---
title: "Complex power and its significance"
subject: "Power System Analysis I"
chapter: 3
key: "semester_4/power_system_analysis_i:complex-power-significance"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/complex-power-significance
site: BCT Engineering
---

# Complex power and its significance

Power System Analysis I · Chapter 3: Computational Technique

- Asked in: 2070 Magh, 4 marks; 2069 Bhadra, 4 marks (asked 2 times)
- Video: [Complex Power & Reactive Power](https://www.youtube.com/watch?v=kngph7wjuBk) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/complex-power-significance
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-3

## Question

Discuss complex power and its significance in circuit analysis.

## Short answer

Complex power S = VI* = P + jQ combines active and reactive power. It simplifies analysis through conservation (sum of S over all elements is zero), directly yields power factor and compensation size, and determines equipment VA ratings.

## Answer

Complex power $S$ is the product of the voltage phasor and the conjugate of the current phasor. Its real part is the active power and its imaginary part is the reactive power:

$$
S = VI^* = |V||I|\angle\phi = P + jQ
$$

where:
- $P = |V||I|\cos\phi$ (W)
- $Q = |V||I|\sin\phi$ (var)
- $|S| = |V||I|$ (VA) is the apparent power
- $\phi = \theta_v - \theta_i$

For a three-phase system, $S_{3\phi} = 3V_{ph}I_{ph}^* = \sqrt3 V_LI_L\angle\phi$.

#### Power triangle
```text
        S (VA)
       /|
      / |  Q (var)
     /phi|
    /____|
      P (W)
```

#### Significance in circuit and power system analysis
- One complex number gives both $P$ and $Q$, together with their signs, so power flow is described completely.
- **Sign convention:** With $S = VI^*$, an inductive load absorbs $+Q$ and a capacitor absorbs $-Q$ (that is, it supplies Q). A source delivers power when $S$, computed with current leaving its + terminal, is positive.
- **Conservation:** The sum of complex power over all elements is zero (Tellegen's theorem), so $\sum P$ and $\sum Q$ each balance. This is the basis of load flow equations.
- It gives the power factor directly, $\cos\phi = P/|S|$, and so the size of the compensation needed.
- Equipment such as generators, transformers and cables is rated in VA (that is, $|S|$), because heating depends on $|I|$.
- With $S = I^2Z = |V|^2/Z^*$ it gives the losses and the reactive drop in lines directly.

**Example:** If $V = 230\angle0^\circ$ V and $I = 10\angle-36.87^\circ$ A, then $S = 2300\angle36.87^\circ = 1840 + j1380$ VA. The load absorbs 1840 W and 1380 var at a power factor of 0.8 lagging.

## Check yourself

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

1. Complex power S equals
   A. VI*, with P as real and Q as imaginary part
   B. V*I
   C. V/I
   D. V + I
   Answer: A. S = VI* = P + jQ. (Key point: Definition)

2. With S = VI*, a capacitor
   A. Absorbs +Q
   B. Supplies Q (absorbs -Q)
   C. Has zero Q
   D. Absorbs P only
   Answer: B. Inductive loads absorb +Q, capacitors absorb -Q. (Key point: Sign of Q)

3. Generators and transformers are rated in VA because
   A. Heating depends on current magnitude
   B. Voltage is constant
   C. P is always zero
   D. Q is never present
   Answer: A. Heating depends on |I|, so rating is on |S|. (Key point: Equipment rating)

4. For V = 230 angle 0 V, I = 10 angle -36.87 A, the load is
   A. 1840 W, 1380 var, 0.8 lagging
   B. 1380 W, 1840 var
   C. 2300 W, 0 var
   D. 1840 W, 1380 var, 0.8 leading
   Answer: A. S = 2300 angle 36.87 = 1840 + j1380, a lagging load. (Key point: Example)

## Related in Power System Analysis I

- [Derivation and phasor diagram of complex power](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Acomplex-power-derivation-phasor) — Power System Analysis I, chapter 3
- [Complex power and sign conventions](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Acomplex-power-sign-conventions) — Power System Analysis I, chapter 3
- [Complex power and why VI* rather than V*I](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Acomplex-power-vi-conjugate-why) — Power System Analysis I, chapter 3
- [Complex power, impedance and power triangles](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Acomplex-power-impedance-power-triangle) — Power System Analysis I, chapter 3
- [Generator and load complex power conventions](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Agenerator-load-complex-power-convention) — Power System Analysis I, chapter 3
- [Receiving end complex power and power circle](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Areceiving-power-circle) — Power System Analysis I, chapter 6

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- [Complex power and power triangle](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Acomplex-power-rl-power-triangle) — Basic Electrical Engineering, chapter 7
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