---
title: "Sequence impedance matrix of star load"
subject: "Power System Analysis II"
chapter: 4
key: "semester_5/power_system_analysis_ii:sequence-impedance-star-load-diagonal"
source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/sequence-impedance-star-load-diagonal
site: BCT Engineering
---

# Sequence impedance matrix of star load

Power System Analysis II · Chapter 4: Unbalanced System Analysis

- Asked in: 2075 Chaitra, 8 marks
- Video: [Sequence Network Of Impedance Load | Sequence Impedance | Part 2 II PSA](https://www.youtube.com/watch?v=Dd5duuZ8UFA) — University Concept
- 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-diagonal
- Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-4

## Question

Define Sequence impedances. For the star connected load show that impedance matrix has non-zero elements on its diagonal matrix.

## Short answer

Sequence impedances are Z1 = Va1/Ia1, Z2 = Va2/Ia2, and Z0 = Va0/Ia0. For a balanced star load with neutral impedance Zn, the sequence impedance matrix is diagonal: Z0 = Zs +3Zn, Z1 = Z2 = Zs. All off-diagonals are zero, so networks are decoupled.

## 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Sequence impedance Z1 of an element is defined as:
   A. Va1/Ia0
   B. Va0/Ia1
   C. Va/Ia of phase a
   D. Va1/Ia1 when only positive sequence is present
   Answer: D. Each sequence impedance relates voltage and current of the same sequence. (Key point: Definition)

2. For the star load with Zn, the phase impedance matrix has diagonal and off-diagonal elements:
   A. Zs+Zn on diagonal, Zn off diagonal
   B. Zs on diagonal, Zn off diagonal
   C. Zs+Zn everywhere
   D. Zn on diagonal, Zs off diagonal
   Answer: A. Each phase voltage includes Zs Ia plus Zn(Ia+Ib+Ic). (Key point: Phase matrix)

3. 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)

4. 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)

## Close variants set in other exams

- [Sequence impedances and zero sequence comment](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Asequence-impedance-star-load-zero) — 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
- [Equal positive and negative sequence impedances](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Astatic-circuit-positive-negative-equal) — Power System Analysis II, chapter 4
- [Sequence networks of Y load, neutral current](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Astar-load-sequence-networks-neutral) — Power System Analysis II, chapter 4, asked 2×
- [No zero sequence current in delta lines](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Adelta-zero-sequence-absent) — Power System Analysis II, chapter 4, asked 2×

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