polarised mho relay

Polarised mho distance relayNew approach to the analysis of practical characteristics

L. M. Wedepohl, Ph.D., B.Sc.(Eng.), Graduate I.E.E.

Synopsis

The use of the polarised mho distance relay for the protection of high-voltage lines has become widespread.Up to the present time, the relay has been thought to be of limited use in the protection of short lines,owing to its relatively small reach for arcing faults. However, recent practical tests have shown that theactual performance is considerably better than that predicted by theory. A new analysis is thereforedeveloped in this paper which shows that the polarised mho relay has an offset characteristic, in the caseof unbalanced faults, which encloses the origin and hence enhances the relay reach in the direction of theresistive axis. The degree of offset is a function of the source/line impedance ratio of the system to whichthe relay is connected. It is shown that the theory developed is in good agreement with results obtained inpractice.

It is shown in an Appendix that the theory also covers the cases of crosspolarised directional relays andpolyphase impedance relays, both classes of relay having an offset characteristic.

The paper concludes by discussing the implication of the results. It is noted that the polarised mho relayhas most of the benefits of the reactance relay, while retaining the advantages of being inherently directionaland insensitive to load currents and power swings. It is also noted that, by using this method of analysis, thereach for lines with series capacitance may be predicted.

List of symbolsVR> Vy> VB ~ phase-neutral voltages of red, yellow and

blue phases, respectively, at relaying point'/?» W> h = phase currents

E = phase-neutral generated voltage on red phase/j = positive-sequence current/2 = negative-sequence current/0 = zero-sequence current

K, KX,K2 = relay constantsZn, Znl, Znl = relay impedance constants

6 = angle of ZnZL = positive-sequence line impedance

ZLO = zero-sequence line impedanceZs = positive-sequence source impedance

Zs0 = zero-sequence source impedanceP = zsolzsq = ZL0\ZL

120°

1 IntroductionIn the past two decades, the use of polarised mho

distance relays for the protection of high-voltage transmissionlines has become widespread, because of their inherentproperty of being simultaneously an impedance and adirectional measuring element.

This type of relay is associated with a number of advan-tages and drawbacks, and these have in the past been usedas a basis for assessing its merits relative to other schemesof feeder protection. It is inherently directional and has thevirtue that of all distance relays it is least sensitive to powerswings.1 On the other hand, by virtue of its constrainedcharacteristic, it is rather insensitive to resistive componentsin the fault impedance and is, for this reason, of limited usein the protection of short lines, when resistance due to faultarcs may be appreciable compared with the line impedance.

Paper 4665 P, first received 2nd June and in revised form 27th October1964Dr. Wedepohl is with A. Reyrolle and Co. Ltd.

PROC. FEE, Vol. 112, No. 3, MARCH 1965

In these applications, it is customary to specify reactancerelays2 or differential schemes of protection. The fault-arcproblem is further aggravated .by the fact that the polarisingvoltage, derived from an unfaulted phase or a tuned circuit,may be out of phase with the fault voltage.

Recent measurements have been made to investigate thesensitivity of polarised mho relays to faults with simulatedarc resistance, and it has been found that the results are notconsistent with the present theory. The relays are found tobe capable of operating in the presence of fault-arc resistanceswhich considerably exceed the values predicted by simpletheory; the situation improves as the source/line impedanceratio increases. As a result of these measurements, a newanalysis of the polarised mho relay was developed, and it isthe purpose of this paper to describe this, together withpresentation of results and consideration of their practicalimplication.

2 Simplified theory of polarised mho relayIt is well known that the characteristics of all distance-

relay functions may be obtained by using either an amplitude-or phase-comparing measuring element. The relationships inthe polarised mho relay are more readily understood byconsidering the operation of the phase comparator. Identicalcharacteristics may be obtained from both comparators ifthe following transformations are observed:

sx =sy =

+ s2)- s2)

or = Sx + Sy

= Sx — Sy

and Sy are the operate and restraint input signalsS2

where Sx y

to the amplitude comparator, and St and S2 are the two inputsto the phase comparator. The criterion for operation of thetwo relays is

and -7r/2 < <f> < n/2where <f> is the phase angle between 5", and S2.

525

The basic phase-comparator input quantities for a polarisedmho relay are

Sy = Vp

S2=IZn-V

where Vp is the polarising voltage and V and / are voltageand current at the relaying point.

