- 2071 Bhadra · 8 marks
Universal torque equation and relay characteristics
Short answer
Universal torque equation is T = K1 I^2 + K2 V^2 + K3 V I cos(theta - tau) + K4. Setting K3=0, K2<0 gives impedance relay, a circle centred at origin; setting K2=0, K3<0, tau=90 deg gives reactance relay, a line parallel to R; setting K1=0, K2<0 gives mho relay, a circle through origin.
What is Universal Torque Equation? Using this equation derive the following characteristics (i) impedance relay (ii) reactance relay (iii) mho relay.
The universal torque equation is the general expression for the torque of an electromagnetic relay that has current, voltage and directional (voltage–current) elements together with a spring:
T = K₁ I² + K₂ V² + K₃ V I cos(θ − τ) + K₄
- K₁ I²: torque of the current element
- K₂ V²: torque of the voltage element
- K₃ V I cos(θ − τ): torque of the directional element (θ = angle between V and I, τ = maximum torque angle)
- K₄: mechanical (spring) torque
Each constant is given a sign (+ for operating, − for restraining), and some may be set to zero to obtain each relay type. In the derivations below the spring torque K₄ is neglected.
(i) Impedance relay
Operating torque from current, restraining torque from voltage: K₃ = 0, K₂ negative.
T = K₁ I² − K₂ V²
At balance (T = 0): K₁ I² = K₂ V²
V/I = Z = √(K₁/K₂) = constant
Operates when Z < √(K₁/K₂)
Since Z² = R² + X² = constant, the characteristic on the R-X diagram is a circle centred at the origin. It is non-directional.
X
│ .-~~~-.
│ / \
───┼( O )─── R
│ \ /
│ '-...-' trip inside circle
(ii) Reactance relay
Operating torque from current, restraining torque from a directional element with τ = 90°: K₂ = 0, K₃ negative.
T = K₁ I² − K₃ V I cos(θ − 90°)
= K₁ I² − K₃ V I sin θ
At balance: K₁ I² = K₃ V I sin θ
(V/I) sin θ = K₁/K₃
Z sin θ = X = K₁/K₃ = constant
Operates when X < K₁/K₃
The characteristic is a straight line parallel to the R axis at X = K₁/K₃. It measures only reactance, so it is not affected by arc resistance; it needs a directional (mho) starting unit.
X
│
──────┼────────────── X = K₁/K₃
│ trip below line
───┼────────────── R
(iii) Mho (admittance) relay
Operating torque from the directional element, restraining torque from voltage: K₁ = 0, K₂ negative.
T = K₃ V I cos(θ − τ) − K₂ V²
At balance: K₃ V I cos(θ − τ) = K₂ V²
V/I = (K₃/K₂) cos(θ − τ)
Z = (K₃/K₂) cos(θ − τ)
Operates when Z < (K₃/K₂) cos(θ − τ)
This is the equation of a circle passing through the origin with diameter K₃/K₂ along the line at angle τ. The relay is inherently directional.
X
│ .-~-.
│ / τ \ diameter K₃/K₂
│ ( ↗ ) at angle τ
│ \ /
O───'-.-'────── R
(circle passes through origin)
Summary
| Relay | Constants | Characteristic on R-X |
|---|---|---|
| Impedance | K₃ = 0, K₂ < 0 | Circle centred at origin |
| Reactance | K₂ = 0, K₃ < 0, τ = 90° | Line parallel to R axis |
| Mho | K₁ = 0, K₂ < 0 | Circle through origin |
✓ Answer checked by Claude Opus 5.5 (high effort)
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