Transcript RL Circuits - Humble ISD

R-L Circuits
R-L Circuits?
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.
V L
di
dt
R-L Circuits
The set up and initial conditions
Assume an ideal
source (r=0)
ε
An R-L circuit is any
circuit that contains both
a resistor and an inductor.
S1
S2
a
R
b
L
c
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)
Initial conditions: At time t = 0…when S1 is closed…i = 0 and

 di 

 
 dt initial L
R-L Circuits
Current
Growth
S
ε
1
Note: we will use lower-case letters
to represent time-varying quantities.
S2
i
a
R
b
Vab  iR
L
Vbc  L dtdi
c
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…
  iR  L
di
dt
di
dt
  LiR  L  iRL
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 Circuits
Steady-State Current
ε
S1
When the current reaches
its final “steady-state”
value, I, then di/dt = 0.
S2
i
a
R
b
L
c
 R
 di 
0  I
 
L L
 dt  final
Solving this equation for I…

I
R
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 steadystate current does NOT depend on L!
R-L Circuits
Current as a function of time during Growth
The calculus and the algebra!
ε
S
Let’s start with the equation we
derived earlier from Kirchoff’s Law…
1
S
2
i
a
b
R
L
c
di
dt
L

iR
L

R
L
i  R 

Rearrange and integrate…

i
di

0 i R
ln

t
R
L 0
 
i  R
 R
i  R
 R
e
 dt
R
L
 RL t
t
Solve for i…
i (t ) 

1 e 
R

 RL t
R-L Circuits
The time constant!
ε
S1
i (t ) 
S2
i
a
b
R

L
R
L
c

1 e 
R

 RL t
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.
R-L Circuits
Energy and Power
ε
Pbattery  i
S1
S2
Presistor  i R
2
i
a
R
b
c
L
Pinductor  Li
i  i R  Li
2
di
dt
di
dt