-i

bU.UUU

bU .UUU

bU .UUU

bU.UUU

oU .UUU

bU.UUU

bu.UOO

bU.UOU

6u .UUU

PULbLb ba;

.^5U0Hl

.^buU4i

.*boo4i

.c:bOOHi

.^buom

.^bUOM-1

• idbuO^l

.^bo04i

.^oli9u

.247742

.<i'+7b94

.2«+79bb

.^79bb

.^479bb

,^n79bb

.2>+79bb

.247955

PERCENT

Ekr°h

PCnJ . PCNT. TIDING FREQUENCY

HArM. NO 1st Erh Or HZ

•ED On HtrtlUu l.UUU

-,*b97x7 1. .0100 lo0.$ bO.

.9194b2 1. .U1U0 luO.$ 60.

.tibo428 1. .010U lUU.$ 60.

.804017 i. .UlUU 100.4 60.

.ri34017 1. .OlUO 100.$ oO.

.8.34017 1. .0100 luO.$ bO.

.8b4017 1. .OlUU luO.$ 60.

.bO4017 1. .0100 100.$ bO.

.634017 1. .0100 luO.$ bO.

PERCENT LONGER

THAN PEkIuu Oh FUiml)mMENI'AI_ bioNAL FkEuuEinCY

4 bU.UOU .dbuO'+i

8 bu.UOO -.25u04l

16 bu.uOu .<ib004l

32 bu.UOu .<ibuOHl

b4 bU.UUU .^buOHi

Xci8 bu.U.UU . <dbU04l

<ibb bU.UUU .<ib004l

Old bU.UOU .<;bu04i

1U24 bU.UUU .cbOOll

1-TIlvllNG PULbtb bAbLQ ON PEkIOu lU.OOu PEKCEN

THAN PErtlOL) Oh huiMiW'.ENTAi. biuN^L FkEUUENCY

•2oblbb

.2b872b

.^7172

.2^6697

•dobti5b

• 2bbh3t>

•2bb83o

•^obBOb

• t_.obhUb

-^.H'+91H0

-3.4743b4

-^.8bl9ub

-<^.7420bl

-<i.7x7bbl

-2.7l7bbl

-2.7l7bbl

-il.70b44b

-2.70544b

1.

1.

1.

1.

1.

1.

1.

1.

1.

LONGER

.OlUO

.0100

.0100

.0100

.0100

.0100

.OlUU

.0100

.0100

luU.$

100.$

luU.$

100.$

100.3>

100.%

100.$

100.$

100.$

bO.

bO.

bO.

bO«

60.

60.

bO.

bO<

bO<

4 bu . OOu .<ibU04i .<ibbUUb -1.9nb349 1. .0100 IGO. bO.

8 bU .UOU .<ibu041 .^bOlibi -.008102 1. .0100 loo. 80.

lb bU.UUU .*:bOU4l .*iJ003i .UU4074 1. .0100 luu. bO.

4 bu . UUU .^bU041 .2bb00b -1.9ttb349 1. .0100 100. 400.

8 bU.UOU .<ibo04l ,<1DUU5L .UU4074 1. .0100 100. 4U0.

4 bU . UOU .^bU041 .^bu3b -I.y97bb3 1. .0100 1U0. 10 0.

8 bU . UUU .tbO04I .^bU18b -.Oob9bl 1. .0100 100. 1000.

lb bU.UUU .<ibUOHl .^oOUix .0U4U74 1. .0100 100. 1000.

4 bu.UOO .^bUOHI .2bblr>7 -2.046373 1. .oluu luu. 2000.

8 bO • UUU .^buOnl .^UUbl -.006132 1. .oloo luu. <iOOO.

ib bU.UOO .<cb004i .^H976o .101714 1. .0100 luu. 2000.

52 bU .UUU .^bu041 ,*;h9S»39 .U40bt39 1. .OlUU lUO. 2000.

b4 b\J .UOU .^bu04i .,c;'t99bS .0^8464 1. .UlOO luo. 2000.

X28 bu.UOU .^bU04l .^^OUbi -.006132 1. .0100 loo. 20U0.

£ibb bU .UUU .<;bU04l . .r.bOUUu , UXo2oU 1. .0100 luo. 2000.

bl2 bU .UUU .^buUnl .2uUu3l .UU407^ 1. .0100 luo. 2000.

