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

# Series RLC resonance, half power, bandwidth

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

- Asked in: 2075 Asoj, 8 marks; 2074 Asoj, 8 marks; 2068 Chaitra, 8 marks (asked 3 times)
- 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-half-power-bandwidth
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-1

## Question

What do you mean by resonance in an RLC series circuit? Define half power frequencies and bandwidth of an RLC series circuit and obtain expressions for them.

## Short answer

Resonance in a series RLC circuit occurs at ω0 = 1/√(LC) where XL=XC and impedance is purely R. Half-power frequencies are ω1,2 = ∓R/2L + √[(R/2L)² + 1/LC] where current falls to I0/√2. Bandwidth is BW = ω2−ω1 = R/L rad/s = ω0/Q.

## Answer

**Resonance** in a series RLC circuit is the condition in which the inductive reactance equals the capacitive reactance (X_L = X_C), so the net reactance is zero, the impedance is purely resistive (Z = R) and the supply voltage and current are in phase.

#### Resonance condition

```text
Z = R + j(ωL − 1/ωC)
Resonance: ωL = 1/ωC
ω0 = 1/√(LC) rad/s      f0 = 1/(2π√(LC)) Hz
```

At resonance: Z is minimum (= R), current is maximum I0 = V/R, power factor = 1, and V_L = V_C in magnitude (equal and opposite), each equal to Q·V.

```text
 I
 I0 |          ***
    |         *   *
I0/√2|- - - -*- - - *
    |       *|     |*
    |     **|     | **
    |_____*_|_____|_*____ ω
          ω1  ω0  ω2
          |<-BW->|
```

#### Half power frequencies

The frequencies at which the power delivered to the circuit falls to half of its maximum value (P = I0²R/2) are called half power frequencies (ω1 lower, ω2 upper). At these points the current is I0/√2 = 0.707 I0, so |Z| = √2 R. This needs the net reactance to equal the resistance:

```text
|X| = |ωL − 1/ωC| = R
Above resonance:  ωL − 1/ωC = +R  →  ω²LC − ωRC − 1 = 0
Below resonance:  ωL − 1/ωC = −R  →  ω²LC + ωRC − 1 = 0
```

Taking the positive roots:

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

Also ω1·ω2 = 1/LC = ω0², so ω0 is the geometric mean of ω1 and ω2.

#### Bandwidth

Bandwidth is the width of the band of frequencies between the half power points.

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

Using the quality factor Q = ω0L/R = 1/(ω0CR):

```text
BW = ω0/Q     (or f0/Q in Hz)
```

For high Q (Q > 10): ω1 ≈ ω0 − BW/2 and ω2 ≈ ω0 + BW/2.

**Answer: resonance at ω0 = 1/√(LC); half power frequencies ω1,2 = ∓R/2L + √[(R/2L)² + 1/LC]; bandwidth BW = R/L rad/s = ω0/Q.**

## Check yourself

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

1. What is true of a series RLC circuit at resonance?
   A. X_L = X_C, so Z = R and V and I are in phase
   B. X_L = R, so Z is maximum
   C. X_L = X_C, so the current lags V by 90°
   D. R = X_C, so the power factor is zero
   Answer: A. At resonance the reactances cancel, leaving a purely resistive impedance Z = R with current in phase with voltage. (Key point: Resonance condition)

2. At a half power frequency of a series RLC circuit, what are the current and impedance?
   A. I = I0/2 and |Z| = 2R
   B. I = I0/√2 and |Z| = √2 R
   C. I = I0/√2 and |Z| = R/√2
   D. I = 0.5 I0 and |Z| = √2 R
   Answer: B. Power is half when I^2 is half, so I = 0.707 I0, which needs |Z| = √2 R, i.e. net reactance equal to R. (Key point: Half power point)

3. What is the bandwidth of a series RLC circuit in rad/s?
   A. L/R
   B. 1/(RC)
   C. R/L
   D. R/(2πL)
   Answer: C. ω2 − ω1 = R/L rad/s; dividing by 2π gives R/(2πL) in Hz. (Key point: Bandwidth expression)

4. How are ω0, ω1 and ω2 related?
   A. ω0 = (ω1 + ω2)/2 exactly
   B. ω1·ω2 = R/L
   C. ω2 − ω1 = ω0^2
   D. ω1·ω2 = ω0^2, so ω0 is the geometric mean
   Answer: D. Multiplying the two roots gives 1/LC = ω0^2, so ω0 is the geometric mean of the half power frequencies. (Key point: Geometric mean)

## 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
- [Bandwidth of series resonant circuit with resonance curve](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Abandwidth-series-resonance-curve) — 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
- [Half power points, bandwidth and effect of Q](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aseries-resonance-bandwidth-q-effect) — Electrical Circuit Theory, chapter 1
- [Parallel RLC resonance and resonance frequency](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aparallel-rlc-resonance-frequency) — Electrical Circuit Theory, chapter 1, asked 2×