The corresponding inputs to the amplitude comparator togive identical characteristics are

Sx = +1ZH-V)

~ (IZn - V)]

Fig. 1 shows the basic input arrangement for a mho-connected phase-angle comparator. The two quantities whichare compared in phase are

EZL

52 = lZn - V

where V =

/ =

St =

s2 =*s + *L

The criterion for operation is that

- 7 T / 2

E

zs +EZ

zs +E(Zn

ZL

ZL

L

ZL

-ZL)

—vw- I-D-

(» n (i

•A/Wz

'

s,-vphase

comparator s2-izn-v

Fig. 1Basic connection for mho relay

The relative phase angle between St and S2 is not disturbedif they are multiplied by the same quantity, i.e. {Zs + Zj)\E.The two vectors to be compared in phase are therefore

S[ = ZL

S-i = Zn — ZL

The vector diagram is shown in Fig. 2, and it is clear thatthe locus of ZL is a circle with Zn as diameter.

In practice, the mho relay is not suitable as a directionalelement, since a finite value of Sx is required in order to effect526

operation, so that the origin is outside the relay characteristic,and there is no protection against terminal faults. Theproblem is solved in the polarised mho relay by makingS\ = Vp, where V is in phase with V but not proportional

Fig. 2Mho-relay characteristic

to it, so that for terminal faults, when V = 0, phase com-parison can be effected. In this case,

S'2 = Zn - ZL

where Zp is a vector of constant magnitude but in phase withZL. The vector diagram is shown in Fig. 3, from which it is

Z..-Z,

crFig. 3Basic polarised-mho-relay characteristic

clear that phase comparison of Zn — ZL with Zp is the sameas ZL, because these latter two impedances are in phase;consequently, the 'polarised mho' characteristic is identicalwith the 'mho' characteristic, except that the origin in thiscase is a well defined point.

The problem is in selecting a suitable polarising voltageVp, and three basic solutions are adopted in practice. Vp iseither derived from the fault voltage V through a resonantcircuit tuned to system frequency (memory) or from anunfaulted phase through a suitable phase-shifting circuit(sound-phase polarising); alternatively, a combination ofpart sound-phase and part faulted-phase polarising is used.The last two methods do not solve the problem in the caseof 3-phase faults, when an unpolarised mho characteristic isobtained, and operation for close faults once more becomesindeterminate.

In practice Vp and V are not in phase for terminal faults,because of the characteristics of the system, principallyunequal source-impedance/line-impedance angles. By con-sidering a number of boundary conditions, Ellis3 has shown

PROC. IEE, Vol. 112, No. 3, MARCH 1965

that, in most cases, a suitable choice of sound-phase polarisingvoltage gives rise to errors in phase of less than 15° betweenVp and V. The effect of phase shifts between these twovoltages modifies the relay-input equations to the following:

Zp and ZL have the same phase, and the angle between Vpand V is accounted for by the additional rotation a. Fromthe relationship in the vector diagram shown in Fig. 4, it is

Fig. 4Characteristic of polarised mho relay with phase shift between Zpand ZL

seen that Zn is a chord of the mho characteristic, and thediameter D lags Zn by a and has a magnitude |£>| = \Zn\ sec a.For a = 15°, sec a is 1 035, which is a negligible increase. Thepolar equation of the mho circle is

\Z\ = \Zn\ cos (<f> — 6 + a) sec a

where <f> and 6 are the angles of ZL and Zn, respectively.The value of Z when (j> = 0, i.e. the relay reach in the resistiveaxis for terminal faults, is

R = \Zn\ cos (6 — a) sec a

Engineers concerned with the application of polarisedmho relays to systems are interested in the maximum negativevalue which a can attain, since this corresponds to a minimumvalue of R. Typically, d = 75°, and, if a = - 15°, R = 0.This case is illustrated in Fig. 5, together with a typical rangeof system impedances superimposed on the diagram, includingthe effect of fault-arc resistance. It may be seen that the relaycoverage under arc-resistance conditions is rather poor. The

boundary of systemfaults

Fig. 5Polarised mho characteristic6 = 75° a = - 15°


effect shown in Fig. 5 is most severe in the case of short linesand low fault currents, corresponding to high source/lineimpedance ratios, and has detracted considerably from theappeal of these relays in this case.

Warrington1 has shown that, in these circumstances, areactance relay is more suitable as a distance-relay element,despite the added complexity of the arrangement, sinceseparate directional elements must be provided.

In order to verify these conditions in practice, a series ofmeasurements was made on a practical polarised mhodistance relay, and marked disparities between theory andpractice were noted. The reach in the resistive axis forterminal faults was found to be greater than expected, and itincreased as the source/line impedance ratio increased. Theseresults are presented and discussed in Section 7.

