L/IUC

TIT

347

A Standard for Accurate Phase-Angle

Measurements at Audio Frequencies

WINSTON W. SCOTT, JR.

<** °F

cci

\ *^tAO Of

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards
1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.

20234.

2 Located at Boulder, Colorado, 80302.

3 Located at 5285 Port Royal Road, Springfield, Virginia, 22151.

THE NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards1 provides measurement and technical information services

essential to the efficiency and effectiveness of the work of the Nation's scientists and engineers. The

Bureau serves also as a focal point in the Federal Government for assuring maximum application of

the physical and engineering sciences to the advancement of technology in industry and commerce. To

accomplish this mission, the Bureau is organized into three institutes covering broad program areas of research and services: THE INSTITUTE FOR BASIC STANDARDS . . . provides the central basis within the United

States for a complete and consistent system of physical measurements, coordinates that system with the measurement systems of other nations, and furnishes essential services leading to accurate and uniform

physical measurements throughout the Nation's scientific community, industry, and commerce. This

Institute comprises a series of divisions, each serving a classical subject matter area: —Applied Mathematics—Electricity—Metrology—Mechanics—Heat—Atomic Physics—Physical

Chemistry—Radiation Physics—Laboratory Astrophysics2—Radio Standards Laboratory,2 which

includes Radio Standards Physics and Radio Standards Engineering—Office of Standard Refer- ence Data. THE INSTITUTE FOR MATERIALS RESEARCH . . . conducts materials research and provides

associated materials services including mainly reference materials and data on the properties of materials. Beyond its direct interest to the Nation's scientists and engineers, this Institute yields services which are essential to the advancement of technology in industry and commerce. This Institute is or- ganized primarily by technical fields: —Analytical Chemistry—Metallurgy—Reactor Radiations—Polymers—Inorganic Materials—Cryogenics2—Materials Evaluation Laboratory—Office of Standard Reference Materials.

THE INSTITUTE FOR APPLIED TECHNOLOGY . . . provides technical services to promote the use of available technology and to facilitate technological innovation in industry and government. The

principal elements of this Institute are: —Building Research—Electronic Instrumentation—Textile and Apparel Technology Center

Technical Analysis—Center for Computer Sciences and Technology—Office of Weights and Measures—Office of Engineering Standards Services—Office of Invention and Innovation—Clearinghouse for Federal Scientific and Technical Information.3
UNITED STATES DEPARTMENT OF COMMERCE • John T. Connor, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

NBS TECHNICAL NOTE 347

ISSUED OCTOBER 14, 1966

A STANDARD FOR ACCURATE PHASE -ANGLE

MEASUREMENTS AT AUDIO FREQUENCIES

WINSTON W. SCOTT, JR.

Radio Standards Laboratory

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means tor making available scientitic data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402

Price: 25 cents

Table of Contents

Page

1. Introduction 1

2. Plan 3

3. Voltage Supply and Detector Systems 4

4. Phase-Angle Measurements, 0° to ±60° 5

5. Phase-Angle Measurements, ±60° to ±170° 9

6. Conclusion 11

7. References 11

8. Additional References 12

9. Appendix 1. Shielding for Control of Capacitance Leakage

Effects. 12

2

10. Appendix 2. Experimental Setup, Calibration Procedure,

and Summary of typical measurements for phase -angle cj> approximately 70° and frequency 400 Hz. 13

11. Appendix 3. Phase-Angle Errors for Periodic Waveforms

other than Sinusoidal 15

Table 1. How Phase-Angle Uncertainty Restrictions on cj> < 60°

Affect Allowable Voltage -Ratio Errors 21

Figure 1. Three -Voltmeter Method 22

Figure 2. Voltage Phasor Diagram 22

Figure 3. Voltage Supply System 23

Figure 4. Detection System 24

Figure 5. Schematic for Phase Measurements ±6° to ±60° 25

Figure 6. Schematic for Phase Measurements ±60° to ±170° 26

Figure 7. Phase -Angle vs Maximum Phase -Angle Error for

Several Voltage Comparison Error Allowances 27

Figure 8. Shielding Details for Phase Measurements ±60° to ±170° 28

Figure 9. Wiring Diagram of Experimental Setup 29

Figure 10. Schematic Diagram of Experimental Setup 30

in
Abstract

A method is described for the measurement of phase

angles from 0° to 360° (except for a band ±10° about the

180° point) with an uncertainty at audio frequencies of

±0. 05° to ±0. 01°, depending on the phase-angle. The

method utilizes readily available equipment which may be

assembled and connected to serve as a standard. The

principal advantage of this standard is in its potentially

broad frequency-response; namely, 10 Hz to 10 kHz or higher. It makes use of inductive voltage dividers and

thermal voltage converters to compare voltage magnitudes, which are related to phase -angle by the law of

cosines. The thermal voltage converters are compared

with each other before each phase -angle measurement.

