Step Response of Second-order RLC Circuits

The problem – find the response for t ≥ 0. Note that there may or may not be initial energy stored in the inductor and capacitor!

The circuit for the parallel RLC step response is repeated here. Consider how this circuit behaves as t ∞. Which component’s final value is non-zero?

A. The resistor

B. The inductor

C. The capacitor

D. All of the above

Step Response of a Parallel RLC Circuits

As t ∞:

response) natural the of form (the FL Iti )(

The only component whose final value is NOT zero is the inductor, whose final current is the current supplied by the source. We now have to construct a response form that can satisfy two initial conditions and one non-zero final value. We can satisfy the final value directly if we specify the inductor current as the response we will solve for:

Step Response of a Parallel RLC Circuit

The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.

To begin, find the initial conditions and the final value. The initial conditions for this problem are both zero; the final value is found by analyzing the circuit as t ∞.

Step Response of a Parallel RLC Circuit

The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.

t ∞:

mA 24FI

Step Response of a Parallel RLC Circuit

The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.

Next, calculate the values of and 0 and determine the form of the response:

rad/s nm

rad/s n

000,40)25)(25(11

000,50)25)(400(2

1

2

1

0

LC

RC

We just calculated = 50,000 rad/s and 0 = 40,000 rad/s, so the form of the response is

A. Overdamped

B. Underdamped

C. Critically damped

Once we know the response form is overdamped, we know we have to calculate

A. d

B. s1 and s2

C. Nothing additional

Step Response of a Parallel RLC Circuit

The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.

Since the response form is overdamped, calculate the values of s1 and s2:

0024.0)(

000,80000,20

000,30000,50

000,40000,50000,50

000,80

2

000,20

1

21

222

0

2

2,1

teAeAti

ss

s

tt

L A,

rad/s and rad/s

rad/s

Step Response of a Parallel RLC Circuit

The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.

Next, set the values of i(0) and di(0)/dt from the equation equal to the values of i(0) and di(0)/dt from the circuit.

0)0(

024.0)0(

0

21

Ii

AAi

L

L

:circuit the From

:equation the From

0024.0)( 000,80

2

000,20

1 teAeAti tt

L A,

Step Response of a Parallel RLC Circuit

The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.

0)0()0(

000,80000,20)0(

0

21

L

V

L

v

dt

di

AAdt

di

LL

L

:circuit the From

:equation the From

0024.0)( 000,80

2

000,20

1 teAeAti tt

L A,

Step Response of a Parallel RLC Circuit

The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.

083224)(

832

0000,80000,20

0024.0

000,80000,20

21

21

21

teeti

AA

AA

AA

tt

L mA,

mA mA;

and

:Solve

0024.0)( 000,80

2

000,20

1 teAeAti tt

L A,

We can check this result at t = 0 and as t ∞; from the equation we get

A. iL(0)=32 mA, iL(∞)=0

B. iL(0)=0, iL(∞)=0

C. iL(0)=0, iL(∞)=24 mA

083224)( 000,80000,20 teeti tt

L mA,

If we now want to find v(t) for t ≥ 0, we need to

A. Find the derivative of the current and multiply by L

B. Find the integral of the current and divide by C

C. Multiply the current by R

083224)( 000,80000,20 teeti tt

L mA,

Step Response of RLC Circuits – Summary

Use the Second-Order Circuits table:1. Make sure you are on the Step Response side.2. Find the appropriate column for the RLC circuit topology.3. Find the values of the two initial conditions and the one

non-zero final value.Make sure the initial conditions and final value are defined exactly as shown in the figure!

4. Use the equations in Row 4 to calculate and 0.5. Compare the values of and 0 to determine the

response form (given in one of the last 3 rows).6. Write the equation for vC(t), t ≥ 0 (series) or iL(t), t ≥ 0

(parallel), leaving only the 2 coefficients unspecified.7. Use the equations provided to solve for the unknown

coefficients.8. Solve for any other quantities requested in the problem.

Step Response of a Series RLC Circuit

The problem –find vC(t) for t ≥ 0.

Find the initial conditions by analyzing the circuit for t < 0:

A

V

V k k

k

0

50

)80(159

15

0

0

I

V

Step Response of a Series RLC Circuit

The problem –find vC(t) for t ≥ 0.

Find the final value of the capacitor voltage by analyzing the circuit as t ∞:

V 100FV

Step Response of a Series RLC Circuit

The problem –find vC(t) for t ≥ 0.

Use the circuit for t ≥ 0 to find the values of and 0:

rad/s

dunderdampe

rad/s

rad/s

0

6000

8000000,10

000,10

)2)(005.0(11

8000)005.0(2802

2222

0

2

0

2

d

LC

LR

Step Response of a Series RLC Circuit

The problem –find vC(t) for t ≥ 0.

Write the equation for the response and solve for the unknown coefficients:

06000sin67.666000cos50100)(

67.6650

060008000)0(

50100)0(

06000sin6000cos100)(

80008000

21

210

21

101

8000

2

8000

1

ttetetv

BB

BBC

IBB

dt

dv

BVBVv

tteBteBtv

tt

C

dC

FC

tt

C

V,

V V,

V,