5.4: RL Circuit without Source
When a DC source is suddenly disconnected from an RL (Resistor-Inductor) circuit, the circuit becomes source-free. Assuming the inductor has an initial current denoted as I0, the initial energy stored in the inductor can be determined.
Applying Kirchhoff's voltage law around the loop of the circuit and substituting the voltages across the inductor and resistor yields a first-order differential equation. A logarithmic equation is obtained by rearranging the terms in this equation, integrating it, and applying the limits. Taking the exponential on both sides of this equation yields the final expression of the circuit's natural response.
If the current is plotted versus time, an exponential decrease is observed in the initial current. This behavior can be expressed in terms of the time constant, which for an RL circuit is the ratio of inductance to resistance. This time constant represents the speed at which the circuit responds to changes in the input signal.
By using the current expression, the voltage across the resistor and the power that gets dissipated in the resistor can be calculated. The power dissipated is essentially the rate at which energy is lost in the form of heat.
To find the total energy absorbed by the resistor, the power dissipated is integrated over time. As time approaches infinity, the energy absorbed by the resistor approaches the initial energy stored in the inductor. This implies that the initial energy stored in the inductor gradually dissipates in the resistor until the inductor's energy is depleted.
In conclusion, understanding the behavior of RL circuits when the DC source is removed offers valuable insights into the transient response of these circuits. This knowledge is fundamental for designing and analyzing circuits in applications such as power electronics and communication systems, where inductors are used extensively to filter or shape signals.
