Equal positive and negative sequence impedances

A symmetrical static circuit (transmission line, transformer, static load) has equal self impedances $Z_s$ in all phases and equal mutual impedances $Z_m$ between all pairs of phases; for such a circuit $Z_1 = Z_2 = Z_s - Z_m$.

Mathematical proof

Voltage drops across the three phases:

$$\begin{aligned} V_a &= Z_s I_a + Z_m I_b + Z_m I_c \\ V_b &= Z_m I_a + Z_s I_b + Z_m I_c \\ V_c &= Z_m I_a + Z_m I_b + Z_s I_c \end{aligned} \qquad\Rightarrow\qquad Z_p = \begin{bmatrix} Z_s&Z_m&Z_m\\Z_m&Z_s&Z_m\\Z_m&Z_m&Z_s \end{bmatrix}$$

Transform with $V_p = AV_s$, $I_p = AI_s$: $V_s = A^{-1}Z_pA\,I_s$.

Write $Z_p = (Z_s - Z_m)U + Z_mJ$ ($U$ = unit matrix, $J$ = matrix of ones). Since $A^{-1}UA = U$ and $A^{-1}JA = \text{diag}(3, 0, 0)$ (column sums of $A$ are $3$, $1+a+a^2 = 0$, $0$):

$$Z_{seq} = A^{-1}Z_pA = \begin{bmatrix} Z_s+2Z_m & 0 & 0 \\ 0 & Z_s-Z_m & 0 \\ 0 & 0 & Z_s-Z_m \end{bmatrix}$$

Hence

$$Z_0 = Z_s + 2Z_m, \qquad Z_1 = Z_2 = Z_s - Z_m$$

Physically, in a static circuit the impedance does not depend on the order in which the phase currents reach their peaks (a-b-c or a-c-b); reversing the phase sequence only interchanges b and c, which gives the same circuit. So $Z_1 = Z_2$. Zero sequence currents are in phase, so the mutual effects add ($+2Z_m$) and $Z_0$ is different, usually larger.

Why generator sequence impedances are not equal

A synchronous generator is a rotating machine, so the sequence of the current changes the field it produces relative to the rotor:

• Positive sequence: the armature MMF rotates at synchronous speed in the same direction as the rotor, so it is stationary relative to the rotor. The reactance is $X_d''$, $X_d'$ or $X_d$ depending on the time after the fault.
• Negative sequence: the MMF rotates opposite to the rotor at relative speed $2\omega_s$. It induces double-frequency currents in the field and damper windings, which oppose the flux, so the reactance is low, about $X_2 \approx \frac12(X_d'' + X_q'')$.
• Zero sequence: the three in-phase currents produce almost no air-gap MMF (they cancel in space), so only leakage flux remains and $X_0$ is the smallest.

So for a generator $X_1$ (which changes with time from $X_d''$ to $X_d$) is not equal to $X_2$, and $X_0$ is different again and usually the smallest. Hence $Z_1 \neq Z_2 \neq Z_0$.