--- title: "Universal torque equation and relay characteristics" subject: "Switchgear and Protection" chapter: 7 key: "semester_6/switchgear_and_protection:universal-torque-equation" source: https://www.bctengineer.com/q/semester_6/switchgear_and_protection/universal-torque-equation site: BCT Engineering --- # Universal torque equation and relay characteristics Switchgear and Protection · Chapter 7: Protective Relays - Asked in: 2071 Bhadra, 8 marks - Video: [SGP313 Universal Relay Torque Equation](https://www.youtube.com/watch?v=lG7vOnaB-sw) — Lectures in Electrical Engineering - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/switchgear_and_protection/universal-torque-equation - Whole chapter: https://www.bctengineer.com/md/semester_6/switchgear_and_protection/chapter-7 ## Question What is Universal Torque Equation? Using this equation derive the following characteristics (i) impedance relay (ii) reactance relay (iii) mho relay. ## 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. ## Answer 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: ```text 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. ```text 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. ```text 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. ```text 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. ```text 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. ```text 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. ```text 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 | ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In T = K1 I^2 + K2 V^2 + K3 V I cos(θ − τ) + K4, which term represents the directional element? A. K1 I^2 B. K2 V^2 C. K3 V I cos(θ − τ) D. K4 Answer: C. The V-I cosine term depends on the phase angle between voltage and current, so it is the directional element. K1 is current, K2 voltage, K4 spring. (Key point: Terms of the universal torque equation) 2. Which constants give an impedance relay, and what does its R-X characteristic look like? A. K3 = 0, K2 negative; circle centred at the origin B. K2 = 0, K3 negative; straight line parallel to R axis C. K1 = 0, K2 negative; circle through the origin D. K1 = 0, K3 = 0; straight line through the origin Answer: A. With K3 = 0 and a restraining voltage term, K1 I^2 = K2 V^2 gives Z = constant, so R^2 + X^2 is constant: a circle centred at the origin. (Key point: Impedance relay constants) 3. For the reactance relay (K2 = 0, τ = 90°), the balance equation K1 I^2 = K3 V I sin θ leads to which result? A. Z = constant B. R = K1/K3 = constant C. X = Z sin θ = K1/K3 = constant D. Z = (K3/K1) cos θ Answer: C. Dividing by I^2 gives Z sin θ = K1/K3, and Z sin θ is the reactance X, so the locus is a line parallel to the R axis. (Key point: Reactance relay derivation) 4. For the mho relay (K1 = 0, K2 negative), Z = (K3/K2) cos(θ − τ). What is the resulting characteristic? A. Circle centred at the origin B. Straight line parallel to X axis C. Circle passing through the origin, inherently directional D. Straight line parallel to R axis Answer: C. The equation is a circle through the origin with diameter K3/K2 at angle τ, so the relay is inherently directional. 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