Their outputs are connected so that short-time power

supply variations do not affect the measurements. The

repeatability of measurements with this standard at a phase-angle of 60° is within a range of 0. 001°.

A model has been built which shows good measurement precision, but it has not been thoroughly tested or

studied at many frequencies and phase-angles for sources

of error. This report serves as a record of the development work completed on this standard to date.

Key Words: Phase -Angle

Standard

Audio Frequency

Thermal Voltage Converter

Inductive Voltage Divider

Measurement

IV
A Standard for Accurate Phase -Angle Measurements

at Audio Frequencies

Winston W. Scott, Jr.

1. Introduction

The phase -angle standard is used for the accurate determination of

phase -angle between two alternating voltages supplied externally and for

the calibration of precise phase -angle standards. The standard has a

broad frequency response, a feature lacking in a previously described

standard of phase-angle [Park and Cones, I960], by using thermal voltage converters and appropriate inductive voltage dividers, and is useful

within the frequency range 10 Hz to 10 kHz. Tests were made only to

10 kHz, but there seems little reason, with an appropriate power supply

and high frequency inductive voltage divider, why these circuits cannot

be useful up to 100 kHz. Measurement uncertainty depends upon the

phase-angle being measured and the uncertainty varies from less than

±0. 01° to ±0. 05° except for phase-angles between 170° and 190° where

the sensitivity of the standard is low.

The basis for operation of the standard is the "three -voltmeter

method" described by Kinnard [ 1956], The three -voltmeter method is a scheme for determining the phase -angle between two periodic, steadystate, alternating voltages by measuring the magnitude of three voltages

between terminals of the source. Kinnard' s method utilizes three voltmeters to measure the three voltages. Figure 1 shows how the three

voltmeters are connected. The three terminals are marked 1, 2, and 3.

If an alternating voltage supply is connected between terminals 1 and 2,

and if another supply, operating at the same frequency, is connected

between terminals 3 and 2, then three voltages designated E , E ,
and E exist between terminals. From Kir chhoff s Voltage Law, the

vector sum of the voltages must equal zero, i. e. , E. _ + E_, + E_ . = 0.

Figure 2 shows the phasor voltage relationship for 9 approximately

60°. From the trignometric identity for oblique triangles for any 9

from 0° to ±180° ,

.l l E 12l

2

*l=a|

2 -.|S

3 l|

2

9 = cos . (1. 1)

2 |E

12 ||E23 |

or expressing voltages in rms and in terms of voltage ratio,

-1 /

E

12

E

23

E

31

E

31

' ^z7 +2E

!2

2E

!2

E

23

Equation (1. 1) gives the phase-angle between voltages E and E in

terms of the voltage magnitudes between each pair of terminals. Since

the equation is one of voltage ratio, the voltage level is not of primary

interest. In other words, the observer is not really interested in measuring the voltage levels E , E , and E accurately, but rather is

interested in determining the ratios between the voltages accurately.

The measurement of voltage ratio is accomplished through the use of an

inductive voltage divider and several thermal voltage converters.

The standard is used to determine the phase -angle between two voltages generated by a power supply. In order to calibrate a phase meter,

connections can be made directly to the standard. The phase meter

indication can then be compared with the indication from the standard.

A phase -shift device can be calibrated by using impedance compensation

[Park and Cones, I960] and voltage E. as the source voltage.
Appendix 1 describes shielding for the control of capacitance effects

in connection with circuitry to be discussed in section 5. Appendix 2

gives information related to typical measurements for phase -angle while

appendix 3 discusses phase-angle errors for periodic waveforms other

than sinusoidal.

2. Plan

When voltages E and E are made equal in magnitude (=E) by

a technique to be described later, then (1. 1) is simplified as follows:

4» = cos

-1

1 - 31

(2.1)

But voltage E can be expressed in terms of voltage E 1 as the result

of two different inductive voltage divider arrangements. The first

arrangement is for phase angles from 0° to ±60° , where

E

31

=nE < n < 1

and therefore,

(j> = cos

-1

1 - n

(2.2)

The second arrangement is for phase -angles from ±60° to ±180°, where

E =-E . 31 n1 5» n ^ 0. 5

and therefore,

cj> = cos

-1

1 - 2n J

(2.3)
However, for values of cj) near 180° the- usefulness of (2. 3) is poor,

because n is near 0. 5 where a change in n affects the value of § by

a relatively large amount.

