Transcript Step Response of an RL Circuit ECE 201 Circuit Theory 1

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
6
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
8
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
9
ECE 201 Circuit Theory 1
10
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
12
ECE 201 Circuit Theory 1
13