---
title: "Symmetrical components and complex power"
subject: "Power System Analysis II"
chapter: 4
key: "semester_5/power_system_analysis_ii:symmetrical-components-complex-power"
source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/symmetrical-components-complex-power
site: BCT Engineering
---

# Symmetrical components and complex power

Power System Analysis II · Chapter 4: Unbalanced System Analysis

- Asked in: 2080 Bhadra, 2+6 marks
- Video: [Lec 63 - Power in terms of Symmetrical Components | Power Systems](https://www.youtube.com/watch?v=9UJIsz6j3nU) — Hiatuz -GATE ESE EE ECE
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/symmetrical-components-complex-power
- Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-4

## Question

What do you understand by symmetrical components? Derive an expression of 3-phase complex power in terms of symmetrical components of voltages and currents.

## Short answer

Symmetrical components decompose any unbalanced three-phase set into three balanced sets: positive (abc), negative (acb), and zero (all in phase). The total complex power is S = 3 (Va0 Ia0* + Va1 Ia1* + Va2 Ia2*), with no cross-terms between sequences.

## Answer

#### Symmetrical components
**Symmetrical components** (Fortescue's theorem, 1918) state that any set of three unbalanced phasors can be replaced by the sum of three balanced sets:
- **Positive sequence:** three equal phasors 120° apart with the same phase sequence as the original (a-b-c).
- **Negative sequence:** three equal phasors 120° apart with opposite sequence (a-c-b).
- **Zero sequence:** three equal phasors in phase with each other.

With operator $a = 1\angle120^\circ$:
$$
\begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix} = \begin{bmatrix} 1&1&1\\1&a^2&a\\1&a&a^2 \end{bmatrix} \begin{bmatrix} V_{a0} \\ V_{a1} \\ V_{a2} \end{bmatrix}, \quad\text{i.e. } V_p = A V_s
$$
The same relation holds for currents, $I_p = A I_s$. This lets an unbalanced problem be solved as three balanced, independent problems.

#### Complex power in terms of symmetrical components
Total complex power in phase quantities:
$$
S = V_a I_a^* + V_b I_b^* + V_c I_c^* = V_p^T I_p^*
$$
Substituting $V_p = AV_s$, $I_p = AI_s$:
$$
S = (AV_s)^T (AI_s)^* = V_s^T A^T A^* I_s^*
$$
$A$ is symmetric ($A^T = A$), and since $a^* = a^2$:
$$
A^* = \begin{bmatrix} 1&1&1\\1&a&a^2\\1&a^2&a \end{bmatrix}
$$
$$
A^T A^* = \begin{bmatrix} 3 & 1+a+a^2 & 1+a^2+a \\ 1+a^2+a & 1+a^3+a^3 & 1+a^4+a^2 \\ 1+a+a^2 & 1+a^2+a^4 & 1+a^3+a^3 \end{bmatrix} = \begin{bmatrix} 3&0&0\\0&3&0\\0&0&3 \end{bmatrix}
$$
using $1+a+a^2 = 0$, $a^3 = 1$, $a^4 = a$. Hence
$$
S = 3V_s^T I_s^* = 3\left(V_{a0}I_{a0}^* + V_{a1}I_{a1}^* + V_{a2}I_{a2}^*\right)
$$

So total power is three times the sum of the sequence powers of phase a, with no cross products between different sequences. For a balanced system only the positive sequence term remains: $S = 3V_{a1}I_{a1}^*$.

## Check yourself

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

1. Symmetrical components represent an unbalanced three-phase set as:
   A. Sum of three single phase sets of equal magnitude 120° apart only
   B. Two balanced sets, positive and negative
   C. Three sets all rotating in the same direction at different speeds
   D. Sum of positive, negative and zero sequence balanced sets
   Answer: D. Fortescue: any unbalanced set of three phasors = positive + negative + zero sequence sets. (Key point: Fortescue's theorem)

2. The key matrix result used in the power derivation is:
   A. A^T A* = 3U (diagonal matrix)
   B. A^T A* = U
   C. A^T A* = A
   D. A^T A* has non-zero off-diagonal terms
   Answer: A. Off-diagonal elements vanish by 1+a+a^2=0, giving 3 times the unit matrix. (Key point: Key step)

3. Total complex power in sequence components is:
   A. S = Va0 Ia0* + Va1 Ia1* + Va2 Ia2*
   B. S = 3(Va0 Ia0* + Va1 Ia1* + Va2 Ia2*)
   C. S = 3 Va1 Ia1* + Va2 Ia2* only
   D. S = 3(Va0 Ia1* + Va1 Ia2* + Va2 Ia0*)
   Answer: B. Factor 3 comes from A^T A* = 3U, with no cross products between sequences. (Key point: Power expression)

4. For a balanced system the complex power reduces to:
   A. S = 3 Va0 Ia0*
   B. S = Va1 Ia1*
   C. S = 3 Va1 Ia1*
   D. S = 3 Va2 Ia2*
   Answer: C. A balanced system has only the positive sequence. (Key point: Balanced case)

## Related in Power System Analysis II

- [3-phase power in symmetrical components](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Athree-phase-power-symmetrical-components) — Power System Analysis II, chapter 4, asked 3×
- [Derive symmetrical components of currents](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Asymmetrical-components-currents-derivation) — Power System Analysis II, chapter 4
- [Symmetrical components of unbalanced voltages](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aunbalanced-voltages-components-2070) — Power System Analysis II, chapter 4
- [Significance of components; phasor voltages](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Asymmetrical-components-significance-voltages) — Power System Analysis II, chapter 4
- [L-G fault through Zf: currents and voltages](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Alg-fault-with-zf-derivation) — Power System Analysis II, chapter 5
- [Sequence components of load phase voltages](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aload-phase-voltages-components) — Power System Analysis II, chapter 4

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