31.8: Current Growth And Decay In RL Circuits

The current growth and decay in RL circuits can be understood by considering a series RL circuit consisting of a resistor, an inductor, a constant source of emf, and two switches. When the first switch is closed, the circuit is equivalent to a single-loop circuit consisting of a resistor and an inductor connected to a source of emf. In this case, the source of emf produces a current in the circuit. If there were no self-inductance in the circuit, the current would rise immediately to a steady value of ε/R. However, from Faraday's law, the increasing current produces an emf across the inductor, which has opposite polarity. In accordance with Lenz’s law, the induced emf counteracts the increase in the current. As a result, the current starts at zero and increases asymptotically to its final value. Thus, as the current approaches the maximum current ε/R, the stored energy in the inductor increases from zero and asymptotically approaches a maximum value. The growth of current with time is given by

When the first switch is opened, and the second switch is closed, the circuit again becomes a single-loop circuit but with only a resistor and an inductor. Now, the initial current in the circuit is ε/R. The current starts from ε/R and decreases exponentially with time as the energy stored in the inductor is depleted. The decay of current with time is given by the relation

The quantity inductance over resistance is given by

measures how quickly the current builds toward its final value; this quantity is called the time constant for the circuit. When the current is plotted against time, It grows from zero and approaches ε/R asymptotically. At a time equal to time constant, the current rises to about 63%, of its final value, but during decaying, at the same time constant, it decreases to about 37%, of its original value.

Consider an RL circuit consisting of a resistor, an inductor, a constant source of emf, and switches S1 and S2.

When switch S1 is closed, the current in the circuit increases, generating emf across the resistor and inductor. These emf's are used in Kirchhoff's loop rule to find the current growth rate.

On rearranging and integrating the equation, the current in the RL circuit with emf is obtained.

After attaining the steady state, switch S2 is closed while S1 is opened, forming a single loop bypassing the emf source. This results in current decay through the resistor and inductor. The decayed current is obtained using Kirchhoff's rule.

The quantity inductance over resistance is called the inductive time constant.

The current versus time graph shows that, when time equals time constant, the current grows to 63 percent of its final value, whereas during decaying, at the same value of the time constant, the current decays to 37 percent of its original value.

Hence, the current increases gradually from zero to a steady state but decays exponentially with time.