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The application of a particular relay is governed by its characteristics and other factors such as accuracy, operating time, reliability, burden, method of setting adjustment etc.

The function of a protective relay is to sense any abnormal condition in the system and send a signal to the breaker which in turn isolates the faulty section of the feeder from the healthy one. The relay does all this by comparing two quantities either in amplitude or in phase. The amplitude or phase relation depends on the conditions of the system and for a predetermined value of this relation, indicative of a particular type and location of fault, the relay operates.

Except in relays, such as overcurrent relays, where only one electrical quantity overcomes a mechanical quantity such as the restraint from a spring, usually two electrical quantities are compared. The device that makes such comparison is called the comparator and forms the heart of a protective of any relay is governed by the comparator.

**General Equation for Comparators: **

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Let us first derive the general threshold equation assuming that there are two input signals S_{1} and S_{2} such that when the phase relationship or magnitude relationship fulfills pre-determined threshold conditions, tripping is initiated. The input signals are derived from the system through instrument transformers (CTs and/or PTs). In case the two quantities to be compared are different (i.e., voltage and current), some form of mixing device, such as current voltage transactor, is required.

Let the two input signals be represented as –

Where, A and B are the primary system quantities, K_{1} and K_{3} are the scalar numbers and |K_{2}|and | K_{4}| are the complex numbers with angles θ_{2} and θ_{4} respectively. Taking A as the reference phasor and phasor B to lag A by an angle ɸ. Then the above equation can be rewritten as –

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S_{1} = K_{1} |A| + |K_{2}||B| {cos (θ_{2} – ɸ) + j sin (θ_{2} – ɸ)}

and S_{2} = K_{3}|A| + |K_{4}||B| {cos (θ_{4 }– ɸ) + j sin (θ_{4 }– ɸ } …(3.2)

**Analysis for Amplitude Comparator****: **

If the operating criterion is given by |S_{1}| __>__ |S_{2}| then at the threshold of operation-

**Analysis for Phase Comparator****: **

The two quantities to be compared are S_{1} and S_{2}. Let the phase angles of S_{1} and S_{2} with respect to a reference axis be α and β respectively. The relay operates when the product of S_{1 }and S_{2} is positive. The product of S_{1} and S_{2} is maximum when they are in phase and the threshold condition i.e., positive torque will be obtained when α – β = ± 90° or ± /2 radians.

Under this condition tan (α – β) = ± ∞

or tan α – tan β/1 + tan α tan β

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or 1 + tan α tan β = 0

Substituting for tan α and tan β from phasor diagram shown in Fig. 3.1

Dividing the above equation by |K_{2}||K_{4}| |A|^{2} cos (θ_{2} – θ_{4}), we have –

as given in Fig. 3.3.

In most of the relays, at least one of the constants K_{1}, K_{2}, K_{3} and K_{4} is zero and two of them are often equal. Also the angle of the two phasor constants is usually the same. This makes the practical case relatively simple.

**If θ _{2} = θ_{4}, the values of r and c in the two cases are tabulated below: **

**Duality between Amplitude and Phase Comparators****: **

It can be shown with the help of phasor diagrams that an inherent amplitude comparator becomes a phase comparator and vice-versa if the input quantities to the comparator are changed to the sum and difference of the original two input quantities.

Consider the operation of an amplitude comparator with input signals S_{1} and S_{2} such that it operates when |S_{1}| > |S_{2}|.

If the inputs are changed to |S_{1} + S_{2}| and |S_{1} – S_{2}| so that it operates when |S_{1} + S_{2}| > |S_{1 }– S_{2}|.

If these quantities are fed to an amplitude comparator, the comparator essentially compares the phase relation between S_{1} and S_{2}. This is illustrated in Fig. 3.4.

It is observed that the requirement |S_{1} + S_{2}| < |S_{1} – S_{2}| puts a condition on the phase relation between S_{1} and S_{2} i.e. unless the phase difference between original phasors S_{1} and S_{2} exceeds 90°, |S_{1} + S_{2}| cannot be less than |S_{1} – S_{2}|. So the original amplitude comparator with inputs now |S_{1} + S_{2}| and |S_{1} – S_{2}| is a phase comparator i.e., a converted phase comparator.

It is to be noted that the phase comparator in case of static circuits is a cosine comparator as opposed to a sine comparator in case of electromechanical relays.

The amplitude comparison using a phase comparator is explained with the help of phasor diagram shown in Fig. 3.5.

From the phasor diagram it is clear that if the original inputs S_{1} and S_{2} to the phase comparator are with such a phase relation that they will operate the relay and if now the inputs are changed to |S_{1} + S_{2}| and |S_{1} – S_{2}| and supplied to the same comparator, the comparator essentially compares the amplitude relation between S_{1} and S_{2}. Unless |S_{1}| > |S_{2}|, the phase relation between |S_{1} + S_{2}| and |S_{1 }– S_{2}| will not be less than 90° and hence the phase comparator with inputs |S_{1} + S_{2}| and |S_{1} – S_{2}| will now be an amplitude comparator.

Though a given relay characteristic can be obtained using either of the two comparators, consideration of the constants computed for required characteristics would indicate which type of comparator is preferable. In general an inherent comparator is better than the converted type because if one quantity is very large in comparison with the other, a small error in the larger quantity may cause an incorrect comparison when their sum and difference are fed as inputs to the relay.