Source: University Physics, Electromagnetism Handout (UIUC)
Tags: alternating current, AC circuits, phasors, reactance, capacitive reactance, inductive reactance, impedance, RLC series circuit, phase angle, resonance, AC generator, forced oscillations, PHY 142
Difficulty: Intermediate | Prerequisites: Section 31.2 (damped RLC), trigonometric identities, basic phasor concepts (Sec. 16.6)
Sections 31.3 and 31.4 move from the "dying oscillation" of the free RLC circuit to what happens when you continuously pump energy in with an AC source. This is the physics behind every wall outlet, every transformer, and every radio tuner. You first study what AC does to each component individually (resistor, capacitor, inductor), then combine them into the series RLC circuit driven at a frequency ω. The central result is impedance, which generalises resistance to AC circuits and determines how much current flows for a given voltage.
An AC generator forces sinusoidal current through a circuit. Resistors, capacitors, and inductors each respond differently: resistors keep current in phase with voltage, capacitors let current lead by 90°, and inductors make current lag by 90°. When all three sit in series, the combined opposition to current is called impedance, and the phase relationship between total voltage and current depends on the balance between inductive and capacitive reactance.
Alternating current (AC)
Current that reverses direction sinusoidally with time. The standard form is ℰ = ℰ_max sin(ωt), where ω is the angular frequency of the source.
In simple terms, the voltage swings positive and negative in a repeating wave, unlike a battery's steady DC.
Forced oscillation
When an external AC source drives a circuit, the circuit oscillates at the driving frequency ω regardless of its own natural frequency ω₀. The natural frequency still matters for resonance, but the steady-state response locks to ω.
Think of it as: pushing a child on a swing at your chosen rhythm rather than letting the swing find its own.
Phasor
A rotating vector in the complex plane used to represent the amplitude and phase of a sinusoidal quantity. Phasors let you add voltages that are out of phase with each other using geometry rather than trigonometric algebra.
In simple terms, phasors turn messy trig into simple vector addition.
Capacitive reactance (X_C)
The opposition a capacitor offers to AC current. Defined as X_C = 1/(ωC). High frequency means low reactance (capacitors pass high-frequency signals easily). Measured in ohms.
Think of it as: at high frequencies the capacitor barely has time to charge before the voltage reverses, so current flows freely.
Inductive reactance (X_L)
The opposition an inductor offers to AC current. Defined as X_L = ωL. High frequency means high reactance (inductors resist rapid changes in current). Measured in ohms.
Think of it as: the faster you try to change the current, the harder the inductor pushes back.
Impedance (Z)
The total opposition to current in an AC circuit containing R, L, and C. For a series RLC circuit: Z = √(R² + (X_L − X_C)²). Measured in ohms. Impedance is the AC generalisation of resistance.
In simple terms, impedance is to AC what resistance is to DC, but it also accounts for the energy-storing behaviour of capacitors and inductors.
Phase angle (φ)
The angle by which the current leads or lags the applied voltage. Defined by tan φ = (X_L − X_C)/R. When φ > 0 the circuit is inductive (current lags); when φ < 0 it is capacitive (current leads); when φ = 0 the circuit is at resonance.
Voltage and current are in phase: both go as sin(ωt).
ℰ = ℰ_max sin(ωt), so i_R = I_max sin(ωt) with I_max = ℰ_max / R.
Voltage across the resistor: v_R = I_max · R · sin(ωt).
On a phasor diagram, the voltage and current phasors point in the same direction.
Current leads voltage by 90° (π/2 radians).
If v_C = ℰ_max sin(ωt), then i_C = ωC · ℰ_max · cos(ωt) = ωC · ℰ_max · sin(ωt + π/2).
Capacitive reactance: X_C = 1/(ωC), and I_max = ℰ_max / X_C.
Mnemonic: "ICE" (I leads C in an E circuit, i.e. current I leads voltage E in a capacitor C).
Current lags voltage by 90° (π/2 radians).
If v_L = ℰ_max sin(ωt), then i_L = −(ℰ_max / ωL) cos(ωt) = (ℰ_max / ωL) sin(ωt − π/2).
Inductive reactance: X_L = ωL, and I_max = ℰ_max / X_L.
Mnemonic: "ELI" (voltage E leads current I in an inductor L).
Combine both mnemonics: "ELI the ICE man."
ELI: in an inductor (L), E (voltage) leads I (current).
ICE: in a capacitor (C), I (current) leads E (voltage).
This is one of the most reliable exam shortcuts in AC circuit problems.
An AC generator ℰ = ℰ_max sin(ωt) drives a series combination of R, L, and C.
The same current I(t) flows through all three components (series circuit).
Each component has its own voltage:
v_R = V_R sin(ωt), where V_R = I_max · R
v_C = −V_C cos(ωt), where V_C = I_max · X_C
v_L = V_L cos(ωt), where V_L = I_max · X_L
The three voltage phasors add as vectors (not scalars, because they are out of phase).
V_R is along the current direction; V_L points 90° ahead; V_C points 90° behind.
The net voltage amplitude is: ℰ_max = √(V_R² + (V_L − V_C)²).
Dividing through by I_max gives: Z = √(R² + (X_L − X_C)²).
The maximum current is: I_max = ℰ_max / Z.
The full time-dependent current: I(t) = (ℰ_max / Z) sin(ωt − φ).
tan φ = (X_L − X_C) / R
φ > 0: inductive circuit, current lags voltage.
φ < 0: capacitive circuit, current leads voltage.
φ = 0: resonance, X_L = X_C, and Z = R (minimum impedance, maximum current).
