Question

At \( t = \infty \) in an RL circuit with DC excitation, the inductor behaves as:

• A. Open circuit
• B. Capacitor
• C. Short circuit
• D. Dependent source

Hint:

To remember the behavior of energy storage elements under DC excitation at steady state (\(t = \infty\)): - Inductors behave as a Short Circuit (\(v = 0\)). - Capacitors behave as an Open Circuit (\(i = 0\)).

Answer 1

The Correct Option is C

Concept: The fundamental voltage-current relationship for an ideal inductor is expressed by Faraday's law of induction in terms of its inductance value \(L\): \[ v_L(t) = L \frac{di(t)}{dt} \] When a circuit involving an inductor is excited by a constant Direct Current (DC) source for a very long period of time (\(t \to \infty\)), the circuit enters a steady-state condition. In this DC steady state, all currents and voltages settle down to fixed, time-invariant values. Because the current becomes completely constant, its rate of change with respect to time reduces identically to zero: \[ \frac{di(t)}{dt} = 0 \] Substituting this zero derivative back into the governing differential relationship gives: \[ v_L(\infty) = L \times 0 = 0\text{ V} \] An electrical component that carries a non-zero current while maintaining an absolute potential difference of zero volts across its terminals is defined as an ideal short circuit.

Step 1: Analyzing transient behavior vs steady-state behavior.
When the DC source is initially switched into the RL network at \(t = 0^+\), the current cannot change instantaneously due to the conservation of magnetic flux linkage. Thus, the inductor opposes the sudden change by acting as an open circuit. As time progresses exponentially according to the time constant, the transient terms decay away completely.

Step 2: Mathematical verification of the steady-state impedance.
Alternatively, we can analyze the circuit using Laplace domain concepts (s-domain). The operational impedance of an inductor is given by: \[ Z_L(s) = sL \] For a steady-state DC excitation, we evaluate the frequency response at the continuous-current frequency, which corresponds to the angular frequency \(\omega = 0\). Substituting \(s = j\omega = j(0) = 0\): \[ Z_L(0) = 0 \times L = 0\ \Omega \] An impedance of zero ohms signifies a perfect path of conduction with no opposition to current flow, which is structurally equivalent to a short circuit.