# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## RL Circuits


### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the current in an RL circuit after a specified number of characteristic time steps.
2. Calculate the characteristic time of an RL circuit.
3. Sketch the current in an RL circuit over time.
We know that the current through an inductor  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?  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 ), the current is zero and it eventually rises to , where 
 
   is the total resistance of the circuit. The opposition of the inductor  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  with a characteristic time constant

 
  
for an RL circuit, given by
where  has units of seconds, since

 
  .
 In the first period of time , the current rises from zero to , since . The current will go 0.632 of the remainder in the next time . 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 . In just a few multiples of the time , the final value is very nearly achieved, as the graph in (b) illustrates.
The characteristic time  depends on only two factors, the inductance  and the resistance . The greater the inductance , the greater  is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance , the greater  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  and small  —more energy is stored in the inductor and more time is required to get it in and out.
When the switch in (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, , 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 . Once again the behavior is exponential, and

 is found to be
(See (c).) In the first period of time  after the switch is closed, the current falls to 0.368 of its initial value, since . In each successive time , the current falls to 0.368 of the preceding value, and in a few multiples of , the current becomes very close to zero, as seen in the graph in (c).
In summary, when the voltage applied to an inductor is changed, the current also changes, but the change in current lags the change in voltage in an RL circuit. In Reactance, Inductive and Capacitive, we explore how an RL circuit behaves when a sinusoidal AC voltage is applied.

### Section Summary

1. When a series connection of a resistor and an inductor—an RL circuit—is connected to a voltage source, the time variation of the current is
 
 where 
  is the final current.
2. The characteristic time constant  is  , where 
 
    is the inductance and  
 
   is the resistance.
3. In the first time constant , the current rises from zero to , and 0.632 of the remainder in every subsequent time interval .
4. When the inductor is shorted through a resistor, current decreases as
        

Here  is the initial current.
5. Current falls to  in the first time interval , and 0.368 of the remainder toward zero in each subsequent time .

### Problem Exercises
