---
title: "Half power points, bandwidth and effect of Q"
subject: "Electrical Circuit Theory"
chapter: 1
key: "semester_3/electrical_circuit_theory:series-resonance-bandwidth-q-effect"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/series-resonance-bandwidth-q-effect
site: BCT Engineering
---

# Half power points, bandwidth and effect of Q

Electrical Circuit Theory · Chapter 1: Network Analysis of AC circuit & Dependent Sources

- Asked in: 2068 Shrawan, 8 marks
- Video: [Series Resonance | Half power frequency and Bandwidth | By Vidit Sir](https://www.youtube.com/watch?v=9VIp08RwOQ4) — LESSONUP
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/series-resonance-bandwidth-q-effect
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-1

## Question

Define half power points and bandwidth for a series RLC circuit and derive the expression for them. How is the bandwidth affected by the quality factor of the circuit?

## Short answer

Half-power points are frequencies where current drops to I0/√2 and |Z| = √2 R; bandwidth BW = f2 − f1 = R/(2πL) = f0/Q. So bandwidth is inversely proportional to Q: higher Q gives narrower bandwidth and sharper resonance.

## Answer

**Half power points** are the two frequencies f1 and f2 (one below and one above the resonant frequency f0) at which the power in the circuit is half of the maximum power at resonance, i.e. P = I0²R/2. The current is then I0/√2 = 0.707 I0, and the voltage-to-current ratio |Z| = √2 R.

**Bandwidth** is the width of the frequency band between them: BW = f2 − f1.

#### Derivation

At a half power point |X_L − X_C| = R.

```text
ωL − 1/ωC = ±R   →   LCω² ∓ RCω − 1 = 0

ω2 = R/2L + √[(R/2L)² + 1/LC]      (upper)
ω1 = −R/2L + √[(R/2L)² + 1/LC]     (lower)

BW = ω2 − ω1 = R/L rad/s  =  R/(2πL) Hz
ω1 ω2 = 1/LC = ω0²
```

#### Effect of quality factor

Q = ω0L/R, so R/L = ω0/Q and

```text
BW = ω0/Q    (or f0/Q)
```

Bandwidth is inversely proportional to Q.

| Q | Bandwidth | Resonance curve | Selectivity |
|---|---|---|---|
| High | Narrow | Sharp, tall | High (good tuner) |
| Low | Wide | Flat, broad | Poor |

Example: f0 = 1 kHz; Q = 10 gives BW = 100 Hz; Q = 100 gives BW = 10 Hz.

So increasing Q (by decreasing R for fixed L, C) narrows the bandwidth and increases the selectivity.

**Answer: BW = f2 − f1 = R/(2πL) = f0/Q, so bandwidth decreases as Q increases.**

## Check yourself

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

1. What is the power at the half power points?
   A. I0^2R/√2
   B. I0^2R/4
   C. I0^2R
   D. I0^2R/2, half the maximum
   Answer: D. Power is half of the maximum value I0^2R at resonance. (Key point: Half power points)

2. What condition holds at a half power point?
   A. |X_L − X_C| = R
   B. X_L − X_C = 0
   C. |X_L − X_C| = 2R
   D. X_L = R only
   Answer: A. Then |Z| = √2 R and I = I0/√2. (Key point: Condition on reactance)

3. How does bandwidth depend on Q?
   A. BW = f0·Q
   B. BW = f0/Q, inversely proportional
   C. BW is independent of Q
   D. BW = Q/f0
   Answer: B. Higher Q gives narrower bandwidth and higher selectivity. (Key point: Bandwidth and Q)

4. For f0 = 1 kHz, what is BW when Q = 100?
   A. 100 Hz
   B. 1 kHz
   C. 10 Hz
   D. 100 kHz
   Answer: C. BW = f0/Q = 1000/100 = 10 Hz. (Key point: Example)

## Related in Electrical Circuit Theory

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