---
title: "Resonance of coil in parallel with capacitor"
subject: "Electrical Circuit Theory"
chapter: 1
key: "semester_3/electrical_circuit_theory:coil-parallel-capacitor-resonance"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/coil-parallel-capacitor-resonance
site: BCT Engineering
---

# Resonance of coil in parallel with capacitor

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

- Asked in: 2069 Chaitra, 4 marks
- Video: [PARALLEL ( coil (R-L) in parallel with Capacitor C) RESONACE PART -1](https://www.youtube.com/watch?v=uPdX_zUacB4) — KK CONCEPTS OF ELECTRICAL ENGINEERING
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/coil-parallel-capacitor-resonance
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-1

## Question

Derive an expression with necessary diagrams for the resonance frequency of a circuit consisting of a coil in parallel with a capacitor excited by a sinusoidal AC voltage.

## Short answer

For a practical coil (R in series with L) in parallel with C, resonance occurs when total susceptance is zero: ω0 = √(1/LC − R²/L²). At resonance, the circuit presents a maximum dynamic impedance Z0 = L/(CR). If R is small, f0 ≈ 1/(2π√LC).

## Answer

A practical coil has inductance L and a series resistance R. When it is connected in parallel with a capacitor C across a sinusoidal supply, the circuit is at resonance when the supply voltage and the line current are in phase (the total admittance is real).

```text
         +--------+---------+
    I -->|        |         |
         |      (R+jωL)    (1/jωC)
  V      |      coil        C
         |        |         |
         +--------+---------+
 I_coil = V/(R + jωL)    I_C = jωCV
```

#### Derivation

The admittance of the coil branch is

```text
Y_coil = 1/(R + jωL) = (R − jωL)/(R² + ω²L²)
```

Total admittance with the capacitor:

```text
Y = R/(R² + ω²L²) + j[ ωC − ωL/(R² + ω²L²) ]
```

Resonance requires the imaginary part to be zero:

```text
C = L/(R² + ω0²L²)
R² + ω0²L² = L/C
ω0² = 1/(LC) − R²/L²

ω0 = √(1/LC − R²/L²) rad/s
f0 = (1/2π)√(1/LC − R²/L²) Hz
```

#### Remarks

- Resonance is possible only if R < √(L/C). Otherwise ω0 is imaginary and no resonance occurs.
- If R << ω0L (high Q coil), f0 ≈ 1/(2π√LC).
- At resonance the impedance is maximum: Z0 = (R² + ω0²L²)/R = L/(CR) (dynamic impedance), and the line current is minimum.

**Answer: f0 = (1/2π)√(1/LC − R²/L²), dynamic impedance L/(CR).**

## Check yourself

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

1. How is resonance of a coil in parallel with a capacitor found?
   A. Set R = ωL
   B. Set the imaginary part of the total admittance to zero
   C. Set the real part of Y to zero
   D. Set the coil current equal to the capacitor current in magnitude only
   Answer: B. Resonance means V and line current are in phase, i.e. Y is real. (Key point: Resonance condition)

2. What is the resonant frequency?
   A. 1/√(LC) exactly
   B. √(1/LC + R^2/L^2)
   C. √(1/LC − R^2/L^2)
   D. R/(2πL)
   Answer: C. From C = L/(R^2 + ω0^2L^2) we get ω0^2 = 1/LC − R^2/L^2. (Key point: Resonant frequency)

3. When does resonance occur?
   A. Only if R > √(L/C)
   B. For any R
   C. Only if R = 0
   D. Only if R < √(L/C)
   Answer: D. Otherwise ω0^2 is negative and no real resonance frequency exists. (Key point: Existence of resonance)

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