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

# Complex power and sign conventions

Power System Analysis I · Chapter 3: Computational Technique

- Asked in: 2073 Bhadra, 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-sign-conventions
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-3

## Question

Define the meaning of complex power in power system. Explain the sign conventions of power for sources and loads.

## Short answer

Complex power S = VI* = P + jQ, where P is real power (W) and Q reactive power (var). In load convention (current enters positive), positive P/Q means absorbed; in source convention (current leaves positive), positive P/Q means delivered.

## Answer

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

$$
S = VI^* = |V||I|\angle(\theta_v - \theta_i) = P + jQ
$$

- $P = |V||I|\cos\phi$ (W), the average power converted to work or heat.
- $Q = |V||I|\sin\phi$ (var), the power that oscillates between the source and the fields of L and C.
- $|S| = \sqrt{P^2+Q^2}$ (VA), the apparent power. Equipment is rated in VA.
- $\phi = \theta_v - \theta_i$ is the power-factor angle. Q is positive for a lagging (inductive) current.

#### Sign conventions

**Load (motor) convention:** the current enters the positive terminal. $S = VI^*$ is then the power **absorbed**.
- P > 0: the element absorbs real power (resistor, motor).
- Q > 0: it absorbs reactive power (inductor, lagging load).
- Q < 0: it supplies reactive power (capacitor).

**Source (generator) convention:** the current leaves the positive terminal. $S = VI^*$ is then the power **delivered**.
- P > 0: the source delivers real power (generator action). P < 0 means it absorbs P (motor action).
- Q > 0: it delivers reactive power (over-excited machine). Q < 0 means it absorbs Q.

```text
 Load convention          Source convention
   I -->  +                 +  --> I
   +------o                 o------+
   | Load |  S absorbed     | Src  |  S delivered
   +------o                 o------+
          -                 -
```

| Element | P | Q (load convention) |
|---|---|---|
| Resistor | + | 0 |
| Inductor | 0 | + (absorbs) |
| Capacitor | 0 | − (supplies) |
| Induction motor | + | + |

Example: $V = 100\angle0^\circ$ V and the current leaving the + terminal is $I = 5\angle-30^\circ$ A. Then $S = 500\angle30^\circ = 433 + j250$ VA, so the source delivers 433 W and 250 var.

## Check yourself

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

1. What is complex power?
   A. S = V I
   B. S = V I* = P + jQ
   C. S = V/I
   D. S = V* I*
   Answer: B. S is voltage phasor times current conjugate, real part P, imaginary Q. (Key point: Definition)

2. In load convention, what does Q > 0 mean?
   A. Element supplies reactive power
   B. Element absorbs reactive power (inductive)
   C. Element absorbs real power only
   D. Capacitor
   Answer: B. Positive Q is absorbed, e.g. inductors and lagging loads. (Key point: Load convention)

3. Source with current leaving + terminal has S = VI* = 433 + j250. Interpretation?
   A. Absorbs 433 W and 250 var
   B. Delivers 433 W and absorbs 250 var
   C. Delivers 433 W and 250 var
   D. Absorbs 433 W and delivers 250 var
   Answer: C. In source convention positive P and Q are delivered. (Key point: Source convention)

## Related in Power System Analysis I

- [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
- [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 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
- [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
- [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

## Related in other subjects

- [Work and heat transfer; sign conventions](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Awork-heat-sign-conventions) — Fundamental of Thermodynamics & Heat Transfer, chapter 2
- [Symmetrical components and complex power](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Asymmetrical-components-complex-power) — Power System Analysis II, chapter 4
- [Star system with three loads, neutral current](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Astar-three-loads-400v-neutral-current) — Basic Electrical Engineering, chapter 8
- [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