Chapter 23 Electromagnetic Induction, AC Circuits, and Electrical Technologies
Chapter 23 Electromagnetic Induction, AC Circuits, and Electrical Technologies
23.10 RL Circuits
- 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 LL size 12{L} {} 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? [link] 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 t=0t=0 size 12{t=0} {}), the current is zero and it eventually rises to I0=V/RI0=V/R size 12{I rSub { size 8{0} } = ital “V/R”} {}, where RR is the total resistance of the circuit. The opposition of the inductor LL size 12{L} {} 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 I0=V/RI0=V/R size 12{I rSub { size 8{0} } = ital “V/R”} {} with a characteristic time constant
ττ
for an RL circuit, given by
where ττ size 12{τ} {} has units of seconds, since
1H=1Ω·s1H=1Ω·s.
In the first period of time ττ size 12{τ} {}, the current rises from zero to 0.632I00.632I0 size 12{0 “.” “632”I rSub { size 8{0} } } {}, since I=I0(1−e−1)=I0(1−0.368)=0.632I0I=I0(1−e−1)=I0(1−0.368)=0.632I0 size 12{I=I rSub { size 8{0} } ( 1 – e rSup { size 8{ – 1} } ) =I rSub { size 8{0} } ( 1 – 0 “.” “368” ) =0 “.” “632”I rSub { size 8{0} } } {}. The current will go 0.632 of the remainder in the next time ττ size 12{τ} {}. 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 ττ size 12{τ} {}. In just a few multiples of the time ττ size 12{τ} {}, the final value is very nearly achieved, as the graph in [link](b) illustrates.
The characteristic time ττ size 12{τ} {} depends on only two factors, the inductance LL size 12{L} {} and the resistance RR size 12{R} {}. The greater the inductance LL size 12{L} {}, the greater ττ size 12{τ} {} is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance RR size 12{R} {}, the greater ττ size 12{τ} {} 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 LL size 12{L} {} and small RR size