# RL Circuits

## An RL circuit consists of an inductor and a resistor, in series or parallel with each other, with current driven by a voltage source.

#### Key Points

• The energy stored in an inductor is $E=\frac{1}{2}LI^{2}$. It takes time to build up stored energy in a conductor and time to deplete it.

• When a resistor and an inductor in series are connected to a voltage source, the time-dependent current is given by $I=I_{0}(1-e^{\frac{-t}{\tau}})$ ("turning on"). The final current after a long time is .

• The characteristic time constant is given by $\tau=\frac{R}{L}$, where R is resistance and L is inductance. This represents the time necessary for the current in a circuit just closed to go from zero to 0.632I0, and 0.632 of the remainder in each interval τ.

• When the voltage source is disconnected from the inductor, the current will decay according to $I=I_{0}e^{\frac{-t}{\tau}}$("turning off"). In the first time interval τ the current falls by a factor of 1/e to 0.368I0, and to 0.368 of the remainder each step τ.

#### Terms

• Denoted by τ, in RL circuits it is given by $\tau=\frac{R}{L}$ where R is resistance and L is inductance. When a switch is closed, it is the time it takes for the current to decay by a factor of 1/e.

• A device or circuit component that exhibits significant self-inductance; a device which stores energy in a magnetic field.

#### Figures

1. ##### Current in an RL Circuit

(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.

## RL Circuits

A resistor-inductor circuit (RL circuit) consists of a resistor and an inductor (either in series or in parallel) driven by a voltage source.

### Review

Recall that induction is the process in which an emf is induced by changing magnetic flux. Mutual inductance is the effect of Faraday’s law of induction for one device upon another, while self-inductance is the the effect of Faraday’s law of induction of a device on itself. An inductor is a device or circuit component that exhibits significant self-inductance.

### Energy of an Inductor

We know from Lenz’ law that inductances oppose changes in current. We can think of this situation in terms of energy. Energy is stored in a magnetic field. It takes time to build up energy, and it also takes time to deplete energy; hence, there is an opposition to rapid change. In an inductor, the magnetic field is directly proportional to current and to the inductance of the device. It can be shown that the energy stored in an inductor Eind is given by:

$E=\frac{1}{2}LI^{2}$.

### Inductors in Circuits

We know that the current through an inductor L cannot be turned on or off instantaneously. The change in current changes flux, inducing an emf opposing the change (Lenz’ law). How long does the opposition last? Current will flow and can be turned off, but how long does it take? Figure 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 t=0), the current is zero and it eventually rises to I0=V/R, where R is the total resistance of the circuit. The opposition of the inductor 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

$I=I_{0}(1-e^{\frac{-t}{\tau }})$ (turning on),

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/R with a characteristic time constant τ for an RL circuit, given by:

$\tau=\frac{R}{L}$,

where τ has units of seconds, since 1H=1Ω·s. In the first period of time τ, the current rises from zero to 0.632I0, since I=I0(1−e−1)=I0(1−0.368)=0.632I0. The current will go 0.632 of the remainder in the next time 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. In just a few multiples of the time τ, the final value is very nearly achieved, as the graph in Figure 1(b) illustrates.

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

When the switch in Figure 1(a) is moved to position 2 and cuts the battery out of the circuit, the current drops because of energy dissipation by the resistor. However, 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)LI02, 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 I2R. Once again the behavior is exponential, and I is found to be

$I=I_{0}e^{\frac{-t}{\tau}}$(turning off).

In Figure 1(c), in the first period of time τ=L/R after the switch is closed, the current falls to 0.368 of its initial value, since I=I0e−1=0.368I0. In each successive time τ, the current falls to 0.368 of the preceding value, and in a few multiples of τ, the current becomes very close to zero.

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.

#### Key Term Glossary

battery
A device that produces electricity by a chemical reaction between two substances.
##### Appears in these related concepts:
capacitor
An electronic component capable of storing an electric charge, especially one consisting of two conductors separated by a dielectric.
##### Appears in these related concepts:
characteristic time constant
Denoted by τ, in RL circuits it is given by $\tau=\frac{R}{L}$ where R is resistance and L is inductance. When a switch is closed, it is the time it takes for the current to decay by a factor of 1/e.
##### Appears in this related concept:
circuit
A pathway of electric current composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. T
##### Appears in these related concepts:
Component
A part of a vector. For example, horizontal and vertical components.
##### Appears in these related concepts:
current
The time rate of flow of electric charge.
##### Appears in these related concepts:
energy
A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.
##### Appears in these related concepts:
flux
The rate of transfer of energy (or another physical quantity) through a given surface, specifically electric flux or magnetic flux.
##### Appears in these related concepts:
induction
The generation of an electric current by a varying magnetic field.
##### Appears in these related concepts:
inductor
A passive device that introduces inductance into an electrical circuit.
##### Appears in these related concepts:
Inductor
A device or circuit component that exhibits significant self-inductance; a device which stores energy in a magnetic field.
##### Appears in these related concepts:
instantaneous
(As in velocity)—occurring, arising, or functioning without any delay; happening within an imperceptibly brief period of time.
##### Appears in these related concepts:
Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
##### Appears in these related concepts:
magnetic field
A condition in the space around a magnet or electric current in which there is a detectable magnetic force, and where two magnetic poles are present.
##### Appears in these related concepts:
magnetic flux
A measure of the strength of a magnetic field in a given area.
##### Appears in these related concepts:
mutual inductance
The ratio of the voltage in a circuit to the change in current in a neighboring circuit.
##### Appears in these related concepts:
parallel
An arrangement of electrical components such that a current flows along two or more paths.
##### Appears in these related concepts:
period
The duration of one cycle in a repeating event.
##### Appears in these related concepts:
Period
The period is the duration of one cycle in a repeating event.
##### Appears in these related concepts:
position
A place or location.
##### Appears in these related concepts:
resistance
The opposition to the passage of an electric current through that element.
##### Appears in these related concepts:
resistor
An electric component that transmits current in direct proportion to the voltage across it.
##### Appears in these related concepts:
series
A number of things that follow on one after the other or are connected one after the other.
##### Appears in these related concepts:
voltage
The amount of electrostatic potential between two points in space.