The particular inductive voltage divider arrangements are described

in detail in sections 4 and 5.

3. Voltage Supply and Detector Systems

The supply and detector systems are an important part of the phase

angle standard. A supply fixture provides convenient physical terminals

to which voltages can be supplied, voltage comparisons can be made,

and phase -angle standards can be connected for calibration. Figure 3

indicates how the supply fixture is provided with voltages of variable

phase. The voltage level for each phase is most conveniently controlled

by a potentiometer (voltage divider) connected between the generator and

the power amplifier. The variable -phase generator indicates the approximate phase difference between the reference and variable phases so that

any ambiguity in angle may be resolved. In the experimental setup,

transformers T and T had 100 -volt primaries and tapped secondaries of 1, 2, 3, and 4 volts. Rated output power was 50 watts. The

voltage supply system was designed to have low impedance outputs, so

that different loads at the supply fixture did not significantly change the

voltage levels.

To help establish definite relationships between the three voltages

E , E , and E , thermal voltage converters are used. A thermal

voltage converter (TVC) consists of a resistive multiplier in series with

the heater circuit of a thermoelement. The thermocouple circuit of the

thermoelement is the output of the thermal voltage converter. If the outputs of two thermal voltage converters are differentially connected and

if the inputs are connected in parallel as shown in figure 4, then shorttime amplitude effects tend to be eliminated. The outputs at rated input
voltage are typically about 7 mV dc. The Lindeck potentiometer adds a few microvolts of the correct polarity as required for galvanometer null.

A null indication then represents equal input voltages applied to the two

thermal voltage converters.

A major source of short-time amplitude instabilities at the supply

fixture is within the oscillator section of the phase generator. Both

reference phase and variable phase voltages at the supply fixture fluctuate alike, because the oscillator is common to both voltages. So, instead

of connecting the inputs of the thermal voltage converters in parallel (see

fig. 4), assume the two inputs were connected to the proper supply fixture

voltages. Then because of the differential connection in the thermal voltage converter outputs, the galvanometer indication will tend to remain

steady in the presence of changes affecting both outputs, although sensitive to changes in either output alone.

Some variable -phase oscillators of older desig'n do not have their

outputs independent of phase or frequency setting. For convenience, the

operator does not want to change to a new phase -angle and then drastically

change the oscillator voltage outputs back to their former level. Good

practice in the use of a thermoelement dictates keeping the applied heater

voltage somewhat constant in magnitude not only to avoid possible burnout

to its heater, but also to avoid undesired thermocouple voltage drift. The

easiest solution is to use modern oscillators which have specifications

limiting this kind of interaction.

4. Phase -Angle Measurements, 0° to ±60°

The circuit used to obtain accurate phase -angle measurements from

0° to ±6° will be described later. An inductive voltage divider is used in

the circuit shown in figure 5 to obtain accurate phase -angle measurements

between ±6° and ±60°.
To determine phase-angle (J> , the following calibration procedure

is used:

1. Connect TVC in parallel with TVC., apply reference -phase

voltage (E ) and adjust Lindeck potentiometer #1 for galvanometer

null (G n ). The null now indicates equality of input voltage (magnitudes only) applied to the thermal voltage converters.

2. Reconnect TVC as shown in figure 5 and adjust the variablephase voltage (E ) for the same null. Now E = -^ = E.

3. Connect TE#1 in parallel with TE#4 and adjust Lindeck

potentiometer #2 for a null in galvanometer G . This null now

identifies the condition when the voltage across TE#1 equals E .

4. Reconnect TE#1 as shown in figure 5 and adjust the inductive

voltage divider and the decade resistance box for a minimum in the

a-c detector indication and a null in galvanometer G_. The best

measurement is realized when the two galvanometers are at null at

the same time. The value n is read directly from the inductive

voltage divider and used with formula (2. 2) to determine phase -angle

*2=

<f>2 ~ cos

-1

2

'-T

The current for Thermoelement #1 is supplied through the decade

resistance branch of the bridge circuit to avoid a loading error on the

inductive voltage divider. The heater circuit of TE#4 is above ground,

but a measurement error due to stray capacitance is not expected. At

worst (<j> = 60°), TE#4 will be above ground by only about one volt rms.