Tii'tlNb EKKOr< iN NAi-iObt-CONuS

21
uiGIIAL KywtK htAbuRtMENI PAGE 5 A

No. OH HrlMbE INTEGRAL bUi«l PERCENT PCNT. PCNT. TIMING FREQU

PuiN'Tb ANbl_E EKROK HARM. NOISE ERkOR HZ

4 6u .uOu .«ibuO<+l .253937 -1.538171 1. .0100 100. 10000.

rt bu.OOU .«ibuO t+l .2»+8fa8ti .341095 1. .0100 100. 10000.

in bit .ulli) .250041 .249847 .077304 1. .0100 loo. 10000.

$Z bu.OOU .^btlOHi .2^0122 -.032541 1. .oloo 100. 10000.

o4 h U . U U U .<;buUHi •24975b .113919 1. .0100 100. 10000.

xd* hu . UOU .«ibG041 .230092 -.020336 1. .0100 100. loooo.

t:.)o bU.UOU •250041 .250092 -.020356 1. .0100 100. 100O0.

^l^ bo • OUu .^bUOHl .*:dUU92 -.020356 1. .0100 100. 10000.

lu24 bU.UUU .^buO'+l .200U31 .u 04074 1. .0100 100. 10000.

Ti.iiUNo tRKOK iN NMNOSEGONUb

22
DIGITAL POWER MEASUREMENT PAGE 1 B

MO. OF r>HAS£ INTEGRAL SUM PERCENT Pe^JT. PCNT. TIMING FREQOENCY

3 OINTS ANGLE ERROR HARM. NOISE ERROR HZ

512 .000 .5001 39 .500092 .009*498 3. .0100 50. 60.

102*4 .000 .5001 39 .500122 .003396 3. .0100 50. 60.

20*48 .000 .5001 39 .500122 .003396 3. .0100 50. 60.

512 .000 .5001 39 .5001 22 .003396 3. .0100 50. *400.

102*4 .000 .5001 39 .5001 22 .003396 3. .0100 50. *400.

20*48 .000 .5001 39 .5001 22 .003396 3. .0100 50. *4U0.

51 2 .000 .5001 39 .500092 .009*498 3. .0100 50. 1000.

102*4 .0 00 .5001 39 .5001 22 .003396 3. .0100 50. 1000.

20*48 .000 .5001 39 .500122 .003396 3. .0100 50. 1000.

512 .000 .500139 .500 9 2 .009*498 3. .0100 50. 2000.

1024 .0 00 .500139 .5001 22 .003396 3. .0100 50. 2000.

20*48 . OGO .500139 .5001 22 .003396 3. .0100 50. 2000.

51 2 .000 .500000 , .*499939 .01 2207 0. .0100 50. 5000.

1024 .0 30 •500000 .'499969 .00610*4 0. .0100 50. 5000.

2o*4e .oou .50001)0 .*499969 .00610*4 0. .0100 50. 5000.

512 6 . J .2501 39 .2501 22 .006789 3. .0100 50. 60.

102*4 6 . J .250139 .2501 22 .006789 3. .0100 50. 60.

20*48 6 0.000 .250139 .250122 .006789 3. *01Q0 50. 60.

51 2 6 0.030 .250 1 39 .2501 22 .006789 3. .0100 50. *400.

102*4 6 U . .2501 39 .2501 22 .006789 3. .0100 50. *400.

20*48 6 . D Li .2501 39 .250122 .006789 3. .0100 50. *400.

51 2 6 G . .250139 .2501 22 .006789 3. .0 100 50. 1000.

1 02*4 6 . G .250139 .250122 .006769 3. .0100 50. 1000.

20*4R 6 0.000 .2501 39 .2501 22 .006789 3. .0100 50. 1000.

TIMING ERROR IN NANOSECONDS

COMPUTER PRINTOUT "b"

23
DIGITAL POWER MEASUREMENT PACE 2 B

NO. OF

POI NTS

PHASE

ANGLE

INTEGRAL SUM PERCENT PCNT. PCNT. TIMING FREQUENO

ERROR HARM. NOISE ERROR HZ

512

1 024

2048

60.000 .2S0139 .250122 .006789 3. .0100 SO. 2000.

60.000 .250139 .250122 .006789 3. .0100 50. 20U0.

