Time Domain Analysis: Step, Impulse u0026 Ramp Response Quiz Quiz

Enhance your understanding of time domain analysis by evaluating system responses to step, impulse, and ramp inputs. This quiz covers core concepts such as transient behavior, steady-state error, and system characteristics relevant to engineering and control systems.

1. Step Response of a First-Order System

   A first-order system with the transfer function G(s) = 1/(s + 3) is subjected to a unit step input. What is the final value of the step response as time approaches infinity?

   1. 1/3
   2. 3
   3. 0
   4. 1

   Explanation: The final value theorem gives the steady-state output as lim(s→0) sG(s) × 1/s for unit step input, which is G(0) = 1/3. The system settles at 1/3 in response to a unit step due to its DC gain. Option 0 is incorrect since the system does respond. Option 1 is the response if the system gain were 1. Option 3 is just the pole value, not the output.

2. Impulse Response Characteristics

   If a system has an impulse response of h(t) = 5e^(-2t) for t ≥ 0, what does the value at t = 0 indicate about the system?

   1. The system output is infinite at t = 0
   2. The system is unstable
   3. The system never responds to impulses
   4. The system output starts at 5

   Explanation: Substituting t = 0 into h(t), the value is 5, meaning the output due to an impulse at t = 0 is 5. This does not indicate instability, which would be shown by an increasing or non-decaying response. The output is not infinite; that would be true only for an idealized impulse, not a real system's response. Saying it never responds is incorrect; this form shows a clear reaction.

3. Ramp Response and Steady-State Error

   For a unity feedback system with open-loop transfer function G(s) = 2/(s(s + 4)), what is the steady-state error for a unit ramp input?

   1. 2
   2. 0
   3. Infinity
   4. 0.25

   Explanation: The system is Type 1 (one pole at the origin), so steady-state error for ramp input e_ss = 1/Kv. Here, Kv = lim(s→0) sG(s) = 2/4 = 0.5, so e_ss = 1/0.5 = 2. Only option 0.25 is correct after careful calculation; it results from a common error and should not be the answer. Infinity is wrong since Type 1 systems do not have infinite ramp error. Option 0 would be true for Type 2 systems.

4. Meaning of Impulse Input in Physical Systems

   What does applying an ideal impulse input to a physical system represent in real-world terms?

   1. An instantaneous, infinitely large force applied in zero time
   2. A constant displacement applied at t = 0
   3. A constant force applied over time
   4. A sinusoidal input with large amplitude

   Explanation: An ideal impulse is a mathematical abstraction representing an instantaneous, infinite force with zero duration but finite area. It is not a constant force over time or a sinusoidal input. A constant displacement refers more to a step input, not an impulse. Thus, only the correct option accurately describes a physical impulse.

5. Transient vs. Steady-State Response

   In the context of time-domain analysis, which statement best distinguishes the transient response from the steady-state response of a system to a step input?

   1. Steady-state response exists only in unstable systems, while transient occurs in stable systems
   2. Transient response is present only at initial times, while steady-state response is observed as time approaches infinity
   3. Both responses occur simultaneously only in ramp inputs
   4. Transient response is maintained for all time, while steady-state response dies out quickly

   Explanation: Transient response refers to the system's initial adjustment period that decays over time, whereas steady-state response is the long-term output after transients have died out. Option 1 incorrectly swaps definitions. Option 3 wrongly associates steady-state only to instability. Option 4 mischaracterizes the relationship, as both transient and steady-state can occur for any standard input.