This error will not be significant provided the thermoelement is constructed with an insulating bead between the heater and thermocouple circuits.
Providing they are reasonably small, a-c response errors in the

thermal voltage converters do not affect the determination of phase-angles.

The reason is that when the thermal voltage converters are connected in

parallel and their outputs equated, any a-c error differences (which is a function of frequency) between them are, in effect, eliminated by the

adjustment of potentiometer #1. Likewise, any differences in a-c error

between TE#1 and TE#4 is accommodated in the adjustment of potentiometer #2. Ordinarily, the a-c response of the decade resistance box does

not affect the measurements because it serves only to supply current to

TE#1. If however, the decade resistance box is excessively poor at high

frequencies, an a-c detector null will not be obtained resulting in an

imprecise inductive voltage divider indication.

The resistance of the thermal voltage converters (including leads,

multiplier resistor, and thermoelements) need not be known, but they

must remain constant over the period of calibration, or at least change

in the same proportion and direction. Multiplier resistors constructed

at NBS have shown excellent short-time resistance stability (typically

less than 0. 001% in ten minutes) and have a very flat frequency response

from dc to 100 kHz. Thermoelements have a high temperature coefficient

but if TE#2 and TE#3 as well as TE#1 and TE#4 are placed in close physical proximity in a thermal lag enclosure (such as a massive copper block

or a thermistatically controlled oven), then temperature changes, if they

do occur, will tend to affect the thermoelements equally and therefore

create no galvanometer difference indications.

Unfortunately, a limitation is reached with thermoelements for values

of n between and 1/10 or in terms of phase angle, between 0° and

±6°. This limitation is reached for small phase angles because voltage

E . is not large enough to energize TE#4 (and TE#1) with resulting good

sensitivity. Fortunately, as table 1 indicates, the "maximum phase-angle

error" is nearly constant with "resultant allowable voltage -ratio error"
(expressed in microvolts) over the phase -angle cj> range of 60° to 0°.

This condition results from a study of (1. 1). The uncertainty in setting

voltage E = E = E is easily done to within ±0. 001% (which is negligible) and is independent of phase-angle. The major source of uncertainty

affecting phase -angle occurs within the determination of the ratio E /E

as identified by a study of (2. 1). As the phase-angle cj> decreases from

60°, voltage E decreases (E remains constant). As E decreases,

a larger uncertainty in E ../E is allowable for the same phase -angle

uncertainty because of the action of the cosine term near 0°. For example, at a phase-angle of about 6°, E /E need be only within ±0. 3%

for a maximum phase -angle error of ±0. 017° (see table 1). The ratio

uncertainty restriction of ±0. 3% can easily be met by using an a-c differential voltmeter to give the same indication at the output of the inductive

voltage divider as was obtained across terminals 1-3. Since the a-c

differential voltmeter is a high impedance device and takes very little

current to operate, the loading on the inductive voltage divider is not

significant and the decade resistance box and a-c detector can be removed.

An ac-dc differential voltmeter with a null sensitivity better than 1 mV ac

is recommended.

When the phase angle cf> equals 60°, the inductive voltage divider,

decade resistance box, and a-c detector are no longer needed and can be

removed from the circuit shown in figure 5. For this phase-angle, TE#1

and TE#4 will be operating at 1. volt, which is 67% over the 0. 6 volt

recommended operating voltage for the thermoelements. A small possibility of heater burnout exists, especially for transient overloads above

this voltage level, but burnouts do not ordinarily occur except for maintained overloads in excess of 100% of 0. 6 volt. If a thermoelement must

be replaced, it can be easily done with no preliminary calibration, such

as d-c reversal, insulating bead efficiency, ac-dc difference, or sensitivity curve determination, such as is necessary for ac-dc transfer

8
standard calibration. In other words, the thermoelements are referenced

against each other just before each determination of phase -angle.

5. Phase-Angle Measurements, ±60° to ±170°

The circuit used to obtain accurate phase-angle measurements

between ±60° and ±170° is shown in figure 6.

In order to obtain phase-angle measurements, the following calibration procedure is used:

1. Connect TVC_ in parallel with TVC., apply reference-phase

voltage and adjust Lindeck potentiometer #1 for galvanometer null.

The null now indicates equality in input voltage applied to the thermal

voltage converters, TVC, and TVC .

2. Reconnect TVC_ as shown in figure 6 and adjust the variable

phase voltage for the same null. Now E = E = E.

3. Connect the inductive voltage divider and associated circuitry

input in parallel with TVC. and adjust Lindeck potentiometer #2 for

galvanometer null. The decade resistance box must be set to zero

and the inductive voltage divider set to unity. This null now identifies the condition when the voltage across the inductive voltage

divider equals E. and corresponds with n = 1.

