---
title: "Sequence impedance of load with mutual coupling"
subject: "Power System Analysis II"
chapter: 4
key: "semester_5/power_system_analysis_ii:mutually-coupled-load-sequence-impedance"
source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/mutually-coupled-load-sequence-impedance
site: BCT Engineering
---

# Sequence impedance of load with mutual coupling

Power System Analysis II · Chapter 4: Unbalanced System Analysis

- Asked in: 2076 Asoj, 6 marks
- Video: [PSA Module 3 Lecture 9 - Sequence impedance of a symmetrical circuit](https://www.youtube.com/watch?v=hlH9q-32hyE) — Sumitha T L
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/mutually-coupled-load-sequence-impedance
- Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-4

## Question

A three phase balanced Y-connected load with self and mutual elements is shown in figure below. The load neutral is grounded with Zn = 0.0. Determine the sequence impedance. [Figure: phases R, Y, B carrying IR, IY, IB through self impedances ZR, ZY, ZB, with mutual impedances ZRY, ZYB and ZRB between phases; star point returned to ground carrying In; phase voltages VR, VY, VB]

## Short answer

For the balanced Y-load with solidly grounded neutral (Zn=0), the sequence impedances are Z0 = Zs + 2Zm and Z1 = Z2 = Zs − Zm, obtained by diagonalizing the phase impedance matrix via symmetrical component transformation.

## Answer

For a balanced star load with self impedance $Z_s$ in each phase and equal mutual impedance $Z_m$ between phases, solidly grounded ($Z_n = 0$), the sequence impedances are $Z_0 = Z_s + 2Z_m$ and $Z_1 = Z_2 = Z_s - Z_m$.

**Since the load is balanced:** $Z_R = Z_Y = Z_B = Z_s$ and $Z_{RY} = Z_{YB} = Z_{RB} = Z_m$.

#### Phase equations
With $Z_n = 0$ the star point is at ground potential:
$$
\begin{aligned}
V_R &= Z_sI_R + Z_mI_Y + Z_mI_B \\
V_Y &= Z_mI_R + Z_sI_Y + Z_mI_B \\
V_B &= Z_mI_R + Z_mI_Y + Z_sI_B
\end{aligned}
\qquad
Z_p = \begin{bmatrix} Z_s&Z_m&Z_m\\Z_m&Z_s&Z_m\\Z_m&Z_m&Z_s \end{bmatrix}
$$

#### Transformation
$V_p = AV_s$ and $I_p = AI_s$ give $V_s = A^{-1}Z_pA\,I_s$.

Multiply $Z_p$ by each column of $A$, using $1 + a + a^2 = 0$:
- Column $[1\ 1\ 1]^T$: each row gives $Z_s + 2Z_m$
- Column $[1\ a^2\ a]^T$: row 1 gives $Z_s + Z_m(a^2+a) = Z_s - Z_m$, and the other rows give the same factor times $a^2$, $a$
- Column $[1\ a\ a^2]^T$: factor $Z_s - Z_m$

So
$$
Z_{seq} = A^{-1}Z_pA = \begin{bmatrix} Z_s+2Z_m & 0 & 0 \\ 0 & Z_s-Z_m & 0 \\ 0 & 0 & Z_s-Z_m \end{bmatrix}
$$

#### Result
$$
Z_0 = Z_s + 2Z_m, \qquad Z_1 = Z_s - Z_m, \qquad Z_2 = Z_s - Z_m
$$
- The matrix is diagonal, so the sequence networks are decoupled.
- If the neutral were grounded through $Z_n$, every element of $Z_p$ would gain $Z_n$ and only $Z_0$ would change: $Z_0 = Z_s + 2Z_m + 3Z_n$.

## Check yourself

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

1. Balanced load means:
   A. ZR=0
   B. All mutuals zero
   C. ZR=ZY=ZB=Zs and ZRY=ZYB=ZRB=Zm
   D. Zs = Zm
   Answer: C. Equal self and mutual impedances. (Key point: Balanced assumptions)

2. Zero sequence impedance (Zn = 0) is:
   A. Zs - Zm
   B. Zs + Zm
   C. Zs
   D. Zs + 2Zm
   Answer: D. In-phase zero sequence currents make both mutuals add. (Key point: Z0)

3. Positive and negative sequence impedances are:
   A. Zs - Zm
   B. Zs + 2Zm
   C. Zs + Zm
   D. Zs
   Answer: A. Because a+a^2 = -1. (Key point: Z1 and Z2)

4. If grounded through Zn, Z0 becomes:
   A. Zs + 2Zm + Zn
   B. Zs + 2Zm + 3Zn
   C. Zs - Zm + 3Zn
   D. Zs+3Zn
   Answer: B. Zn adds only to zero sequence, as 3Zn. (Key point: With Zn)

## 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 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
- [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×
- [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

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