PW-C1 · Transmission and per-unit

The line model: a wire is an impedance

Treat a line as a perfect resistance-free wire and you can explain neither voltage drop nor loss. A real line is a series impedance Z = R + jX, and at high voltage the reactance X from the magnetic field dominates the resistance R. Drag the line length to see the impedance grow, and learn why a wire behaves more like a reactor than a resistor.

Drag the line length and grow the impedance

Drag the handle to change the line length. The base is resistance R, the height is reactance X, the hypotenuse is the impedance Z = R + jX. The longer the line the larger all three, but X rises far faster than R, so the hypotenuse stands steep — the impedance angle is near 80 degrees.

Drag the handle to set the line length.

Line impedance (from typical values)

|Z| ≈ 45.7 Ω   θ ≈ 80°

100 km   R ≈ 8.0 Ω   X ≈ 45.0 Ω   (X/R ≈ 5.6)

A wire is not perfect

A conductor carrying current has resistance R and loses power as heat. The magnetic field around the current opposes its change, giving inductance, while the gap between conductor and earth gives capacitance. A short line of around 80 km lumps just the series resistance and inductance into a single impedance Z = R + jX.

Reactance dominates resistance

A high-voltage overhead line is thick and conductive, so its resistance per length is small. But reactance is set not by conductor thickness but by geometry — the spacing and arrangement of conductors — and is comparatively large. So X per length is five or six times R, and the impedance angle θ = atan(X/R) is near 80 degrees. This is why transmission work often neglects resistance and keeps only reactance.

ObserveZ = R + jX

The line impedance is resistance and reactance in series.

Choose|Z| = √(R² + ?)

The magnitude is the root of the sum of the squares.

Fill inθ = atan(X / ?)

The angle is the arctangent of reactance over resistance.

On your ownVs = Vr + ?

Current through Z gives sending = receiving + I·Z.

Longer lines, capacitance counts

As a line gets longer, the capacitance between conductor and earth can no longer be ignored. A medium line over about 100 km is modeled as a pi network with half of this shunt capacitance at each end, and a long line of hundreds of km is treated with distributed parameters where the impedance spreads continuously along the line. But the starting point is always the same — a wire is a path with impedance.

Back to the first screen

The longer you dragged it the larger the impedance triangle grew, yet its angle stayed near 80 degrees. The angle is fixed by the ratio of R to X per length, while length only scales the magnitude. This steep triangle, with the height X dominating the base R, is the essence of a line — a wire is not a plain conductor but a path with impedance, and current through that Z creates the voltage drop Vs = Vr + I·Z.

The line model — a transmission line is not a perfect wire but a series impedance Z = R + jX. R is conductor resistance, X the reactance from the magnetic field (inductance), and at high voltage usually X > R. |Z| = √(R²+X²), impedance angle θ = atan(X/R). A longer line adds the capacitance to earth (shunt) and is modeled as a pi network. Current through Z creates the drop Vs = Vr + I·Z.

The next step

A power system scatters the impedances of generators, transformers and lines across different voltage levels, so raw ohms must be converted every time you cross a transformer. The next unit (PW-C2) looks at the per-unit system. Fix a base voltage and base power, normalize every quantity to a ratio of that base, and both sides of a transformer share the same per-unit impedance, unifying the whole system into one clean number system.