ECE 201 Circuit Theory 1 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.
Jan 18, 2016
ECE 201 Circuit Theory 1 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 2
s
s s
s
s
diV Ri L
dtRi V Vdi R
idt L L R
Vdi Rdt i dt
dt L R
VRdi i dt
L R
Write KVL around the loop
Solve for the highest-order derivative
Differential change in current
ECE 201 Circuit Theory 1 3
0
( )
0
0
0
0
( )ln
( )
( )
s
i t t
sI
s
s
sRt
L
s
Rt
Ls s
di Rdt
V LiR
dx Rdy
V LxRV
i tRRt
V LI
R
Vi t
Re
VI
R
V Vi t I e
R R
Separate the variables
Introduce dummy variables x and y
Integrate to ln
Invert to exponential
Final expression for i(t)
ECE 201 Circuit Theory 1 4
Take a closer look at the current
1
( )
( ) 0.6321
Rt
Ls s
s s s
V Vi t e
R RV V V
i eR R R
When the initial energy is equal to zero
Plot shown on the next slide
ECE 201 Circuit Theory 1 5
ECE 201 Circuit Theory 1 6
Look at the derivative at t = 0
1 1
(0)
t ts s
ts
s
V Vdie e
Ldt R RR
Vdie
dt LVdi
dt L
ECE 201 Circuit Theory 1 7
If the current continued at this rate
@ ,
( )
s
s s s
Vi tLt
V V VLi
L L R R
Plot shown on next slide
ECE 201 Circuit Theory 1 8
ECE 201 Circuit Theory 1 9
What About the Voltage Across the Inductor?
Rt
Ls s
R Rt t
L Lss
div L
dt
V Vdv L e
dt R R
V Rv L e V e
R L
ECE 201 Circuit Theory 1 10
ECE 201 Circuit Theory 1 11
Look at the derivative of the voltage
(0)
Rt
Ls
Rt
Ls
s
v V e
dv RV e
dt Ldv R
Vdt L
ECE 201 Circuit Theory 1 12
If the voltage continued at this rate
@ ,
s
s s
Rv V t
Lt
R Lv V V
L R
ECE 201 Circuit Theory 1 13