The reason for the disparity between theory and practiceis in assuming that Vp and V are in phase or separated by afixed angle a. In practice, this only applies when ZL and Znare in phase. Deviations become progressively more severe asZL moves around the polar diagram, and it is possible undercertain conditions for a to equal 180°. In the Sections tofollow, a more rigorous analysis of the operation of thepolarised mho relay is presented, in order to take this effectinto account.

In Section 3 it will be seen that, in the case of the polarisedmho relay, the input quantities to the relay take the mostgeneral form, i.e.

51 = Zn2 + KZL

52 = Znl — ZL

Jt is shown in Appendix 12.1 that the locus of ZL at theboundary of operation of the relay is a circle, and a simpleconstruction is developed which relates the position of thecircle in the complex plane to the three constants Z;il, Zn2and K.

3 Analysis of polarised-mho-relay charac-teristic for phase-to-phase faultsThe system is shown schematically in Fig. 6, together

with the sequence impedance diagram. The operation of arelay connected between yellow and blue phases is considered.

•AAAA-Zen

•AW

-A/W 1z,

•AAAA- AA/V 1z-S ^2 *-L

Fig.6Equivalent circuits for system with phase fault

527

The voltages and currents in each phase are

VR=E

^Y = ^ E ^ d^ZL - Zs)2(Z? + ZL)

VB = 1(7 I 7 S^ZL ~ Zs)2(ZS + ZjJ

VYR = E[(a2 - \)ZL -

VYB = E{a2 - a)Zj(Zs + ZL)

IY = E(a2 - a)l2(Zs + ZL)

IB = — Iy

IY-JB = E(a2 - a)l(Zs + ZL)

The measuring signal for a polarised mho phase-fault relay is

S2 = Uy ~ lB)Zn — VYB

which, in this case, is

S2 = E{a2 - a)(Zn - Zj)l{Zs + ZL)

There are three possible practical alternative choices forpolarising voltage:

(a) combination of VYB and VR

(b) combination of VYB and VYR

(c) memory circuit associated with VYB.

VBR is not used in practice, because the vector position issuch that inductive phase shift is required to achieve thecorrect phase relation with VYB, and this raises practicalproblems. There are no further advantages to be gained bythis choice, and it will not be considered.

3.1 Derivation of polarising voltage for phase-fault relayelementThe three practical cases are considered below for the

derivation of the polarising signal S}.

This is a case of mixed polarising, where K2 is complexwith an angle of approximately —90°. For later simplification,we write K2 = —j\/3K2. Kl is generally real and approxi-mately equals 1.

Substituting for VYB and VR and simplifying,

5, = E(a2 - a)[KxZL + K&Z, + ZL)]I(ZS + ZL)

(b) 5, = K,VYB + K2VYR

Kx is as before. For convenience in this case we writeK2 = K2/_60°.Substituting for the voltages and simplifying, 5", becomes

E(a2 - a)[KxZL + K'2ZL

+ (V3/2)A^3O^ZJ/(Z, + ZL)

(c) S{ = K{ VYB

This is the case of a memory relay, where K{ VYB is initiallythe interphase voltage prior to the fault, which then decaysexponentially to the fault voltage. Kx may have a smallangle, owing to the resonant frequency of the tuned circuitnot coinciding with the system frequency. Tn this case,

5, = KxE{a2 - d){Zs + ZL)I(ZS + ZL)

which is the signal just prior to fault occurrence.528

These three cases cover those generally used in practice.The general characteristics for the three types of mho relaymay be obtained in the manner detailed in Appendix 12.1.The signal S2 in each is the same; 51, takes the three alternativeforms described in (a), (b) and (c) above. In order to obtainthe general form of input signal S[ and S2, all input signalswill be multiplied by the vector

(Zf + ZL)lE(a2 - a)

The input signals for the three cases then become

(a) [ = K'2ZS K2)ZL

K'2)ZL

Fig. 7Polarised mho phase-fault-relay characteristicsa Polarised Kx KYB + ^2^Rb Polarised /CIKYB + A^YRc Polarised KjEya (memory)


The relationship between the three general constants Znl,Zn2 and K is given in Table 1.

Table 1RELATIONSHIP BETWEEN VECTORS

Case

ift)

(b)

(c)

Znl

Zn

Zn

Zn

Znz

K'2ZS

(V3/2)A^3£Z,

K

K\ + K'2

K\ -\- K2

3.2 Characteristic of polarised mho phase-fault relay

The relay characteristics for the forward direction ofpower flow in the three cases are shown in Fig. 7. In allcases, the origin is enclosed by the relay characteristic, thedegree of offset of the relay in the third quadrant beingprincipally a function of the source-impedance vector Zs andthe constant K2. When Zs = 0, the characteristic alwayspasses through the origin. The construction for this specialcondition for case (a) is shown in Fig. 8. By virtue of theconstruction for the relay characteristic, the diameter subtendsan angle of 90° at the origin, which must therefore lie on therelay characteristic. Also shown in Fig 8 for the same case is

K 2 Z S

I 1Fig. 8Characteristics of polarised mho relay for case {a) of Section 3.1a Z. = 0b Z,= oo

the construction for the special condition Zs = oo in case (a).It is evident that the characteristic is a straight line throughZn perpendicular to K2ZS.