60. ODD .25CI39 .250122 .006789 3. .0100 50. 2000.

51 2 1024

2048

60.000 .250049 .250061 -.004919 1. .0100 50. bUOO.

60.000 .250049 .250000 .019491 1. .0100 50. 500.0.

60.000 .25U049 .250061 -.004919 1. .0100 50. 5000.

512

1 024

2048

89 .999

89.999

89.999

000 1 48

000148

OOUl 48

.000 1 22

.000122

.000122

-.U05 I 42*

-. 005142*

-.005142*

.0100

.0100

.0100

50.

50.

50.

60

60.

60.

512

1 024

2048

89.999

89.999

89.999

000 148

0001 48

.000 1 48

.0001 22

.000122

.000122

-.0 05142*

-.0 05 1 42*

-.005142*

3.

3.

3.

.0100

.0100

.0100

50.

50,

50,

4U0,

4U0,

400.

51 2

1 024

2048

89.999 .000148 .000122 -.005142* 3. .0100 50. 1000.

89.999 .000148 .000122 -.005142* 3. .0100 50. 10U0.

89.999 .000148 .000122 -.005142* 3. .0100 50. 1 U .

51 2

1 024

2048

89.999

89.999

89. 999

OOUl 48

oojite

U U 1 4 8

.000122

.OOu 1 53

.000122

.0051 42*

.00096 1 * .0051 42*

.0100

.0100

.0 100

50.

50.

50.

20J0.

2JU0.

2000.

5 1 2

1 024

204 8

89.999 .000057 .000000 -.011492* 1. .0100 50. bOuO.

89.999 .00U057 .000031 -.005388* 1. .0100 50. 5000.

89.999 .000057 .000031 -.005388* 1. .0100 50. 5UU0.

51 2

1 024

2048

60.00 .25J139 .250122 .0067 8 9 3 . .0000 50. 4u0.

6 0.000 .250139 .25J122 .006789 3. .0000 50. 4d0.

60.000 .250139 .25U122 .0 06789 3. .0000 50. 4U0.

TIMING ERROR IN NANOSECONDS

• = ERROR CALCULATED AS PERCENT OF FULL SCALE

2U
DIGITAL PO'NER MEASUREMENT page 3 b

512

102 4 2048

5 I 2

1 024

2 04 8

5 I 2

I 02 4 20 M8

5! 2 102 4 2048

PHASE INTEGRAL SUM Pt^v-L... . PCNT. PCNT. T I M IN G FkEUU

ANGLE ERROR HARM. NOISE ERROR HZ

6 0.000 .250139 .250092 .01 8989 3. .0300 50. 4J0.

6 0.000 .2501 39 . 2501 22 .006789 3. .0300 50. 400.

6 0.030 .2501 39 . 25U1 22 .006789 3. .0300 50. 400.

60.000 .250 I 39 .2501 53 -.0054 1 1 3. . 1 000 50. 400.

6 0.000 . 250 1 39 .250153 -.00541 1 3. . I 000 50. 400.

6 0.0 .250139 .250092 . 18 9 8 9 3. . 1 000 50. 400.

6 0.0 . 25U 1 39 .2501 22 .006789 3. .0000 1 00. 400.

6.0. 00 .25J1 39 . 2501 22 .006789 3. .0000 1 00. 400.

6 0.000 .250 1 39 . 2501 22 .006789 3. .0000 100. 400.

6 0.000 . 25U1 39 .250122 .006789 3. .0100 1 00. 400.

6 0.0 J .2501 39 . 250 1 22 .006789 3. .0100 1 00. 400.

6U.0UJ .250139 .25U 1 22 .006789 3* .0100 1 00. 400.

6 0.0 U .250139 .25U1 22 .006789 3. .0300 1 00. 400.

6 0.000 .250139 .2501 22 .006709 3. .0300 100. 4dO.

6 0.0 1.) U . 250 1 39 . 250 1 22 .006789 3. .0300 1 00. 4U0.

6 . J .250139 .250092 .018989 3. . 1 000 100. 400.

6 0.000 .25u 1 39 .250092 .01 8989 3. .10 1 00. 400.

6 0.000 .250 1 39 . 2501 22 .006789 3 . . 1 000 1 00. 400.

6 .000 . 25ul 39 .250 1 22 .006789 3. .oooc 50. 2000.

6 u . .250139 .25U1 22 .006789 3. .0000 50. 2000.

