Difficulty: Intermediate | Prerequisites: RL and LC circuit basics, phasor concepts, trigonometric identities (especially cos²).
Tags: AC circuit, RLC circuit, impedance, reactance, resonance, average power, power factor, transformer, turns ratio, PHYS
Once you add an AC voltage source to an RLC circuit, the behaviour shifts from transient decay to steady-state oscillation. The circuit's response depends on the frequency of the driving source relative to the natural frequency of the LC components. This is where impedance, reactance, and phase angles become essential. Resonance, the condition where the circuit draws maximum current, sits at the centre of the topic. Transformers then apply these AC principles to step voltages up or down for practical power distribution. If RL and LC transient circuits are the "what happens when you flip the switch" chapter, this is the "what happens when the switch flips back and forth continuously" chapter.
In an AC-driven RLC circuit, you find the reactances of L and C, combine them with R into an impedance Z, and use Z to find the maximum current. The phase angle tells you whether voltage leads or lags the current. At resonance, inductive and capacitive reactances cancel, impedance is minimised, and current is maximised. Transformers trade voltage for current using turns ratios.
Reactance (X)
The opposition to current flow from a capacitor or inductor in an AC circuit, measured in ohms. Unlike resistance, reactance depends on frequency. In simple terms, reactance is how much a capacitor or inductor "pushes back" against alternating current, and that pushback changes with how fast the current is alternating.
Inductive reactance (X_L)
X_L = ωL. Increases with frequency. At high frequencies the inductor strongly opposes current changes; at DC (ω = 0) it is zero, and the inductor is just a wire.
Capacitive reactance (X_C)
X_C = 1/(ωC). Decreases with frequency. At high frequencies the capacitor passes current easily; at DC it blocks entirely (X_C → ∞).
Impedance (Z)
The total effective opposition to current in an AC circuit, combining resistance and net reactance: Z = √(R² + (X_L − X_C)²). Think of it as the AC version of resistance. It determines the amplitude of the current for a given driving voltage.
Phase angle (φ)
The angle by which the voltage leads or lags the current: tan φ = (X_L − X_C)/R. When φ > 0, the circuit is inductive (voltage leads current). When φ < 0, the circuit is capacitive (current leads voltage). At resonance, φ = 0.
Resonance
The condition where X_L = X_C, so the inductive and capacitive effects cancel. Impedance drops to its minimum value (Z = R), current reaches its maximum, and the phase angle is zero. In simple terms, the circuit is "tuned" to the driving frequency.
RMS (root mean square)
The effective value of an oscillating quantity: V_rms = V_max/√2 and I_rms = I_max/√2. RMS values are what you use to compute average power, and they are what household voltage ratings refer to (e.g., 120 V in the US is an RMS value).
Power factor (cos φ)
The fraction of apparent power that actually does work. At resonance, cos φ = 1 and all power is delivered to the resistor. In simple terms, it tells you how efficiently the circuit converts source power into useful dissipation.
Transformer
A device that uses mutual inductance between two coils (primary and secondary) to step voltage up or down. The key principle is that an ideal transformer conserves power: what goes up in voltage comes down in current, and vice versa.
The standard approach for any AC RLC problem is four steps:
Step 1 – Find the reactances:
X_L = ωL
X_C = 1/(ωC)
Step 2 – Find the impedance:
Z = √(R² + (X_L − X_C)²)
Step 3 – Find the maximum (or RMS) current:
I_max = V_max / Z, or equivalently I_rms = V_rms / Z
Step 4 – Find the phase angle:
tan φ = (X_L − X_C) / R
Positive φ: inductive (voltage leads)
Negative φ: capacitive (current leads)
The average power delivered to the circuit is:
⟨P⟩ = ε_rms × I_rms × cos φ
Only the resistor dissipates power on average. The inductor and capacitor alternately store and return energy, contributing nothing to average power.
At resonance, cos φ = 1, and ⟨P⟩ = ε_rms × I_rms = (ε_rms)² / R. This is the maximum possible average power for a given source.
Resonance occurs when X_L = X_C, i.e., ωL = 1/(ωC).
Solving for the resonant frequency: ω₀ = 1/√(LC), the same natural frequency as the LC circuit.
At resonance:
Z = R (minimum impedance)
I_max = V_max / R (maximum current)
φ = 0 (voltage and current in phase)
Power delivery is maximised
The sharpness of the resonance peak depends on R: lower resistance gives a sharper, taller peak (higher "quality factor" Q = ω₀L/R).
A transformer has a primary coil (N_p turns) and a secondary coil (N_s turns) wound around a shared core.
Voltage ratio: V_s / V_p = N_s / N_p
If N_s > N_p, the transformer steps voltage up.
If N_s < N_p, it steps voltage down.
