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 initia…
When a DC source is suddenly disconnected from an RL circuit, it becomes source-free.
Assuming the inductor has an initial current i0, the initial energy stored in the inductor can be determined.
Applying Kirchhoff's voltage law around the loop and substituting the voltages across the inductor and resistor yields a first-order differential equation.
Rearranging terms, integrating, and applying the limits gives a logarithmic equation.
By taking exponential on both sides, the final expression of the circuit's natural response is determined.
The current versus time graph shows an exponential decrease in the initial current.
The current response can be expressed in terms of the time constant, which is the ratio of inductance to resistance.
The current expression is used to determine the voltage and power dissipated across the resistor.
Integration of dissipated power over time provides the expression for the energy absorbed by the resistor.
As the time approaches infinity, the energy absorbed by the resistor approaches the initial energy stored in the inductor, implying that the initial energy is gradually dissipated in the resistor.
Q1: What happens to an RL circuit when the DC source is suddenly disconnected?
When a DC source is disconnected from an RL circuit, the circuit becomes source-free. The inductor, which initially stores energy based on its current, begins to release this energy. Applying Kirchhoff's voltage law around the loop yields a first-order differential equation that describes how the circuit responds to the sudden removal of the source.
Q2: How is the natural response of a source-free RL circuit expressed mathematically?
The natural response is derived by applying Kirchhoff's voltage law and solving the resulting first-order differential equation. After rearranging, integrating, and applying limits, a logarithmic equation is obtained. Taking the exponential of both sides yields the final expression showing current as an exponential function of time.
Q3: What is the time constant in an RL circuit and why does it matter?
The time constant is the ratio of inductance to resistance (L/R) and represents the speed at which the circuit responds to changes. A larger time constant means the current decays more slowly, while a smaller time constant indicates faster decay. This parameter is fundamental for predicting how quickly the circuit reaches steady state.
Q4: Why does current decrease exponentially in a source-free RL circuit?
When the source is removed, the inductor's magnetic field collapses, driving current through the resistor. The resistor dissipates energy as heat, reducing the current exponentially over time. The rate of decay depends on the time constant; larger resistance or smaller inductance causes faster exponential decay of the initial current.
Q5: How can you calculate the power dissipated in the resistor of a source-free RL circuit?
Power dissipated in the resistor is calculated using the current expression derived from the natural response. Since power equals I²R, substituting the exponential current function gives the instantaneous power. This power represents the rate at which energy stored in the inductor is converted to heat in the resistor.
Q6: What is the relationship between initial inductor energy and energy absorbed by the resistor?
The initial energy stored in the inductor equals one-half LI₀². As time approaches infinity, the total energy absorbed by the resistor approaches this initial value. This demonstrates energy conservation: all magnetic energy initially stored in the inductor is gradually dissipated as heat in the resistor until the inductor's energy is depleted.
Q7: How does a source-free RL circuit differ from other first-order circuits?
Like other first-order circuits, the source-free RL circuit exhibits exponential response governed by a single time constant. However, RL circuits store energy in magnetic fields, while RC circuits store energy in electric fields. Understanding source-free RL behavior provides insights applicable to analyzing first-order circuits across various applications.