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UY1: Resonance & Power In A.C. Circuits

Find resonance in series LRC circuits and compute AC average power using phase angle and power factor.

  • University Physics Year 1
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Learning objectives

  • Use vector and differential or integral calculus to express physical change and accumulation.
  • Use complex numbers, linear algebra, and differential equations to solve coupled physical models.
  • Choose an efficient mathematical method and validate the result physically.
Why this matters + quick links

This page gives the UY1 working model/result for Resonance & Power In A.C. Circuits. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

1) At a glance

In a series LRC circuit:

  • Resonance occurs at ω₀ = 1/square root of LC.
  • Average AC power is P_avg = VᵣₘₛIᵣₘₛ cos φ.
  • At resonance: X_L = X_C, so Z = R, φ = 0, and power factor cos φ = 1.
  • Modelling context: sinusoidal steady state (phasors); reactive elements exchange energy with the source, but only R dissipates net energy.

Prerequisites: L-R-C Series Circuit With A.C.
Next uses: Electromagnetic Spectrum & Sinusoidal EM Plane Waves

2) Setup

Take source voltage as reference:

v(t) = V₀ cos ω t

Current is

i(t) = I₀ cos(ω t-φ)

with impedance

Z = square root of (R² + (ω L-1/(ω C))²), I₀ = V₀/Z

Phase convention used:

tan φ = (ω L-1/(ω C))/R

Common traps (omega vs f, RMS vs peak)
  • Resonance is in angular frequency: ω₀ = 1/square root of LC. Convert to f₀ using f₀ = ω₀/(2π).
  • Don’t mix peak and RMS. The power formula P_avg = VᵣₘₛIᵣₘₛ cos φ is RMS-based.
  • At resonance, current is not infinite in real circuits because R ≠ 0 (and there are always losses).

3) Core derivation/explanation

Resonance

At resonance, inductive and capacitive reactances cancel:

ω₀L = 1/ω₀C ⇒ ω₀ = 1/(square root of LC)

Then Z = R (minimum), so current amplitude is maximum:

I_(0, max) = V₀/R

Power in AC circuit

Instantaneous power:

p(t) = v(t)i(t) = V₀I₀ cos ω t cos(ω t-φ)

Using product-to-sum:

p(t) = (V₀I₀/2)[cos φ + cos(2ω t-φ)]

Average over one cycle:

P_avg = V₀I₀/2 cos φ = VᵣₘₛIᵣₘₛ cos φ

where cos φ is the power factor.

Checks (sanity)

  • If φ = 90° (purely reactive), then cos φ = 0 and average power is zero.
  • At resonance, φ = 0 so P_avg = VᵣₘₛIᵣₘₛ = Vᵣₘₛ²/R.

4) Worked example(s)

Given R = 20 Ω, L = 0.20 H, C = 50 μF, Vᵣₘₛ = 120 V:

  • ω₀ = 1/square root of LC = 316 rad s⁻¹.
  • f₀ = ω₀/(2π) = 50.3 Hz.
  • At resonance, Z = R = 20 Ω, so

Iᵣₘₛ = 120/20 = 6.0 A

and φ = 0, hence

P_avg = VᵣₘₛIᵣₘₛ = 720 W

5) Practice set (with hints + answers)

  1. L = 0.10 H, C = 25 μF. Find ω₀. Hint: ω₀ = 1/square root of LC. Answer: ω₀ = 632 rad s⁻¹.

  2. If Vᵣₘₛ = 230 V, Iᵣₘₛ = 4.0 A, and cos φ = 0.80, find average power. Hint: P = VᵣₘₛIᵣₘₛ cos φ. Answer: 736 W.

  3. At resonance in an ideal series LRC (fixed R), is power factor leading, lagging, or unity? Hint: compare reactances. Answer: unity (φ = 0).

6) Summary + next steps

Resonance is a reactance-cancellation condition, while real power depends on phase alignment between total voltage and current through the power factor.

Next: Electromagnetic Spectrum & Sinusoidal EM Plane Waves Previous: L-R-C Series Circuit With A.C. Back To Electromagnetism (UY1)