6 0.00 .250139 . 250 1 22 . 6 7 8 9 3. .0000 50. 2000.

6 . J .250139 .250122 .006789 3. . 1000 50. 2U00.

6 . J .253139 .250092 . 1 8989 3. . 1000 50. 2UG0.

6 . . 2501 39 .250122 .006789 3. . 1 000 50. 2000.

T I M I N G E R K fl R I M N A N S E C M 1) S

25
DIGITAL POWER MEASUKEMENT PAGE H B

NO. OF

POI NTS

PHASE

ANGLE

INTEGRAL SUM PERCENT PCNT. PCNT. TIMING FREQUENCY

ERROR HARM. NOISE ERROR HZ

51 2 1024

2 0M8

60.0U0 .250139 .250092 .018989

63.033 .250139 .250122 .006789

6 3.303 .250139 .253122 .006789

3. .0000 100. 2 3 3 0.

3. .0000 100. 2030.

3. .0000 100. 2030.

51 2 1024

20*48

60.003 .253139 .250122 .006789 3. .0100 100. 2000.

60.000 .250139 .250092 .018989 3. .0100 100. 2000.

60.003 .25 J 139 .250122 .006789 3. .0103 130. 2030.

51 2 132M

20HH

6 0.000 .253139 .250122 .006789 3. .1000

63.03d .253139 .253122 .006789 3. .1000

6 3.303 .25U139 .25 3 122 .036789 3. .1030

TIMING ERROR IN NANOSECONDS

NO

100. 2033.

100. 2330.

10 0. 2333.

26
Digital Wattmeter Using a Sampling Method

RAYMOND S. TURGEL, member, ieee

Abstract—Average electric power can be measured by a system

that samples voltages and currents at predetermined intervals. The

sampled signals are digitized and the result is computed by numerical

integration. The response of the system agrees with that of a stand- ard electrodynamic wattmeter within 0.02 percent from dc to 1 kHz,

with the possible exception of zero power factor measurements. Measurements up to 5 kHz can be made with somewhat greater

uncertainties.

I. INTRODUCTION

BECAUSE BASIC electrical standards are defined in terms of dc quantities, the measurement of ac quantities requires an ac-dc transfer process. Underlying all such transfer processes are certain assumptions about the

limits of uncertainty associated with the measurement that cannot be determined precisely. Corroborative evidence from other independent methods of measurement is therefore the only means to increase the confidence in the method

used and in the assumptions made.

The digital wattmeter described here is based on a method that differs from the electrodynamic, electrostatic, thermal, and Hall effect wattmeters, all of which

make use of analog multiplication of current and voltage

performed by a physical process in the device. In the

digital instrument the multiplication involves discrete numbers and in itself introduces no experimental errors (other than predictable rounding errors)

Additionally, the digital method opens up possibilities of performing power measurements very rapidly, which

may be of value in power control applications, and also

of measuring nonsinusoidal waveforms with greater ac- curacy. Both of these advantages, however, have so far not been fully exploited.

II. EXPERIMENTAL METHOD

The fundamental equation for calculating the average

electric power is P = (l/T) f eidt. (1)

•'o For sinusoidal quantities, T is chosen to be a multiple of the period of e or i, T = n/f (2) where n is a positive integer and / is the frequency of the

sine wave. The present method is closely related to the fundamental formula ( 1

) . Instantaneous values of voltage

Manuscript received July 3, 1974; revised August 22, 1974. This

paper is a contribution of the National Rureau of Standards and is not subject to copyright. The author is with the Electricity Division, National Bureau of Standards, Washington, D. C. 20234.

and current are measured simultaneously, and from sets of these measurements the average power is computed

using digital methods. Because the number of measure- ments is necessarily finite, the integral of (1) is replaced

by a summation and the time inteval T by the number of measured points N,

P-U/JV) £«/»*, j = i,.- vv. (3)

For dc power, where e and i are constant, the summation

will obviously lead to the correct result. For sinusoidal

quantities, however, we i^ed to define conditions further. In (2) the time over which the integration extends is related to the period of the voltage or current waveforms.

Similarly, the time taken for N samples must be such that the interval between samples is an integral fraction of the

period, and all such intervals must be equal to one another.

