---
title: "Series RLC resonance: f0, bandwidth, half power, Q"
subject: "Electrical Circuit Theory"
chapter: 1
key: "semester_3/electrical_circuit_theory:series-resonance-derive-f0-bw-q"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/series-resonance-derive-f0-bw-q
site: BCT Engineering
---

# Series RLC resonance: f0, bandwidth, half power, Q

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

- Asked in: 2069 Chaitra, 6 marks
- Video: [Band Width and Q  factor in resonant  circuit (Unit 2 AC circuit) BEE](https://www.youtube.com/watch?v=GxcmW98i7PI) — Gautam Varde
- 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-derive-f0-bw-q
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-1

## Question

Explain the phenomenon of resonance in an RLC series circuit. Derive the expression for resonant frequency, bandwidth, half power frequencies and quality factor.

## Short answer

Resonance: XL=XC at f0 = 1/(2π√LC). Quality factor Q = ω0L/R = 1/(ω0CR). Half-power frequencies: ω1,2 = ∓R/2L + √[(R/2L)² + 1/LC]. Bandwidth BW = ω2−ω1 = R/L rad/s = ω0/Q.

## Answer

**Resonance** in a series RLC circuit is the state in which X_L = X_C, so the impedance is purely resistive and the current is in phase with the supply voltage, with the maximum value V/R.

#### Resonant frequency

```text
Z = R + j(ωL − 1/ωC)
Im(Z) = 0  →  ω0L = 1/(ω0C)
ω0 = 1/√(LC) rad/s,   f0 = 1/(2π√(LC)) Hz
```

#### Quality factor

Q is 2π × (maximum energy stored)/(energy lost per cycle), or the voltage magnification at resonance.

```text
Q = ω0L/R = 1/(ω0CR) = (1/R)√(L/C)
V_L = V_C = Q·V at resonance
```

#### Half power frequencies

The half power points are where I = I0/√2, i.e. |Z| = √2 R, i.e. |X| = R:

```text
ωL − 1/ωC = ±R
ω2 = R/2L + √[(R/2L)² + 1/LC]
ω1 = −R/2L + √[(R/2L)² + 1/LC]
ω1 ω2 = ω0²
```

#### Bandwidth

```text
BW = ω2 − ω1 = R/L rad/s   (R/2πL Hz)
BW = ω0/Q
```

| Quantity | Expression |
|---|---|
| Resonant frequency | 1/(2π√LC) |
| Quality factor | ω0L/R |
| Half power frequencies | ∓R/2L + √[(R/2L)² + 1/LC] |
| Bandwidth | R/L = ω0/Q |

**Answer: f0 = 1/(2π√LC), Q = ω0L/R, BW = R/L, ω1,2 = ∓R/2L + √[(R/2L)² + 1/LC].**

## Check yourself

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

1. Which condition gives the series resonant frequency?
   A. Re(Z) = 0
   B. Im(Z) = 0, so ω0L = 1/(ω0C)
   C. ωL = R
   D. ω0C = R
   Answer: B. Resonance needs zero net reactance, giving ω0 = 1/√(LC). (Key point: Resonant frequency)

2. Which expression gives Q of a series RLC circuit?
   A. R/(ω0L)
   B. ω0RC
   C. ω0L/R = 1/(ω0CR)
   D. BW/ω0
   Answer: C. Q = ω0L/R, which equals the voltage magnification V_L/V at resonance. (Key point: Quality factor)

3. What are V_L and V_C at series resonance?
   A. Each equal to V/Q
   B. Both zero
   C. V_L = V and V_C = 0
   D. Each equal to Q × V
   Answer: D. At resonance I = V/R, so V_L = V_C = I·X = (X/R)V = QV. (Key point: Voltage magnification)

4. How are BW, ω0 and Q related?
   A. BW = ω0/Q
   B. BW = ω0·Q
   C. BW = Q/ω0
   D. BW = ω0/Q^2
   Answer: A. BW = R/L and Q = ω0L/R, hence BW = ω0/Q. (Key point: Bandwidth and Q)

## Related in Electrical Circuit Theory

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