From the foregoing, it would appear that the directionalfeature of the relay has been lost, since the origin is enclosedPROC. IEE, Vol. 112, No. 3, MARCH 1965

by the relay characteristic. This interpretation follows fromthe fact that negative impedance in the forward sense andpositive impedance with reverse power flow are normallyidentified. This assumption is not valid. In the case of reversepower flow, the relay-input equations change, owing to thenew vector relationship between voltage and current.Typically, in case (a), the equations become

S[ = K'2ZS + {K{ + K2)ZL

S2— — Zn — ZL

The vector construction for this case is shown in Fig. 9. Itmay be seen that the characteristic is totally different from

Fig. 9Polarised-relay characteristics, case (a) of Section 3.1;reverse-power-flow conditions

that for forward power flow. In particular, the origin liesoutside the relay characteristic, which is almost entirely inthe third or negative-impedance quadrant. Only in the specialcase of Zs = 0 is it permissible to identify negative impedanceand reverse power, since the characteristics for both directionsof power flow are then identical.

4 Analysis of polarised mho relay for earthfaultsIn this case, operation of a relay connected between

the red phase and earth is considered. The sequence diagramis shown in Fig. 10.

Fig. 10Sequence diagram for phase-earth fault

529

/, = E/[(2 + p)Zs + (2 + q)ZL]

Where Zs0 = /;ZS and ZL0 = ?ZL

K* = £ - Z,/, - Z / 2 - />Z,/0

= E[(2 + q)ZL]l[{2 + p)Zs +

Ky = £[(2 + /,>**, + (2 £

+ (1 - Jp)Zj/[(2+ />)Z5

VB = £[(2 + p)aZs + (2

i.e. (a) S{ =

S'2=Zn-ZL

(b) S{ = A/3^[ (2 + P)ZJ(2 + q)] + (Kx

(2

(2 + q)ZL]

(2 + q)ZL)

VYB = E(a2 - a)

IR=3E/[(2+p)Zs + (2 + q)ZL]

In the case of an earth-fault relay, the measuring current isa combination of phase and zero sequence to give correctmeasurement impedance, i.e.

Im = IR + [(Zi0/ZL) - l ] / 0

- E{2 + q)l[(2 + p)Zs + (2 + q)ZL]

The measuring signal in the case of an earth-fault relay is

S2 = ImZn - VR

= E(2 + q){Zn - ZJI[(2 +p)Zs + (2

S'2=Zn- ZL

(c) S[ = Kx[(2 + p)Zs/(2 + q)] + KXZL

s*> = zn — ^- 6 0 °

In this case, there are four practical cases of polarising-voltage•signal 5", to be considered: KXVR + K2VB, KXVR + K2VYB,Kx VR (memory) and Kt VR + K2 VRB.

4.1 Derivation of polarising voltage for earth-fault relays

(a) Sx = KXVR + K2VS

Writing for convenience K2 = K2^— 120°,

It may be seen that cases (a) and id) are almost identicalif K2 in the second case has a leading angle of 30°, whilecases (Jb) and (c) are similar.

The characteristics for cases (a) and (Jb) are shown inFig. 11. The general appearance is similar to that for thephase-fault relays. The condition for reverse power is similarto that previously described for the phase-fault elements,and the characteristics are not plotted for this reason. Thesame arguments regarding extreme limits of Zs apply, i.e.zero and infinity.

In the former case, simple mho-relay characteristics areobtained and, in the latter case, reactance-relay characteristics.

5 Relay characteristics under 2-phase-to-earth fault conditionsOwing to the complexity of the voltage and current

relationships, it is not possible to describe the characteristicsin terms of the simple basic quantities as has been done inother cases. However, the following general observations maybe made:(i) When Zs = 0, all characteristics are simple 'mho' circles

through the origin.

