Var Measrmnt

Accuracy Standards for Phase-Shifting Transformers for VAR

Measurement

JAMES M. VANDERLECK, SENIOR

MEMBER, IEEE

Summary-New accuracy standards are proposed for phase-shifting transformers generally applied in polyphase circuits for var (reactive volt-amperes) measurement. Limiting values of ratio correction factor (RCF) and phase angle defect (PD) are given for three proposed standard accuracy levels, namely, 0.3 per cent, 0.6 per cent, and 1.2 per cent. These levels represent the maximum error in var measurement owing to the phase-shifting transformer. The limiting values appear as circle diagrams. The proposals bring the standards up to the same level of development as those for instrument current and potential transformers.

EXISTING STANDARDS The American Standards Association (ASA) Code for Electricity Meters' requires the components of output voltage, expressed as a percentage of input voltage of the corresponding autotransformer across which the respective components are measured, to be within one per cent of the theoretically correct values. This requirement does not theoretically limit the RCF and PD of the output voltage, but depends on the fact that in INTRODUCTION practical designs of phase-shifting transformers, satisflpMHE AIMl of the paper is to propose new accuracy factory results are usually achieved. Obviously, it is | standards for those phase-shifting transformers impossible to make a definite statement regarding the that are used as auxiliary transformers in con- limit of error in var measurement based on the ASA junction with wattmeters for var measurement. Fig. 1 requirements. Unusual designs could cause intolerable is an example and shows a typical phase-shifting trans- error in var measurement and yet conforim to ASA former for application to three-phase, three-wire cir- requirements. The one accuracy standard that is mandatory in cuits. The proposed standards apply without restriction on the classification of the circuit in respect to number Canada for phase-shifting transformers for revenue metering purposes is Specification No. 4 for Approval of phases and wires. of Type of Instrument Transformers.2 Fig. 2 is the geometric figure defining the accuracy requirements of 9P0" this specification. This figure is diamond shaped and has limiting values of 1.012 and 0.988 for RCF and +31.2 minutes and -31.2 minutes for PD. Consequently, the maximum error in var measurement depends on the power factor (PF) of the load. For example, at 0.8 PF the limits of error are +0.7 per cent, and at zero PF, the limits of error are + 1.2 per cent [with R error expressed as a percentage of the volt-amperes (VA) in the circuit]. These results are derived in the ApFig. 1 Schematic diagram of typical phase-shifting-transformer OUTPUT VOLTAGE 6- 7 (PHASE B LAGGED 7 6

OIUTPUT VOLTAGE 4- 5 (PHASE A LAGGED 90°) 4 5

2

PHASE A

INPUT VOLTAGE

2

PHASE e

INPUT VOLTAGE 3

-

2

PHASE ROTATION 1-2-3

pendix. No other ASA or Canadian Standards Association Present accuracy standards are poor because they are (CSA) standards appear to cover specifically the accunot based on the concept of permitting as much ratio racy of phase-shifting transformers. error and PD as possible within the confines of an arbiPROPOSED STANDARDS trary limit of error in the measurement of var. The standards recommended herewith are similar to those The recommended geometrical figure defining the in existence for potential and current transformers, in limits of RCF and PD is a circle based on rectangular that a geometric figure is given within which the RCF co-ordinates for which the unit of phase defect, the and PD must fall. The bounds of the geometric figure radian, is equal in length to unit value of RCF. Fig. 3 limit the error in var measurement to a standard value, is constructed this way. as explained subsequently. To conform with the pattern of existing ASA and three-phase, three-wire circuit.

Manuscript received July 16, 1964. Presented at the IEEE Summer General Meeting & Nuclear Radiation Effects Conf., Toronto, Ont., Canada, June, 1963. The author is with Hydro-Electric Power Commission of Toronto, Ont., Canada.

l "Code for Electricity Meters," American Standards Association, New York, N. Y., ASA C12, sec. 561; 1941. 2 "Specification No. 4 for Approval of Type of Instrument Transformers," Dept. of Trade and Commerce, Standards Branch, Ottawa, Canada; effective April 1, 1952.

