---
title: "Complex power and why VI* rather than V*I"
subject: "Power System Analysis I"
chapter: 3
key: "semester_4/power_system_analysis_i:complex-power-vi-conjugate-why"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/complex-power-vi-conjugate-why
site: BCT Engineering
---

# Complex power and why VI* rather than V*I

Power System Analysis I · Chapter 3: Computational Technique

- Asked in: 2072 Magh, 4 marks
- Video: [Why conjugate for complex power S=VI*  Interesting insights](https://www.youtube.com/watch?v=2AgAwOoGSwc) — The PiSquare 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-vi-conjugate-why
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-3

## Question

What is complex power? Why is complex power taken as VI* rather than V*I in power systems? Explain.

## Short answer

Complex power is S=P+jQ=VI*, with magnitude in VA. VI* gives positive Q for lagging inductive loads, matches the accepted convention that inductors absorb vars, and equals |I|^2Z. V*I would reverse the reactive sign and disagree with the standard power triangle.

## Answer

**Complex power** is $S = P + jQ$, the phasor quantity whose real part is the active power (W), whose imaginary part is the reactive power (var) and whose magnitude is the apparent power (VA). It is defined as $S = VI^*$.

#### Why $VI^*$
Let $V = |V|\angle\theta_v$ and $I = |I|\angle\theta_i$. The power-factor angle is $\phi = \theta_v - \theta_i$.

$$
\begin{aligned}
VI^* &= |V||I|\angle(\theta_v - \theta_i) = |V||I|\cos\phi + j|V||I|\sin\phi = P + jQ\\
V^*I &= |V||I|\angle(\theta_i - \theta_v) = |V||I|\cos\phi - j|V||I|\sin\phi = P - jQ
\end{aligned}
$$

1. **Both give the same P.** Only the sign of Q differs.
2. **Sign agreement with the accepted convention.** With $VI^*$, a lagging (inductive) load ($\theta_i < \theta_v$, $\phi > 0$) gives $Q > 0$. Inductive loads "absorb" vars and capacitors "supply" vars. This is the convention adopted by IEEE/IEC and used in load-flow studies, generator capability charts and var meters. With $V^*I$, an inductive load would show negative Q, which conflicts with the convention.
3. **Consistency with impedance.** With $S = VI^* = |I|^2Z = |I|^2(R + jX)$, inductive reactance (+X) gives +Q. The power triangle is then the impedance triangle scaled by $|I|^2$. With $V^*I$ the triangle would be the mirror image of the impedance triangle.
4. **Independence from reference.** Taking the conjugate removes the absolute phase angles and keeps only the difference $\theta_v - \theta_i$. So S does not depend on which phasor is chosen as reference. ($VI$ without a conjugate would give the angle $\theta_v + \theta_i$, which changes with the reference and has no meaning.)

```text
 Im(Q)
  ^        S = VI*
  |      /
  |    /  Q = |V||I| sin(phi) > 0
  |  / phi    (lagging load)
  +----------> Re (P)
```

Example: $V = 100\angle0^\circ$ V and $I = 10\angle-30^\circ$ A (lagging). Then $VI^* = 866 + j500$ VA, which is correct: the load absorbs 500 var. But $V^*I = 866 - j500$ VA would wrongly suggest that the inductive load supplies vars.

## Check yourself

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

1. Which statement correctly describes complex power S = P + jQ?
   A. Real part is reactive power and imaginary part is active power
   B. Real part is active power (W), imaginary part is reactive power (var), magnitude is apparent power (VA)
   C. Magnitude is the active power in W
   D. Real part is apparent power and imaginary part is power factor
   Answer: B. S = P + jQ with |S| the apparent power. (Key point: Definition of complex power)

2. Compared with VI*, what does V*I give for a lagging load?
   A. The same P but the opposite sign of Q (P - jQ)
   B. The same Q but the opposite sign of P
   C. Both P and Q with reversed signs
   D. Exactly the same P + jQ
   Answer: A. V*I = |V||I|(cos phi - j sin phi), so only Q changes sign. (Key point: Effect of conjugating V instead of I)

3. Why is VI* preferred in power systems?
   A. It makes P negative for loads
   B. It gives larger P than V*I
   C. It makes the capacitor absorb vars
   D. A lagging inductive load then gives Q > 0, matching the convention that inductive loads absorb vars
   Answer: D. With VI*, inductive loads absorb vars (Q > 0), the IEEE/IEC convention. (Key point: Sign convention of Q)

4. Why can S not be defined simply as VI (no conjugate)?
   A. Because VI has no magnitude
   B. Because VI is always real
   C. The angle would be theta_v + theta_i, which changes with the chosen reference phasor
   D. Because VI gives the wrong P value only at unity pf
   Answer: C. The conjugate leaves only the difference theta_v - theta_i, which is reference independent. (Key point: Reference independence)

## Related in Power System Analysis I

- [Complex power and its significance](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Acomplex-power-significance) — Power System Analysis I, chapter 3, asked 2×
- [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
- [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, 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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