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When a unit step input voltage is applied to a simple RC lag network, what is the time-domain behavior of the output response?

Difficulty: Easy

Correct Answer: increases exponentially from zero to the final value

Explanation:


Introduction / Context:
Lag networks (first-order RC circuits) are widely used in control and communication systems for shaping frequency response. Understanding their time-domain step response helps correlate time and frequency domain behavior.


Given Data / Assumptions:

  • Unit step input u(t) applied at t = 0.
  • Lag network: RC with transfer function H(s) = 1 / (1 + sRC).
  • Output across capacitor.


Concept / Approach:
The step response of a first-order low-pass network is exponential. It starts at 0 V and asymptotically approaches the final steady-state value equal to the input amplitude (here, 1 V).


Step-by-Step Solution:

Transfer function H(s) = 1 / (1 + sRC).Laplace of input: 1/s.Output: Y(s) = (1/s)(1 / (1 + sRC)).Inverse Laplace: y(t) = 1 − e^(−t/RC).Hence, y(t) rises exponentially from 0 to 1 as t → ∞.


Verification / Alternative check:

Compare with capacitor charging curve; matches physical intuition.


Why Other Options Are Wrong:

Constant from start: wrong, no system can instantaneously jump to final.Decreasing from 1 to 0: describes discharge, not step input.Oscillation: absent in first-order lag networks.


Common Pitfalls:

Confusing lag network with lead or resonant circuits; forgetting exponential charging law.


Final Answer:

increases exponentially from zero to the final value
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