89

Authorized licensed use limited to: VELLORE INSTITUTE OF TECHNOLOGY. Downloaded on March 17, 2009 at 05:41 from IEEE Xplore. Restrictions apply.

°-rt7Y-C~LAS

90 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT June-Sept. LAGGING

variation of voltage and burden. That is, the limits defined by the circle diagrams shall apply from ten per cent below to ten per cent above rated input voltage at rated frequency, and from zero burden on the transformer to rated burden at the nominal factor PF of rated burden. This requirement is similar to that in the ASA Standard.3 It is recommended that the limits defined by the circle diagrams be specified to apply when all output phases are burdened equally, because such burdening is normally encountered in practice. It is proposed that the standard rated burdens for phase-shifting transformers be identical to those for instrument transformers in ASA and CSA standards for instrument transformers. Three of these are designated W, X, and Y and have VA values of 12.5, 25, and 75, respectively.

LEADING

L012

L1008

:

0

U 1,004 z

0

.--_

1.000

W

§ 0.996

Q992

0.98I

30

20 10 0 10 20 30 PHASE DISPLACEMENT

ERROR (MINUTES)

Fig. 2-Accuracy limits for phase-shifting transformersCanadian Department of Trade and Commerce.

10

2

1.2

-0.J

-18

PHASE

0

t0.4

-34.4 -41.3

-20.6 -27.0

+1.2

LEADING

LAGGING

-48.1

+0.S

DEFECT-CENTIRADIANS -6.9

+6.9

1|3.7 PHASE

+20.6

+13.7

0

DEFECT-

+A1,3

MINUTES

COR?RECTION FACTO)R AND PHASE DEFECT PHASE-SHIFTING TRANSFORMER SHALL SE WITHIN

THE RATIO

INDICATED

CIRCLES

+48.1

±34.

+27.5

FOR THE RESPECTIVE

ACCURACY

OF THE THE

CLASSES

Fig. 3-Standard accuracy classes for phase-shifting transformers, limits for 0.3, 0.6, and 1.2 accuracy classes.

GSA

different

standards, 3 '4

accuracy

instrument-transformer

three

accuracy classes would

established

be

to limit the error in var measurement, expressed as a 0.6 per

percentage of VA in the circuit, to 0.3 per cent,

cent, and 1.2 per cent, respectively. Each accuracy class would be represented by a circle, defining the limits of RCF and PD as shown in Fig. 3. It is likely, that the

however,

1.2-per cent accuracy class would be almost

always specified

in

practice.

As shown

in

Fig.

3,

the

limiting values of RCF and PD of the 1.2-per cent circle are

1.012

and

0.988

for

RCF

and

41.3

minutes

(±0.012 radian) for PD. The proof that a circle gives the limits of RCF and PD for a given percentage error in var measurement when load PF varies is given in the Appendix. It

is suggested

that

any

new

standard

shifting transformers might well copy the

for

phase-

GSA Stand-

ard for instrument voltage transformers4 in respect of

3 "American

Standard

Requirements,

Terminology,

and

Code for Instrument Transformers," American Standards

C57.13-

tion, New York, N. Y., ASA and Test

4"Specification

1954. Instrument

Code for

Test

Associa-

Transformers,"

Canadian Standards Association, Ottawa, Canada, CSA C13; 1958.

TEST METHODS The proposed standards require test methods comparable to those for instrument potential and current transformers. The test method in the ASA Code for Electricity Meters' would be unsatisfactory because the method does not yield RCF and PD data. A testing bridge for measuring the RCF and phase angle of instrument potential transformers, however, can be used to determine the RCF and PD of phase-shifting transformers. Tests may be conducted with either singlephase or polyphase voltage supply to the phase-shifting transformer, but polyphase supply is recommended. With polyphase voltage supply, the determination of RCF and PD is simplest when the transformer is burdened on all output phases simultaneously with impedors having standard-burden values. The polyphase voltages need be balanced only approximately. The potential transformer testing bridge can be connected to each single-phase transformer in turn to measure the ratio and phase angle of each tap voltage. For example, in Fig. 1 the testing bridge can measure directly the RCF and phase angle of the voltage across terminals 4-2 relative to the voltage across terminals 1-2, the nominal ratio being 1: V3. A simple mathematical operation will convert these bridge indications of RCF and phase angle to values of RCF and PD to be applied to the output voltage across terminals 4-5. A similar test involving terminals 2, 3, and 5 and a similar mathematical operation yields additional values of RCF and PD to be applied to the output voltage across terminals 4-5. The sum of the two values for each of RCF and PD yields the required final values of RCF and PD for the output voltage 4-5. Similarly, required values can be determined for any output voltage of any phase-shifting transformer for var measurement. With single-phase supply, the testing bridge is applied in the same manner as for polyphase voltage. A standard-burden impedor, however, will not apply the correct burden for testing purposes. Any known burden can be connected to the same pair of terminals (4-5 in

