--- title: "Sequence impedances and zero sequence comment" subject: "Power System Analysis II" chapter: 4 key: "semester_5/power_system_analysis_ii:sequence-impedance-star-load-zero" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/sequence-impedance-star-load-zero site: BCT Engineering --- # Sequence impedances and zero sequence comment Power System Analysis II · Chapter 4: Unbalanced System Analysis - Asked in: 2080 Bhadra, 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/sequence-impedance-star-load-zero - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-4 ## Question Define sequence impedances. For the star connected load, verify that impedance matrix has non-zero elements on its diagonal matrix. Comment on zero sequence impedance. ## Short answer Sequence impedances are defined as Z1, Z2, Z0 as above. For a Y load with Zn, the matrix is diagonal: Z1=Z2=Zs, Z0=Zs+3Zn. Zero-sequence impedance is larger because neutral carries 3Ia0; solid grounding gives Z0=Zs, ungrounded shows infinite Z0, blocking zero-sequence current. ## Answer **Sequence impedance** of an element is the impedance it offers to the flow of currents of one particular sequence: $Z_1 = V_{a1}/I_{a1}$ (positive), $Z_2 = V_{a2}/I_{a2}$ (negative) and $Z_0 = V_{a0}/I_{a0}$ (zero sequence), when only that sequence is present. #### Star connected load Consider a balanced star load with impedance $Z_s$ per phase and neutral grounded through $Z_n$ (solid grounding: $Z_n = 0$). #### Circuit and phase equations ```text Ia --> Zs a o-----/\/\---+ Ib --> Zs | b o-----/\/\---+ n Ic --> Zs | c o-----/\/\---+ | In = Ia+Ib+Ic Zn | ground ``` Voltage of each phase to ground = drop in $Z_s$ + drop in $Z_n$: $$ \begin{aligned} V_a &= Z_s I_a + Z_n (I_a + I_b + I_c) \\ V_b &= Z_s I_b + Z_n (I_a + I_b + I_c) \\ V_c &= Z_s I_c + Z_n (I_a + I_b + I_c) \end{aligned} $$ In matrix form $V_p = Z_p I_p$ with $$ Z_p = \begin{bmatrix} Z_s+Z_n & Z_n & Z_n \\ Z_n & Z_s+Z_n & Z_n \\ Z_n & Z_n & Z_s+Z_n \end{bmatrix} $$ #### Transformation to sequence quantities Put $V_p = A V_s$ and $I_p = A I_s$: $$ A V_s = Z_p A I_s \;\Rightarrow\; V_s = A^{-1} Z_p A\, I_s = Z_s^{seq} I_s $$ with $A = \begin{bmatrix} 1&1&1\\1&a^2&a\\1&a&a^2 \end{bmatrix}$, $A^{-1} = \frac13\begin{bmatrix} 1&1&1\\1&a&a^2\\1&a^2&a \end{bmatrix}$. Write $Z_p = Z_s U + Z_n J$, where $U$ is the unit matrix and $J$ is the matrix of all ones. Then $A^{-1}(Z_sU)A = Z_sU$. For $J$: every row of $J A$ is $[3\ \ 0\ \ 0]$ (since column sums of $A$ are $3, 1+a^2+a=0, 1+a+a^2=0$), and $A^{-1}$ times that gives $$ A^{-1} J A = \begin{bmatrix} 3&0&0\\0&0&0\\0&0&0 \end{bmatrix} $$ Hence $$ Z_{seq} = A^{-1} Z_p A = \begin{bmatrix} Z_s+3Z_n & 0 & 0 \\ 0 & Z_s & 0 \\ 0 & 0 & Z_s \end{bmatrix} $$ The sequence impedance matrix has **non-zero elements only on its diagonal**: $$ Z_0 = Z_s + 3Z_n, \qquad Z_1 = Z_s, \qquad Z_2 = Z_s $$ All off-diagonal elements are zero, which means a current of one sequence produces a voltage drop of the same sequence only. So the three sequence networks are **decoupled** and can be solved separately, which is the main advantage of symmetrical components. The same holds if the load also has equal mutual impedances $Z_m$ between phases: then $Z_1 = Z_2 = Z_s - Z_m$ and $Z_0 = Z_s + 2Z_m + 3Z_n$, still diagonal. #### Comment on zero sequence impedance - $Z_0 = Z_s + 3Z_n$ is **larger** than $Z_1$ and $Z_2$, because zero sequence currents of all three phases are in phase and add up to $3I_{a0}$ in the neutral; the drop $3I_{a0}Z_n$ is felt by each phase. - **Solidly grounded star** ($Z_n = 0$): $Z_0 = Z_s$. - **Ungrounded star** ($Z_n = \infty$): $Z_0 = \infty$; zero sequence current cannot flow, and the zero sequence network is open. - **Delta connected load**: no return path, so $Z_0$ seen from the lines is infinite. - In a fault study, the method of neutral grounding therefore mainly changes the zero sequence network and hence the single line to ground fault current; this is why resistance or reactance grounding is used to limit earth fault current. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The sequence impedance matrix A^-1 Zp A for the star load is: A. diag(Zs, Zs+3Zn, Zs) B. diag(Zs+3Zn, Zs, Zs) C. diag(Zs+Zn, Zs+Zn, Zs+Zn) D. Full matrix with Zn in off-diagonal places Answer: B. Zp = Zs U + Zn J and A^-1 J A = diag(3,0,0). (Key point: Diagonalisation) 2. A diagonal sequence impedance matrix means: A. All sequence impedances are equal B. The load is unbalanced C. Each sequence current produces voltage drop of its own sequence only, so networks are decoupled D. Zero sequence current is absent Answer: C. No off-diagonal terms means no coupling between sequences. (Key point: Meaning of diagonal) 3. Z0 of a star load is larger than Z1 because: A. Zero sequence current is larger than positive B. Zs is bigger for zero sequence C. Zn is in series with each phase only for Z1 D. Neutral carries 3Ia0, so each phase feels a drop 3Ia0 Zn Answer: D. In-phase zero sequence currents add in the neutral, giving Z0 = Zs + 3Zn. (Key point: Z0 comment) 4. For an ungrounded star load the zero sequence impedance is: A. Infinite, zero sequence current cannot flow B. Zs C. Zero D. 3Zs Answer: A. No return path means Zn = infinity and the zero sequence network is open. (Key point: Ungrounded star) ## Close variants set in other exams - [Sequence impedance matrix of star load](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Asequence-impedance-star-load-diagonal) — Power System Analysis II, chapter 4, asked 2× ## Related in Power System Analysis II - [Sequence impedances of star load with Zn](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Astar-load-sequence-zn) — Power System Analysis II, chapter 4, asked 2× - [Sequence impedance of load with mutual coupling](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Amutually-coupled-load-sequence-impedance) — Power System Analysis II, chapter 4 - [Zero sequence Thevenin impedance at bus 2](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Azero-sequence-thevenin-bus2) — Power System Analysis II, chapter 4 ## Related in other subjects - [Star three impedances four-wire system](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Astar-three-impedances-4-wire-400v) — Basic Electrical Engineering, chapter 8