Circuit Analysis in S-domain

8/11/2019 Circuit Analysis in S-domain

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Circuit Analysis in Laplace Domain

SDomain Analysis


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The Laplace Transform

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

0

)()()]([ dtetfsFtfL st

The Inverse Laplace Transform is defined by

j

j

tsdsesF

jtfsFL

)(2

1)()]([1

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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). If we can remember thePair relationships between approximately 10 of the

Laplace transform pairs we can go a long way.3


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Laplace Transform of the unit step.

|0

0

11)]([

stst es

dtetuL

stuL 1)]([

The Laplace Transform of a unit step is:s

1

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Time Differentiation:

Making the previous substitutions gives,

0

00

)()0(0

)()( |

dtetfsf

dtsetfetfdt

dfL

st

stst

So we have shown:

)0()()(

fssFdt

tdfL

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Final Value Theorem:

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

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

and the exists, then)(lim ssFs

)()(lim)(lim ftfssF0s t

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

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Circuit Element Modeling in SDomain

1.0 Energy Sources

7

i(t) I(s)

+_ +_v(t) V(s)


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2.0 Resistance

Time Domain

Complex Frequency Domain

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3.0 Inductor

di(t)

v t = LL dt

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

Mesh-Current Model

Nodal-Analysis Model

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4.0 Capacitor

1( ) ( ) (0)

0

tv t i t dt vc c

C

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Circuit Analysis in theSDomain

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Summary

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Example

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

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Solution

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The Transfer function

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.

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The Transfer function

The transfer function is also known as the network function.

there are four possible

transfer functions:16


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Application

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

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Application

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

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

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(b)

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SINUSOIDAL FREQUENCY ANALYSIS

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

Circuit represented bynetwork function

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Application

solution

a) Calculate the transfer function Vo/Vi

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

expression of Vo.

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Thats all Folks ! 22

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Circuit Analysis in S-domain