It can be shown that under these conditions, with at

least 3 samples per cycle, the summation will theoretically

be exact [1]. In practice, of course, more samples are needed to reduce the effects of random noise and fluctua- tions in sample timing. The important point is that samples

are taken at definite intervals and not at random. Random

sampling does not produce rapid convergence in the numerical integration process and therefore is not suited

for high accuracy measurements. To determine how many samples would be needed for a power measurement, a computer simulation of the measurement process was carried out. The study took into account the effects of input noise, timing uncertainty, presence of harmonics, and errors due to finite resolution. The known value of the integral given by (1) and (2) was compared with the computed summation of (3). The acceptable tolerance limit was set at 100 ppm and

the presence of random noise with a peak value of 100 ppm was postulated. From this study, which is being

published as a separate National Bureau of Standards

(NBS) report, the following conclusions can be drawn.

1) Not more than 512 sample points are required.

2) A timing uncertainty of 50 ns is tolerable.

3) The percentage error due to systematic timing errors

is proportional to the percentage deviation of the timing

base period from the period of the input waveform.

4) A resolution corresponding to 15 bits for the measured

values of current and voltage is adequate.

5) The sampling rate must be at least 40 kHz to resolve frequency components up to 10 kHz.

The measurement process can now be summarized as

follows. Simultaneous samples of the current and voltage

waveforms are taken at predetermined intervals. The

sampling interval is an integral fraction of the period

27
1 I

PLL • FREQ OIV

TIMING PULSE GEN fH

1 AMPL S/H • ADC

j $ f=

AMPL S/H ADC

CM

P

U MUX

T

ER

Fig. 1. Functional block diagram of digital wattmeter.

of the waveform and the time at which sampling occurs is defined with an uncertainty less than 50 ns. The sampling

process is implemented by sample-and-hold amplifiers controlled by a special timing pulse generator. The values

of the instantaneous signal voltage "captured" by the S/H circuits are digitized using analog-to-digital converters (ADC), with a resolution of 1 in 16384. The resulting

digital output is then used in the numerical calculation

of the power.

III. HARDWARE

A . Signal Conditioning

The functional blocks of the system are shown in Fig. 1. The voltage and current waveforms can be sampled either

in a circuit with a real load, or with phantom loading.

The "current" signal is obtained from a specially constructed bifilar shunt with a time constant of 2 ns.

1 At 5 A, the voltage drop across the shunt is 0.5 V, thus keeping the dissipation to 2.5 W. Forced air cooling is used to minimize the temperature rise of the resistive element. The "voltage" signal is obtained from a two-stage

step-down transformer [2] with a 200-to-l ratio which

has negligible ratio and phase corrections. Two similar transformers are used, one for the frequency range of 50

to 400 Hz, the other for the range of 400 to 2000 Hz. The secondary voltage is 0.5 V. This value was chosen so that the current and voltage signals are comparable in magnitude and so that input channels can easily be inter- changed. For dc measurements a precision volt-box accurate to within 2 ppm is used instead of a transformer.

B. Preamplifiers

The two input signals are fed into high-impedance

instrumentations amplifiers2 which permit the shunt output

and the transformer secondary to work essentially on

open circuit. The amplifier gain is fixed at a nominal

value of 13.0 and is fiat within 50 ppm from dc to 5 kHz.

The gain raises the 0.5-V inputs to the level required by the sample-and-hold circuitry, that is, ±10-V peak. An im-

' Designed by T. M. Souders, Electricity Division, National Bureau of Standards, Washington, D. C.

2 Designed by L. Marzetta, Measurement Engineering Division, National Bureau of Standards, Washington, D, C.

portant consideration in power measurement is that the phase shift be the same in both channels. At 10 kHz the

difference is phase shift of the two amplifiers was less than

600 ^rad. Moreover, the effects of small remaining

differences in phase shift can be canceled by averaging measurements with the two channels interchanged [3].

C. Timing Pulse Generator A critical part of the measurement process is the timing

of the sampling and conversion command. The timing

pulse changes the S/H amplifiers from the "sample" to the "hold" state in which they retain at their outputs the

values of the input voltages present at the time of the

transition. The timing pulse also initiates the analog-todigital conversion, which begins after a short delay to allow for settling of the S/H amplifiers.

