Questions: First-Order System Response: Time Constant and Behavior

Question 1

A first-order RC circuit has τ = 10 ms. An engineer wants to reduce the settling time (to within 2% of final value) by a factor of 4. Which change accomplishes this?

• A. Apply a larger step input voltage — more voltage drives the capacitor to charge faster
• B. Reduce the time constant to τ = 2.5 ms, for example by reducing resistance or capacitance
• C. Double the resistance while keeping capacitance the same
• D. Apply a sinusoidal input at the corner frequency to accelerate the transient

Answer: (B)

Settling time is approximately 4τ. To cut settling time by a factor of 4, you need to cut τ by a factor of 4 — from 10 ms to 2.5 ms. The time constant τ = RC, so reducing R or C (or both) achieves this. Applying a larger input voltage does not change the settling time — it changes the final value but the system still takes 4τ to reach 98% of that value. The shape of the exponential approach (and thus τ) is a property of the system, not the input magnitude.

Question 2

A control engineer examines the step response of a closed-loop system and observes clear overshoot — the output rises above the commanded setpoint before settling. What can she conclude about the system's order?

• A. Nothing definitive — first-order systems can overshoot if the input step is large enough or the gain is too high
• B. The system must be at least second-order, since a first-order system cannot overshoot
• C. The system is first-order with an unusually large time constant and a high-gain controller
• D. The system has a right-half-plane zero, which causes overshoot regardless of system order

Answer: (B)

A first-order system governed by y(t) = 1 − e^(−t/τ) approaches its final value monotonically — it can never exceed it. This is a mathematical fact: the exponential term is always positive, so the output always falls short of 1, approaching from below. Overshoot requires a system with at least two energy-storage elements (two poles) — enough complexity to produce oscillatory behavior. The moment you observe overshoot, you know the system has at least two poles. This is a powerful diagnostic: overshooting responses are never first-order.

Question 3

A first-order system with time constant τ = 5 s has a bandwidth (corner frequency) of 0.2 rad/s, meaning sinusoidal inputs above this frequency are attenuated at −20 dB/decade.

• T. True
• F. False

Answer: True

The transfer function of a first-order system is H(s) = 1/(τs + 1), which has its pole at s = −1/τ. The corner frequency (−3 dB bandwidth) is ω_c = 1/τ = 1/5 = 0.2 rad/s. Below this frequency, the system passes inputs with near-unity gain. Above it, the magnitude rolls off at −20 dB/decade. A system with τ = 5 s is a slow system: it faithfully tracks inputs that change on timescales longer than about 5 seconds but heavily attenuates faster variations.

Question 4

After 2 time constants (t = 2τ), a first-order step response has completed approximately 95% of its total change and can be considered essentially settled.

• T. True
• F. False

Answer: False

At t = 2τ, the response is 1 − e^(−2) ≈ 0.865, meaning about 86.5% complete — not 95%. The 95% threshold is reached at approximately t = 3τ (since 1 − e^(−3) ≈ 0.950). The conventional 2% settling criterion is reached at t ≈ 4τ (since 1 − e^(−4) ≈ 0.982). The common confusion is between the 63% mark at τ, the 95% mark at 3τ, and the 98% settling at 4τ. Misidentifying the settling time leads to significant control design errors — a system that looks settled at 2τ still has 13.5% of its transient remaining.

Question 5

Explain why the time constant τ appears in both the step response formula and the frequency-domain bandwidth, and what this tells you about the relationship between response speed and bandwidth in a first-order system.

Model answer: Both come from the same underlying transfer function H(s) = 1/(τs + 1). In the time domain, the inverse Laplace transform gives y(t) = 1 − e^(−t/τ), so τ sets the timescale of the exponential decay. In the frequency domain, the magnitude |H(jω)| falls to 1/√2 (−3 dB) at ω = 1/τ, defining the bandwidth. They are two representations of the same parameter. A small τ means fast exponential rise AND large bandwidth — the system responds quickly to steps AND passes high-frequency inputs. A large τ means slow rise AND narrow bandwidth. Speed and bandwidth are not independent choices; they are two faces of the same coin.

This duality is a fundamental property of linear time-invariant systems: the step response and the frequency response are Fourier-transform pairs. Fast systems (small τ) have high bandwidth because they can faithfully track rapidly changing inputs; slow systems (large τ) behave as low-pass filters, smoothing out rapid variations. In design, increasing bandwidth (to get faster response) always comes at a cost — typically sensitivity to high-frequency noise. This tradeoff between speed and noise rejection runs throughout control systems design.