---
title: "Impedance relay from universal torque equation"
subject: "Switchgear and Protection"
chapter: 7
key: "semester_6/switchgear_and_protection:impedance-relay-torque-equation"
source: https://www.bctengineer.com/q/semester_6/switchgear_and_protection/impedance-relay-torque-equation
site: BCT Engineering
---

# Impedance relay from universal torque equation

Switchgear and Protection · Chapter 7: Protective Relays

- Asked in: 2070 Magh, 8 marks; 2077 Chaitra, 6 marks (asked 2 times)
- 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/impedance-relay-torque-equation
- Whole chapter: https://www.bctengineer.com/md/semester_6/switchgear_and_protection/chapter-7

## Question

What is an impedance relay? Starting from the universal torque equation, explain its operating principle, torque equation and operating characteristics (as used for the protection of feeder).

## Short answer

An impedance relay measures V/I = Z and operates when Z drops below the set value. Its torque equation is T = K₁I² − K₂V² − K₄; at balance K₁I² = K₂V² so Z = √(K₁/K₂) = Zₛ. It trips when Z < Zₛ. The characteristic is a circle centred at the origin on the R-X diagram, making it non-directional.

## Answer

An **impedance relay** is a distance relay that measures the ratio of voltage to current (V/I = Z) at the relay point and operates when this impedance falls below a set value. Since line impedance is proportional to length, it measures the distance to the fault.

#### Universal torque equation
For a general electromagnetic relay with current, voltage and directional elements:

```text
T = K₁ I² + K₂ V² + K₃ V I cos(θ − τ) + K₄
```
where K₁, K₂, K₃ are constants, θ is the angle between V and I, τ is the maximum torque angle, and K₄ is the mechanical restraining (spring) torque.

#### Torque equation of impedance relay
In the impedance relay the current element produces operating torque and the voltage element produces restraining torque; there is no directional element (K₃ = 0):

```text
Put K₂ = −K₂ (restraint), K₃ = 0:
T = K₁ I² − K₂ V² − K₄
```

At the balance point (T = 0), neglecting spring torque K₄:

```text
K₁ I² = K₂ V²
V² / I² = K₁ / K₂
Z = V / I = √(K₁ / K₂) = constant (setting Zs)
```

The relay operates when operating torque > restraining torque:

```text
K₁ I² > K₂ V²   ⟹   Z = V/I < √(K₁/K₂) = Zs
```

#### Operating principle
- Construction: a balanced beam; the current coil (fed by CT) pulls one end down, the voltage coil (fed by PT) pulls the other end.
- **Normal condition:** voltage is high and current is normal, so Z_seen (load impedance) is large; voltage pull wins and contacts stay open.
- **Fault within the zone:** voltage at the relay falls and current rises, so Z_seen = impedance up to the fault point < Zs; current pull wins and the relay trips.
- **Fault beyond the zone:** Z_seen > Zs, so the relay does not operate.

```text
        Voltage coil       Current coil
          (PT) ▼              ▼ (CT)
     ──────────────△───────────────
               pivot     ▲ trip contacts
```

#### Operating characteristic (R-X diagram)
Since operation depends only on |Z|, the characteristic is a **circle centred at the origin** with radius Zs.

```text
          X
          │   .-~~~-.
          │ /  Trip  \
        ──┼─(  zone   )── R
          │ \  Z<Zs  /
          │   '-...-'
          │ outside: no trip
```

#### Properties for feeder protection
- **Non-directional:** it operates for faults in all four quadrants (behind the relay too), so it is used with a **directional unit** (mho or directional relay) in series.
- Affected by **arc resistance** moderately and by **power swings**, since the circle covers a large area.
- Used in three zones with time steps: Zone 1 ≈ 80–90% of the line (instantaneous), Zone 2 ≈ 120–150% (about 0.3–0.5 s), Zone 3 covers the next line (about 1 s).
- Its reach does not depend on fault current level, so it gives faster and more selective protection than overcurrent relays on long lines.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does an impedance relay measure and when does it operate?
   A. It measures V/I and operates when the measured impedance falls below the set value
   B. It measures V x I and operates above a set power
   C. It measures current only and operates above a set current
   D. It measures phase angle between V and I
   Answer: A. Line impedance is proportional to length, so Z = V/I below the setting means the fault is within the set reach. (Key point: Impedance relay principle)

2. Starting from the universal torque equation, which terms give the impedance relay torque equation T = K1 I^2 - K2 V^2 - K4?
   A. K3 is kept and K1 is made zero
   B. K2 is made negative (restraint) and K3 = 0 (no directional element)
   C. K1 is made negative and K2 = 0
   D. K4 is made the operating term
   Answer: B. The current element is operating, the voltage element is restraining (K2 negative), and the VI cos(θ-τ) term K3 is zero. (Key point: Torque equation)

3. At the balance point (neglecting spring), what is the setting impedance Zs?
   A. K1/K2
   B. K2/K1
   C. sqrt(K2/K1)
   D. sqrt(K1/K2)
   Answer: D. K1 I^2 = K2 V^2 gives V^2/I^2 = K1/K2, so Zs = sqrt(K1/K2). (Key point: Setting impedance)

4. What is the R-X characteristic of an impedance relay and what follows from it?
   A. A circle through the origin; it is inherently directional
   B. A straight line; it detects only resistance
   C. A circle centred at the origin with radius Zs; it is non-directional so needs a directional unit
   D. An ellipse; it is unaffected by power swings
   Answer: C. Operation depends only on |Z|, so the characteristic is a circle at the origin, covering all quadrants, and a directional unit must be added. (Key point: Characteristic and directionality)

## Related in Switchgear and Protection

- [Universal torque equation and relay characteristics](https://www.bctengineer.com/question.md?key=semester_6%2Fswitchgear_and_protection%3Auniversal-torque-equation) — Switchgear and Protection, chapter 7
- [Impedance type distance relay](https://www.bctengineer.com/question.md?key=semester_6%2Fswitchgear_and_protection%3Aimpedance-distance-relay-application) — Switchgear and Protection, chapter 7
- [IDMT characteristics and induction disc relay](https://www.bctengineer.com/question.md?key=semester_6%2Fswitchgear_and_protection%3Aidmt-induction-disc-relay) — Switchgear and Protection, chapter 7, asked 2×
- [IDMT settings for 20 MVA transformer and feeder](https://www.bctengineer.com/question.md?key=semester_6%2Fswitchgear_and_protection%3Atransformer-feeder-idmt-20mva) — Switchgear and Protection, chapter 7, asked 2×
- [Distance protection relay and applications](https://www.bctengineer.com/question.md?key=semester_6%2Fswitchgear_and_protection%3Adistance-protection-relay) — Switchgear and Protection, chapter 7

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