4. Reconnect the inductive voltage divider as shown in figure 6

and adjust its dials for galvanometer null and the decade resistance

box for an a-c detector minimum indication. The value n is read

directly from the inductive voltage divider and used with formula

(2. 3) to determine phase-angle (j> :

Cm

4> = cos

-1 l~i

2n

2
In this setup, a-c response errors in the thermal voltage converters

do not create a measurement error because when the Lindeck potentiometers are adjusted, any differences in a-c error with frequency are

accommodated. The best measurement precision would be realized if a fourth thermal voltage converter were connected permanently in parallel

with TVC and the resulting thermocouple output compared with TE#4.

This change would allow E,_ to equal E at the same time E

equals nE . as a consequence of nulling both galvanometers at the same

time. Shielding is used around the circuitry connected to E . and is

described in appendix 1. As the phase-angle c}> approaches 170°, the requirement for precision voltage comparisons relative to a specific phase -angle uncertainty

rapidly becomes more difficult to achieve. In fact, for a phase -angle

error of ±0. 017° near 170°, the total voltage comparison uncertainty

must be within ±0. 0013%. Proportionally, for a phase-angle error of

±0. 05° , the total voltage uncertainty cannot be larger than ±0. 0038%.

For phase-angles 170° to 180°, some other method is required for good

precision. Figure 7 shows the maximum phase -angle error as a function

of phase -angle for different allowed voltage -ratio errors.

Appendix 2 summarizes data obtained for a phase -angle of approximately 70°. The data show good measurement repeatability as well as

high sensitivity.

A disadvantage with the standard, apparent at this stage of its development, is its "blind spot" near 180°. The balancing procedure could be

simplified by the use of automatic balancing equipment to keep E = E

and by careful planning of plug -in components for use when changing

setups.

The standard is used with sinusoidal or slightly distorted sinusoidal

waveforms. Appendix 3 shows that the phase-angle <J> , determined by

the standard, is independent of the phases of the harmonics. It also

10
shows that if the maximum magnitudes of the harmonics remain constant

during the measurement, the standard can be used to calibrate a narrowband phase -angle device. If the maximum magnitudes of the harmonics

in the two power supplies are equal, measurement error due to waveshape

will not occur. If the harmonics are not equal and the difference is small

compared with the fundamental, a corrected phase -angle equation is used

as derived in the appendix. If the phase -angle device has the same bandwidth as the standard, or all significant harmonics are responded to by

both the standard and the device, then the restrictions of equal harmonic

magnitudes are eliminated.

6. Conclusion

The principal advantages of this phase -angle standard are its broad

frequency range --potentially 10 Hz to 100 kHz, its self -alignment features,

its high accuracy, and its ability to be used with sinusoidal or slightly

harmonically-distorted waveforms. If the measurement errors that have,

been identified are accounted for, there seems little reason why more

sensitive galvanometers cannot be used and even higher accuracies realized in the measurements of phase -angle.

7. References

Kinnard, I. F. (1956), Applied Electrical Measurements, pp. 219-220

(John Wiley and Sons, New York, N. Y. ). Lawrence, R. R. (1922), Principles of Alternating Currents, pp. 82-85,

(McGraw-Hill Book Company, Inc. , New York, N. Y. ). Park, J. H. , and H. N. Cones (I960), Phase angle master standard for

400 c/s, J. Res. NBS 64C (Eng. Instr. ), No. 3, pp. 229-240.

11
8. Additional References

Gertsch Products Inc. (I960), Use of Gertsch standard ratio transformers

for low impedance voltmeter calibration, Gertsch Engineering

Bulletin No. 2.

Hermach, F. L. (1952), Thermal converters as AC-DC transfer standards

for current and voltage measurements at audio frequencies, J. Res.

NBS 48, No. 2, pp. 121-138.

Zapf, T. L. , and R. V. Lisle (1964), Comparison calibration of inductive

voltage dividers, ISA Trans. 3, No. 3, pp. 238-242.

9. Appendix 1

Shielding for Control of Capacitance Leakage Effects

Proper shielding is a requirement for accurate phase-angle measurements from ±60° to ±170° (see fig. 8). Shielding is required in order to

make certain that capacitance leakage paths do not change when terminal

"a" is grounded, then ungrounded at the supply fixture. For configuration

1, terminal" a" is grounded and distributed capacitances C_ and C are

across voltage source E . Distributed capacitance C, does not exist.