Authorized licensed use limited to: VELLORE INSTITUTE OF TECHNOLOGY. Downloaded on March 17, 2009 at 05:41 from IEEE Xplore. Restrictions apply.

1964

Vanderleck: Standards for IPhase-Shifting I

_

t,

the example) as connected to the testing bridge, and then the measured results of RCF and phase angle must be corrected to account for the difference between the applied burden and a standard-burden connected polyphase. Because this correction is involved, polyphase testing is preferred. ERROR IN VA Because var readings are frequently used along with watt readings to determine VA for billing purposes, it is of interest to consider the error in VA owing to the error in var measurement. Eq. (7) developed in the Appendix reveals that the percentage error in VA is the sum of two components, one related to watt measurement and the other related to var measurement. The latter component is equal to the percentage error in var measurement times the sine of the phase angle of the load. Percentage values are expressed as a percentage of the VA in the circuit. From the foregoing relationship it follows that by selecting a standard accuracy class to limit the error in var measurement to 1.2 per cent of the VA, for example, the percentage error in VA owing to the error in var measurement will never exceed 1.2 per cent, and must be less than 0.72 per cent if the load PF is greater than 80 per cent, for example. If there is a need to limit the effect of the error in var measurement to half as much, that is, to 0.36 per cent in the determination of VA, then the 0.6-per cent accuracy class can be specified for the phase-shifting transformer.

CONCLUSION 1) New limits of RCF and PD have been recomImended for phase-shifting transformers for var measurement. 2) Unlike existing standards, the new limits permit maximum values of RCF and PD without exceeding a fixed error in var measurement, expressed as a percentage of VA in the circuit, irrespective of the PF of the load. 3) The proposed limits are shown as three circle diagrams in Fig. 3. The limiting errors in var measurement are 1.2 per cent, 0.6 per cent, and 0.3 per cent, expressed as a percentage of VA in the circuit. 4) The effect of the error in var measurement on the determination of VA is simple; the percentage error in VA equals the error in var measurement, expressed as a percentage of the VA, times the sine of the phase angle of the load. 5) Accuracy test methods have been suggested using a testing bridge for instrument potential transformers. APPENDIX

A. Definitions The RCF and PD of a phase-shifting transformer are terms applied to the vectorial relationship between output and input voltages. The RCF and PD of an output voltage are the RCF and phase angle with reference

-

7

-

r

t,

Transformers

91

trr

P-"

to the theoretically correct output voltage, when balanced polyphase voltages are applied to the input terminals. Definitions of RCF and phase angle are given in ASA C42.30-1957, 30.81.110 and 30.81.120, respectively. In applying the ASA terminology to phase-shifting transformers, the marked ratio is normally taken as exactly unity, the secondary voltage is taken as the output voltage, and the primary voltage is taken as the input voltage.

B. Development of Circle Diagrams The circle diagrams are developed for conventional phase-shifting transformers as applied to two common three-phase circuits with balanced voltages and currents. The same results can be obtained by similar methods for other polyphase circuits and phase-shifting transformers. The effect of unbalanced voltages and currents is beyond the scope of this paper.

