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S-plane Explained

Introduction

Motivation for s-plane analysis

When I say s-plane, what do I mean?

How to use the s-plane

Circuit analysis/design: the Integrator

Signal analysis/design: FM

A (P)REVIEW OF LAPLACE TRANFORMS AND

HOW TO USE THEM IN EE122

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Motivation for s-plane analysis

Understanding circuits in another

domain (the frequency domain) The more ways we can understand our

circuit, the better because...

The time domain is hard for some things

filtering = convolution (Doh!)

The s-plane (often) makes things easier

filtering = multiplication (Woohoo!)

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Math behind the s-plane

The actual Laplace Transform*:

h(t) could be a signal, or a system As a signal:

As a system:

H(s): Laplace Transform of h(t)

s = + j: Complex Frequency Variable

*: not necessary for EE122, but interesting (maybe)

=

0)()( dtethsH st

)16.015sin()( += tth

)(

)()(

tv

tvth

in

out=

The Laplace Transform of a time-domain function gives us

a function of the complex frequency variable s

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A note about transforms

They make life easier (or should)

If youve taken E40 (and/or Math 53),

you have already used transforms

E40 uses phasor analysis of linear circuits

Complex impedances are just a tricky way ofgetting around differential equations so that we

can take C as a resistor of value 1/ jC.

Indeed, they are simply Laplace (or Fourier)transforms in disguise--as you will see.

Math 53 uses Laplace Transforms to solve

differential equations

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What do I mean when I say s-plane?

Example:

Tent pole plot of the

magnitude of a

Chebychev filters

response to inputs ofcomplex frequency s

is what you usually

think of as radianfrequency (=2f)

is decay rate*

ANSWER: THE REAL () AND IMAGINARY () PARTS OF s

ARE THE x and y ON A GRAPH

*: Not important for EE122, but used in stability analysis

-axis axis s-plane

|H(s)|

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Why should I use the s-plane?

Turns differential equations (hard) into

algebraic equations (easy)

Turns convolution (hard) into multiplication

(easy)

THIS IS THE SAME QUESTION AS:WHY IS THE LAPLACE TRANSFORM USEFUL?

inoutinoutout CvvsBAtCvdtdvBtAv =+=+ )()()(

sBACsH

vv

in

out

+== )(

)()()()()()( sGsHsFtgthtf ==

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How to use s-plane analysis

Examples to follow:

Systems Simple RC circuit

Integrator

Signals

Multiplication by a sinusoid: Heterodyning

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RC circuit (simple)

Straight from EE122 lecture notes:

sRCsH

vv

vsRCv

TransformLaplaceMagictvdtdvRCtv

dtdvCi

iRtvtverivation

in

out

inout

inoutout

out

inout

+==

=+

=+

=

=

11)(

)1(

)("")()(

)()(:

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Integrator (slightly less simple)

121

2

2211

21

1

2

)(

)(

)//)(()(

/)(

/)()(

))()(()(

:

RCRsR

RsH

v

v

RvvCRsRR

RdtCdvRtvtv

dtCdvti

Rtvti

Rtititv

AnalysisC

in

out

inout

outinout

outC

inin

Cinout

+==

=+

+=

=

=

=

1

2

2

1

R

R

CR2

1

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A familiar slide

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Another familiar slide...

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A last familiar slide

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Heterodyning (for FM radio)

Motivation: Base-band txn is trouble Efficiency

/4 Antenna for kHz ~ length > kilometers Interference

All of the signals would muddle together

Hence, we need distinct carrierfrequencies (regulated by the FCC)

92.3 MHz (KSJO)

88.5 MHz (KQED)

1070 kHz (KNX)

710 kHz (KABC)

FM

AM

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The frequency domain

All base-band:

Heterodyned

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Back to heterodyning

Because convolution in the time domain is multiplicationin the frequency domain, when we multiply by a sinusoid

in the time domain, it convolves the frequency content of

our signal up to the frequency of the sinusoid.(Whew! That was a mouthful)

Base-band signal...

convolved with pure tone...

yields heterodyned signal!

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