E E 2415

Lecture 10 - Step Response of Series and Parallel RLC Circuits

Step Response

• DC Sources are present after switches change

• Transient parts have same form as in natural response

• Use Steady-state response and initial conditions to determine constants

• In steady-state with DC sources:– inductors are as short circuits– capacitors are as open circuits

Series RLC Step Response (1/5)

Initial Conditions:

Switch has been closed for a long time.

Steady-state Conditions:

Since series RLC, start with capacitor voltage:

Series RLC Step Response (2/5)

Now find current in capacitor as:

Use initial conditions:

then

Series RLC Step Response (3/5)

0 0.02 0.04 0.06 0.08 0.140

60

80

100

120Voltage Across Capacitor

vc t( )

t

Series RLC Step Response (4/5)

0 0.02 0.04 0.06 0.08 0.10

5

10

15Inductor Current

i t( )

t

Series RLC Step Response (5/5)

Parallel RLC Step Response (1/6)

Initial Conditions:

Switch has been open for a long time.

Steady-state Conditions:

0 0 0 20Lv V i A

From experience, we know this is a damped sinusoid. Therefore, start with that form.

Parallel RLC Step Response (2/6)

Evaluate at initial conditions.

Parallel RLC Step Response (3/6)

201 2

2 1

20 100 cos 100

40 20 100 sin 100

t k k tev t

k k t

Evaluate at initial conditions.

Parallel RLC Step Response (4/6)

201 2

2 1

20 100 cos 100

40 20 100 sin 100

t k k tev t

k k t

0 0.05 0.1 0.15 0.220

40

60

80

100Inductor Current

iL t( )

t

Parallel RLC Step Response (5/6)

0 0.05 0.1 0.15 0.250

0

50

100Capacitor Voltage

v t( )

t

Parallel RLC Step Response (6/6)