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Distance relays can be either single phase or polyphase using multi-input comparators. Such multi-input comparators may be instantaneous or integrating type and compare either phase or amplitude or both.

The characteristics of conventional two-input comparators are in the forms of straight lines, circles or sectors of circle on complex plane. Multi-input comparators may have elliptical, conical or quadrilateral characteristics on complex plane.

Distance relays, have some limitations for long lines which are heavily loaded and are operating at or near the stability limit, these may cause mal-operation because they may fail to distinguish between power swings and faults. This can be avoided if we use a relay with elliptic characteristics. Such characteristics can be had if we employ a three-input amplitude comparator or hybrid comparator.

**1. Three-Input Amplitude Comparator****: **

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The basic circuit for a three-input comparator is shown in Fig. 4.30. It is an amplitude comparator and comprises three rectifier bridges.

Three inputs A, B and C are –

A = [V_{r}/(Z_{1} + Z_{2})] – {I [Z_{1}/(Z_{1} + Z_{2})]}

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B = [V_{r}/(Z_{1} + Z_{2})] – {I [Z_{2}/(Z_{1} + Z_{2})]}

and C = I

where, Z_{1} and Z_{2} are the design or replica impedances, V_{r} is the voltage at the relay point during fault and I is the fault current.

If Z_{1} and Z_{2} represent the phasors, the tips of which coincide with the foci of the ellipse drawn on complex plane and if 2 a represents the major axis of ellipse then –

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|Z_{L }– Z_{1}| + |Z_{L }– Z_{2}| = 2a … (4.9)

Where, Z_{L} is the line impedance.

If the characteristic is passing through origin, as shown in Fig. 4.31, then –

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|Z_{L }– Z_{1}| + |Z_{L} – Z_{2}| = |Z_{1} + Z_{2}|

Multiplying above equation by I we have –

|I Z_{L }– I Z_{1}| + |I Z_{L }– I Z_{2}| = |I (Z_{1} + Z_{2})|

Now since I Z_{L} = V_{r}, the above equation can be written as –

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|V_{r }– I Z_{1}| + |V_{r }– I Z_{2}| = |I Z_{1} + Z_{2}|

The above Eq. (4.10) represents the operating characteristic of an elliptic relay. The relay operating region is inside the ellipse.

We will recall that when V/I measured by the relay is beyond the characteristic, the relay does not operate. During power swings, the elliptical characteristic with narrow coverage across R-axis is less liable for tripping than circular characteristic.

**2. Hybrid Comparator****:**

The hybrid comparator is a combination of amplitude comparator and phase comparator. The hybrid comparators are usually multi-input comparators. The three (or more) inputs are derived from output of CT and PT by means of replica impedances, mixing transformers, auxiliary CT’s and PT’s. The characteristic of hybrid comparator depends upon the three inputs.

In the case of a hybrid comparator the general equation for a conic characteristic is –

Z = Z_{R}/1 – K cos (ɸ – θ) … (4.11)

The above equation represents an ellipse if K < 1, a parabola if K = 1, a hyperbola if K > 1 and circle if K = 0.

The inputs required for having the elliptical characteristic represented by Eq. (4.11) are I Z_{R}, V and K V cos (ɸ – θ). The first two inputs are supplied directly to the amplitude comparator but the third input is obtained from an auxiliary phase comparator which makes use of voltage I Z_{R} to polarize the voltage V.

**3. Four Input Phase Comparator with Quadrilateral Characteristics****: **

The quadrangular (or quadrilateral) characteristic can be obtained by using four relays having straight line characteristics.

**The static arrangement for obtaining such a characteristic is very flexible and is easily realizable in two ways viz.: **

(i) By combination of two phase comparators and

(ii) By multi-input comparator.

The drawback of the former arrangement is that the output of each comparator with two inputs has to be prolonged for a short time because they may not occur simultaneously. This will cause wrong tripping. This difficulty is overcome in a single multi-input comparator, which trips immediately when all the conditions are satisfied simultaneously.

**The four inputs necessary for obtaining quadrilateral characteristic are:**

where V is the line voltage, I is the line current, ɸ is the phase angle between V and I and θ_{1}, θ_{2} and θ_{3} are the phase angles of the impedances Z_{1}, Z_{2} and Z_{3} respectively which are connected in the current circuit, α_{1} and α_{4} are the phase shifts of the voltage where required for locating the impedance characteristic.

To enclose the fault area, let –

Z_{2} = X_{R}, Z_{3} = R_{R} and Z_{1} = R_{R} + j X_{R} = Z_{R}

and let α_{1} = α_{4} = 0 and K_{1} = K_{4} = 1.

The above inputs become –

S_{1} = I Z_{R} – V; S_{2} = I X_{R}, S_{3} = I R_{R} and S_{4} = V.

These are given to AND gate.

The mho circle caused by the intersection of S_{1} and S_{0} will not interfere with the rectangular tripping area if Z_{R} = R_{R} + j X_{R}, because the circle of diameter Z_{R} goes through the corners of the rectangular bounded by R_{R} and X_{R}.

Tripping occurs if all the equations resulting from comparison of all the inputs in pairs are simultaneously satisfied for the length of time set by the delay unit.

The unwanted mho circle resulting due to intersection of V and (I Z_{R} – V) can be eliminated by converting at least one of them into a pulse.