Unbalanced Systems and Symmetrical Components

Fortescue's method, with the operator $a = 1\angle120°$:

$\begin{bmatrix}V_0\\V_1\\V_2\end{bmatrix} = \frac{1}{3}\begin{bmatrix}1&1&1\\1&a&a^2\\1&a^2&a\end{bmatrix}\begin{bmatrix}V_a\\V_b\\V_c\end{bmatrix}$

• Positive sequence — the normal rotation.
• Negative sequence — opposite rotation; heats motor rotors.
• Zero sequence — in phase in all three; flows through neutral/earth.

This is the standard tool for analysing unsymmetrical faults (line-to-ground, line-to-line) and designing earth-fault protection.