Chapter 14. Inductance
14.4 RL Circuits
Learning Objectives
By the end of this section, you will be able to:
- Analyze circuits that have an inductor and resistor in series
- Describe how current and voltage exponentially grow or decay based on the initial conditions
A circuit with resistance and self-inductance is known as an RL circuit. Figure 14.12(a) shows an RL circuit consisting of a resistor, an inductor, a constant source of emf, and switches [latex]{\text{S}}_{1}[/latex] and [latex]{\text{S}}_{2}.[/latex] When [latex]{\text{S}}_{1}[/latex] is closed, the circuit is equivalent to a single-loop circuit consisting of a resistor and an inductor connected across a source of emf (Figure 14.12(b)). When [latex]{\text{S}}_{1}[/latex] is opened and [latex]{\text{S}}_{2}[/latex] is closed, the circuit becomes a single-loop circuit with only a resistor and an inductor (Figure 14.12(c)).
We first consider the RL circuit of Figure 14.12(b). Once [latex]{\text{S}}_{1}[/latex] is closed and [latex]{\text{S}}_{2}[/latex] is open, 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 [latex]\text{ε}\text{/}R.[/latex] However, from Faraday’s law, the increasing current produces an emf [latex]{V}_{L}=\text{−}L\left(dI\text{/}dt\right)[/latex] across the inductor. In accordance with Lenz’s law, the induced emf counteracts the increase in the current and is directed as shown in the figure. As a result, I(t) starts at zero and increases asymptotically to its final value.
Applying Kirchhoff’s loop rule to this circuit, we obtain
which is a first-order differential equation for I(t). Notice its similarity to the equation for a capacitor and resistor in series (See RC Circuits). Similarly, the solution to Equation 14.23 can be found by making substitutions in the equations relating the capacitor to the inductor. This gives
where
is the inductive time constant of the circuit.
The current I(t) is plotted in Figure 14.13(a). It starts at zero, and as [latex]t\to \infty[/latex], I(t) approaches [latex]\text{ε}\text{/}R[/latex] asymptotically. The induced emf [latex]{V}_{L}\left(t\right)[/latex] is directly proportional to dI/dt, or the slope of the curve. Hence, while at its greatest immediately after the switches are thrown, the induced emf decreases to zero with time as the current approaches its final value of [latex]\text{ε}\text{/}R.[/latex] The circuit then becomes equivalent to a resistor connected across a source of emf.
The energy stored in the magnetic field of an inductor is
Thus, as the current approaches the maximum current [latex]\epsilon \text{/}R[/latex], the stored energy in the inductor increases from zero and asymptotically approaches a maximum of [latex]L{\left(\epsilon \text{/}R\right)}^{2}\text{/}2.[/latex]
The time constant [latex]{\tau }_{L}[/latex] tells us how rapidly the current increases to its final value. At [latex]t={\tau }_{L},[/lat