The timing of the command pulse is brought into a

definite relationship with the input signal by means of a

phase-lock circuit. 3 The phase-locked loop (PLL) operates

in a conventional manner with only minor modifications. Part of the input voltage signal is fed to a high-impedance instrumentation amplifier with good common

mode rejection which acts as a buffer to isolate the analog

measuring circuit from the digital circuitry. The amplifier

gain is chosen so that the phase lock will work reliably over an input voltage range from 0.1 to 1.0 V. The frequency of the voltage signal appearing at the output of the amplifier is then compared to a subharmonic

of the pulse repetition frequency in a phase sensitive detector which in turn controls the frequency of the pulse

oscillator. The subharmonic is obtained by means of a binary fre- quency divider which has 9 manually selectable ratios. These range from a ratio of 4 to a ratio of 512. Therefore,

since the oscillator operates over a band from 20 kHz to 40 kHz the subharmonic generated by the highest ratio setting (512) ranges from 40 to 80 Hz. Similarly, for the

lowest ratio setting (4) the subharmonic varies from 5 kHz to 10 kHz. Thus there is a definite ratio setting for eVery analog input frequency. It follows that, for instance,

for an input of 60 Hz there are 512 sampling points per

cycle, while for an input of 6 kHz there are only 4 samples

per cycle. The ADC's however, always operate at the

3 Designed by O. Laug, Measurement Engineering Division, National Bureau of Standards, Washington, D. C.

28

mm msm "«TM—TM«
maximum possible rate consistent with the phase-lock

requirements from 20 000 to 40 000 conversions per second. There is a tradeoff between capture range and stability in phase-lock circuits. Since in this application stability

is of overriding importance, the loop filter circuits have

been selected to provide high stability. As a consequence

the capture range is fairly narrow and a manual frequency

adjustment of the oscillator is required to bring it into lock. A 10-turn variable resistor serves as the tuning element. To determine the correct adjustment point, an indicator

light signals the out-of-lock condition and a meter con- nected to the output of the phase comparator provides a

reading for fine adjustment. An alternate method to deter- mine proper adjustment uses the digitized data and a

special software routine in the computer which will be

described later.

D. Analog-to-Digital Converters (ADC)

The outputs of the preamplifiers are fed to two parallel

S/H circuits that are physically part of the converter unit. The sampled "instantaneous" voltages are retained at the outputs of the S/H amplifiers within the specified ac- curacy limits long enough for the analog-to-digital conversion to be completed. The two resulting digital numbers are then multiplexed to a common 15-bit parallel

output.

The configuration described is preferable to the per- haps more usual one where an analog multiplexer feeds

signals to a single channel ADC. It avoids the additional

errors introduced by analog multiplex switching circuits and, at least in this case, at no additional cost. The output

digital multiplexer, which is not subject to disturbance by noise and offset voltages, does not contribute to the

error of the measurement. Each of the ADC's converts the analog input by the method of successive approximations to a 15-bit (14

bits plus sign) two's complement binary number. The

actual conversion time is about 15 /xs. Taking into account,

various settling times and the time required to transfer the output data, a minimum of 25 us is required between

conversion cycles. The maximum throughput rate is therefore 40 kHz. Binary signals control the output- channel selection, output-inhibit function, and indicate when the conversion is completed.

E. Computing Circuitry At the time the project was planned, intergrated circuits

for high-speed digital arithmetic were not as easily avail- able as they are now. Building of special circuits for real time computation, therefore, did not appear advisable. The selection of a minicomputer instead of special hardware, in retrospect, certainly seems to have been the better

choice, even though calculations eannot be done in real time. The flexibility afforded by a general purpose computer makes possible not only a relatively easy change of experimental procedures, but also provides the opportunity

for using diagnostic routines to detect malfunction or input errors.

The computer chosen had a 16-bit word and originally

had 4096 (4K) words of memory. The memory was later expanded to 12K to permit the use of a "basic" (programing language) interpreter. In order to link_the computer with the ADC's, a special interface is required which must generate a signal to interrogate the two ADC channels alternately. The time

between samples is insufficient for the normal inputoutput (I/O) channels of the computer and the ADC,

data are therefore transferred through a direct memory

access (DMA) channel. Since the ADC data occupy only

15 bits, the sixteenth bit of the computer word is used as a channel indicator. Channel identification is helpful in verifying the proper functioning of the interface and in correlating corresponding ADC outputs.

With the 4K memory, 1024 data words (corresponding

to 512 sample points), as well as the machine language program, could be accommodated, however, with little room to spare. On the other hand, in the 12K environment,

the "basic" interpreter together with additional special routines and data storage do not fill all of the available memory.