Nomenclature V1 = input voltage of phase-shifting transformer 12= output voltage of phase-shifting transformer I=phase current 0 =phase angle of load (I relative to X1) a=-PD, radians -= var error, per unit of VA. Three-Phase Four- Wire Wye Circuits: Conventionally, the phase-shifting transformer is supplied with three polyphase voltages representing line-to-neutral circuit voltages. The phase-shifting transformer supplies the end device with three corresponding voltages shifted 90 degrees in phase. The var reading of an ideal end device is equal to the sum of the vars in each of the three phases. Owing to one phase, var reading = V2I1 sin (0 + a) VARs True vars = V11 sin 0 V21 sin (0 + a) - V1I sin 0 Vil =

sin 0 cos a + cos 0 sin a

RCF

-sin 0.

For values of a within 0.012 radian, cos a = 1 within 75 parts in 10l sin a = a within 30 parts in 106.

Thus, ,E=

1 -[(1- RCF) sin0+ a cosG]. RCF

(1)

Assuming that the RCF and PD associated respectively with each of the three transformer output voltages are the same because of symmetry and balanced burdens, then (1) applies to each phase separately and all three phases in total

Authorized licensed use limited to: VELLORE INSTITUTE OF TECHNOLOGY. Downloaded on March 17, 2009 at 05:41 from IEEE Xplore. Restrictions apply.

92 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT June-Sept. is a maximum when dc/dO

e

0,

=

Let the errors in VA, P, and Q be

when

(1 - RCF) cos 0 = Let maximum value of var =m. Substitute (2) in (1)

a

=

error

/a sin2

1 =

(m

a

RCF

1I cosO

sin 0. in

(2)

per

Cos 0

+

AVA, AP, and AQ, respectively, (VA + AVA)2-= (p + Ap)2 + (Q + AQ)2.

unit of VA

Expand (6), neglect values of small quantities squared, and substitute (5); then,

\

AVA (VA)

Per units

(RCF)em cos a.

error

Substitute (3) in (2) 1

=

RCF

-

=em

(4)

sin

Since em is to be constant, (3) and (4) describe a circle for values of and (1-RCF), when 0 is the independent variable. Three-Phase Three-Wire Circuits: Conventionally, the phase-shifting transformer is supplied with two voltages representing two of the three line-to-line circuit voltages. The phase-shifting transformer supplies the end device with two corresponding voltages shifted 90 degrees in phase, as in Fig. 1, for example. It is assumed that the RCF and PD associated respectively with each of the two transformer output voltages are the same because of symmetry and balanced burdens. The var reading of an ideal varmeter is equal to the sum of the contributions from the two elements. With an ideal phase-shifting transformer, the contributions are5 First element, a

V1I

+\-3sin

cos

)

VARs

VjI t- 2

+

\23

VARs.

var

(6+a)

reading

True

vars

for 0. =

=

/3 V2I sin (6 + a) V3 V1i sin 0.

vars

Eq. (1) and the circle diagrams follow as in the foregoing part.

C. Error in VA

Volt-amperes are determined from measured values of watts P and vars Q from the relation (VA) 2 I

=

VA

sin

(7)

6,

reduces to e

=

0.6(1 - RCF) + 0.8 a approximately.

The equation for the lower-right-hand side of Fig. 2 is RCF

0.988 + 1.324a.

Thus, e-

is

a

=

maximum when

0.0072 + 0.006a. a

is +0.009 radian

+ 0.00725 per cent,

approximately.

Similarly, em= -0.7 per cent when is -0.009 radian. Thus, limits of error are +0.7 per cent at 0.8 PF. Similarly, it can be shown that at zero PF the maximum errors occur when RCF equals 0.988 and 1.012. Thus, limits of error are + 1.2 per cent at zero PF. a

For a phase-shifting transformer with errors, substitute V2 for VI and

AQ -

D. Limits of Error-Specification No. 4 for Approval of Type of Instrument Transformers It will be shown the application of Fig. 2 gives limits of error of +0.7 per cent at 0.8 PF of load. Referring to Section B of the Appendix, at 0.8 PF (1)

-+ 0.7 sin 0)

+

cos

where AQ/VA is the error in var measurements expressed as per unit of VA. 0 is the phase angle of the load.

