Question

In a series RL circuit excited by DC, the current reaches 63.2% of its final value at:

• A. \( t = RC \)
• B. \( t = \frac{1}{RC} \)
• C. \( t = \frac{R}{L} \)
• D. \( t = \frac{L}{R} \)

Hint:

The time constant \(\tau\) is universally defined as the time required for a transient signal to change by approximately \(63.2%\) of its total dynamic swing. - For an RL circuit: \(\tau = \frac{L}{R}\) - For an RC circuit: \(\tau = RC\)

Answer 1

The Correct Option is D

Concept: When a series combination of a resistor \(R\) and an inductor \(L\) is connected across a constant DC voltage source \(V_s\) at \(t = 0\), the growth of current \(i(t)\) through the circuit is governed by a first-order linear differential equation derived from Kirchhoff's Voltage Law (KVL): \[ V_s = R \cdot i(t) + L \frac{di(t)}{dt} \] Solving this differential equation with the initial condition that the inductor is unenergized at \(t = 0^-\) (\(i(0^-) = 0\)) yields the standard exponential rising transient response: \[ i(t) = \frac{V_s}{R} \left( 1 - e^{-\frac{t}{\tau}} \right) \] where:
• \(\frac{V_s}{R} = I_{\text{final}}\) represents the maximum steady-state current reached as \(t \to \infty\).
• \(\tau\) represents the characteristic time constant of the system. For a series RL circuit, this time constant is defined as \(\tau = \frac{L}{R}\).

Step 1: Setting up the target threshold calculation.
The problem statement asks for the specific instance of time \(t\) at which the instantaneous current reaches exactly \(63.2%\) of its final maximum steady-state value. Expressing this percentage as a mathematical fraction gives: \[ i(t) = 0.632 \cdot I_{\text{final}} \] Substituting our theoretical formula for \(i(t)\) and \(I_{\text{final}}\) into this condition, we obtain: \[ I_{\text{final}} \left( 1 - e^{-\frac{t}{\tau}} \right) = 0.632 \cdot I_{\text{final}} \]

Step 2: Solving for time \(t\).
Since the steady-state current \(I_{\text{final}} \neq 0\), we can divide both sides of the equation by \(I_{\text{final}}\): \[ 1 - e^{-\frac{t}{\tau}} = 0.632 \] Rearranging the algebraic terms to isolate the exponential component on one side: \[ e^{-\frac{t}{\tau}} = 1 - 0.632 \] \[ e^{-\frac{t}{\tau}} = 0.368 \] We recognize that the value \(0.368\) is an approximation for the mathematical constant \(\frac{1}{e}\): \[ \frac{1}{e} = e^{-1} \approx 0.367879 \] Equating the exponents directly from both sides of the relation: \[ -\frac{t}{\tau} = -1 \quad \implies \quad t = \tau \] Since the time constant for a series RL configuration is explicitly \(\tau = \frac{L}{R}\), the current achieves this specific threshold precisely at time \(t = \frac{L}{R}\).