Printed output from the computer is obtained through

a conventional teleprinter.

F. Input Circuit Switching

Input connections to the two preamplifiers are made

through a patchboard using British Post Office type

coaxial connectors. These provide very reliable lowresistance connections and show little signs of wear. The

choice of a patchboard type input was a matter of ex- pediency. In the long run, however, a system of computer

controlled switches would be more useful. In the present

arrangement each of the two channels can be connected,

singly or together, to the following "sources":

1) "zero" ( 100-ft internal resistance)

2) calibration voltage;

3) shunt output voltage;

4) transformer output voltage; and

5) volt-box output (for dc voltage).

All of these are required for the measurement procedure.

IV. SOFTWARE

With the limited amount of memory available initially,

all routines were written in assembly language using

integer arithmetic. Because the ADC output has only

15 significant bits, a 16-bit computer word is adequate

for the calculation of the results. Later, after the memory

was expanded, some of the integer computations were

incorporated as special subroutines into the "basic" system in the interest of speed. As an example, the computation time required for a typical power measurement

using unmodified "basic" is roughly 55 s. With integer subroutines the time taken is reduced to less than 1 s. It is usually desirable to provide at least some internal checks to guard against malfunction or improper inputs.

29
TABLE I

Difference from Uncertainty

Comparison of Standard Standard Power (percent of

full scale) Comparison (percent of Frequency Factor Standard full scale)

dc -0.004 dc potenti- ometer and shunt

0.002

50 1 0.014 wattmeter 0.01 0.007 DTVC° 0.005 -0.004 rms^ 0.005

0.5 lead 0.018 wattmeter 0.02

0.5 lag 0.014 wattmeter 0.02 60 1 0.008 wattmeter 0.01 -0.018 rmsd 0.005

0.5 lead 0.018 wattmeter 0.02

0.5 lag 0.010 wattmeter 0.02 400 1 0.026 wattmeter 0.02 0.016 DTVC 0.005 -0.018 rmsd 0.005

0.5 lead 0.015 wattmeter 0.03

0.5 lag 0.008 wattmeter 0.03 Olead 0.035 wattmeter 0.03

1000 1 -0.009 wattmeter 0.03 -0.017 DTVC 0.005 -0.019 rmsd 0.005

0.5 lead 0.012 wattmeter 0.06

0.5 lag 0.024 wattmeter 0.06 Olead -0.046 wattmeter 0.06 Olag -0.018 wattmeter 0.06 2000 1 -0.038' wattmeter 0.04 -0.016 DTVC 0.005

0.5 lead 0.083 wattmeter 0.11 0.5 lag 0.013 wattmeter 0.11 5000 0.016b DTVC 0.005

a Amplifier response correction 0.141 ± 0.01 percent.

b Amplifier response correction 1.14 ± 0.1 percent.

" DTVC indicates differential thermocouple voltage converter.

d See text.

Diagnostic routines check the ADC channel number, the sequence in which data is stored in memory, and also whether the ADC output exceeded maximum positive

or negative values. One indication of faulty connections

in the analog signal path is the noise superimposed on dc measurements. Such noise can be detected from the stand- ard deviation of a set of measurements performed at the 40-kHz sampling rate. Certain test instruments, such as oscilloscopes, when

attached at critical points modify the system response because of loading, ground loops, or pickup. To prevent such interference it is possible to use the computer itself to test the system by means of special software routines. For instance, offset adjustment of the data amplifiers can be checked by displaying the ADC output on the computer front panel indicator lights. Similarly, such a

display can be used to locate the transition points when

calibrating the ADC.

Under some conditions the sensitivity of the phase-lock

null meter is insufficient for optimum adjustment of the

oscillator. Finer adjustment can be obtained with the help

of a routine that calculates the differences between

successive peak-to-valley amplitudes of the input waveform as measured by the sampling circuit and the ADC.

The phase lock can then be optimized by adjusting the

oscillator so that the differences, which are displayed on

the panel, are minimized. This method of adjustment pre- supposes that the ac source has sufficient cycle-to-cycle

stability.