For configuration 2, terminal "a" is ungrounded and distributed capacitance C is across voltage source E . Likewise, C~ is across E

and C is across E . These capacitances are all across the low

impedance power supply, where the leakage current does not affect the

measurement. With efficient shielding, capacitances within the shields

are fixed and independent of whether terminal 1

' a" is grounded or ungrounded. This independence obviates a source of systematic error when the

circuitry is referenced in configuration 1 and used for phase-angle measurements in configuration 2.

12
10. Appendix 2

Experimental Setup, Calibration Procedure, and Summary of

typical measurements for phase -angle cf> approximately 70°

and frequency 400 Hz.

The calibration procedure used was as follows:

1. Turn on the two voltage supplies and adjust each to about

0. 5 volt with a vacuum tube voltmeter. Connect leads "A" to "Z"

terminals (refer to fig. 9 and fig. 10) and with the switch in #1

position, adjust Lindeck potentiometer A for a galvanometer null

indication.

2. Reconnect leads "A" to "X" terminals and adjust the E

supply for an exact balance in the galvanometer. Voltage E 1 is

now equal to voltage E .

3. Connect leads "B" to "Z" terminals and with the switch

in #2 position adjust Lindeck potentiometer B for a null indication

on the galvanometer. The inductive voltage divider must be at setting 0. 9999X and the decade resistor at setting 0. ohms.

4. Set phase -angle for about 70° , reconnect leads "B" to "Y"

terminals and adjust the decade resistor for a null indication on the

galvanometer. Then adjust the inductive voltage divider for a minimum indication on the a-c detector and readjust the decade resistor.

The formula used is (2. 3):

(j> _ = cos

-1 i—

1

2a

2

-.

where n is the inductive voltage divider indication.

13
A direct indication of reading precision was obtained when the switch

was placed in the #2 position and the phase was changed from 60° to 59°.

The galvanometer, which indicates the difference between E and E , changed from a reading of 43 cm to about 3 cm. Therefore:

Al° = 400 divisions on the galvanometer scale (40 cm)

A0. 1° = 40 divisions (4 cm)

A0. 01° =4 divisions (4 mm)

No fluctuations in the galvanometer indications was observed. The interconnection of the thermoelements is effective in eliminating the effects

of power supply variations. A slight drift of galvanometer indication

with time was observed, however, with all measurements. It is believed

it could be largely eliminated if all thermoelements are placed in a massive thermal conductor such that changes in ambient temperature tends

to increase each thermoelement output in the same direction and magnitude.

The calibration procedure was then repeated three additional times.

The data indicated that variations in voltage ratio are less than ±0. 00001

in the region where n = 0. 84222. This is equivalent to a variation in

phase-angle of ±0. 001° in the region of 7 3°. This excellent repeatability

would not be expected at phase -angles near 180°. The data indicate

stability in phase -angle for both the variable -phase generator and the

standard. It was considered highly unlikely that the generator was changing phase-angle (drifting) at the same rate and direction as the standard.

However, a source of systematic error exists, in the use of an unshielded

isolation transformer; its effective voltage-ratio changes depending upon

whether its input is grounded or not. The isolation transformer was used

to avoid shorting E because of the ground in the inductive voltage

divider circuit. This source of error is avoided by using a shielded

isolation transformer, but the problem is better solved by the proper

use of circuit shielding as shown in appendix 1.

14
1 1 . Appendix 3

Phase -Angle Errors for Periodic Waveforms other than Sinusoidal

Harmonic voltages may be expressed algebraically in terms of the

scale of angles for the harmonics, as follows:

e = E . sin(cot+0 .) + E _ sin 2(oot+9 ,) + E _ sin 3(cot+9 ,) + ml 1 mc 2 mi i

+ E sin n(oot+9 ) . (A-l) mn n

To determine the effective or root -mean -square value as responded to

by the thermal voltage converters, (A-2) must be used:

!/2 T

E -:

! ; e

2

dt IH (A-2)

where T represents the waveform period. Reference [Lawrence, 1922]

proves by derivation that the effective value of a non- sinusoid is the

following:

E.+E_+E + + E N ml m2 mi mn \ . . _. E = r . (A-3)

This equation for rms voltage is a function of the relative magnitudes of

the fundamental and harmonics, and is independent of the phases of the

harmonics relative to the fundamental.

