CIRCUIT ANALYSIS IN LAPLACE DOMAIN

“S” DOMAIN ANALYSIS

2

Pierre Simon Laplace (1749–1827),

The Laplace Transform

The Laplace Transform of a function, f(t), is defined as;

0

)()()]([ dtetfsFtfLst

The Inverse Laplace Transform is defined by

j

j

tsdsesF

jtfsFL

)(2

1)()]([1

3

The Laplace Transform

An important point to remember:

)()( sFtf

• The above is a statement that f(t) and F(s) are transform

pairs.

• What this means is that for each f(t) there is a unique F(s)

and for each F(s) there is a unique f(t).

4

The Laplace Transform

Laplace Transform of the unit step.

|0

0

11)]([

stste

sdtetuL

stuL

1)]([

The Laplace Transform of a unit step is:s

1

5

The Laplace Transform

Time Differentiation: Making the previous substitutions gives,

0

00

)()0(0

)()( |

dtetfsf

dtsetfetfdt

dfL

st

stst

So we have shown:

)0()()(

fssFdt

tdfL

6

The Laplace Transform

If the function f(t) and its first derivative are Laplace transformable

and f(t) has the Laplace transform F(s),

and the exists, thens

)()(lim)(lim ftfssF0s t

Again, the utility of this theorem lies in not having to take the inverse

of F(s) in order to find out the final value of f(t) in the time domain.

This is particularly useful in circuits and systems.

7

Final Value Theorem:

sF(s)lim

8

9

APPLICATION TO CIRCUITS

In the AC sinusoidal circuits we follow the Phasor theory:

But in Sinusoidal and Non-Sinusoidal circuits we follow

Laplace’s Theory:

Circuit Element Modeling in “S” Domain

i(t) I(s)

+_ +

_v(t) V(s)

1.0 Energy Sources

10

2.0 Resistance

Time Domain

Complex Frequency Domain

11

3.0 Inductor

di(t)

v t = LL dt

L [VL(t)] = VL(S)

Mesh-Current Model

Nodal-Analysis Model

12

4.0 Capacitor

1( ) ( ) (0)

0

tv t i t dt vc cC

13

14

CIRCUIT ANALYSIS IN THE “S” DOMAIN

Summary

15

Example 1

16

• Find vo(t) in the circuit, assuming zero initial conditions

Solution

17

18

Example 2

Find vo(t) in the circuit of Fig. shown. Assume vo(0) = 5 V.

Solution

19

20

We transform the circuit to the s domain as shown. The initial condition is included

in the form of the current source Cvo(0) = 0.1(5) = 0.5 A.

Example 3

21

In the circuit in Fig. shown, the switch moves

from position a to position b at t = 0. Find i(t)

for t > 0.

Solution

The initial current through the inductor is i(0) = Io.

For t > 0, Fig shows the circuit transformed to the s

domain. The initial condition is incorporated in the

form of a voltage source as Li(0) = LIo.

Using mesh analysis,

22

Example 4

Find Vo(t) assuming Vc(0) = 5 V

23

24

The Transfer function

25

The transfer function is a key concept in signal processing

because it indicates how a signal is processed as it passes

through a network.

It is a fitting tool for finding the network response, determining (or

designing for) network stability, and network synthesis.

The transfer function of a network describes how the output

behaves in respect to the input.

It specifies the transfer from the input to the output in the s

domain, assuming no initial energy.

The Transfer function

26

• The transfer function is also known as the network function.

There are four possible transfer

functions:

Application

27

• Determine the Transfer Function H(s) = Vo(s)/Io(s) of the circuit

Application

28

find: (a) the transfer function H(s) = Vo/Vi ,

(b) the response when vi (t) =u(t) V

(b)

29

SINUSOIDAL FREQUENCY ANALYSIS

30

)cos(0 tB )(cos|)(|0 jHtjHB )(sH

Circuit represented by

network function

Application

31

a) Calculate the transfer function Vo/Vi

b) if Vi = 2cost(400t) V, what is the steady state

expression of Vo

solution

Good Luck

32