Difficulty: Intermediate to Advanced | Prerequisites: Capacitance (Part 2), inductance and Faraday's Law (Part 3), trigonometry.
Circuits with resistors, capacitors, and inductors are where electrostatics and magnetism become practical engineering. RC and RL circuits describe how voltages and currents change over time when you flip a switch, governed by exponential growth and decay with characteristic time constants. AC circuits extend this to sinusoidal driving voltages, where capacitors and inductors introduce frequency-dependent opposition to current (reactance). The series RLC circuit is the electrical analogue of a mass-spring system: it has a resonant frequency where the impedance is minimised and power transfer is maximised. Transformers, which rely on mutual inductance and Faraday's Law, let you step voltages up or down for efficient power transmission.
RC circuits charge/discharge exponentially with time constant τ = RC. RL circuits do the same with τ = L/R. In AC circuits, capacitors and inductors have frequency-dependent reactance. At resonance in a series RLC circuit (ω₀ = 1/√(LC)), impedance is purely resistive, the phase angle is zero, and power transfer peaks. Transformers scale voltage by the turns ratio.
Time constant (τ)
The characteristic time for an RC or RL circuit to reach about 63% of its final value (charging) or drop to about 37% of its initial value (discharging). For RC circuits, τ = RC. For RL circuits, τ = L/R.
Reactance
The opposition to AC current from a capacitor or inductor, measured in ohms. Unlike resistance, reactance depends on frequency and does not dissipate power.
Capacitive reactance (X_C)
X_C = 1/(ωC) = 1/(2πfC). At high frequencies, X_C is small (capacitor passes current easily). At low frequencies, X_C is large (capacitor blocks current). In simple terms, a capacitor is a short circuit for fast signals and an open circuit for DC.
Inductive reactance (X_L)
X_L = ωL = 2πfL. At high frequencies, X_L is large (inductor resists rapid changes). At low frequencies, X_L is small (inductor is nearly a short circuit for DC). In simple terms, an inductor is the opposite of a capacitor.
Impedance (Z)
The total opposition to AC current in a circuit, combining resistance and net reactance: Z = √(R² + (X_L − X_C)²). Measured in ohms. Think of it as the AC version of resistance.
Phase angle (φ)
The angle by which the current leads or lags the voltage: tan φ = (X_L − X_C) / R. When X_L > X_C, the circuit is inductive and current lags voltage. When X_C > X_L, the circuit is capacitive and current leads voltage.
Resonance
The condition where X_L = X_C, so the impedance reduces to Z = R. The resonant angular frequency is ω₀ = 1/√(LC). At resonance, φ = 0 and the circuit behaves as if it were purely resistive.
RMS (root mean square)
For a sinusoidal quantity, the RMS value is the peak value divided by √2: I_rms = I_max / √2, V_rms = V_max / √2. RMS values are what AC meters read and what you use in power calculations.
Transformer
Two coils (primary and secondary) sharing a magnetic core. The voltage ratio equals the turns ratio: V_s / V_p = N_s / N_p. For an ideal transformer, power in equals power out.
At t = 0, the switch closes. The capacitor is uncharged, so V_C = 0 and it acts like a wire.
Initial current: I(0) = ε/R (the full battery EMF drives current through the resistor alone).
As the capacitor charges, V_C rises and current drops: I(t) = (ε/R) e^(−t/τ), where τ = RC.
Voltage across capacitor: V_C(t) = ε(1 − e^(−t/τ)).
After one time constant (t = τ), V_C ≈ 0.63ε and I ≈ 0.37(ε/R).
At t = 0, the switch closes. The inductor opposes sudden changes in current, so I(0) = 0.
Current grows: I(t) = I_max(1 − e^(−t/τ)), where I_max = ε/R and τ = L/R.
After one time constant (t = τ = L/R), the current reaches about 63% of its maximum value.
The inductor's back-EMF is ε_L = −L(dI/dt), which starts at −ε and decays to zero.
In a purely resistive AC circuit, voltage and current are in phase.
In a purely capacitive circuit, current leads voltage by 90°.
In a purely inductive circuit, current lags voltage by 90°.
