seegongsik
PW-C1 · Transmission and per-unit

The line model: a wire is an impedance

If a transmission line is treated as a perfect conductor with no resistance, neither voltage drop nor loss can be explained. A real line is a series impedance Z = R + jX, and at high voltage the reactance X coming from the magnetic field far exceeds the resistance R; drag the line length yourself, watch how the impedance grows, and learn why a line is closer to a reactor than to a resistor.

Drag the line length to grow the impedance

Dragging the handle changes the line length. The base is the resistance R, the height the reactance X and the hypotenuse the impedance Z = R + jX; all three grow with length, but X rises much faster than R, so the hypotenuse stands steeply. The impedance angle is close to 80°.

Drag the handle to change the line length between 10 and 400 km. Typical values of R 0.08 Ω and X 0.45 Ω per km are used, and the figure draws only the series R and X whatever the 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. On top of that, a magnetic field forms around the current and opposes changes in it, giving inductance, and capacitance forms between the conductor and the ground. A short line of up to about 80 km is modeled by bundling only the series resistance and inductance into one impedance Z = R + jX.

Reactance dominates resistance

High-voltage overhead conductors are thick and highly conductive, so their resistance per unit length is small. Reactance, on the other hand, is set by the logarithm of the ratio of conductor spacing to conductor radius, so it does not drop much even with thicker conductors and stays relatively large. So X per unit length reaches five or six times R and the impedance angle θ = atan(X/R) comes close to 80°, which is the basis for often neglecting resistance and using reactance alone when handling transmission lines.

ObserveZ = R + jX
Line impedance is resistance and reactance in series.
Choose|Z| = √(R² + ?)
Its size is the square root of the sum of the squares of the two sides.
Fill inθ = atan(X / ?)
Its angle is the arctangent of reactance divided by 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 the conductors and the ground can no longer be ignored. Medium lines from beyond 80 km up to around 250 km are modeled as a π, with this shunt capacitance split in half at each end, and longer lines are treated as distributed parameters, with impedance and capacitance spread evenly along the line. The starting point is always the same, though: a line is a path with impedance.

Back to the first screen

As the length was dragged out the impedance triangle grew, yet its angle always stayed around 80°, because the angle is set by the ratio of R to X per unit length while length only scales the size. This steep triangle, with height X far exceeding base R, is what a transmission line really is: not a mere wire but a path with impedance, and current flowing through that Z produces the voltage drop Vs = Vr + I·Z.

Transmission line model: a transmission line is not a perfect conductor but a series impedance Z = R + jX. R is the conductor resistance and X the reactance from the magnetic field (inductance), and on high-voltage lines usually X > R. |Z| = √(R²+X²), impedance angle θ = atan(X/R). For long lines the capacitance to ground (shunt) is added in a π model, and current flowing through Z produces the voltage drop Vs = Vr + I·Z.
The next step

The impedances of generators, transformers and lines are scattered across different voltage levels in a system, so using ohm values as they are means converting every time a transformer is crossed. The next unit (PW-C2) works out the per-unit system: choose a base voltage and a base power and express every quantity as a ratio to that base, and the impedances on both sides of a transformer take the same per-unit value, tying the whole system into one tidy set of numbers.