RL transient — time constants: Is it accurate to say that current in an inductor “reaches its maximum value” exactly after five time constants in a first-order RL step response?

Introduction / Context:First-order RL circuits respond to a DC step with an exponential current rise toward a final value. Engineers often use “time constants” to estimate when the response is essentially complete. However, exponential functions are asymptotic; they approach the final value but reach it only at infinite time. Thus, “exactly at five time constants” is not correct, though 5τ is a common practical benchmark (~99.3%).

Given Data / Assumptions:

• Series RL circuit driven by a DC step.
• Time constant τ = L / R for an RL network.
• Ideal inductor and resistor.

Concept / Approach:The current response is i(t) = I_final * (1 − e^(−t/τ)). At t = τ, the response is ~63.2%; at 2τ, ~86.5%; 3τ, ~95.0%; 4τ, ~98.2%; 5τ, ~99.3%. The current never truly equals I_final at any finite t; it only gets arbitrarily close as t increases. Therefore, the claim that it “reaches its maximum” at 5τ is false in the strict sense.

Step-by-Step Solution:

Verification / Alternative check:Oscilloscope measurements show the current trace flattening after several τ but never mathematically touching the final value. Designers often treat 5τ as “essentially final” to within about 0.7% error.

Why Other Options Are Wrong:“Correct” contradicts exponential behavior. Restrictions about core type, R = 0, or waveform do not change the asymptotic nature for the DC step case.

Common Pitfalls:Equating “practically complete” with “exactly complete.” Always state tolerances when using time-constant heuristics.

Final Answer:Incorrect