In a series RLC circuit driven at angular frequency ω:
X_L = ωL
X_C = 1/(ωC)
Z = √(R² + (X_L − X_C)²)
Peak current: I_max = V_max / Z
Phase angle: tan φ = (X_L − X_C) / R
X_C = 1/(2πfC): as frequency f increases, X_C decreases.
At very high frequencies, the capacitor is essentially a short circuit (low reactance).
At very low frequencies (including DC), the capacitor is essentially an open circuit (high reactance).
Resonance occurs when X_L = X_C, i.e. ωL = 1/(ωC).
Solving: ω₀ = 1/√(LC), or equivalently f₀ = 1/(2π√(LC)).
At resonance: Z = R (minimum impedance), I_max is at its maximum, φ = 0° (voltage and current are in phase), and average power dissipated is maximised.
For sinusoidal signals: I_rms = I_max / √2, V_rms = V_max / √2.
If I_rms = 5 A, then I_max = 5√2 ≈ 7.07 A.
Average power dissipated: P_avg = I_rms² R = ½ I_max V_max cos φ.
Voltage transformation: V_s / V_p = N_s / N_p.
If N_s > N_p: step-up transformer (voltage increases).
If N_s < N_p: step-down transformer (voltage decreases).
Example: N_p = 100, N_s = 500, V_p = 120 V → V_s = 120 × (500/100) = 600 V.
Given: R = 200 Ω, L = 0.5 H, C = 10 μF, V_max = 100 V, f = 60 Hz.
Step 1, angular frequency: ω = 2π(60) = 377 rad/s.
Step 2, reactances: X_L = ωL = 377 × 0.5 = 188.5 Ω. X_C = 1/(ωC) = 1/(377 × 10 × 10⁻⁶) = 265.3 Ω.
Step 3, impedance: Z = √(200² + (188.5 − 265.3)²) = √(40000 + 5900) ≈ 214.2 Ω.
Step 4, peak current: I_max = V_max / Z = 100 / 214.2 ≈ 0.467 A.
Step 5, phase angle: tan φ = (188.5 − 265.3) / 200 = −76.8/200 = −0.384, so φ ≈ −21°. (Negative means the circuit is capacitive and current leads voltage.)
Step 6, average power: P_avg = ½ I_max V_max cos φ = ½(0.467)(100) cos(−21°) ≈ 23.35 × 0.933 ≈ 21.8 W.
Quantity | Expression |
|---|---|
RC time constant | τ = RC |
RL time constant | τ = L/R |
RC charging current | I(t) = (ε/R) e^(−t/RC) |
RL charging current | I(t) = (ε/R)(1 − e^(−tR/L)) |
Capacitive reactance | X_C = 1/(ωC) |
Inductive reactance | X_L = ωL |
Impedance (series RLC) | Z = √(R² + (X_L − X_C)²) |
Peak current | I_max = V_max / Z |
Phase angle | tan φ = (X_L − X_C) / R |
Resonant frequency | ω₀ = 1/√(LC) |
RMS to peak | I_max = √2 × I_rms |
Average power | P_avg = ½ I_max V_max cos φ = I_rms² R |
Transformer voltage ratio | V_s / V_p = N_s / N_p |
The RC time constant governs how quickly a camera flash charges. RL circuits appear in relay coils and motor start-up transients. RLC resonance is the principle behind radio tuning: you adjust C (or L) until ω₀ matches the broadcast frequency, maximising the signal. Power grids use transformers to step voltage up for long-distance transmission (reducing I²R losses) and step it down for safe household use.
Students often confuse the RC and RL time constants. For RC, τ = RC (seconds = ohms × farads). For RL, τ = L/R (seconds = henrys / ohms). Mixing them up is a favourite exam trap.
A common error is thinking that at t = 0 in an RC charging circuit the current is zero. It is not: the capacitor is uncharged and acts like a short circuit, so I(0) = ε/R, the maximum current.
Students sometimes think resonance means zero current. The opposite is true: at resonance, Z is minimised and current is maximised.
When computing average power, students sometimes use P = ½ I_max V_max and forget the cos φ factor. The power factor cos φ accounts for the phase difference; omitting it overestimates the power.
