--- title: "Sequence networks decoupled for static circuit" subject: "Power System Analysis II" chapter: 4 key: "semester_5/power_system_analysis_ii:static-circuit-sequence-decoupled" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/static-circuit-sequence-decoupled site: BCT Engineering --- # Sequence networks decoupled for static circuit Power System Analysis II · Chapter 4: Unbalanced System Analysis - Asked in: 2068 Chaitra, 8 marks - 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-sequence-decoupled - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-4 ## Question For a 3-phase symmetrical static circuit, justify that sequence networks are decoupled. ## Short answer The sequence impedance matrix of a symmetrical static circuit is diagonal, with zero off-diagonal terms, because transforming the balanced Zp yields Z0 = Zs+2Zm, Z1 = Z2 = Zs−Zm. Thus each sequence current produces voltage drop only in its own network, leaving them fully decoupled. ## Answer For a three-phase **symmetrical static circuit** (line, transformer, static load), the impedance matrix in sequence quantities is diagonal, so a current of one sequence produces voltage drops of that sequence only; this means the three sequence networks are **decoupled**. #### Phase equations Each phase has self impedance $Z_s$ and equal mutual impedance $Z_m$ to each other phase: $$ \begin{bmatrix} V_a\\V_b\\V_c \end{bmatrix} = \begin{bmatrix} Z_s&Z_m&Z_m\\Z_m&Z_s&Z_m\\Z_m&Z_m&Z_s \end{bmatrix}\begin{bmatrix} I_a\\I_b\\I_c \end{bmatrix}, \qquad V_p = Z_pI_p $$ #### Transformation With $V_p = AV_s$, $I_p = AI_s$: $$ V_s = A^{-1}Z_pA\,I_s = Z_{seq}I_s $$ where $$ A = \begin{bmatrix} 1&1&1\\1&a^2&a\\1&a&a^2 \end{bmatrix}, \qquad A^{-1} = \frac13\begin{bmatrix} 1&1&1\\1&a&a^2\\1&a^2&a \end{bmatrix} $$ First find $Z_pA$, column by column (using $1 + a + a^2 = 0$): - Column 1: each row $= Z_s + 2Z_m$, so column $= (Z_s+2Z_m)[1\ 1\ 1]^T$ - Column 2: rows $= Z_s + Z_m(a^2+a) = Z_s - Z_m$ times $[1\ a^2\ a]^T$ - Column 3: rows $= (Z_s - Z_m)$ times $[1\ a\ a^2]^T$ So $Z_pA = A\,\text{diag}(Z_s+2Z_m,\ Z_s-Z_m,\ Z_s-Z_m)$; the columns of $A$ are eigenvectors of $Z_p$. Therefore $$ Z_{seq} = A^{-1}Z_pA = \begin{bmatrix} Z_s+2Z_m & 0 & 0 \\ 0 & Z_s-Z_m & 0 \\ 0 & 0 & Z_s-Z_m \end{bmatrix} $$ #### Result $$ V_{a0} = Z_0I_{a0}, \qquad V_{a1} = Z_1I_{a1}, \qquad V_{a2} = Z_2I_{a2} $$ with $Z_0 = Z_s + 2Z_m$ and $Z_1 = Z_2 = Z_s - Z_m$. All off-diagonal (mutual) terms between sequences are zero, so: - Positive sequence current causes only positive sequence voltage drop, and similarly for negative and zero. - Each sequence network can be drawn and solved **independently** as a single-phase circuit. - The networks are connected to each other only at points of unbalance (fault points), which is the basis of unsymmetrical fault analysis. If the circuit is not symmetrical (e.g. unequal mutual impedances in an untransposed line), off-diagonal terms appear and the sequence networks become coupled. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The impedance matrix of a symmetrical static circuit has: A. Zm on diagonal and Zs elsewhere B. Zs - Zm on diagonal C. Zs on diagonal and Zm elsewhere D. Zero off-diagonal Answer: C. Equal self and mutual impedances. (Key point: Phase matrix) 2. Z_p A = A diag(...) means: A. A is singular B. Zp is diagonal C. A is unitary D. Columns of A are eigenvectors of Zp Answer: D. So A^-1 Zp A is diagonal. (Key point: Eigenvector idea) 3. Resulting sequence impedances are: A. Z0 = Zs+2Zm, Z1 = Z2 = Zs-Zm B. Z0 = Zs-Zm, Z1 = Z2 = Zs+2Zm C. All equal Zs D. Z0 = Zs+Zm Answer: A. Zero sequence mutuals add (+2Zm), the others subtract. (Key point: Result) 4. When mutual impedances are unequal, e.g. an untransposed line: A. Networks stay decoupled B. Off-diagonal terms appear and networks are coupled C. Z0 becomes zero D. Z1 = Z2 = 0 Answer: B. Diagonalisation depends on symmetry. 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