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