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 sou…
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.
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Q1: Why does current not rise immediately to its final value when an RL circuit is first closed?
When the circuit closes, the increasing current produces an induced emf across the inductor that opposes the applied emf, following Lenz's law. This opposing emf counteracts the current increase, forcing the current to start at zero and rise gradually toward its steady-state value ε/R rather than jumping instantly.
Q2: What is the inductive time constant and what does it tell us about circuit behavior?
The inductive time constant is the ratio of inductance to resistance (L/R). It measures how quickly current builds toward its final value. At time equal to one time constant, current grows to 63% of its final value during growth, or decays to 37% of its original value during decay.
Q3: How does the energy stored in an inductor change as current grows in an RL circuit?
As current increases from zero toward its steady-state value ε/R, the energy stored in the inductor increases asymptotically from zero to a maximum value. This stored energy in the magnetic field grows proportionally with the increasing current until the circuit reaches steady state.
Q4: What happens to current when the emf source is disconnected from an RL circuit?
When the emf source is disconnected and the circuit forms a single loop with only the resistor and inductor, the initial current ε/R decreases exponentially with time. The energy stored in the inductor is gradually depleted through the resistor until the current reaches zero.
Q5: How do you apply Kirchhoff's loop rule to find current growth in an RL circuit?
Kirchhoff's loop rule states that the sum of emfs around a closed loop equals zero. In an RL circuit with applied emf, the applied emf equals the sum of the emf across the resistor and the induced emf across the inductor. Rearranging and integrating this equation yields the current growth equation.
Q6: Why does current decay exponentially rather than linearly when an RL circuit is disconnected from its emf source?
During decay, the inductor's induced emf is proportional to the rate of current change. As current decreases, the induced emf also decreases, resulting in a slower decay rate at each moment. This proportional relationship produces exponential decay rather than a constant linear decrease.
Q7: How can you compare the behavior of RL circuits to RC circuits in terms of current response?
Both RL and RC circuits exhibit exponential responses to switching, but with different time constants. RL circuits have time constant L/R, while RC circuits have time constant RC. Both show asymptotic approach to steady state, though the physical mechanisms differ—inductors oppose current change while capacitors oppose voltage change.