For purpose of illustration, assume a power supply generates two

voltages described by

e,_=E . sin(cot+9 .) + E . sin j(cot+9 ) 12 ml 1 mj 3

15
where j represents the j harmonic and,

e„„ = E' , sinfcot+0. 1 ) + E' . sin j(cot+9 ' ) 23 ml 1 mj 3

which as a result of (A-3) reduce to

v

2

»

2Nl/2

E . + E . E = -^ *—£ (A-4)

12 n/T

and

E' _ 2 + E' ml mj

1/2

E = \^-l "iUZ . (A-5)

23 N^

The standard described in this paper uses the law of cosines to determine

the phase -angle cj> between voltages E ] and E . In the procedure

we force E__ to equal E 1n and E„, to equal either nE.. or —

E

10 , 23 12 31 12 n 12

depending on the inductive voltage divider configuration. In this case,

because of the fact that the law of cosines is an equation of ratio, the

components of E and E >

_ need not be known. It is necessary however, that the components remain constant in magnitude during the

measurement.

In order to use this standard to calibrate another phase-angle device,

no measurement error due to waveform will result providing the device

is also broad-band because it responds to the same harmonics of significancei However, if the device is narrow-band, it will not respond to

some harmonics and a measurement error due to waveform could result.

16
Consider the narrow-band device as equivalent to another standard

which uses internal filters to block all harmonics above the fundamental.

When the adjustment of voltages E 17 = E is made with the broad-band

standard, the narrow-band device may see an error voltage e (assuming

e represents the error voltage due to harmonic voltages) defined by:

v '«* e E

12 = E23

+

-N/T

where

E

12

E / + ml

1/2

and

E

23

E' , 2 + ml

1/2

or

e = E . - E ' . (A-6) ml ml

Referring to the balance with the broad-band standard and using (A-4)

and (A-5),

E

12

= E

23

or

2 Z\ 1/Z

f , 2 Z- 1/Z

E . + E . ) = ( E* . + E1

. ml mj / V ml mj

17
or

2 2 2 2

E , - E 1 . = E' . - E . ml mi mj mj

If the i harmonics in both E._ and E_ supplies have the same J 12 23

magnitude, then E . = E . ' and mj mj

2 . 2 • E -E' = ml ml

(E -E ')(E + E ') = ml ml ml ml

Since E , + E ' - is trivial, then the error voltage e = E , - E .

ml ml ml ml

must equal zero. In other words, if the magnitudes of the harmonics are

equal in the power supplies, then the device may be calibrated without

error due to waveshape, even though the device is narrow-band. On the

other hand, if the harmonics are unequal, then a corrected phase -angle

equation must be used. The equation is developed as follows:

*

-1 rE

COS

12

*2 „ *2 „ *2

+ E - E„.,

23 31

2E

12

E

23

where E., =E ./*fz, E^ " = E VnTz, E * = kE . ' I^Z where k

12 ml7 23 ml7 31 ml ' may be n or 1/n depending upon the inductive voltage divider connection:

4> ' = cos

-1 rEml

2 ,2 2,2 + E' - k E' ml m.

2E E ' ml mi

18
but from (A-6), E ' =E . -e: ml ml

(j> ' = cos

-1 r E

2 + (E -e)

2 -k2

(E -e)

2

ml ml ml

2E (E .-e) ml ml

-1

= cos

E (E -e) (1-k ) ml ml

+ 2(E -e) ml 2E

mJ

If e is sufficiently small compared with E , such that the e term ml

may be neglected, then the term E /Z(K -e) can be multiplied by mi m

E . + e/E + e to obtain E . + e/2E . , so ml ' ml ml ' ml

*z

-1

= cos

E _ + e + E ml -k2E

mJ ml - e + k e-, 2Eml

-1

= cos 1 k A __e

2 V Eml /J

Substituting k = n or 1/n gives us formulas similar in form to (2. 2)

and (2. 3):

<j> ' = cos

-1

1 a( X __!.

2 V E N m

for 0° to ± 60 c (A-7)

1 /J

and

j _

d>T

" = cos

2

-1

2n N ml

for ± 60° to ± 170°. (A-8)

Voltage E ^ can be obtained closely enough by measuring and using the

E value while voltage e can be obtained using a harmonic wave analyzer

19
Equations (A-7) and (A-8) are mathematically valid and phase-angle is

meaningful only if e is small compared with E .In other words, the

distorting harmonics must be small compared with the fundamental. For

example, if <j> - 60° (k = 1), a 0. 1% distortion difference between the

two power supplies would be equivalent to a 0. 01% voltage error which

is, using figure 7, about 0. 005°. Note that these equations also predict

the effect on phase -angle that an offset from galvanometer null would

create. (E 1 = -^?o + e » where e is the difference voltage in the supplies which creates an offset galvanometer deflection.

