Electromagnetic Induction, AC Circuits, and Electrical Technologies
Electromagnetic Induction, AC Circuits, and Electrical Technologies
55 RL Circuits
Learning Objectives
- Calculate the current in an RL circuit after a specified number of characteristic time steps.
- Calculate the characteristic time of an RL circuit.
- Sketch the current in an RL circuit over time.
We know that the current through an inductor [latex]L[/latex] cannot be turned on or off instantaneously. The change in current changes flux, inducing an emf opposing the change (Lenz’s law). How long does the opposition last? Current will flow and can be turned off, but how long does it take? Figure 55.1 shows a switching circuit that can be used to examine current through an inductor as a function of time.
When the switch is first moved to position 1 (at [latex]t=0[/latex]), the current is zero and it eventually rises to [latex]{I}_{0}=\text{V/R}[/latex], where [latex]R[/latex] is the total resistance of the circuit. The opposition of the inductor [latex]L[/latex] is greatest at the beginning, because the amount of change is greatest. The opposition it poses is in the form of an induced emf, which decreases to zero as the current approaches its final value. The opposing emf is proportional to the amount of change left. This is the hallmark of an exponential behavior, and it can be shown with calculus that
is the current in an RL circuit when switched on (Note the similarity to the exponential behavior of the voltage on a charging capacitor). The initial current is zero and approaches [latex]{I}_{0}=\text{V/R}[/latex] with a characteristic time constant
[latex]\tau[/latex]
for an RL circuit, given by
where [latex]\tau[/latex] has units of seconds, since
[latex]\text{1}\phantom{\rule{0.25em}{0ex}}\text{H}\text{=}\text{1}\phantom{\rule{0.25em}{0ex}}\text{Ω}\text{·}\text{s}[/latex].
In the first period of time [latex]\tau[/latex], the current rises from zero to [latex]0\text{.}\text{632}{I}_{0}[/latex], since [latex]I={I}_{0}\left(1-{e}^{-1}\right)={I}_{0}\left(1-0\text{.}\text{368}\right)=0\text{.}\text{632}{I}_{0}[/latex]. The current will go 0.632 of the remainder in the next time [latex]\tau[/latex]. A well-known property of the exponential is that the final value is never exactly reached, but 0.632 of the remainder to that value is achieved in every characteristic time [latex]\tau[/latex]. In just a few multiples of the time [latex]\tau[/latex], the final value is very nearly achieved, as the graph in Figure 55.1(b) illustrates.
The characteristic time [latex]\tau[/latex] depends on only two factors, the inductance [latex]L[/latex] and the resistance [latex]R[/latex]. The greater the inductance [latex]L[/latex], the greater [latex]\tau[/latex] is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance [latex]R[/latex], the greater [latex]\tau[/latex] is. Again this makes sense, since a small resistance means a large final current and a greater change to get there. In both cases—large [latex]L[/l