⚠️ The initial current in an RC charging circuit (I = ε/R at t = 0) is a classic 3-mark multiple-choice question.
⚠️ The RL time constant τ = L/R, and the fact that current reaches 63% of maximum after one τ, is tested repeatedly.
⚠️ The resonant frequency ω₀ = 1/√(LC) appears in multiple-choice. Students sometimes confuse it with ω₀ = √(LC) or 1/(LC).
⚠️ The full RLC impedance calculation (find X_L, X_C, Z, I_max, φ, P_avg) is a 10-mark long-answer question. Practise all six steps.
⚠️ At resonance, φ = 0. This is a commonly tested multiple-choice item.
⚠️ RMS-to-peak conversion (I_max = √2 × I_rms) is straightforward but often missed. If I_rms = 5 A, the peak is 7.07 A, not 10 A.
⚠️ The transformer turns-ratio calculation is a direct 3-mark question. Identify which coil is primary and which is secondary before applying V_s/V_p = N_s/N_p.
Fill in the blank: At t = 0 in an RC charging circuit, the current through the resistor is _______.
True or false: In an RL circuit, the time constant is τ = RC.
Fill in the blank: The resonant angular frequency of a series RLC circuit is ω₀ = _______.
True or false: At resonance in a series RLC circuit, the phase angle between voltage and current is 90°.
Fill in the blank: If I_rms = 5 A, then I_max = _______ A.
Answers: 1. ε/R. 2. False (τ = L/R). 3. 1/√(LC). 4. False (φ = 0°). 5. 7.07 A (5√2).
Q: In a simple RC circuit being charged by battery ε, what is the current at t = 0?
A: I = ε/R. At t = 0 the capacitor is uncharged (V_C = 0), so by Kirchhoff's loop rule, ε − IR = 0.
Q: In an RL circuit, how long does it take for the current to reach 63% of its maximum?
A: One time constant, τ = L/R. At t = τ, I = I_max(1 − e⁻¹) ≈ 0.63 I_max.
Q: What is the resonant angular frequency of a series RLC circuit?
A: ω₀ = 1/√(LC). This is the frequency where X_L = X_C and the impedance equals R.
Q: At resonance, what is the phase angle φ between voltage and current?
A: φ = 0°. The circuit is purely resistive, so voltage and current are in phase.
Q: If the RMS current in an AC circuit is 5 A, what is the peak current?
A: I_max = I_rms × √2 = 5 × 1.414 ≈ 7.07 A.
Q: As frequency increases, what happens to the capacitive reactance X_C?
A: It decreases. X_C = 1/(2πfC), so higher f means lower X_C. The capacitor passes more current at higher frequencies.
Q: A transformer has 100 primary turns and 500 secondary turns. If the input is 120 V AC, what is the output voltage?
A: V_s = V_p(N_s/N_p) = 120 × (500/100) = 600 V.
Q: For a series RLC circuit with R = 200 Ω, L = 0.5 H, C = 10 μF, driven at 60 Hz with V_max = 100 V, calculate the average power dissipated.
A: ω = 377 rad/s, X_L = 188.5 Ω, X_C = 265.3 Ω, Z ≈ 214.2 Ω, I_max ≈ 0.467 A, φ ≈ −21°. P_avg = ½(0.467)(100)cos(21°) ≈ 21.8 W.
RC and RL circuits connect to the capacitance and inductance material in Parts 2 and 3. The RLC resonance condition reappears in electromagnetic wave theory: the speed of light c = 1/√(μ₀ε₀) has the same mathematical structure as ω₀ = 1/√(LC). Transformers rely on Faraday's Law (Part 3) and are the practical bridge between AC circuit theory and the power grid.
RC circuit, RL circuit, RLC circuit, time constant, τ, exponential decay, charging, discharging, reactance, capacitive reactance, inductive reactance, impedance, phase angle, power factor, resonance, resonant frequency, RMS, root mean square, peak current, average power, transformer, turns ratio, step-up, step-down, Kirchhoff's loop rule, AC circuits, PHYS 212, University Physics electricity and magnetism