14.4 RL Circuits

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},[/latex] the current in the circuit is, from Equation 14.24,

which is [latex]63\text{%}[/latex] of the final value [latex]\text{ε}\text{/}R[/latex]. The smaller the inductive time constant [latex]{\tau }_{L}=L\text{/}R,[/latex] the more rapidly the current approaches [latex]\text{ε}\text{/}R[/latex].

We can find the time dependence of the induced voltage across the inductor in this circuit by using [latex]{V}_{L}\left(t\right)=\text{−}L\left(dI\text{/}dt\right)[/latex] and Equation 14.24:

The magnitude of this function is plotted in Figure 14.13(b). The greatest value of [latex]L\left(dI\text{/}dt\right)\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}\epsilon ;[/latex] it occurs when dI/dt is greatest, which is immediately after [latex]{\text{S}}_{1}[/latex] is closed and [latex]{\text{S}}_{2}[/latex] is opened. In the approach to steady state, dI/dt decreases to zero. As a result, the voltage across the inductor also vanishes as [latex]t\to \infty .[/latex]

The time constant [latex]{\tau }_{L}[/latex] also tells us how quickly the induced voltage decays. At [latex]t={\tau }_{L},[/latex] the magnitude of the induced voltage is

The voltage across the inductor therefore drops to about [latex]37\text{%}[/latex] of its initial value after one time constant. The shorter the time constant [latex]{\tau }_{L},[/latex] the more rapidly the voltage decreases.

After enough time has elapsed so that the current has essentially reached its final value, the positions of the switches in Figure 14.12(a) are reversed, giving us the circuit in part (c). At [latex]t=0,[/latex] the current in the circuit is [latex]I\left(0\right)=\epsilon \text{/}R.[/latex] With Kirchhoff’s loop rule, we obtain

The solution to this equation is similar to the solution of the equation for a discharging capacitor, with similar substitutions. The current at time t is then

The current starts at [latex]I\left(0\right)=\epsilon \text{/}R[/latex] and decreases with time as the energy stored in the inductor is depleted (Figure 14.14).

The time dependence of the voltage across the inductor can be determined from [latex]{V}_{L}=\text{−}L\left(dI\text{/}dt\right)\text{:}[/latex]

This voltage is initially [latex]{V}_{L}\left(0\right)=\epsilon[/latex], and it decays to zero like the current. The energy stored in the magnetic field of the inductor, [latex]L{I}^{2}\text{/}2,[/latex] also decreases exponentially with time, as it is dissipated by Joule heating in the resistance of the circuit.

Example

An RL Circuit with a Source of emf

In the circuit of Figure 14.12(a), let [latex]\epsilon =2.0V,R=4.0\phantom{\rule{0.2em}{0ex}}\text{Ω},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}L=4.0\phantom{\rule{0.2em}{0ex}}\text{H}\text{.}\phantom{\rule{0.2em}{0ex}}[/latex] With [latex]{\text{S}}_{1}[/latex] closed and [latex]{\text{S}}_{2}[/latex] open (Figure 14.12(b)), (a) what is the time constant of the circuit? (b) What are the current in the circuit and the magnitude of the induced emf across the inductor at [latex]t=0,\phantom{\rule{0.2em}{0ex}}\text{at}\phantom{\rule{0.2em}{0ex}}t=2.0{\tau }_{L}[/latex], and as [latex]t\to \infty[/latex]?

Strategy

The time constant for an inductor and resistor in a series circuit is calculated using Equation 14.25. The current through and voltage across the inductor are calculated by the scenarios detailed from Figure 14.24 and Equation 14.32.

Solution

Show Answer

1. The inductive time constant is
   
   [latex]{\tau }_{L}=\frac{L}{R}=\frac{4.0\phantom{\rule{0.2em}{0ex}}\text{H}}{4.0\phantom{\rule{0.2em}{0ex}}\text{Ω}}=1.0\phantom{\rule{0.2em}{0ex}}\text{s}.[/latex]
2. The current in the circuit of Figure 14.12(b) increases according to Equation 14.24:
   
   [latex]I\left(t\right)=\frac{\text{ε}}{R}\left(1-{e}^{\text{−}t\text{/}{\tau }_{L}}\right).[/latex]

   
   At [latex]t=0,[/latex]
   

   [latex]\left(1-{e}^{\text{−}t\text{/}{\tau }_{L}}\right)=\left(1-1\right)=0;\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}I\left(0\right)=0.[/latex]

   
   At [latex]t=2.0{\tau }_{L}[/latex] and [latex]t\to \infty ,[/latex] we have, respectively,
   

   [latex]I\left(2.0{\tau }_{L}\right)=\frac{\text{ε}}{R}\left(1-{e}^{-2.0}\right)=\left(0.50\phantom{\rule{0.2em}{0ex}}\text{A}\right)\left(0.86\right)=0.43\phantom{\rule{0.2em}{0ex}}\text{A},[/latex]

   
   and
   

   [latex]I\left(\infty \right)=\frac{\text{ε}}{R}=0.50\phantom{\rule{0.2em}{0ex}}\text{A}.[/latex]

   
   From Equation 14.32, the magnitude of the induced emf decays as
   

   [latex]|{V}_{L}\left(t\right)|=\text{ε}{e}^{\text{−}t\text{/}{\tau }_{L}}.[/latex]

   
   [latex]\text{At}\phantom{\rule{0.2em}{0ex}}t=0,t=2.0{\tau }_{L},\phantom{\rule{0.2em}{0ex}}\text{and as}\phantom{\rule{0.2em}{0ex}}t\to \infty ,[/latex] we obtain
   

   [latex]\begin{array}{ccc}\hfill |{V}_{L}\left(0\right)|& =\hfill & \epsilon =2.0\phantom{\rule{0.2em}{0ex}}\text{V},\hfill \\ \hfill |{V}_{L}\left(2.0{\tau }_{L}\right)|& =\hfill & \left(2.0\phantom{\rule{0.2em}{0ex}}\text{V}\right)\phantom{\rule{0.2em}{0ex}}{e}^{-2.0}=0.27\phantom{\rule{0.2em}{0ex}}\text{V}\hfill \\ & \text{and}\hfill & \\ \hfill |{V}_{L}\left(\infty \right)|& =\hfill & 0.\hfill \end{array}[/latex]

Significance

If the time of the measurement were much larger than the time constant, we would not see the decay or growth of the voltage across the inductor or resistor. The circuit would quickly reach the asymptotic values for both of these. See Figure 14.15.