Distance Protection & the Mho Circle

A distance relay can't measure distance directly — it measures impedance. For a fault on the line, the apparent impedance Z = V/I is proportional to the distance to the fault, so the relay reaches a fixed distance regardless of fault current. The mho circle is the relay's trip boundary drawn on the R–X plane. Move the fault, change the system, and watch the impedance point travel through the protection zones.

Transmission Line

Line length

Impedance / length

Line angle θ_L

Source · Generator + GSU

System voltage

Source impedance ratio (SIR)

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Relay & Fault

Zone 1 reach

Fault location

Fault resistance R_f

Remote infeed

Infeed factor (I_remote/I_relay)

One-Line Diagram

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Mho Characteristic · R–X Plane

Zone 1 (instant) Zone 2 (delayed) Zone 3 (backup) Line impedance ● Apparent Z (relay sees)

Fault Current vs. Location

Readout

Fault current at relay

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Apparent impedance

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Relay decision

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Apparent reach error

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The Physics

Impedance is distance. The relay at the sending bus measures voltage and current and computes Z = V/I. For a fault a fraction m along a line of impedance Z_L, the relay sees m·Z_L — the fault current cancels out, so the reach is independent of source strength:

Z_apparent = V_relay / I_relay = m·Z_L + R_f

The mho circle is a circle through the origin whose diameter lies along the line angle θ_L. Its diameter equals the reach setting. The relay trips when the measured Z lands inside the circle. Zones step the reach outward in distance and time:

Zone 1 ≈ 80% Z_L (instantaneous) · Zone 2 ≈ 120% (≈0.3 s) · Zone 3 = remote backup

Zone 1 is set short of the remote bus because relay/CT/PT errors make 100% reach unreliable — you must never overreach the far bus instantaneously. Zone 2 deliberately overreaches to cover the rest of the line.

Fault current falls with distance. A close-in fault sees only the source impedance; a far fault adds line impedance, so the current drops — yet the relay still places it correctly because both V and I scale together.

I_fault = E_LN / |Z_source + m·Z_L + R_f|

📐 Fault resistance bends the truth. Arc or ground resistance adds a real (horizontal) term, pushing the apparent point to the right — toward the circle's edge. A resistive fault near the reach point can fall outside the mho circle and not trip. That's why resistive coverage matters.

🔀 Infeed makes the relay under-reach. When another source feeds the fault at the remote bus, the relay current is smaller than the total fault current, so the remote section looks bigger than it is: Z_app = Z_L1 + (1 + I_infeed/I_relay)·Z_remote. Turn infeed on and watch the apparent point jump outward — a Zone 2 fault can slip out of reach.