Source: PHYS 212 Final Examination, University of Illinois at Urbana-Champaign
Tags: AC circuit, RLC circuit, resonance, resonant frequency, impedance, reactance, inductive reactance, capacitive reactance, phase angle, transformer, turns ratio, power transmission, series RLC
Difficulty: Intermediate to Advanced | Prerequisites: RC circuit basics, inductor behaviour, phasor concepts, basic trigonometry.
AC circuits extend everything you know about DC circuits into the world of oscillating voltages and currents. The crucial new ideas are reactance (the AC equivalent of resistance for capacitors and inductors), impedance (the combined opposition to current from R, L, and C together), and resonance (the special frequency where the circuit responds most strongly). Transformers are the practical payoff: they let us step voltages up or down using electromagnetic induction, which is why electrical power can be transmitted efficiently over hundreds of kilometres. If you are not yet solid on how capacitors and inductors behave with DC sources, review that first (see the RC Circuits notes), because AC behaviour generalises those ideas.
In a series RLC circuit driven by an AC source, current is maximised at the resonant frequency ω₀ = 1/√(LC), where inductive and capacitive reactances cancel and the impedance equals R alone. At resonance the voltage and current are in phase (φ = 0). Away from resonance, either the inductor or capacitor dominates the impedance. Transformers use the turns ratio to convert voltages: V_s = V_p (N_s / N_p).
Impedance (Z)
The total opposition to AC current in a circuit, measured in ohms: Z = √[R² + (X_L - X_C)²]. It generalises resistance to include the frequency-dependent effects of inductors and capacitors. In simple terms, impedance is "AC resistance."
Inductive reactance (X_L)
The opposition to AC current from an inductor: X_L = ωL. It increases with frequency, meaning inductors resist rapid changes in current more than slow ones. Think of it as: the faster the current tries to change, the harder the inductor pushes back.
Capacitive reactance (X_C)
The opposition to AC current from a capacitor: X_C = 1 / (ωC). It decreases with frequency, meaning capacitors pass high-frequency signals more easily. Think of it as: a rapidly alternating signal does not have time to fully charge the capacitor, so current flows easily.
Resonant frequency (ω₀)
The driving frequency at which X_L = X_C, so they cancel and the impedance is at its minimum (Z = R). For a series RLC circuit: ω₀ = 1 / √(LC). At this frequency the current is maximised and the circuit responds most strongly.
Phase angle (φ)
The angle by which the current leads or lags the driving voltage: tan φ = (X_L - X_C) / R. At resonance, X_L = X_C, so φ = 0 and voltage and current are in phase.
Transformer
A device consisting of two coils (primary and secondary) wound around a shared iron core. A changing current in the primary coil induces a voltage in the secondary coil via Faraday's law. The voltage ratio equals the turns ratio: V_s / V_p = N_s / N_p.
Turns ratio
The ratio N_s / N_p of secondary to primary windings. If N_s > N_p, the transformer steps voltage up (and current down). If N_s < N_p, it steps voltage down (and current up). Power is conserved in an ideal transformer: V_p I_p = V_s I_s.
At the resonant frequency ω₀:
X_L = X_C (inductive reactance equals capacitive reactance)
The two reactances cancel in the impedance formula: Z = √[R² + (X_L - X_C)²] = √[R² + 0] = R
Current amplitude is maximised: I_max = V_max / R
Phase angle φ = 0: voltage and current are perfectly in phase
The resonant frequency is found by setting X_L = X_C: ωL = 1/(ωC), so ω² = 1/(LC), giving ω₀ = 1 / √(LC).
Note: the answer X_L = -X_C (option A in the exam) is a distractor. Reactances are defined as positive quantities. The condition is X_L = X_C, not X_L = -X_C.
Since ω₀ = 1 / √(LC), the resonant frequency increases when LC decreases. That means:
Decreasing L increases ω₀ (correct)
Decreasing C increases ω₀ (correct)
Increasing L or C decreases ω₀ (incorrect for the question "which increases ω₀")
When ω >> ω₀ (driving frequency much higher than resonance):
X_L = ωL becomes very large
X_C = 1/(ωC) becomes very small
The inductor dominates the impedance
The circuit is "inductive": current lags voltage
When ω << ω₀ (driving frequency much lower than resonance):
X_C = 1/(ωC) becomes very large
X_L = ωL becomes very small
The capacitor dominates the impedance
The circuit is "capacitive": current leads voltage
φ > 0: X_L > X_C, current lags voltage (inductive behaviour)
φ = 0: X_L = X_C, resonance, current and voltage in phase
φ < 0: X_C > X_L, current leads voltage (capacitive behaviour)
For an ideal transformer (no energy losses):
Voltage relation: V_s = V_p · (N_s / N_p)
Current relation: I_s = I_p · (N_p / N_s)
Power conservation: V_p I_p = V_s I_s
The voltage ratio is linear in the turns ratio (N_s / N_p), not quadratic. The distractor V_s = V_p(N_s² / N_p²) is incorrect.