V. EXPERIMENTAL PROCEDURE

Wherever possible, measurement procedures relied on

external standards in order to be independent of such

circuit parameters as drift, offset, gain, and internal reference source stability. Each measurement sequence

starts with a "zero reading" which is then applied as a

correction to subsequent data. Next is the "calibration voltage" obtained from a Zener reference source stable to within a few parts per million per month. These measure- ments are the average of 512 readings on each channel. Both positive and negative calibration voltages are used

to compensate for a slight asymmetry in the response

of the S/H-ADC combination that varied slowly with time. During the computation of the power, the appropriate

pair of calibration constants is used depending on the sign

of the measured value and the measurement channel. Thus, if it can be assumed that in the brief interval between

calibration and measurement the circuit parameters

30
change only by negligible amounts, the computed value

will be independent of offsets and the value of the gain

in amplifiers and ADC's.

To investigate the performance of the system two types

of measurements were made: 1) comparison with an

electrodynamic wattmeter (Yokogawa APR-2) ; 2) comparison with a differential thermal voltage converter [4]. The results of these measurements are summarized in Table I, which shows the percentage differences between

the readings from the digital system and those from the comparison standard. The intercomparison with the wattmeter was carried out at 5 frequencies at unity and

at half-power factors, and in two cases at zero power

factor. Power measurements were made at a nominal

input of 100 V and 5 A. The results using the differential thermal voltage converter with an input of 100 V to both channels were

calculated either as a simulated power measurement at unity power factor, or by computing the rms value for each channel (indicated by "rms" in the table). In the

first case the calculations were carried out exactly as in the wattmeter comparison using (3) with N = 512. In the

second, the rms value was computed separately for each channel before forming the product. At 2 kHz and 5 kHz a correction was applied for the drop in the response

of the S/H amplifier. This increased the uncertainty by

0.01 percent and 0.1 percent, respectively. Short term

precision of 0.006 percent could be realized, and overall repeatability was estimated by combining all 400-Hz

data, regardless of how they were obtained. This yielded

a standard deviation of 0.02 percent of reading and

0.004 percent of the mean.

VI. CONCLUSIONS

A method was described of measuring average electric power by sampling instantaneous voltages and currents from which the result was computed by numerical inte- gration. Intercomparisons with an electrodynamic watt- meter and a differential thermal voltage converter agreed

within 0.02 percent from dc to 1 kHz, with the possible

exception of zero power factor measurements and with

jomewhat larger uncertainties up to 5 kHz.

ACKNOWLEDGMENT

The author wishes to thank N.M. Oldham for assistance with obtaining the intercomparison data and A.G. Perrey

for help with the construction of the electronic circuitry,

both of the Electricity Division, National Bureau of Standards. REFERENCES

[1] P. J. Davis and R. Rabinowitz, Numerical Integration. Waltham, Mass.: Blaisdell, 1967.

[2] W. C. Sze and F. R. Kotter, "The design of near-perfect instru- ment transformers of simple and inexpensive construction," J. Appl. Meas., vol. 2, pp. 22-27, 1974.

[3] M. Tomata, T. Sugiyama, and K. Yamaguchi, "An electronic multiplier for accurate power measurements," IEEE Trans.

Instrum. Meas., vol. IM-17, pp. 245-251, Dec. 1968.

[4] F. L. Hermach, J. E. Griffin, and E. S. Williams, "A system for accurate direct and alternating voltage measurements," IEEE

Trans. Instrum. Meas., vol. IM-14, pp. 215-224, Dec. 1965.

Reprinted by permission from IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT

Vol. IM-23, No. 4, December 1974, pp. 337-341 Copyright 1975 by the Institute of Electrical and Electronics Engineers, Inc. PRINTED IN THE U.S.A.

31
U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-870

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Sampling Techniques for Electric Power Measurement

5. Publication Date June 1975

6. Performing Organization Code

7. AUTHOR(S)

R. S. Turgel

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) 13. Type of Report & Period CoveredFinal

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 75-600034

16. ABSTRACT (A 2 00- word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

A system is described that determines average electric power by periodically

sampling current and voltage waveforms and calculating the result from digitized

values of measured instantaneous currents and voltages. System performance is modeled on a digital computer to investigate the effects of noise, harmonics and

sampling time errors on the result of a simulated power measurement. With 15

bit analog- to- digital conversion and 512 measured sample points an accuracy of

0. 01% can the^--*-^=>Uv be obtained from dc up to 5 kHz.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Analog- to- digital conversion; digital; electric power; electricity; measurement;

sampling; simulation; wattmeter.

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