1

SECOND ORDERSECOND ORDER

To obtain Coefficient ATo obtain Coefficient A11 and A and A22

General solutionsGeneral solutions

2

02 22 oss

222,1 os

General Characteristic Equation:

LR

2

RC21

LCo1

22 od

for series and for parallel

3

Three types of response:Three types of response:

the underdamped response , If 3. , If 2.

the overdamped response , If 1.

0

0

0

the critically damped response

Once we know the type of the response, we Once we know the type of the response, we can write its general solution.can write its general solution.

4

General Solution for Overdamped Response:tsts eAeAVtv 21

21)()(

tetAAVtv )()()( 21

)sin()cos()()( 21 tAtAeVtv ddt

General Solution for Critically Damped Response:

General solution for Underdamped Response:

5

To solve for ATo solve for A11 and A and A22

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Overdamped Response:

tsts eAeAVtv 2121)()(

21)()0( AAVv

tsts eAseAsdt

tdv21

2211)(

2

1

2211)0( AsAs

dtdv

To solve for ATo solve for A11 and A and A22

Expression for v(t) and dv(t)/dtExpression for v(t) and dv(t)/dt

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Critically Damped Response:

tetAAVtv )()()( 21

2)()0( AVv

21)0( AA

dtdv

1

2

)())(()(221

tt eAetAAdt

tdv

Expression for v(t) and dv(t)/dtExpression for v(t) and dv(t)/dt

To solve for ATo solve for A11 and A and A22

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Underdamped Response:

)sin()cos()()( 21 tAtAeVtv ddt

1)()0( AVv

12)0( AA

dtdv

d

1

2

To solve for ATo solve for A11 and A and A22

)sin()cos(

)cos()sin()(

21

21

tAtAe

tAtAedt

tdv

ddt

ddddt

Expression for v(t) and dv(t)/dtExpression for v(t) and dv(t)/dt

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Problem in mathematics

10)(15)(10)( tx

dttdx

dttdx

A Comparison

2)0(10)0( dt

dxx

Find x(t) for t>0.

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Problem in circuits : Problem 1

A Comparison

Find v(t) for t>0.

24 V

R = 5 1 H

v+

-

i

v (0) = 10Vi (0) = 2A

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What are the differences?

A Comparison

1. The differential equation is not given.

2. v(0+) is given but dv(0+)/dt is not.We have to find dv(0+)/dt from the circuit by usingdv(0+)/dt = i(0+)/C.

We have to find from the circuit, in particular from the source free circuit. For a series circuit, it always has the same form. Memorize it.

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Problem in circuits: Problem 2

A Comparison

The switch has been closed for a long time and it is open at t = 0. Find i(t) for t > 0.

vi

24 V

R = 1 1 H

0.25 F

+

-

1

t = 0

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What are the differences?

A Comparison

1. The differential equation is not given.

2. v(0+) and dv(0+)/dt are not given.We have to find v(0+) and dv(0+)/dt from the circuit. From the given information, the circuit is in steady state at t = 0.

We have to find from the source free circuit.Since this circuit is a series RLC circuit, we can write it directly.

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Problem in circuits: Problem 3

A Comparison

The switch has been opened for a long time and it is closed at t = 0. Find i(t) for t > 0.

vi

24 V

R = 1 1 H

0.25 F

+

-

1

t = 0

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What are the differences?

A Comparison

1. The differential equation is not given.

2. v(0+) and dv(0+)/dt are not given.We have to find v(0+) and dv(0+)/dt from the circuit. From the given information, the circuit is in steady state at t = 0.

We have to find from the source free circuit.Since this circuit is a general second order; like it or not, we have to derive the differential equation from the source free circuit. Then, obtain the characteristic equation.

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(Problem 2)Steps to solve this problem

The switch has been closed for a long time and it is open at t = 0. Find i(t) for t > 0.

vi

24 V

R = 1 1 H

0.25 F

+

-

1

t = 0

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1. Draw the circuit for t = 0-

This statement means that the circuit is in steady state at t = 0-. Therefore, C is open and L is shorted.

“The switch has been closed for a long time and it is open at t = 0”

24 V

1

1

i

v

+

-

Find i(0) and v(0)

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2. Draw the circuit for t = 0+

This is a starting point the circuit to experience transient. Therefore, C is not open and L is not shorted.

We know that i(0-) = i(0+) and v(0-) = v(0+)

24 V

1

i(0+) =0.25 F

+

-

v(0+) =

Find dv(0+)/dt or/and di(0+)/dt

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3. Draw the circuit for t = ∞At t = ∞ the circuit reaches steady state again. Therefore, C is open and L is shorted.

24 V

1

+

-

Find v(∞) or/and i(∞)

i

v

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4. Draw the source free circuit for t >0

1H 1

+

-

Voltage source is shorted and current source is opened.

i

v0.25F

Find the differential equation for the source free circuit. Then its characteristic equation. Since the circuit is RLC series, we can directly write its characteristic equation. Determine the type of the response.

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5. Write the general solution for the circuit for t > 0.

24 V

1

i0.25 F

+

-

v

22

6. Find A1 and A2

24 V

1

i0.25 F

+

-

v

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7. Find other circuit quantities for t > 0.

24 V

1

i0.25 F

+

-

v

+ -vL

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Example 1

A series RLC circuit has R = 10 k, L = 0.1 mH, and C = 10 F. What type of damping is exhibited by the circuit.

Example 2

A parallel RLC circuit has R = 10 k, L = 0.1 mH, and C = 10 F. What type of damping is exhibited by the circuit.

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Example 4The responses of a series RLC circuit are

ttL

ttc

eeti

eetv1020

1020

3040)(

301030)(

Determine the values of R, L, and C.

V

mA

(a) Overdamped (b) Critically damped(c) Underdamped

If R = 20 , L = 0.6 H, what value of C will make an RLC series circuit:

Example 3

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Example 5

Find i(t) in the circuit of Fig. 8.10. Assume that the circuit has reached steady state at t = 0-.Fig 8.10

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Example 8.9

• Find v(t) for t > 0 in Fig. 8.29.

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Example 8.9

• Find v(t) for t > 0 in Fig. 8.29.

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Problem 8.56

• Find i(t) for t > 0 in Fig. 8.102.