--- title: "Equal positive and negative sequence impedances" subject: "Power System Analysis II" chapter: 4 key: "semester_5/power_system_analysis_ii:static-circuit-positive-negative-equal" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/static-circuit-positive-negative-equal site: BCT Engineering --- # Equal positive and negative sequence impedances Power System Analysis II · Chapter 4: Unbalanced System Analysis - Asked in: 2081 Bhadra, 6+2 marks - Video: [Sequence Impedances of Transmission Line - Symmetrical Components - Power System 2](https://www.youtube.com/watch?v=oCGgmL1_qxY) — Ekeeda - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/static-circuit-positive-negative-equal - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-4 ## Question With proper mathematical interpretations, justify that for a symmetrical static circuit, positive and negative sequence impedances are equal. Also discuss why the sequence impedances for a generator are not equal. ## Short answer For a symmetrical static circuit, Z1 = Z2 = Zs − Zm because the phase impedance matrix has equal self and mutual impedances, making the positive and negative sequence terms identical. In a generator, the rotor motion causes positive sequence MMF to be stationary relative to the rotor, while negative sequence MMF rotates opposite, inducing different magnetic reactions, so impedances differ. ## Answer 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 $Z_1 \neq Z_2 \neq Z_0$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For a symmetrical static circuit: A. Z1 = Z2 = Zs + 2Zm B. Z0 = Z1 = Zs - Zm C. Z1 = Z2 = Zs - Zm, Z0 = Zs + 2Zm D. Z1 = Zs - Zm, Z2 = Zs + Zm Answer: C. A^-1 Zp A = diag(Zs+2Zm, Zs-Zm, Zs-Zm). (Key point: Result) 2. Z1 = Z2 in a static circuit because: A. Currents have equal magnitude B. Frequency is the same C. Zero sequence is absent D. Reversing phase sequence just interchanges b and c, giving the same circuit Answer: D. A static circuit's impedance does not depend on phase order. (Key point: Why equal) 3. The generator negative sequence reactance is lower than X1 since: A. Negative MMF rotates at 2ws relative to rotor, inducing opposing currents in damper windings B. It is stationary to the rotor C. There is no armature flux D. Field winding is open Answer: A. X2 ≈ (Xd''+Xq'')/2. (Key point: Generator X2) 4. For a generator, the typical order is: A. X0 > X1 > X2 B. X1 > X2 > X0 C. X2 > X1 > X0 D. X1 = X2 > X0 Answer: B. Zero sequence MMFs cancel in space, leaving only leakage flux. 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