---
title: "Bandwidth of series resonant circuit with resonance curve"
subject: "Electrical Circuit Theory"
chapter: 1
key: "semester_3/electrical_circuit_theory:bandwidth-series-resonance-curve"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/bandwidth-series-resonance-curve
site: BCT Engineering
---

# Bandwidth of series resonant circuit with resonance curve

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

- Asked in: 2072 Chaitra, 8 marks
- Video: [Resonance and Q Factor in Series RLC AC Circuits](https://www.youtube.com/watch?v=AUirtqrm-o0) — Engineers Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/bandwidth-series-resonance-curve
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-1

## Question

What do you understand by the bandwidth of a series resonant circuit? Explain with the help of the resonance curve and also derive its expression both in terms of ω and f.

## Short answer

Bandwidth is the frequency range where current is at least I0/√2. From |X|=R, ω1,2 = ∓R/2L + √[(R/2L)²+1/LC], giving BW = ω2−ω1 = R/L rad/s = R/(2πL) Hz = ω0/Q. A smaller R (higher Q) gives a sharper, narrower resonance curve.

## Answer

**Bandwidth** of a series resonant circuit is the range of frequencies, f2 − f1, over which the circuit current is at least 1/√2 (70.7 %) of its maximum value at resonance, i.e. the power delivered is at least half of the maximum power. f1 and f2 are the lower and upper half power (cut-off) frequencies.

#### Resonance curve

```text
 I
 I0 |          ***
    |        **   **
0.707 I0 |- -*- - - - *-
    |     **|       |**
    |   **  |       |  **
    |__*____|___|___|____*__ f
            f1  f0  f2
            |<-- BW -->|
```

- The current is maximum (I0 = V/R) at f0 and falls on both sides.
- A small R gives a tall, narrow (sharp) curve, so a high Q and a narrow bandwidth. A large R gives a flat, wide curve.
- A narrow bandwidth means a highly selective circuit, as used in radio tuners.

#### Derivation in terms of ω

At the half power points |Z| = √2 R, which requires |ωL − 1/ωC| = R.

```text
ω²LC ∓ ωRC − 1 = 0
ω2 = R/2L + √[(R/2L)² + 1/LC]
ω1 = −R/2L + √[(R/2L)² + 1/LC]

BW = ω2 − ω1 = R/L  rad/s
```

#### In terms of f

```text
BW = f2 − f1 = (ω2 − ω1)/2π = R/(2πL)  Hz
```

#### In terms of Q

Q = ω0L/R, so R/L = ω0/Q:

```text
BW (rad/s) = ω0/Q        BW (Hz) = f0/Q
f0 = √(f1 f2)  (geometric mean)
```

For example, R = 10 Ω, L = 0.1 H gives BW = 100 rad/s (15.9 Hz); with R = 1 Ω the bandwidth drops to 10 rad/s.

**Answer: BW = R/L rad/s = R/(2πL) Hz = ω0/Q. The bandwidth decreases when R is reduced (Q increases).**

## Check yourself

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

1. Bandwidth of a series resonant circuit is the range f2 − f1 over which...
   A. the current is at least half of its resonance value
   B. the voltage is zero
   C. the power is at least 0.707 of maximum
   D. the current is at least 0.707 of its resonance value (half power)
   Answer: D. Half power points correspond to I = I0/√2 = 0.707 I0, and the band between them is the bandwidth. (Key point: Definition of bandwidth)

2. What happens to the resonance curve when R is made small?
   A. It becomes tall and narrow: high Q, small bandwidth
   B. It becomes flat and wide
   C. It shifts to a higher f0
   D. The peak current falls
   Answer: A. BW = R/L, so smaller R means narrower bandwidth, higher Q and better selectivity. (Key point: Effect of R)

3. What is the bandwidth in Hz?
   A. R/L
   B. R/(2πL)
   C. 2πR/L
   D. 1/(2πRC)
   Answer: B. BW (Hz) = (ω2 − ω1)/2π = R/(2πL). (Key point: Bandwidth in Hz)

## Related in Electrical Circuit Theory

- [Series RLC resonance: f0, bandwidth, half power, Q](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aseries-resonance-derive-f0-bw-q) — Electrical Circuit Theory, chapter 1
- [Series RLC: resonant frequency, bandwidth limits, L and C](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aseries-resonance-q-factor-numerical) — Electrical Circuit Theory, chapter 1
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