Mini Physics
Part of Mini EducationSwitch subject

UY1: L-R-C Series Circuit With A.C.

Solve steady-state AC series RLC circuits using impedance, phase angle, and phasor voltage relations.

  • University Physics Year 1
On this page

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 L-R-C Series Circuit With A.C.. 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 AC steady state, series RLC current is the same through all elements.
  • Impedance magnitude:
Z = square root of (R² + (X_L-X_C)²) .
  • Phase angle (source voltage relative to current):
tan φ = (X_L-X_C)/R.
  • Modelling context: this is sinusoidal steady state. Phasors assume a single angular frequency ω and that switch-on transients have died out.

Prerequisites: Phasors & Alternating Currents, Resistors, Inductors & Capacitors In A.C. Circuits
Next uses: Resonance & Power In A.C. Circuits

2) Setup

For source

v(t) = V₀ cos ω t,

series elements have:

X_L = ω L, X_C = 1/(ω C).

Phasor reference: take current phasor along positive real axis.

Common traps (RMS vs peak, and phase sign)
  • Don’t mix peak and RMS. If you use V₀ and I₀, keep everything in peak form; if you use Vᵣₘₛ and Iᵣₘₛ, keep everything in RMS form.
  • The net reactance is (X_L-X_C), not (X_C-X_L). The sign controls whether the circuit is inductive (φ > 0) or capacitive (φ < 0).
  • Near resonance, V_L and V_C can each be larger than the source voltage even though the source is not “violated” (phasors cancel).

3) Core derivation/explanation

Element voltage amplitudes:

V_R = I₀R, V_L = I₀X_L, V_C = I₀X_C.

Phasor sum gives source amplitude:

V₀ = square root of (V_R² + (V_L-V_C)²) = I₀Z.

So

I₀ = V₀/Z, Z = square root of (R² + (X_L-X_C)²) .

Phase meaning:

  • φ > 0 (inductive): voltage leads current.
  • φ < 0 (capacitive): current leads voltage.
  • φ = 0 at resonance (X_L = X_C).

Average power:

Pₐᵥ = VᵣₘₛIᵣₘₛ cos φ.

Checks (sanity)

  • Low frequency: X_C is large and X_L is small, so the circuit is strongly capacitive (φ < 0).
  • High frequency: X_L is large and X_C is small, so the circuit is strongly inductive (φ > 0).
  • Resonance (X_L = X_C): Z = R and φ = 0.

4) Worked example(s)

Given R = 40 Ω, L = 0.20 H, C = 100 μF, f = 50 Hz, Vᵣₘₛ = 120 V.

ω = 2π f = 314 rad s⁻¹, X_L = 62.8 Ω, X_C = 31.8 Ω.
Z = square root of (40² + (62.8-31.8)²) = 50.6 Ω.
Iᵣₘₛ = Vᵣₘₛ/Z = 2.37 A.
φ = tan⁻¹ (31.0/40) = 37.8°

(inductive, so current lags).

5) Practice set (with hints + answers)

  1. If X_L = X_C, what is Z for a series RLC? Hint: net reactance is zero. Answer: Z = R.

  2. For fixed R,L,C, increasing frequency from very low values usually moves circuit toward which behavior first: capacitive or inductive? Hint: compare X_C ∝ 1/ω and X_L ∝ ω. Answer: it starts strongly capacitive at very low frequency, then approaches resonance.

  3. Why can V_L and V_C each exceed source V₀ near resonance while current remains finite? Hint: they are nearly opposite phasors. Answer: large opposing reactive voltages partially cancel in vector sum.

6) Summary + next steps

  • AC series RLC problems reduce to impedance geometry in the phasor plane.
  • Phase sign tells you whether the net behavior is inductive or capacitive.
  • Resonance is the key special point where reactive parts cancel.

Next: Resonance & Power In A.C. Circuits Previous: Resistors, Inductors & Capacitors In A.C. Circuits Back To Electromagnetism (UY1)