RL Circuit with Source

When an RL (Resistor-Inductor) circuit is connected to a DC source, the complete response of the circuit can be divided into two parts: the transient response and the steady-state response.

The transient response of the circuit is its temporary reaction to the sudden application of the DC source. This response is characterized by a current that exponentially decays to zero as time approaches infinity. During this transitional period, the inductor behaves like a short circuit, causing the source voltage to drop across the resistor.

Once the transient response has decayed to zero, the circuit enters its steady-state phase. In this phase, the current in the circuit becomes stable, equaling the ratio of the source voltage to the resistance of the circuit. This is the steady-state response.

The constant term in the transient response can be determined by substituting the initial current through the inductor at the time the switch is closed (t=0). This gives us the initial condition for the transient response.

Plotting the complete step response of the circuit shows that the initial current decreases exponentially until it reaches a steady-state value. From this current response, the voltage response is derived, which also follows an exponential decay from a maximum value to zero.

If the initial current in the circuit is zero, the complete step response shows the current increasing exponentially until it reaches the steady-state value. Correspondingly, the voltage response starts at the source voltage and decreases exponentially to zero.

In conclusion, understanding the complete response of an RL circuit to a DC source provides valuable insights into how these circuits react to sudden changes in input voltage. This knowledge is essential for designing and analyzing electronic circuits, particularly in applications such as power supply filtering and signal processing, where inductors are used extensively.