Question

The response of a first-order instrument to a step input is:

• A. Instantaneous
• B. Linear with time
• C. Exponential with time
• D. Sinusoidal

Hint:

Logic Tip: Imagine pulling a cold drink out of the fridge. It warms up really fast at first, but takes hours to finally reach exact room temperature. That classic fast-then-slow curve is an exponential response.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A first-order system (like a mercury-in-glass thermometer) has a "lag" in its response due to its internal capacity to store energy or mass.
Step 2: Key Formula or Approach:
The governing differential equation for a first-order system is: $\tau \frac{dy}{dt} + y = K x(t)$. For a step input of magnitude $M$, the response $y(t)$ is: \[ y(t) = KM(1 - e^{-t/\tau}) \]
Step 3: Detailed Explanation:
When you subject a first-order instrument to a sudden change (a step), the output does not change instantly. Instead, it follows an exponential curve, rising quickly at first and then slowing down as it approaches the new steady-state value. The speed of this response is dictated by the time constant ($\tau$).
Step 4: Final Answer:
The response is exponential with time.