Every element you've commissioned so far asked one of two questions: “is the current too high?” (overcurrent) or “do the currents balance?” (differential). Distance asks a completely different question: how far away is the fault?

The trick is Ohm's law. The relay continuously divides the voltage at its terminals by the current flowing out of them to get an apparent impedance, Z = V/I. Under normal load that ratio is large (little current, full voltage). When a fault occurs, the voltage collapses and the current surges, so Z drops sharply — and crucially, it drops to a value that depends only on the impedance of the line between the relay and the fault. Since a transmission line has a near-constant impedance per kilometre, that measured impedance is a direct proxy for distance. The relay never needs to talk to the other end of the line to know roughly where the fault is. That self-contained reach is why distance protection is the backbone of transmission protection.

Impedance is a complex number — it has resistance (R) and reactance (X) — so the relay lives on a 2-D map called the R-X plane: resistance along the horizontal axis, reactance up the vertical. Everything the relay cares about is a point or a shape on this map.

The protected line itself is a vector from the origin: its length is the line's impedance and its tilt is the line angle (typically 75–85°, because a line is mostly reactance). A fault somewhere along the line shows up as a point sitting on that vector — near the origin for a close-in fault, far out for a remote one. Healthy load, by contrast, is a point well off to the side: a large impedance at a low angle (load is mostly resistive). Faults and load live in different neighbourhoods of the plane, and that separation is what lets the relay tell them apart.

The relay draws zones — closed shapes — on the R-X plane. If the measured impedance lands inside a zone, the relay trips after that zone's time delay. A line is protected by three:

Zone 1 is instantaneous but deliberately set to reach only ~80% of the line. Why not 100%? Because CT errors, VT errors, and uncertainty in the line impedance mean the last 10–20% is a grey area — if Z1 reached the full line it could occasionally over-reach past the remote bus and trip for a fault on the next line, which isn't yours. Under-reaching guarantees security. Zone 2 covers the remaining ~20% of the line (and a bit into the next), but it waits ~0.3–0.4 s so the next line's Zone 1 gets first crack at faults near that boundary. Zone 3 is slow remote backup. Plotted as trip-time versus fault-distance, this becomes the famous stepped staircase — instant near in, stepping up in time as the fault moves away. Drag the fault below and watch the step.

Two characteristic shapes dominate. The mho is a circle that passes through the origin with its diameter along the line angle. Its great virtue is that it's inherently directional — being anchored at the origin, it only looks forward — and it's simple and robust. Its weakness is resistive coverage: a fault with significant arc resistance pushes the measured point to the right, and a round circle pinches in there, so the mho can under-reach resistive faults, especially on short lines.

The quadrilateral solves that by setting the reactive reach (how far up the line) and the resistive reach (how wide a blinder)independently, giving a box that hugs the line yet still catches high-resistance ground faults. The cost is that a quad isn't inherently directional, so it needs a separate directional element to supervise it. Toggle mho/quad above and slide the arc-resistance: you'll see a resistive fault fall out of the mho circle but stay inside the quad.

For phase-to-phase faults the loop impedance is just the positive-sequence line impedance, so Z = (fraction) × Z1 and life is simple. Ground faults are different: the current returns through the earth and the neutral, and that return path has a different impedance (the zero-sequence impedance Z0, usually 2.5–3.5× Z1). If the relay naively used Z1 for a ground fault, it would compute an impedance that's too large — the fault would appear farther away than it really is, and the relay would under-reach and miss faults near its reach point.

The fix is the residual (earth-return) compensation factor $k_0 = \dfrac{Z_0 - Z_1}{3\,Z_1}$. The relay adds k0 × (residual current) to the phase current before dividing, which mathematically collapses the ground-fault loop back onto the same Z1-based reach as a phase fault. For this line k0 works out to 0.67∠-7°. Set it wrong (or leave it at zero, as many do on first commissioning) and your ground-fault reach is off by tens of percent. Switch the interactive above to Ground and watch the fault point slide out along the line when k0 is missing.

Load is an impedance too, and on a heavily loaded long line the load point can creep toward the zones — especially the big Zone 3. If a zone reaches far enough to swallow the load point, the relay trips the line for nothing. The defence is a load-encroachment blinder: a wedge carved out of the characteristic along the low-angle resistive region where load lives, so the zone ignores anything that looks like load rather than a fault.

A power swing is the harder cousin. When generators oscillate against each other after a disturbance, the apparent impedance drifts slowly across the plane and can pass straight through the zones — looking, for an instant, exactly like a fault. Power-swing blocking watches the rate: a real fault jumps into a zone in milliseconds, while a swing crawls across a concentric outer band over tens of milliseconds. If the impedance lingers in that band before entering a zone, it's a swing, and the trip is blocked so the system can recover.

All of this is set in the relay as secondary ohms, because the relay sees CT and VT secondaries, not the primary system. The conversion is $Z_{\text{sec}} = Z_{\text{pri}}\times\frac{\text{CTR}}{\text{VTR}}$. For this line the positive-sequence impedance of 8∠80° Ω primary becomes 4.80 Ω secondary, and a Zone 1 reach of 80% is simply 0.8 × that. Get the picture first; the arithmetic is the easy part once you know what you're aiming at.