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  • 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 L 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.

Part a of the figure shows an inductor connected in series with a resistor. The arrangement is connected across a cell by an on and off switch with two positions. When in position one, the battery, resistor, and inductor are in series and a current is established. In position two, the battery is removed and the current stops eventually because of energy loss in the resistor. Part b of the diagram shows the graph when the switch is in position one. It shows a graph for current growth verses time. The current is along the Y axis and the time is along the X axis. The graph shows a smooth rise from origin to a maximum value I zero corresponding to Y axis and value four tau on X axis. Part c of the diagram shows the graph when the switch is in position two. It shows a graph for current decay verses time is shown. The current is along the Y axis and the time is along the X axis. The graph is decreasing curve from a value I zero on Y axis, touching the X axis at a point where value of time equals four tau.
(a) An RL circuit with a switch to turn current on and off. When in position 1, the battery, resistor, and inductor are in series and a current is established. In position 2, the battery is removed and the current eventually stops because of energy loss in the resistor. (b) A graph of current growth versus time when the switch is moved to position 1. (c) A graph of current decay when the switch is moved to position 2.

When the switch is first moved to position 1 (at t = 0 size 12{t=0} {} ), the current is zero and it eventually rises to I 0 = V/R size 12{I rSub { size 8{0} } = ital "V/R"} {} , where R is the total resistance of the circuit. The opposition of the inductor L 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

I = I 0 ( 1 e t / τ )     (turning on), size 12{I=I rSub { size 8{0} } \( 1 - e rSup { size 8{ - t/τ} } \) } {}

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 I 0 = V/R size 12{I rSub { size 8{0} } = ital "V/R"} {} with a characteristic time constant     τ for an RL circuit, given by

τ = L R , size 12{τ= { {L} over {R} } } {}

where τ size 12{τ} {} has units of seconds, since 1 H = 1 Ω · s . In the first period of time τ size 12{τ} {} , the current rises from zero to 0 . 632 I 0 size 12{0 "." "632"I rSub { size 8{0} } } {} , since I = I 0 ( 1 e 1 ) = I 0 ( 1 0 . 368 ) = 0 . 632 I 0 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 τ 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 τ size 12{τ} {} . In just a few multiples of the time τ size 12{τ} {} , the final value is very nearly achieved, as the graph in [link] (b) illustrates.

The characteristic time τ size 12{τ} {} depends on only two factors, the inductance L size 12{L} {} and the resistance R size 12{R} {} . The greater the inductance L size 12{L} {} , the greater τ size 12{τ} {} is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance R size 12{R} {} , the greater τ 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 L size 12{L} {} and small R size 12{R} {} —more energy is stored in the inductor and more time is required to get it in and out.

When the switch in [link] (a) is moved to position 2 and cuts the battery out of the circuit, the current drops because of energy dissipation by the resistor. But this is also not instantaneous, since the inductor opposes the decrease in current by inducing an emf in the same direction as the battery that drove the current. Furthermore, there is a certain amount of energy, ( 1/2 ) LI 0 2 size 12{ \( "1/2" \) ital "LI" rSub { size 8{0} } rSup { size 8{2} } } {} , stored in the inductor, and it is dissipated at a finite rate. As the current approaches zero, the rate of decrease slows, since the energy dissipation rate is I 2 R size 12{ I rSup { size 8{2} } R} {} . Once again the behavior is exponential, and I is found to be

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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