Natural and Step Response of Series & Parallel RLC Circuits (Second-order Circuits)

Objectives:Determine the response form of the circuitNatural response parallel RLC circuitsNatural response series RLC circuitsStep response of parallel and series RLC circuits

Natural Response of Parallel RLC Circuits

The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.

It is convenient to calculate v(t) for this circuit because

A. The voltage must be continuous for all time

B. The voltage is the same for all three components

C. Once we have the voltage, it is pretty easy to calculate the branch current

D. All of the above

Natural Response of Parallel RLC Circuits

The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.

0)(1

)(1)(

0)(1

)(1)(

0)(

)(1)(

2

2

2

2

0

0

dt

tdv

RCtv

LCdt

tvdC

dt

tdv

Rtv

Ldt

tvdC

R

tvIdxxv

Ldt

tdvC

t

:form standard in place to by sides both Divide

:integral the remove to sides both ateDifferenti

:KCL

Natural Response of Parallel RLC Circuits

The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.

0)(1

)(1)(

2

2

dt

tdv

RCtv

LCdt

tvd:equation Describing

This equation isSecond orderHomogeneousOrdinary differential equationWith constant coefficients

Once again we want to pick a possible solution to this differential equation. This must be a function whose first AND second derivatives have the same form as the original function, so a possible candidate is

A. Ksin t

B. Keat

C. Kt2

Natural Response of Parallel RLC Circuits

The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.

0)(1

)(1)(

2

2

dt

tdv

RCtv

LCdt

tvd:equation Describing

The circuit has two initial conditions that must be satisfied, so the solution for v(t) must have two constants. Use

0)]1()1([)]1()1([

0)(1

)(1

)(

)(

21

212121

21

22

2

211

2

1

2122112

2

21

2

1

21

tsts

tstststststs

tsts

eALCsRCseALCsRCs

eAeALC

eAseAsRC

eAseAs

eAeAtv :SubstituteV;

Natural Response of Parallel RLC Circuits

The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.

0)1()1(

)(

)(1)(

2

21

21

2

2

21

LCsRCs

ss

eAeAtv

tvLCdt

tvd

tsts

:EQUATION STICCHARACTERI the for solutions are and Where

:Solution

0dt

dv(t)

RC

1:equation Describing

is called the “characteristic equation” because it characterizes the circuit.

A. True

B. False

0)1()1(2 LCsRCs

Natural Response of Parallel RLC Circuits

The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.

rad/s) in frequency radian resonant (the and

rad/s) in frequency neper (the where

0LC

RC

LCRCRCs

LCRCRCsLCsRCs

1

2

1

)1()21()21(

2

)1(4)1()1(;0)1()1(

2

0

22

2,1

2

2,1

2

The two solutions to the characteristic equation can be calculated using the quadratic formula:

So far, we know that the parallel RLC natural response is given by

A. The value of

B. The value of 0

C. The value of (2 - 02)

and where 0LCRC

s

eAeAtv tsts

1

2

1

)(

2

0

2

2,1

2121

There are three different forms for s1 and s2. For a parallel RLC circuit with specific values of R, L and C, the form for s1and s2 depends on

Natural Response – Overdamped Example

Given V0 = 12 V and I0 = 30 mA, find v(t)for t ≥ 0.

rad/s rad/s,

case! overdamped the is this so

rad/s

rad/s

2

0

000,2050007500000,12

)000,10()500,12(500,12

000,10)2.0)(05.0(

11

500,12)2.0)(200(2

1

2

1

21

222

0

2

2,1

2

ss

s

LC

RC

o

Natural Response – Overdamped Example

Given V0 = 12 V and I0 = 30 mA, find v(t)for t ≥ 0.

circuit the in equation the in

circuit the in equation the in

:circuit the in conditions initial the

satisfy to equation the in tscoefficien the use must we Now

for V,

00

00

000,20

2

5000

1

)()(

)()(

0)(

tt

tt

tt

dt

tdv

dt

tdv

tvtv

teAeAtv

Natural Response – Overdamped Example

Given V0 = 12 V and I0 = 30 mA, find v(t)for t ≥ 0.

21

)0(000,20

2

)0(5000

1

21

0

21

)0(000,20

2

)0(5000

1

000,205000

)000,20()5000()0(

12

12)0(

)0(

AA

eAeAdt

dv

AA

Vv

AAeAeAv

:Equation

V :Circuit

:Equation

Now we need the initial value of the first derivative of the voltage from the circuit. The describing equation of which circuit component involves dv(t)/dt?

A. The resistor

B. The inductor

C. The capacitor

Natural Response – Overdamped Example

Given V0 = 12 V and I0 = 30 mA, find v(t)for t ≥ 0.

000,450000,205000

000,450200

1203.0

2.0

1)0()0(

1)0(

))0()0((1

)0(1)0()(

)(

000,205000

)000,20()5000()0(

21

21

)0(000,20

2

)0(5000

1

AA

R

vi

Cdt

dv

iiC

iCdt

dv

dt

tdvCti

AA

eAeAdt

dv

L

RLCC

V/s

:Circuit

:Equation

Natural Response – Overdamped Example

Given V0 = 12 V and I0 = 30 mA, find v(t)for t ≥ 0.

(OK)

(OK) V :Checks

for V,

Thus,

usly,simultaneo Solving

for V,

0)(

122614)0(

02614)(

26,14

000,450000,205000;12

0)(

000,205000

21

2121

000,20

2

5000

1

v

v

teetv

AA

AAAA

teAeAtv

tt

tt

Natural Response – Overdamped Example

Given V0 = 12 V and I0 = 30 mA, find v(t)for t ≥ 0.

You can solve this problem using the Second-Order Circuits table:

1. Make sure you are on the Natural Response side.2. Find the parallel RLC column.3. Use the equations in Row 4 to calculate and 0.4. Compare the values of and 0 to determine the

response form (given in one of the last 3 rows).5. Use the equations to solve for the unknown coefficients.6. Write the equation for v(t), t ≥ 0.7. Solve for any other quantities requested in the problem.