)

20
r-o

UJ

u_oO

CD

VI 2

CD

rr

O o

r- or o or

hi

ir

i- o rn

UJ

or 5

cr >2 i UJo

<x <

rorUJ

!jo> o2

UJ1 => OQ<

UJ ^ _l o o .1 2 _J < <

UJ

CD<I

0_£oX» % •—

UJ

_iCD<

RESULTANT

ALLOWABLE

VOLTAGE-RATIO

ERROR I 8 8

+1 ro rO 5+1+1

to s° 38

00

O r-> o Jo 9 d d ^

+i +i +i

PHASE

ANGLE

4>2 60

.000°

5.732° 0.573°

VOLTAGE

LEVEL E

3|

(VOLTS) - o - 6 o

VOLTAGE

LEVEL E

(VOLTS) -H

—H -rt

MAXIMUM

PHASE

ANGLE

ERROR

FOR ±

0.017° ±

0.017° ±

0.017 °

21
THREE -VOLTMETER METHOD

VOLTAGE PHASOR DIAGRAM

IMAGINARY AXIS

Figure 2

22
VOLTAGE SUPPLY SYSTEM

VA

O

RIABLE-PHAS

GENERATOR

ERENCE VARIA SE PHAS

E

REF

PHA

BLE

E o

POWER AMPLIFIER

60 WATT

POWER AMPLIFIER

60 WATT

100 v 100 v SAAJ

lv

T, SUPPLY

FIXTURE

YYV

lv \

REFERENCE

PHASE

fE I2 1E 23 v

VARIABLE

PHASE

\^ T r

T

2

, AUDIO VOLTAGE STEP -DOWN TRANSFORMER

Figure 3

23
or

Id

LLl

ITLU>oo

LU

(£> <t <> o> < <

CD 2 O ^ Q £

.. I-u>

LU

f-

(S) >-

if) o

Ld

I-LUQ

-A/WW >

lcvi o>i _i 3C

<2>

UJ < Li Q- 55

24
CO

LU

UJ

q:

co<

uj

<° CO +i <X

Q.

oro

LUIO

CO o

I-o

CD

+1

cr

LU>zo

UJo<o><

CC

UJXo>

(V) O>>

CO

hLU

LU

_lLUO

orLUXo>

CD6

LU

LU

CO

LU

LU

_l

LUO

o:LUXo>

ind

LU

ILU

or

i-LUQQZ<

ceLUu.

Xo

CDLUOz

COen

LU

oro < < Q Q

LU

X ^3 CO

1-. X

or o o m

iCJ IE LU

1-

LUQ

LUO<

<> LU < o

or

LUQ>Q

LU

CDo>

LU>

I-O3QZ

cc

LU3

LUOz

COCO

UJ

C£ > E

t-i or

in3 i I
SCHEMATIC FOR PHASE MEASUREMENTS

± 60° TO ± 170°

DECADE R BOX

1 VOLT THERMAL

VOLTAGE CONVERTERS

TE#2' I 0.6 VOLT

TE# 3'

[ THERMOELEMENTS

TE # 4, '

Figure 6

26
PHASE ANGLE vs. MAXIMUM PHASE ANGLE ERROR

FOR SEVERAL VOLTAGE COMPARISON ERROR ALLOWANCES

i 0.140

± 0.120 —

+ 0.100

(/) UJUJ

cco

UJQ

*— + 0.080

01o

or

orUJS3

+ 0.060 S

X

<

t 0.040

± 0.020

PHASE ANGLE (DEGREES)

Figure 7

27
SHIELDING DETAILS

FOR PHASE MEASUREMENTS

± 60° TO ± 170°

TO LIN DECK

DETECTION SYSTEM

m

•HI—

TVC 3 -VWWW

1 DECADE R BOX *"] 1

1

IT, ISOLATION TRANSFORMER

IVD, INDUCTIVE VOLTAGE

TRANSFORMER

CONFIGURATION i

b 1 E 3I \ SUPPLY /

\ FIXTURE /

E I2\ /E 23

f\ Y?

a -=-

CONFIGURATION 2

Figure 8

28
Q.

Z>

LU

2

a:LUx

O

<

or<

o

o

z

or

o>

3

r-^EEF-r*

29
SCHEMATIC DIAGRAM

OF EXPERIMENTAL SETUP

AAA^AAr

T #2 #1 -T

Figure 10

30

GPO 831-025



U.S. DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20230

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERCE

OFFICIAL BUSINESS