Impedance relay from universal torque equation

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:

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):

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

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

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

The relay operates when operating torque > restraining torque:

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.

        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.

          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.