Sx =KXVR+K2VB =E[KX(2 + q)ZL + K$L + p)Z q)ZL + K'2{\ -p)Zs/ 120°]

(2 + p)Zs + (2 + q)ZL

q)] + (Kx

(2 + p)Zs + (2 + q)ZL

(b) Sx = KXVR + K2VYB

W r i t i n g ^ = - jK'2,

_ E{2 + q){[V3K2V^:p)Zsl(2 + q)] + (Kx •

M(c)

(d) Sx =

and K2 =

s1

- V F

E(2

(2 + p)Zs + (2 + q)ZL

_ _ _ _

- 60° + p)Zs/(2 + q)] + (Kx

(2+p)Zs q)ZL

4.2 Characteristic of polarised mho earth-fault relayelement

The input quantities S'x and S2 for the four cases areobtained by multiplying Sx and S2 by

[(2 + p)Zs + (2 + q)ZL]K2 + q)E530

(ii) When Zs = 00, the characteristics are straight lines whoseangles of inclination are functions of Zs, as before.

The choice of the type of sound-phase polarising is ofsome importance, since the vectors are subject to severe phaseshifts. A danger exists when Zs is large that, if K2ZS is toofar in the fourth quadrant, overreach for arcing faults will be


experienced. The basis for selection of sound-phase polarisingdescribed by Ellis3 is valid in this case, since the phase shiftsdescribed in his paper are in fact related to the effectiveposition of K'2ZS on the mho characteristic. In general, the

K2-K2' /90!

Fig. 11Polarised mho earth-fault-relay characteristicsa Polarised S\b Polarised S,

preferred choice of 'sound phase' for a phase-fault element isVB for a RY relay while the preferred phase for a RE relayis also VB. It is important to note that the RE relay measurescorrectly for both RYg and RBE faults. In the former case,the characteristic encloses the origin as in the case of thesimple earth fault, while in the latter case the origin is inde-terminate, because VB falls to zero with VR, and a simplemho characteristic is obtained.

6 Relay characteristic under balanced-faultconditionsWith the exception of the memory relay, the charac-

teristics will be simple 'mho' circles, the origin being inde-terminate. The diameter may not coincide with Zn if K2 is notreal.

The behaviour of a RE relay polarised from VB and thesame relay with memory are considered below:

(i) VR = EZJ(ZS + ZL)

VB = aEZJ(Zs + ZL)

IR = E\{ZS + ZL)

(ii) S, = E[KXZL + K^ZL]/(ZS + ZL)

S2 = E(Zn - ZL)KZS + ZL)

Sx — KXZL + K2'ZL

s2 = zn — zLPROC. IEE, Vol. 112, No. 3, MARCH 1965

(iii) S, = KXE

S2 = E(Zn - ZL)I(ZS + ZL)

Oj = K.XZS -\- K\Z^

$2 — Zn ~ ZL

The two characteristics are shown in Fig. 12. The angle ofhas purposely been exaggerated to show the lack of

coincidence between Zn and the diameter in this case.K'2

• -R

Fig. 12Polarised mho earth-fault-relay characteristics during 3-phase faultsa Polarised ATi KK + ^ Bb Polarised K\ER (memory)

7 Practical resultsTests were carried out on a polarised mho phase-fault

relay using the rectifier-bridge moving-coil principle. Polarisingwas as for case (a) of Section 4.1, and the constants of therelay were 6 (angle of Zn) = 60°, Kx = 1-42 and K2 —014 / —15°.

A set of polar curves is presented in Fig. 13. These wereobtained by connecting a relay to a 3-phase test bench andvarying the line impedance together with simulated faultresistance. The curves are normalised, in that all vectors aredivided by Zn. It follows that Zs\Zn — y is the system-impedance range factor. The curves are presented for anumber of such factors. The curves are not circles about themajor diameter, since in this particular type of relay thecriterion for operation is that the angle between the twosignals S1, and S2 is 75° rather than 90°, so that the relaycharacteristic consists of the arcs of two circles with themajor diameter becoming a chord. If the reach in the resistiveaxis is critical, this effect could be taken into account. Thetheoretical curves are also given, and, apart from the dis-parity in reach in the resistive direction for the reason stated,the agreement between theory and practice is good.

531

8 Assessment of the capabilities of thepolarised mho relayIn the past, it has been customary to use polarised

mho relays for relatively long feeders, while reactance relayshave been preferred for short lines where arc resistance has

10

0-5

nominalangle 60°

0-5 1 0 1-5

Fig. 13Comparison between theoretical and experimental results

y = z,izn—O— experimental

theoretical

been a problem. This latter solution has not been ideal,because of the need for a directional-control element and animpedance element for preventing undesired operation on loadcurrent. From the analysis presented in this paper, it may beseen that, when the source impedance is large compared withthe relay setting, the polarised mho characteristic is similar tothat of a reactance relay, and the advantage of the latterbecomes marginal. The condition of a very short line with arcresistance usually implies that the source-impedance/line-impedance ratio is high, and it follows that the polarised mhorelay has the virtue of automatically adapting itself to systemconditions. Load current is not a problem, since, in this caseof balanced current flow, the characteristic is the classicalmho circle. Generally the likelihood of a 3-phase arcing faultis small, and the lack of reach in this case would not be aserious drawback.

