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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.
Learning Objectives

Calculate energy that can be stored in an inductor

Describe currentvoltage relationship in the RL circuit

Describe structure of a resistorinductor circuit
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 timedependent current is given by
$I=I_{0}(1e^{\frac{t}{\tau}})$ ("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 0.632I_{0}, 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 0.368I_{0}, and to 0.368 of the remainder each step τ.
Terms

inductor
A device or circuit component that exhibits significant selfinductance; a device which stores energy in a magnetic field.

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.
Full Text
RL Circuits
A resistorinductor 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 selfinductance 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 selfinductance.
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 E_{ind} is given by:
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? 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 I_{0}=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
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 with a characteristic time constant for an RL circuit, given by:
where has units of seconds, since 1H=1Ωs. In the first period of time τ, the current rises from zero to 0.632I_{0}, since I=I_{0}(1−e^{−1})=I_{0}(1−0.368)=0.632I_{0}. The current will go 0.632 of the remainder in the next time . A wellknown 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 , the final value is very nearly achieved, as the graph in (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 is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance R, 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 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 (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)LI_{0}^{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. Once again the behavior is exponential, and I is found to be
In (c), in the first period of time =L/R after the switch is closed, the current falls to 0.368 of its initial value, since I=I_{0}e^{−1}=0.368I_{0}. 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 Reference
 Component
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 Law
 Appears in this related concepts: TwoComponent Forces, Physics and Other Fields, and Models, Theories, and Laws
 battery
 Appears in this related concepts: Fuel Cells, The Battery, and Sources of EMF
 capacitor
 Appears in this related concepts: Combinations of Capacitors: Series and Parallel, Introduction and Importance, and ParallelPlate Capacitor
 circuit
 Appears in this related concepts: Driven Oscillations and Resonance, Electric Potential Energy and Potential Difference, and Microwaves
 conductor
 Appears in this related concepts: Physical Properties and Atomic Size, Semiconductors, and Conductors and Insulators
 current
 Appears in this related concepts: Reporting LongTerm Liabilities, Magnetic Force Between Two Parallel Conductors, and The Junction Rule
 decay
 Appears in this related concepts: Radioactive Decay Series: Introduction, Models Using Differential Equations, and Sensory Registers
 energy
 Appears in this related concepts: Energy Transportation, Introduction to Work and Energy, and The Role of Energy and Metabolism
 flux
 Appears in this related concepts: Applications to Economics and Biology, The Divergence Theorem, and Maxwell's Equations
 induction
 Appears in this related concepts: The Sociological Approach, The Law of Universal Gravitation, and Reasoning
 instantaneous
 Appears in this related concepts: Sinusoidal Nature of Simple Harmonic Motion, Polarization, and Instananeous Velocity: A Graphical Interpretation
 magnetic field
 Appears in this related concepts: Magnetic Force on a CurrentCarrying Conductor, Ampere's Law: Magnetic Field Due to a Long Straight Wire, and Brain Imaging Techniques
 magnetic flux
 Appears in this related concepts: Inductance, Motional EMF, and Transformers
 mutual inductance
 Appears in this related concept: Inductance
 parallel
 Appears in this related concepts: Newton and His Laws, Superposition of Forces, and Combination Circuits
 period
 Appears in this related concepts: The Periodic Table, Number of Periods, and Annuities
 position
 Appears in this related concepts: Damped Harmonic Motion, Longitudinal Waves, and Graphical Interpretation
 resistance
 Appears in this related concepts: Resistors in Parallel, Resisitors in Series, and Ecosystem Dynamics
 resistor
 Appears in this related concepts: Power, Impedance, and The Loop Rule
 selfinductance
 series
 Appears in this related concepts: Taylor Polynomials, Charging a Battery: EMFs in Series and Parallel, and Finding the General Term
 voltage
 Appears in this related concepts: Conductors, The Millikan OilDrop Experiment, and The Nernst Equation
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Source: Boundless. “RL Circuits.” Boundless Physics. Boundless, 28 May. 2015. Retrieved 28 May. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/inductionaccircuitsandelectricaltechnologies22/accircuits162/rlcircuits5816216/