A control engineer reduces a thermostat system's time constant τ from 10 seconds to 1 second to achieve faster response. A colleague warns this change will introduce a new problem. What is it?
AThe system will now overshoot the target temperature and oscillate
BThe system's bandwidth increases tenfold, causing it to pass significantly more high-frequency measurement noise
CThe DC gain K will decrease proportionally, making the steady-state temperature less accurate
DA time constant below 5 seconds causes the step response to become non-exponential
Bandwidth equals 1/τ, so reducing τ from 10s to 1s increases bandwidth from 0.1 rad/s to 1 rad/s. A wider bandwidth means the system responds to — and amplifies — higher-frequency input variations, including sensor noise. First-order systems cannot overshoot (a common misconception); overshoot requires two poles, not one. The DC gain K is independent of τ and is unaffected by this change. The fundamental speed-versus-noise tradeoff is inherent to any dynamic system.
Question 2 Multiple Choice
A first-order system has transfer function G(s) = 5/(2s + 1). A unit step input is applied. What is the output at t = 2 seconds?
AApproximately 3.16, since t = 2s equals the time constant τ = 2 and the output reaches 63.2% of its final value K = 5
BApproximately 4.90, since the 2% settling criterion is met at t = 2s
CExactly 2.50, since t/τ = 1 gives exactly half the final value
DExactly 5.00, since the system has had sufficient time to fully settle
The time constant τ = 2 (from the denominator 2s + 1). At t = τ, the step response y(t) = K(1 − e^{−t/τ}) = 5(1 − e^{−1}) ≈ 5 × 0.632 = 3.16. The 2% settling time is 4τ = 8 seconds — not 2 seconds. The step response never exactly reaches the final value of 5 in finite time; it approaches it asymptotically. The output at t = τ is always 63.2% of the final value, regardless of K or τ.
Question 3 True / False
A first-order system's step response approaches its final value asymptotically and technically never reaches it in finite time, which is why engineers use the 2% settling time as a practical criterion for when the transient is complete.
TTrue
FFalse
Answer: True
The exponential y(t) = K(1 − e^{−t/τ}) approaches K as t → ∞ but only equals K at t = ∞. At t = 4τ, the output is at 98.2% of the final value — within 2% — which engineers accept as 'settled.' This is a practical convention, not a mathematical endpoint. The exponential decay never reaches zero.
Question 4 True / False
The time constant τ of a first-order system is the time at which the step response settles to within 2% of its final value.
TTrue
FFalse
Answer: False
This is a common and consequential misconception. The time constant τ is the time at which the step response reaches 63.2% (= 1 − e^{−1}) of its final value — not 98%. The 2% settling time is approximately 4τ, not τ. Confusing these can lead to significant design errors: a system with τ = 1s does not settle in 1 second; it settles in approximately 4 seconds.
Question 5 Short Answer
Explain why reducing a first-order system's time constant makes it both faster and more noise-sensitive, and why this represents a fundamental tradeoff rather than an engineering oversight.
Think about your answer, then reveal below.
Model answer: The time constant τ is directly linked to bandwidth by ω₋₃dB = 1/τ. A smaller τ means wider bandwidth — the system faithfully tracks faster input changes, which is what makes it 'faster.' But bandwidth does not distinguish between desired signals and noise: a system with wider bandwidth passes high-frequency noise just as readily as high-frequency signals. Since real sensor noise has significant high-frequency content, a faster system amplifies more noise. This tradeoff is fundamental because it is impossible to track fast signals without also responding to fast noise at similar frequencies — the two are physically indistinguishable to the filter. The engineering task is to choose τ to meet speed requirements while keeping noise amplification within acceptable limits.
This speed-versus-noise tradeoff reappears in every control and signal processing design. In higher-order systems it manifests in the gain-bandwidth product; in digital filters it appears as the transition band width. Understanding it at the level of first-order systems provides the conceptual foundation for all subsequent filter and controller design.