Questions: RLC Circuit Applications of Differential Equations

Question 1

In a series RLC circuit, resistance R is very large relative to 4L/C. What behavior does the current exhibit after the circuit is energized?

• A. The current oscillates with exponentially decaying amplitude (ringing)
• B. The current decays exponentially to zero without oscillating
• C. The current oscillates indefinitely at the natural frequency
• D. The current immediately reaches a steady-state constant value

Answer: (B)

When R² > 4L/C, the characteristic equation has two distinct real roots, corresponding to the overdamped case in the spring-mass analogy. The current decays monotonically — like a mass in thick oil that returns slowly to rest without bouncing. The oscillating (ringing) behavior occurs in the underdamped case (R² < 4L/C), where complex roots produce decaying oscillations. The duality to the mechanical system makes this intuitive once the analogy is internalized.

Question 2

In the mechanical-electrical duality, which electrical component corresponds to mass in the spring-mass equation?

• A. Resistance R, because both R and mass resist motion
• B. Capacitance C, because both store energy
• C. Inductance L, because both resist changes in their respective flow (current / velocity)
• D. The voltage source, because it drives both systems

Answer: (C)

The duality maps: L (inductance) ↔ m (mass) — both resist changes in flow; R (resistance) ↔ c (damping coefficient) — both dissipate energy; 1/C ↔ k (spring constant) — both provide a restoring force proportional to accumulated displacement or charge. The equation L·i'' + R·i' + (1/C)·i = 0 is structurally identical to m·x'' + c·x' + k·x = 0. Option A is tempting because damping 'resists motion,' but mass specifically resists *changes* in velocity (inertia), and inductance specifically resists changes in current — that's the correct match.

Question 3

Resonance in an RLC circuit occurs at ω₀ = 1/√(LC) because at that frequency, the inductive and capacitive impedances cancel, leaving only resistance to limit current.

• T. True
• F. False

Answer: True

This is correct and is the direct electrical analog of mechanical resonance. At the natural frequency, the impedance contribution from the inductor (+jωL) and capacitor (1/jωC) have equal magnitude and opposite sign, canceling each other. Only R remains, so current amplitude is maximized. This is why radio tuning works: adjusting C changes ω₀ until it matches the broadcast frequency, maximizing the current response to that station's signal.

Question 4

Increasing resistance R in a series RLC circuit will increase the maximum current amplitude at resonance.

• T. True
• F. False

Answer: False

At resonance, the only impedance limiting current is the resistance R — the inductive and capacitive terms cancel. So current at resonance is V₀/R: it is inversely proportional to R. Larger R means *lower* maximum current at resonance, not higher. This is the opposite of the tempting intuition that 'more R somehow helps.' Larger R increases damping, suppresses oscillatory behavior, and reduces the sharpness (Q-factor) of the resonance peak.

Question 5

Explain why the condition R² < 4L/C produces 'ringing' (oscillating current) in an RLC circuit, using the analogy to the mechanical spring-mass system.

Model answer: When R² < 4L/C, the characteristic equation Lλ² + Rλ + 1/C = 0 has complex conjugate roots, producing a solution of the form e^(–αt)cos(ωt). This is decaying oscillation: the current swings back and forth with decreasing amplitude. In the mechanical analogy, this corresponds to an underdamped spring-mass system where damping is too weak to prevent the mass from overshooting equilibrium — like a spring with light friction that bounces several times before settling. The inductance (like mass) stores energy and drives overshoot; resistance (like damping) gradually dissipates it.

The key is connecting the sign of the discriminant to the nature of the characteristic roots: real roots → exponential decay (overdamped); complex roots → oscillation with decay (underdamped). The mechanical analogy makes this physically intuitive — once you know what underdamped means for a spring, you immediately know what it means for a circuit.