The analysis also enables an assessment of reach to bemade for faults which lie in the fourth quadrant. This may benecessary in lines which have series-capacitor compensation,and in the past it has been difficult to predict the relaybehaviour in this case.

A further important point which should be noted is thatthere is only one point on the polarised-mho-relay charac-teristic which is independent of system conditions, i.e.ZL = Zn. In the past, in certain cases, the setting of apolarised mho relay for line angles other than that of Znhas been specified in terms of the simple trigonometricalequation Zs = Zn cos (9 — <f>), where Zs is the setting for aline angle 9 — <j> displaced from that of Zn. It may be seenfrom the analysis in this paper that this equation is not validand that errors in setting may arise if this approach is used.If an accurate knowledge of the setting is required, theangle 9 — <f> should not exceed 10°.

In the case of lines with series capacitors, this conditioncannot be met and the setting becomes indeterminate.

9 ConclusionsOwing to disparities between theory and practice in

predicting the performance of polarised mho relays, a new532

theoretical analysis was undertaken, the treatment beingpresented in Section 1 of this paper.

The characteristics of the polarised mho relay for a numberof well known connections are shown to have an offset in thenegative-impedance quadrant in the case of unbalancedfaults, thus providing added reach in the direction of theresistive axis. In particular, the reach for arcing-terminalfaults is far greater than would be expected from a simplifiedanalysis.

Negative impedance and reverse power flow should not,in general, be identified, since the characteristic for reversepower flow is different from that for forward flow. It isshown that, for unbalanced faults, the polarised mho charac-teristic for reverse power flow is a circle lying almost entirelyin the negative-impedance quadrant and not enclosing theorigin, so that the relays are directional.

The characteristics of crosspolarised directional relays arein accord with the general theory as shown in Appendix 12.2.For unbalanced faults, the origin is included within the relaycharacteristic for faults in the forward direction and liesoutside it for faults in the reverse direction. In this case, therelay characteristic is a straight line.

The polar characteristics of polyphase directional impe-dance relays may be obtained by the same general method(Appendix 12.3) and are in accordance with the results forsingle elements.

The advantages of the reactance relay for short lines arenot as great as may be expected from a simplified analysis,and the polarised mho relay may be favoured, because of itsability to adapt itself to the system conditions; i.e. increasingits reach in the resistive axis for arcing faults on short lines,whilst retaining the virtue of insensitivity to impedances dueto load currents and power swings.

If an accurate knowledge of the settings of a polarised mhorelay is required, the angles of the nominal impedance Znand the line impedance ZL should not differ by more than 10°.The setting in the case of lines with series capacitance may bedetermined for certain specific plant conditions but cannot bespecified in the general case, since it is a function of thesystem source-impedance/line-impedance ratio.

Finally, it should be noted that the analysis in this paper isbased on the assumption that the faulted line is energisedfrom one end of the system only. The analysis in the moregeneral case does not lend itself readily to a simple geometricalinterpretation. In this case, it would be more appropriate tostudy specific cases with the aid of a digital computer backedby practical results obtained from a test bench. This does notdetract from the analysis in the paper, however, since themain effect of an interconnected system would be to alterslightly the amplitude and phase of the voltage derived fromthe unfaulted phases and to include reactive effects in the arc-resistance voltage, which is purely resistive in the simple case.The general form of the characteristic would remainunchanged. The comparison with earlier analysis is in anyevent valid, because this was invariably based on the assump-tion of a power feed from one end of the system only.

10 AcknowledgmentsThe author wishes to thank A. Reyrolle and Co. Ltd.

for permission to publish this paper.Thanks are expressed to Mr. F. L. Hamilton (Engineer-

in-charge of research), and Mr. J. B. Patrickson (DeputyEngineer-in-charge of research) for helpful discussions duringthe preparation of this paper, and to Mr. T. H. Potts forcarrying out the practical tests.