Current ratio: I_p / I_s = N_s / N_p
Current is inversely proportional to the turns ratio. Stepping voltage up means stepping current down, and vice versa.
The underlying principle is power conservation in an ideal transformer: V_p × I_p = V_s × I_s.
Real transformers have losses (resistive heating, eddy currents, flux leakage), but exam problems usually treat them as ideal.
Quantity | Expression |
|---|---|
Inductive reactance | X_L = ωL |
Capacitive reactance | X_C = 1/(ωC) |
Impedance | Z = √(R² + (X_L − X_C)²) |
Maximum current | I_max = V_max / Z |
Phase angle | tan φ = (X_L − X_C) / R |
Average power | ⟨P⟩ = ε_rms I_rms cos φ |
Resonant frequency | ω₀ = 1/√(LC) |
Transformer voltage | V_s/V_p = N_s/N_p |
Transformer current | I_p/I_s = N_s/N_p |
RMS values | V_rms = V_max/√2, I_rms = I_max/√2 |
The entire electrical power grid relies on transformers. Power stations generate electricity at moderate voltages, step it up to hundreds of thousands of volts for efficient long-distance transmission (lower current means less I²R loss in the wires), then step it back down for household use. Resonance is used in radio tuning: adjusting a variable capacitor changes the resonant frequency of the receiver circuit to match the desired station's broadcast frequency.
"Impedance is just resistance." Resistance is the real part; impedance also includes the reactive part from L and C. A circuit can have high impedance even with low resistance if the reactances are large and unbalanced.
"At resonance, the voltages across L and C are zero." They are not. The voltages across L and C can individually be very large at resonance, but they are equal in magnitude and opposite in phase, so they cancel each other. The net reactive voltage is zero.
"A step-up transformer creates free energy." It does not. Voltage goes up, but current goes down proportionally. Power is conserved (in an ideal transformer).
"Average power depends on V_max and I_max directly." It depends on RMS values and the power factor. Using peak values without dividing by √2 will give you twice the correct answer.
⚠️ The four-step AC circuit method (reactances → impedance → current → phase) is the backbone of nearly every RLC problem. Memorise and practise it until it is automatic.
⚠️ Know the resonance condition (X_L = X_C) and its consequences (Z = R, φ = 0, max current, max power).
⚠️ Transformer problems almost always test whether you can correctly apply V_s/V_p = N_s/N_p and the inverse current relationship. Watch out for which is primary and which is secondary.
⚠️ Power factor questions often ask what happens to average power at resonance vs. off-resonance.
Fill in the blank: At resonance, X_L = ___.
True or false: Impedance is always greater than or equal to resistance.
Fill in the blank: The average power in an AC circuit is ⟨P⟩ = ε_rms × I_rms × ___.
True or false: A step-up transformer increases both voltage and current.
Fill in the blank: The phase angle is zero when the circuit is at ___.
Answers: 1. X_C. 2. True (Z = √(R² + (X_L − X_C)²) ≥ R). 3. cos φ. 4. False (voltage up, current down). 5. Resonance.
Q: An RLC circuit has R = 100 Ω, X_L = 200 Ω, and X_C = 50 Ω. What is the impedance?
A: Z = √(100² + (200 − 50)²) = √(10,000 + 22,500) = √32,500 ≈ 180.3 Ω.
Q: For the same circuit, is the voltage leading or lagging the current?
A: X_L > X_C, so the circuit is inductive and voltage leads current (φ > 0).
Q: A transformer has 500 turns on the primary and 50 turns on the secondary. If the primary voltage is 240 V, what is the secondary voltage?
A: V_s = V_p × (N_s/N_p) = 240 × (50/500) = 24 V. This is a step-down transformer.
Q: At resonance, an AC source with ε_rms = 120 V drives a circuit with R = 60 Ω. What is the average power dissipated?
A: At resonance, cos φ = 1 and Z = R, so I_rms = 120/60 = 2 A. ⟨P⟩ = 120 × 2 × 1 = 240 W.
Q: Why can the individual voltages across L and C exceed the source voltage at resonance?
A: At resonance, V_L and V_C are each equal to IX (which can be large), but they are 180° out of phase and cancel. The source voltage only needs to drive current through R, so V_source = IR, which can be much smaller than either V_L or V_C individually.
The impedance concept extends the idea of resistance from DC circuits into the frequency domain and connects to the broader topic of signal filtering (low-pass, high-pass, band-pass filters). Resonance in RLC circuits is mathematically identical to mechanical resonance (mass-spring-damper systems), so if you have covered that in mechanics, the analogy is direct: L ↔ mass, C ↔ 1/spring constant, R ↔ damping. Transformers link to Faraday's law and mutual inductance, which you studied earlier in the course.
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