Questions: Series and Parallel RC, RL, and RLC Networks

Question 1

A series RLC circuit has a damping ratio ζ = 0.3. After a step voltage is applied, what shape will the voltage response have?

• A. A monotonically increasing curve that slowly approaches steady state without oscillating
• B. A step function that jumps immediately to the final steady-state value
• C. An oscillating waveform that decays in amplitude over time, overshooting the final value
• D. A monotonically increasing curve that reaches steady state as quickly as possible

Answer: (C)

ζ = 0.3 < 1 means the circuit is underdamped. Underdamped systems oscillate at the damped natural frequency ωd = ω₀√(1 − ζ²) while their envelope decays exponentially. The response overshoots the final value, rings back, and gradually settles. Option A (monotonic slow settling) is overdamped (ζ > 1). Option D (fastest non-oscillatory settling) is critically damped (ζ = 1).

Question 2

Critical damping (ζ = 1) is often preferred in control and measurement applications over slight overdamping (ζ = 1.2). Why?

• A. Critically damped circuits consume less power than overdamped ones
• B. Critical damping reaches steady state as fast as possible without overshoot, while overdamped circuits are slower despite also avoiding oscillation
• C. Overdamped circuits oscillate slightly, while critically damped ones do not
• D. Critical damping requires fewer components than overdamped designs

Answer: (B)

Both ζ = 1 and ζ > 1 produce non-oscillatory (monotonic) responses, but they settle at different speeds. An overdamped circuit has two distinct exponential time constants, both slower than the critical case. Critically damped has the fastest possible settling time with no overshoot — it is the 'sweet spot' between the ringing of underdamped and the sluggishness of overdamped. This makes it desirable in applications like galvanometers, electronic amplifier feedback, and anywhere that speed and stability must be simultaneously optimized.

Question 3

In a series RL circuit, increasing the resistance R causes the transient current to reach its final value more quickly.

• T. True
• F. False

Answer: True

The time constant for a series RL circuit is τ = L/R. Increasing R decreases τ, which means the exponential buildup I(t) = (ε/R)(1 − e^{−t/τ}) reaches 63% of its final value sooner. More resistance means more energy is dissipated per unit current, which drains the transient faster. Note this is the opposite of RC circuits, where τ = RC and larger R slows the response — a common source of confusion between the two circuit types.

Question 4

An underdamped RLC circuit's oscillating response means it rarely fully reaches a steady-state value.

• T. True
• F. False

Answer: False

The oscillations in an underdamped response decay exponentially — their amplitude shrinks by a factor of e^{−ζω₀t} with each cycle. Although they theoretically never reach exactly zero in finite time, they become negligibly small within a few time constants (practically within 5τ). The circuit does reach steady state; it just takes a slightly winding path there. 'Oscillating' does not mean 'never settling' — the decay envelope ensures convergence.

Question 5

Explain what the damping ratio ζ tells you about an RLC circuit's transient response, and why critical damping (ζ = 1) is often considered the optimal design target in engineering applications.

Model answer: The damping ratio ζ determines whether the RLC circuit oscillates in its transient response. When ζ < 1 (underdamped), the circuit overshoots and oscillates before settling. When ζ = 1 (critically damped), it settles to steady state as quickly as possible without any oscillation or overshoot. When ζ > 1 (overdamped), it settles monotonically but more slowly than the critically damped case. Critical damping is often the engineering target because it simultaneously minimizes settling time (compared to overdamped) and eliminates overshoot and ringing (compared to underdamped). Applications that cannot tolerate overshoot — mechanical positioning systems, measuring instruments, amplifier feedback — commonly aim for ζ ≈ 1.

The damping ratio ζ = R/(2)·√(C/L) shows that resistance controls damping. Too little R → underdamped (oscillation), too much R → overdamped (sluggish), just right R → critical damping. This is why ζ is a central design parameter in filter, amplifier, and control system engineering.