111

ReferencesWARRINGTON, A. R. VAN c.: 'Application of the ohm and mhoprinciples to protective relays', Trans. Amer. Inst. Elect. Engrs., 1946,65, p. 378

2 WARRINGTON, A. R. VAN c.: 'Reactance relays negligibly affected byarc impedance', Elect. World, 1931, 98, p. 502

3 ELLIS, N. s.: 'Distance protection of feeders', Reyrolle Rev., 1957,(168), p. 16

4 WARRINGTON, A. R. VAN c.: 'Protective relays, their theory andpractice' (Chapman and Hall, 1962) p. 285

12 Appendixes12.1 General distance-relay characteristic

The most general input to a 2-terminal phase-anglecomparator is

5, = AI - BV

S2 = CI + DV

The relative phase angles are not disturbed if both signalsare divided by BI to give

+where

ZnX = A\B

Zn2 = C\B

K= DIB

ZL = VII

and A,B, C, D and K are, in general, complex. The boundaryof relay operation is defined by the condition that S{ and S'2should be displaced in phase by 90°.

The vector diagram is shown in Fig. 14, the vectors S[and S2 being represented by AB and DC, which are at right

Fig. 14General vector diagram for phase comparatorOE = zn2\K

angles on the relay boundary. Since it is the locus of point Bwhich is of interest, a point E is described, so that triangleOCD is similar to triangle OBE. The ratio between sides isOC/OB = K, so that corresponding sides of the two trianglesare in the magnitude ratio K and separated in phase by 8,the angle of K. The corresponding sides EB and CD intersectat X, and the angle BXC is 8. By definition, AB and DC areat right angles; angle XBY is therefore 90° — 8 and angleABE is 90° + 8. Since A and E are points fixed by ZnX, Zn2


and K and are not functions of ZL, AE must be a chord of therelay-characteristic circle. A diameter of the circle must beAF, such that ABF is a right angle, and therefore angle FBEis 8. Since A is also on the characteristic circle, FE mustsubtend the same angle at B as at A, so that angle FAE is 8.Finally, FEA is a right angle, since it is subtended by thediameter.

This diagram provides the basis for a simple constructionfor the general circle. It is noted that ZL — ZnX is a point onthe circle; the vector diagram is drawn for this special casein Fig. 15. Here B and A are coincident, since Z,,, = ZL and

Fig.15General vector diagram for phase comparatorZL = Zn

OC = KZL = KZnX. E is the same as before. The phase of thezero vector AB must be at right angles to DC (= Zn2 + KZnX).The triangles OCD and OAE are similar, as before. A dia-meter is obtained by describing F so that angle FAE is 8and angle FEA is a right angle as before.

A new point M is fixed so that MA is equal and parallelto OC (= KZnl), and G is fixed so that MG is equal andparallel to DO (=. Zn2). OH is drawn perpendicular to OA(= ZnX), and it remains only to show that HF is parallelto MG and GFA lies on a straight line. This is done bynoting that triangles OAH and EAF are similar (equalangles 8 and one right angle), and consequently trianglesAHF and AOE are similar, since there is an equivalence intranslation from H to O and F to E. However, trianglesOAE and MAG are similar, and therefore MAG and HAFare similar, so that F lies on AG. The final construction ofthe general characteristic is shown in Fig. 16. Vector K isalso shown for clarification.

Since the construction will be used repeatedly in the text, itis worth while summarising the steps:

1 Vector Znl is drawn from the origin.2 Vector KZnX is drawn to meet the extremity of the Zni

vector.3 Zn2 is drawn in such a position that the extremity touches

the beginning of vector KZnX.533

4 Vector Zn2 + KZnX is formed by joining the beginning ofZn2 to the extremity of KZnl (and incidentally Znl).

5 A line is drawn from the origin perpendicular to Zrtl tointersect KZnl.

(1,0)

Fig. 16Construction for phase-comparator polar diagram {general case)

6 From the point of intersection with KZnX, a line is drawnparallel to Zn2 to intersect Zn2 -f KZnl.

7 The vector diameter of the relay characteristic is drawnfrom the final point of intersection to the extremity of theZnX vector.

12.2 Directional relays'Crosspolarised' directional relays may be explained

in terms of the analysis described for polarised mho relays.The analysis will be performed for one particular connection,but the extension to other connections is obvious. Consider aphase-comparing relay of the type previously described, inwhich 5, = IRZn and S2 = Ky^/90°, this being the so-called'quadrature' directional-relay connection.

A number of faults need to be considered, i.e. RY, RB, REand RYB.

RY faultSt = E{\ - a2)Zn/2(ZS2 =

S[ =

RB faultSt = E{\ - a)ZJ2(Zs + ZL)S2 = V ^S[ = Zw/-30°/2S^ = V 3 Z ^ - 3 O ^ + ZL

RE faultSx = £Z,,/[(2 + p)Zs + (2 + q)ZL]S2 = {a1 - a

S[ =

RYB fault5, = EZ,,KZS + ZL)S2 = V3EZJ(ZS + ZL)S[ =

The characteristics are shown in Fig. 17, where it can beseen that they are similar to those of the polarised mho relay,except that in this case the diameter is infinite. The generalcharacteristic is a straight line through the modified Zs

Fig. 17Characteristic of crosspolarised directional relaya RY faultb RB fault

vector, perpendicular to the modified Zn vector. Again theorigin is included within the relay-operating characteristic forforward power flow and is outside it for reverse power,except in the case of a 3-phase fault, when the characteristicpasses through the origin. For the cases considered for thisparticular connection, the characteristic rotates through± 30°, depending on the type of fault. This is a well knowneffect and is taken into account when specifying the angleofZ«.

