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

# Sequence impedances of star load with Zn

Power System Analysis II · Chapter 4: Unbalanced System Analysis

- Asked in: 2078 Bhadra, 8 marks; 2070 Asar, 8 marks (asked 2 times)
- 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/star-load-sequence-zn
- Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-4

## Question

Starting from suitable point, derive an expression to determine sequence impedances for a balanced three-phase star-connected load with impedance Zs in each phase and grounded neutral with impedance Zn, and show that Z0 = Zs + 3Zn, Z1 = Zs and Z2 = Zs.

## Short answer

From phase voltages V_a = Z_s I_a + Z_n(I_a+I_b+I_c) etc., transform to sequence via A. The phase impedance matrix Z_p = Z_s U + Z_n J yields diagonal sequence matrix Z_seq = diag(Z_s+3Z_n, Z_s, Z_s). Hence Z_0 = Z_s + 3Z_n, Z_1 = Z_s, Z_2 = Z_s.

## Answer

**Sequence impedance** is the impedance offered by a circuit to currents of one sequence only: $Z_0 = V_{a0}/I_{a0}$, $Z_1 = V_{a1}/I_{a1}$, $Z_2 = V_{a2}/I_{a2}$. We start from the phase-domain voltage equations of the load and transform them with the symmetrical component matrix $A$.

#### 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
$$
Z_{seq} = A^{-1} Z_p A = \begin{bmatrix} Z_s+3Z_n & 0 & 0 \\ 0 & Z_s & 0 \\ 0 & 0 & Z_s \end{bmatrix}
$$

#### Result
Writing the three rows separately:
$$
\begin{aligned}
V_{a0} &= (Z_s + 3Z_n) I_{a0} &\Rightarrow\; Z_0 &= Z_s + 3Z_n \\
V_{a1} &= Z_s I_{a1} &\Rightarrow\; Z_1 &= Z_s \\
V_{a2} &= Z_s I_{a2} &\Rightarrow\; Z_2 &= Z_s
\end{aligned}
$$

#### Physical meaning
- Positive and negative sequence currents are balanced sets; their sum is zero, so **no current flows in $Z_n$** and the impedance is just $Z_s$.
- Zero sequence currents are equal and in phase in all three phases, so the neutral carries $3I_{a0}$. The drop across $Z_n$ is $3I_{a0}Z_n$, which appears as an extra $3Z_n$ in the zero sequence circuit of each phase.
- If the neutral is solidly grounded ($Z_n = 0$), $Z_0 = Z_s$; if it is isolated ($Z_n \to \infty$), $Z_0 = \infty$ and no zero sequence current can flow.
- Since the sequence impedance matrix is diagonal, the three sequence networks are **independent (decoupled)**.

## Check yourself

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

1. Phase voltage to ground for phase a is
   A. V_a = Z_s I_a
   B. V_a = Z_s I_a + Z_n I_a
   C. V_a = Z_s I_a + Z_n (I_a + I_b + I_c)
   D. V_a = (Z_s + Z_n)(I_a + I_b + I_c)
   Answer: C. The neutral impedance carries the sum of all three line currents, so its drop is Z_n(I_a + I_b + I_c). (Key point: Phase equations)

2. Z_p is written as Z_s U + Z_n J. Which result makes the sequence impedance matrix diagonal?
   A. A^-1 J A = diag(3, 0, 0)
   B. A^-1 J A = U
   C. A^-1 J A = 3U
   D. A^-1 J A = J
   Answer: A. Column sums of A are 3, 0, 0, so A^-1 J A has only a 3 in the first position, giving Z0 = Z_s + 3Z_n, Z1 = Z2 = Z_s. (Key point: Matrix decomposition)

3. Why does Z_n appear as 3Z_n in the zero-sequence impedance?
   A. Because three phases have three load impedances
   B. Zero-sequence currents are equal in all phases, so the neutral carries 3I_a0 and drops 3I_a0 Z_n
   C. Because Z_n is in parallel with three branches
   D. It is just a convention with no physical reason
   Answer: B. The neutral drop 3I_a0 Z_n is attributed to phase a's zero-sequence current I_a0, giving an extra 3Z_n. (Key point: Why Z0 includes 3Zn)

4. Why are Z1 and Z2 independent of Z_n?
   A. Z_n is zero
   B. Positive and negative sets sum to zero, so no current flows in the neutral
   C. The neutral is open
   D. They flow only in the ground
   Answer: B. Balanced positive and negative sequence currents add to zero in the neutral, so Z_n has no effect: Z1 = Z2 = Z_s. (Key point: Positive and negative sequence)

## Related in Power System Analysis II

- [Equivalent Z0, Z1, Z2 at bus 3](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Asequence-thevenin-z0-z1-z2-bus3) — Power System Analysis II, chapter 5
- [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
- [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×
- [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×
- [Line currents with isolated and grounded neutral](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Astar-impedance-unbalanced-voltages-neutral) — Power System Analysis II, chapter 4
- [Unbalanced source feeding star load](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aunbalanced-source-star-load-components) — Power System Analysis II, chapter 4

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