---
title: "Complex power and power triangle"
subject: "Basic Electrical Engineering"
chapter: 7
key: "semester_1/basic_electrical_engineering:complex-power-rl-power-triangle"
source: https://www.bctengineer.com/q/semester_1/basic_electrical_engineering/complex-power-rl-power-triangle
site: BCT Engineering
---

# Complex power and power triangle

Basic Electrical Engineering · Chapter 7: Power in AC Circuits

- Asked in: 2069 Asar, 4 marks
- Video: [Power Triangle](https://www.youtube.com/watch?v=1iBr7Ng7tDY) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/basic_electrical_engineering/complex-power-rl-power-triangle
- Whole chapter: https://www.bctengineer.com/md/semester_1/basic_electrical_engineering/chapter-7

## Question

What do you mean by complex power? Explain it with the help of an R-L series circuit and power triangle.

## Short answer

Complex power is S = P + jQ = VI*, where P is active power (W) and Q is reactive power (VAR). In an R-L series circuit, S = VI∠φ, P = I²R, Q = I²X_L, and the power triangle shows S as the vector sum of P and Q, with Q drawn upward for lagging loads.

## Answer

**Complex power** (S) is the complex quantity whose real part is the active power P and whose imaginary part is the reactive power Q:

S = V I* = P + jQ  (volt-ampere, VA)

Here V is the voltage phasor and I* is the conjugate of the current phasor. Its magnitude |S| = VI is the apparent power.

#### R-L series circuit

Take V = V∠0° and Z = R + jX_L = Z∠φ. Then I = V/Z = I∠−φ (lagging).

S = V I* = V∠0° × I∠+φ = VI∠φ = VI cos φ + j VI sin φ

- Active power P = VI cos φ = I²R (watt)
- Reactive power Q = VI sin φ = I²X_L (VAR), positive for an inductive load
- Apparent power S = VI = I²Z (VA)

S = P + jQ, |S| = √(P² + Q²), cos φ = P/S

#### Power triangle

```text
          /|
     S   / |
  (VA)  /  | Q = VI sin φ (VAR)
       /   |
      / φ  |
     /_____|
      P = VI cos φ (W)
```

It is obtained by multiplying the sides of the impedance triangle (R, X_L, Z) by I². For an inductive load, Q is drawn upward (lagging). For a capacitive load, Q is drawn downward (leading).

## Check yourself

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

1. How is complex power defined?
   A. S = V I, the product of the phasors
   B. S = V I* = P + jQ
   C. S = V* I = P - jQ always
   D. S = P x Q
   Answer: B. Complex power is S = V I* (voltage times the conjugate of current) = P + jQ. (Key point: Complex power definition)

2. For an R-L series circuit, the current lags the voltage by φ. What is the sign of Q?
   A. Negative
   B. Zero
   C. Positive, for an inductive load
   D. Equal to P
   Answer: C. Q = VI sin φ is positive for a lagging inductive load. (Key point: R-L circuit result)

3. How is the power triangle obtained from the impedance triangle?
   A. Multiplying its sides (R, X_L, Z) by I²
   B. Dividing its sides by I²
   C. Multiplying its sides by V
   D. Rotating it by 90°
   Answer: A. P = I²R, Q = I²X and S = I²Z, so the power triangle is the impedance triangle scaled by I². (Key point: Power triangle)

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