Transcript Step Response of an RL Circuit

Step Response of an RL Circuit
Find the current in the loop or the voltage across the
inductor after the switch is closed at t = 0.
ECE 201 Circuit Theory 1
1
Write KVL
around the
loop
Solve for the
highest-order
derivative
Differential change
in current
di
dt
di  Ri  Vs
R  Vs 

  i  
dt
L
L R 
di
R  Vs 
dt    i   dt
dt
L R 
Vs  Ri  L
R  Vs 
di    i   dt
L R 
ECE 201 Circuit Theory 1
2
Separate the variables
Introduce dummy
variables x and y
Integrate to ln
Invert to exponential
Final expression for
i(t)
di
R
  dt
Vs
L
i
R
i (t )
t
dx
R
I Vs   L 0 dy
0 x 
R
V 
i (t )   s 
 R  Rt
ln
L
V 
I0   s 
R
 Vs 
i (t )   
R

 R   e  L t
V
I0  s
R
R
Vs 
Vs   L t
i (t )    I 0   e
R 
R
ECE 201 Circuit Theory 1
3
Take a closer look at the current
When the initial energy is equal to zero
R
  t
L
Vs Vs
i (t )   e
R R
Vs Vs 1
Vs
i ( )   e  0.6321
R R
R
Plot shown on the next slide
ECE 201 Circuit Theory 1
4
ECE 201 Circuit Theory 1
5
Look at the derivative at t = 0
Vs  1  t
di
   e
dt
R 


Vs  1  t
   e
R L
 R
di Vs t
 e
dt L
Vs
di
(0) 
dt
L
ECE 201 Circuit Theory 1
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If the current continued at this rate
Vs
i t
L
@t  ,
Vs
Vs L Vs
i ( )   

L
L R R
Plot shown on next slide
ECE 201 Circuit Theory 1
7
ECE 201 Circuit Theory 1
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What About the Voltage
Across the Inductor?
di
vL
dt
R


d Vs Vs  L t 
vL   e



dt  R R

R
 R


t


 t
Vs R  L 
vL  e
  Vs e  L 

R  L

ECE 201 Circuit Theory 1
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ECE 201 Circuit Theory 1
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Look at the derivative of the voltage
v  Vs e
R
  t
L
R
 t
L
dv
R
  Vs e
dt
L
dv
R
(0)   Vs
dt
L
ECE 201 Circuit Theory 1
11
If the voltage continued at this rate
R
v   Vs t
L
@t ,
R L
v   Vs  Vs
L R
ECE 201 Circuit Theory 1
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ECE 201 Circuit Theory 1
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