Em

cosO

'AP

AQ Q VA VA

P

VA VA

VA

e

Second element,

AP

VA

(3)

cos

AP(P) + AQ(Q).

=

in VA is

AVA

If RCF= 1.000 within 0.012, then a =m within 1.2 per cent.

(6)

p2 + Q2.

A. E. Knowlton, "Electric Power Metering," 1st ed., p. 221.

Discussion by G. J. Wey6 Vanderleck is to be commended for his timely paper on a very live subject. His formula for computing the error in terms of VAs is particularly useful. However, the author's proposal to express the error limits for the various accuracy classes as a circle seems to be rather restrictive on the design of the phase-shift-

(5) 6 Assistant to Engineering Manager, Westinghouse Meter Division, Raleigh, N. C.

Authorized licensed use limited to: VELLORE INSTITUTE OF TECHNOLOGY. Downloaded on March 17, 2009 at 05:41 from IEEE Xplore. Restrictions apply.

Vanderleck: Standards for Phase-Shifting Transformers

1964

1.006 1LO0C

I.OQO~ _

l

-

-

~~~

-F

0.997

-

_ 1.2 ACCURACY CLASS

I

LOAD PF LAGGING ERROR IN 'v A

-

ERROR IN V

0.994

0.991

-

0.988

- -

60 -

40 LAGGING

20

I'' FlI 0

20

C.(MINUTES)

A =

[(I-RCF)SIN4- ctCoSe]

_

___

_-

LOAD PF

4ERROR

] 0

0.997

ERROR

0.998

4

L

LEADING

IN

V

A

IN

V

Aa

-[(RCF)SINe+XCOS0]

0.988~-

60

0 40 20 20 a (MINUTES) LAGGING +

40 60 LEADING -

Fig. 2-Limits for RCF and phase angle for phase-shifting transformers used for var measurement, load-PF leading.

ing transformer to meet a specific accuracy class where only lagging or leading vars are to be measured. That is, the circle envelope is applicable only in the case where both leading and lagging var measurements are necessary.

If, however, only lagging or leading var measureare to be made, the more lenient ratio and phase angle error envelopes shown in Fig. 1 or 2 may be used. Thus, an error falling outside of a circle but within the for lagging PF permits corner area as shown in Fig. the use of a smaller transformer for a given accuracy class. ments

In view of the preponderance of applications requiring the measurement of lagging vars only it would appear that the more lenient error-limit envelope of Fig. is preferable. Also, it should be noted that the error parallelograms commonly used with instrument transformers apply only to lagging PFs. Thus, where instrument transformers are to be used on leading load PF it is necessary that the actual ratio and phase angle errors of the transformers be used to determine the over-all accuracy of the installation for measuring either vars or watts.

40 60 LEADING +

Fig. 1-Limits for RCF and phase angle for phase-shiftinig transformers used for var measurement, load PF lagging.

1.006

93

It appears only logical, therefore, that phase-shifting transformers for var measurement be permitted error limits comparable to those placed on instrument transformers. J. M. Vanderleck Whether the standards for phase-shifting transformers should be based on any load power factor or just on lagging PF depends on weighing the pros and cons. The any-power-factor basis would be preferred at metering points such as interchange points between systems, where the PF can change from lag to lead. It also has the advantage that any confusion in assessing whether the PF is lag or lead can be overlooked. The lagging-PF basis has the advantage that the phase-shifting transformer can be smaller, perhaps 30 per cent smaller. For an auxiliary transformer, however, this decrease in size may not be significant. It should be noted that when phase-shifting transformers operate from potential transformers, and varmeters from current transformers, then to assure a certain accuracy of var measurement, the net RCF and net phase angle should fall within the desired accuracy diagram, for example, the 1.2-accuracy-class circle. By net RCF is meant the product of the RCFs of the potential, current, and phase-shifting transformers. By net phase angle is meant the sum of the phase angles of the potential and phase-shifting transformers, minus the phase angle of the current transformers.

Authorized licensed use limited to: VELLORE INSTITUTE OF TECHNOLOGY. Downloaded on March 17, 2009 at 05:41 from IEEE Xplore. Restrictions apply.