R-L CircuitsR-L Circuits

R-L Circuits?R-L Circuits?What does the “L” stand for?What does the “L” stand for?

Good Question! “L” stands for the self-inductance of an inductor measured in Henrys (H).

So…What is an inductor?

• An inductor is an electronic device that is put into a circuit to prevent

rapid changes in current.

• It is basically a coil of wire which uses the basic principles of

electromagnetism and Lenz’s Law to store magnetic energy within

the circuit for the purposes of stabilizing the current in that circuit.

• The voltage drop across an inductor depends on the inductance value L and the rate of change of the current di/dt.

dtdiLV

R-L CircuitsR-L CircuitsThe set up and initial conditionsThe set up and initial conditions

S1

S2

ε

LRa b c

Assume an ideal source (r=0)

An R-L circuit is any circuit that contains both a resistor and an inductor.

Initial conditions: At time t = 0…when S1 is closed…i = 0 and

At time t = 0, we will close switch S1 to create a series circuit that includes the battery. The current will grow to a “steady-state” constant value at which the device will operate until powered off (i.e. the battery is removed)

Ldt

di

initial

R-L CircuitsR-L CircuitsCurrent GrowthCurrent Growth

iRVab dtdi

bc LV

LiR

LLiR

dtdi

dtdiLiR

S1

S2

ε

LRa b c

i

Note: we will use lower-case letters to represent time-varying quantities.

At time t = 0, S1 is closed, current, i, will grow at a rate that depends on the value of L until it reaches it’s final steady-state value, I

If we apply Kirchoff’s Law to this circuit and do a little algebra we get…

As “i” increases, “iR/L” also increases, so “di/dt” decreases until it reaches zero. At this time, the current has reached it’s final “steady-state” value “I”.

R-L CircuitsR-L CircuitsSteady-State CurrentSteady-State Current

RI

S1

S2

ε

LRa b c

i

When the current reaches its final “steady-state” value, I, then di/dt = 0.

Solving this equation for I…

IL

R

Ldt

di

final

0

Do you recognize this?

It is Ohm’s Law!!!

So…when the current is at steady-state, the circuit behaves like the inductor is not there…unless it tries to change current values quickly! The steady-state current does NOT depend on L!

R-L CircuitsR-L CircuitsCurrent as a function of time during GrowthCurrent as a function of time during Growth

The calculus and the algebra!The calculus and the algebra!

RLR

LiR

Ldtdi i

ti

LRi

t

LR

i

idi

LR

R

R

R

R

R

e

t

dt

ln

00

S

1S

2

ε

LRa b c

i

Let’s start with the equation we derived earlier from Kirchoff’s Law…

Rearrange and integrate…

Solve for i… tLReR

ti 1)(

R-L CircuitsR-L CircuitsThe time constant!The time constant!

tLReR

ti 1)(

S1

S2

ε

LRa b c

i

The time constant is the time at which the power of the “e” function is “-1”. Therefore, time constant is L/R

At time t = 2 time-constants, i = 0.86 I,

and at time t = 5 time-constants, i = 0.99995 I

Therefore, after approximately 5 time-constant intervals have passed, the circuit reaches its steady-state current.

RL

R-L CircuitsR-L CircuitsEnergy and PowerEnergy and Power

dtdi

inductor

resistor

battery

LiP

RiP

iP

2

S1

S2

ε

LRa b c

i

dtdiLiRii 2