Integrator time-scale rule: In a practical integrator, when the input pulse width is much shorter than the circuit time constant, the output does not reproduce the input shape; it tends toward a small triangular/linear ramp proportional to the area. True or false?

Introduction / Context:
First-order integrators (RC or RL low-pass forms) approximate mathematical integration only under specific time-scale relationships. Getting this relationship backward is a common source of lab and design errors.

Given Data / Assumptions:

• RC integrator with τ = R*C (the same reasoning applies to RL low-pass).
• Input is a pulse of width Tw.
• “Transient time” refers to the circuit time constant τ.

Concept / Approach:

For good integration, τ should be much larger than Tw (τ ≫ Tw). In this regime, the output cannot follow the rapid input changes; instead, it responds slowly, producing a small-amplitude ramp whose height is proportional to the input area, not the input shape. Therefore, the statement that the output “approaches the shape of the input” is false under τ ≫ Tw.

Step-by-Step Solution:

Verification / Alternative check:

Oscilloscope observation with adjustable τ: as τ increases relative to Tw, the output top rounds into a shallow ramp; as τ decreases (τ ≪ Tw), the circuit behaves more like a follower, not an integrator, and the output resembles the input—opposite of the integrator condition.

Why Other Options Are Wrong:

• “True” confuses integrator and follower regimes; reproducing the input shape requires bandwidth comparable to or greater than the input spectrum, not the limited bandwidth of an integrator.

Common Pitfalls:

Mixing “differentiator” and “integrator” criteria; forgetting that component tolerances and source/load impedances affect effective τ and integration quality.

Final Answer:

False