12.3 Polyphase directional impedance relaysThere is a certain class of relay connection which

gives rise to polyphase directional impedance characteristics.1

The characteristics of one of these will be described below.The two input signals to a phase comparator in this case are

= (IR - Iy)Zn ~ V,RY

and S2 = IB)Zn - VRB

534

The only practical realisation of this relay described in theliterature4 makes use of an induction-cup movement, so thatthe criterion for operation is that Sx and 5"2 should eitherbe in phase or in antiphase at the boundary of operation.The same characteristic could be realised with a cos <f> com-parator, if a relative phase shift of 90° were introducedbetween the two signals. It has been indicated1 that a phasecomparator with this connection gives correct impedancemeasurement for interphase faults between any pair of phasesbut gives no protection against 3-phase faults. However, thepolar characteristic for various fault types is not described.


The polar characteristic is obtained for the possible faulttypes as shown below.

12.3.1 Fault between phases R and Y

- a2)E,5, =

zs + zL(1 -a2)E[Zn

n - zL)- a)(Zs + ZL) Z

+

-(XF - i) - Z L / - 6 0

It is important to note that the criterion for operation is Sjand S2 in phase or antiphase. If S2 is advanced 60° to becomeS2, the criterion for operation becomes

60° < a < 240°

where a is the angle of S'2' relative to S[ and

^i = Zn — ZL

S2 = j Z a / 6 0 o - V 3 Z , / -30° - ZL

The polar characteristic is shown in Fig. 18a. It is seenthat the characteristic is an offset 'mho' circle with the vector

Fig. 18Polar curves for polyphase mho relaya RY faultb RB faultc YB faultd General characteristic, all phase faults, forward power flowe General characteristic, reverse power flow


It is clear that the chord subtends an angle of 60° to theright and 120° to the left of the chord. The diameter musttherefore be the vector D = C/30° sec 30°, where C isthe chord; i.e. D = Zn + Zs. For reverse power, the diameteris D = — Zn + Zs, and, as in previous cases, the charac-teristic lies almost entirely in the third quadrant and theorigin is outside the relay circle.

12.3.2 Fault between phases R and B

In this case, the two signals become, after manipulation,

S\' = \Zn/ -60° - W*Z,/_+30° - ZL

S'2 = Zn- ZL

The criterion for operation is, as before,60° < a < 240°

where a is the angle of S2 relative to S\'. The polar charac-teristic is shown in Fig. \Sb. In this case, the vector chord is

Applying the same reasoning as for a RY fault, the diameteris Zs + Zn, and the characteristics are identical in the twocases.

12.3.3 Fault between phases Y and B

In this case, the two input signals are

5',' = -30° -ZL

-zL

where, as before, the operating criterion is 60° <; a <; 240°.The characteristic is once more an offset circle through Zn,as shown in Fig. 14c. The vector chord in this case is

$Zn(/_6Q° - /JW) - JV3Z,(/-3O° - ^30°)

= iyV3(Zi + Z,,)As before, the vector chord subtends 60° in the major

quadrant, so that the diameter is Zs + Zn, and the charac-teristic is identical with that for the previous two cases.

The characteristic for all phase faults for forward powerflow is shown in Fig. )Sd and for reverse power flow inFig. 18<?.

12.3.4 2-phase-earth faultsAs before, a geometrical presentation of the polar

characteristics is complicated, but it is readily seen that thevector Zn is on the relay characteristic, and it may be assumedthat the main effect in this case is for the degree of offsetin the negative-impedance quadrant to be modified, whileretaining accuracy of setting in the positive-impedancequadrant.

12.3.5 3-phase faultsIn this case, S1, and S2 are equal in magnitude. 5, is

proportional to (1 — a2) and S2 to (1 — a), so that there is apermanent restraint condition; no operation can take place.

12.3.6 Comparison of the relay characteristic for differentfault typesIt has been shown that, in the case of each type of

phase fault, the relay characteristics are identical, withdiameter Zs + Zn. The connection thus yields a true poly-phase polarised 'mho' phase-fault element which is insensitiveto 3-phase balanced conditions.

535

polarised mho relay

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