Determine the current flowing in an RL circuit if the...

Determine the current flowing in an RL circuit if the applied EMF is $E(t)=E_{0} \sin \omega t,$ where $E_{0}$ and $\omega$ are constants. Identify the transient part of the solution and the steady-state solution.

Key Concepts

RL Circuit Differential Equation

The behavior of an RL circuit is modeled using a first order linear differential equation. In such circuits, Kirchhoff's voltage law leads to an equation that accounts for the applied electromotive force (EMF), the voltage drop across the resistor (Ohm's law), and the induced voltage across the inductor (related to the rate of change of current). This equation forms the basis for analyzing and solving for current as a function of time when the circuit is driven by a time-varying source.

Transient Response

The transient response is the part of the circuit's solution that depends on the initial conditions and represents the natural behavior of the circuit as it adjusts from its initial state. In an RL circuit, this response is characterized by an exponential decay governed by the time constant (L/R). Over time, the transient terms decrease and eventually become negligible as the circuit settles into its steady state.

Steady-State Response

The steady-state response describes the long-term behavior of the circuit when the effects of the initial conditions have decayed to zero. Under a sinusoidal input, the steady-state current response is also sinusoidal and can be determined by finding a particular solution of the differential equation that matches the form of the applied signal. This highlights the circuit’s response while respecting the frequency and amplitude of the continuous driving source.

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