| Q.12 The transfer function of a process is G(s) = Kp / (τps + 1), where Kp is the gain and τp is the time constant. This is a ________ process. |
|
| (A) | first order |
| (B) | multi-capacity |
| (C) | purely capacitive |
| (D) | second order |
to mathematically model system behavior. A common MCQ tests whether you can identify
the system order from its transfer function.
Problem Statement
The transfer function of a process is:
G(s) = Kp / (τps + 1)
where Kp is the process gain and
τp is the time constant.
This is a _____ process.
Options include first order, multi-capacity,
purely capacitive, and second order.
Correct Answer
(A) First-Order Process
Why This Is a First-Order Process
A first-order system contains exactly one energy storage element
(such as a tank, capacitor, or thermal mass). This results in a denominator
that is linear in s.
The standard first-order transfer function is:
G(s) = K / (τs + 1)
In the given equation:
- Kp determines steady-state gain
- τp determines speed of response
A smaller time constant implies a faster response.
Time-Domain Behavior
The step response of a first-order system is:
y(t) = Kp(1 − e−t/τp)
This response is smooth, exponential, and non-oscillatory.
Example: Liquid level control in a tank, where inflow changes
cause gradual level rise due to a single capacitance.
Why Not the Other Options?
(B) Multi-Capacity Process
Multi-capacity systems contain multiple energy storage elements and therefore
have multiple time constants.
Typical form:
G(s) = K / [(τ1s + 1)(τ2s + 1)]
These exhibit delayed, multi-stage responses—not a single exponential.
(C) Purely Capacitive Process
A purely capacitive system behaves as an integrator:
G(s) = K / s
Output ramps indefinitely for a step input. There is no stabilizing “+1” term
in the denominator.
(D) Second-Order Process
Second-order systems contain two energy storage elements and a quadratic
denominator:
G(s) = ωn2 /
(s2 + 2ζωns + ωn2)
These systems may show overshoot and oscillations, unlike first-order systems.
Option Comparison Table
| Option | Denominator Form | Key Trait | Example |
|---|---|---|---|
| (A) First order | τs + 1 | Single exponential response | Tank level control |
| (B) Multi-capacity | (τis + 1) | Multiple delays | Heat exchanger |
| (C) Purely capacitive | s | Unbounded ramp | Accumulator |
| (D) Second order | s² + … | Oscillatory potential | RLC circuit |
Real-World Applications & Exam Tips
- Many chemical and thermal systems are approximated as first order
- Bode plot shows −20 dB/decade slope after corner frequency 1/τ
- First-order systems often require PI control for zero steady-state error
- Sketching step responses helps identify system order quickly in exams
Final Answer
The correct answer is: First-Order Process.