This is a short-answer exam question. The reasoning:
Power transmitted: P = IV
Power lost as heat in the transmission line: P_loss = I²R_line
For a fixed power P being transmitted, increasing the voltage decreases the current (since P = IV, I = P/V)
A lower current means less I²R loss in the wires
Transformers step the voltage up before transmission (reducing current and losses) and step it back down near the consumer
The key insight is that losses scale as I² but power scales as IV, so doubling the voltage halves the current and cuts transmission losses by a factor of four.
Inductive reactance: X_L = ωL
Capacitive reactance: X_C = 1 / (ωC)
Impedance: Z = √[R² + (X_L - X_C)²]
Resonant frequency: ω₀ = 1 / √(LC)
Phase angle: tan φ = (X_L - X_C) / R
Current amplitude: I_max = V_max / Z
Transformer voltage: V_s = V_p (N_s / N_p)
Transformer power: V_p I_p = V_s I_s
Transmission loss: P_loss = I²R_line
Every power grid in the world relies on transformers. Power stations generate electricity at moderate voltages (~20 kV), step it up to very high voltages (110-765 kV) for long-distance transmission to reduce I²R losses, then step it back down in stages to the 120 V or 240 V used in homes. RLC resonance is the principle behind radio tuning: by adjusting the capacitor (or inductor) in a receiver circuit, you change ω₀ to match the frequency of the station you want, maximising the signal from that station while rejecting others.
Students confuse the resonance condition X_L = X_C with X_L = -X_C. Reactances are positive quantities. The impedance formula subtracts them (X_L - X_C), and at resonance this difference is zero.
Thinking that at resonance the impedance is zero. It is not; it equals R. The reactive part vanishes, but the resistive part remains.
Believing the transformer voltage scales as the square of the turns ratio. The voltage relationship is linear: V_s / V_p = N_s / N_p, not (N_s / N_p)².
Forgetting which component dominates at high frequency. A useful mnemonic: "high frequency, high X_L" (because X_L = ωL grows with ω), so the inductor dominates.
⚠️ Question 4A tests the resonance condition. The answer is X_L = X_C (option D), not X_L = -X_C.
⚠️ Question 4B asks for the phase angle at resonance. The answer is φ = 0°.
⚠️ Question 6 asks which component dominates when ω >> ω₀. The answer is the inductor.
⚠️ Question 8 tests the transformer equation. The answer is V_s = V_p(N_s / N_p).
⚠️ Question 12 (multi-select) asks what increases ω₀. Both decreasing L and decreasing C are correct.
⚠️ Question 18 (short answer) asks why transformers step up voltage for transmission. Frame your answer around P_loss = I²R and the fact that higher voltage means lower current for the same power delivered.
Fill in the blank: At resonance in a series RLC circuit, X_L ______ X_C.
True or False: The phase angle at resonance is 90°.
Fill in the blank: The resonant frequency of a series RLC circuit is ω₀ = ______.
True or False: When ω >> ω₀, the capacitor dominates the impedance.
Fill in the blank: In an ideal transformer, V_s = V_p · ______.
Answers: 1. equals. 2. False (φ = 0°). 3. 1/√(LC). 4. False (the inductor dominates). 5. N_s / N_p.
Q: At the resonant frequency ω₀ of a series RLC circuit, what is the relationship between X_L and X_C?
A: X_L = X_C. The inductive and capacitive reactances are equal, so they cancel in the impedance expression, leaving Z = R.
Q: What is the phase angle φ between voltage and current at resonance?
A: φ = 0°. The voltage and current are in phase because the net reactance (X_L - X_C) is zero.
Q: If the driving frequency is much greater than the resonant frequency, which component dominates the impedance?
A: The inductor. At high frequencies, X_L = ωL becomes very large while X_C = 1/(ωC) becomes very small, so the inductor's reactance dominates.
Q: In an ideal transformer, the primary coil has N_p turns and the secondary has N_s turns. If the primary voltage is V_p, what is the secondary voltage?
A: V_s = V_p · (N_s / N_p). The voltage ratio equals the turns ratio.
Q: Explain why transformers are used to step up voltage for long-distance power transmission.
A: For a given amount of power P = IV, increasing the voltage reduces the current. Power lost as heat in the transmission lines is P_loss = I²R. By stepping up the voltage (and thus reducing the current), the I²R losses are greatly reduced, making long-distance transmission efficient. At the destination, transformers step the voltage back down for safe consumer use.
Q: Which changes would increase the resonant frequency ω₀ of a series RLC circuit?
A: Decreasing the inductance L or decreasing the capacitance C. Since ω₀ = 1/√(LC), making either L or C smaller increases ω₀.
RLC circuits are the electrical analogue of a mass-spring system with damping: the inductor plays the role of mass (stores kinetic energy as magnetic field energy), the capacitor plays the role of the spring (stores potential energy as electric field energy), and the resistor provides damping. Resonance in RLC circuits connects to resonance phenomena across physics, from mechanical vibrations to optical cavities. Transformers rely on Faraday's law of induction, linking this topic back to the fundamentals of how changing magnetic flux induces an EMF.
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