At the resonant frequency, ω₀ = 1/√(LC), the inductive and capacitive reactances cancel: X_L = X_C.
Impedance drops to its minimum value of R alone.
Current reaches its maximum value ℰ_max / R.
This is the principle behind radio tuning: adjust L or C until ω₀ matches the station's broadcast frequency.
Quantity | Formula |
|---|---|
AC source | ℰ = ℰ_max sin(ωt) |
Resistor current | i_R = (ℰ_max / R) sin(ωt) |
Capacitive reactance | X_C = 1/(ωC) |
Capacitor current | i_C = ωC · ℰ_max sin(ωt + π/2) |
Inductive reactance | X_L = ωL |
Inductor current | i_L = (ℰ_max / ωL) sin(ωt − π/2) |
Impedance (series RLC) | Z = √(R² + (X_L − X_C)²) |
Maximum current | I_max = ℰ_max / Z |
Phase angle | tan φ = (X_L − X_C) / R |
Resonant frequency | ω₀ = 1/√(LC) |
Phasor diagram for the series RLC circuit: draw V_R along the positive x-axis, V_L pointing straight up (+y), and V_C pointing straight down (−y). The resultant from the origin to the tip of V_R + (V_L − V_C) gives the total voltage phasor. The angle between this resultant and the current (along x) is φ.
Every mains-powered appliance deals with AC impedance. Power factor correction (adjusting φ toward zero) saves energy in industrial motors and is required by electrical codes. Radio receivers use a tuneable LC combination to select one broadcast frequency from the many hitting the antenna, exploiting the sharp current peak at resonance.
Students often add the voltage amplitudes V_R, V_L, and V_C as plain numbers. You cannot do this because they are out of phase. You must add the phasors as vectors: ℰ_max = √(V_R² + (V_L − V_C)²).
Confusing "current leads voltage" with "current is bigger." Leading and lagging refer to timing (phase), not amplitude.
Assuming reactance is the same as resistance. Reactance stores and returns energy; resistance dissipates it. A pure capacitor or inductor dissipates no power on average.
Forgetting that X_C decreases with frequency while X_L increases. At low frequencies a capacitor dominates; at high frequencies an inductor dominates.
⚠️ "ELI the ICE man" will save you on almost every AC phase question. Know it cold.
⚠️ Be able to calculate Z and φ for a series RLC circuit given numerical values of R, L, C, and ω.
⚠️ Phasor diagrams are a favourite exam question. Practise drawing them from scratch: V_R horizontal, V_L up, V_C down, resultant at angle φ.
⚠️ At resonance, Z = R and φ = 0. If asked "at what frequency is current maximised?", the answer is ω = 1/√(LC).
⚠️ Know which way the phase shifts: inductive circuits have positive φ (current lags), capacitive circuits have negative φ (current leads).
True or false: In a purely capacitive AC circuit, current and voltage are in phase.
Fill in the blank: Inductive reactance equals ______.
True or false: At resonance in a series RLC circuit, the impedance equals R.
Fill in the blank: The mnemonic for remembering phase relationships is ______.
True or false: Doubling the frequency doubles both X_L and X_C.
Answers: 1. False (current leads voltage by 90°). 2. X_L = ωL. 3. True. 4. ELI the ICE man. 5. False (doubling ω doubles X_L but halves X_C).
Q: A series RLC circuit has R = 200 Ω, L = 0.4 H, C = 5 μF, and is driven at ω = 1000 rad/s. Find the impedance Z.
A: X_L = ωL = 1000 × 0.4 = 400 Ω. X_C = 1/(ωC) = 1/(1000 × 5×10⁻⁶) = 200 Ω. Z = √(200² + (400 − 200)²) = √(40000 + 40000) = √80000 ≈ 283 Ω.
Q: For the same circuit, find the phase angle φ. Does the current lead or lag the voltage?
A: tan φ = (X_L − X_C)/R = (400 − 200)/200 = 1, so φ = 45°. Since φ > 0, the circuit is inductive and the current lags the voltage.
Q: At what angular frequency would this circuit resonate?
A: ω₀ = 1/√(LC) = 1/√(0.4 × 5×10⁻⁶) = 1/√(2×10⁻⁶) ≈ 707 rad/s.
Q: Explain why you cannot simply add V_R + V_L + V_C to get the total voltage amplitude in a series RLC circuit.
A: The three voltages are not in phase with each other. V_R is in phase with the current, V_L leads by 90°, and V_C lags by 90°. Adding them requires vector (phasor) addition, which gives ℰ_max = √(V_R² + (V_L − V_C)²).
Q: A capacitor is connected to a 60 Hz AC source. If the frequency is increased to 120 Hz, what happens to the capacitive reactance?
A: X_C = 1/(ωC). Doubling the frequency doubles ω, which halves X_C. The capacitor offers half as much opposition to current at the higher frequency.
This material builds directly on damped RLC oscillations (Sec. 31.2); the driven case adds the external energy source that sustains the oscillation. Resonance here is the electrical version of mechanical resonance from waves and oscillations (Ch. 15–16). These ideas also feed into power in AC circuits (Sec. 31.5), where you will learn that only the resistive component dissipates average power, and the power factor cos φ determines how efficiently the circuit uses the supplied energy.
AC circuits, alternating current, phasor diagram, reactance, capacitive reactance, inductive reactance, impedance, phase angle, series RLC, resonance, resonant frequency, ELI the ICE man, AC generator, forced oscillations, power factor, PHY 142, UIUC physics, electromagnetism, Kirchhoff's laws AC