Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker [email protected] Signals and...

Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker [email protected]

1

Signals and Systems Introduction

EEE393 Basic Electrical Engineering

K.A.Peker

Bilkent University


2

From

Circuits to

Signals and Systems


3

From Circuits to Signals and Systems

Output(t) = System{ Input(t) } y(t) = H{ x(t) }

+Vin

Iout Iout

+VinIin

Iin

+Vout

-

+Vout

-

H{•}x(t) y(t)


4

Some System Properties

Linear Systems: y1(t) = H{ x1(t) }

y2(t) = H{ x2(t) }

Time-Invariant Systems: y(t) = H{ x(t) } y(t - t0) = H{ x(t -

t0) }

same behavior at all times

H{ a·x1(t) + b·x2(t) }

= a·y1(t) + b·y2(t)


5

Linear Time Invariant Circuits R, L, C linear, time invariant elements

V = R·i linear, time invariant C and L are also linear, time invariant

(diff. is linear):

R-L-C circuits are also linear, time invariant systems

dt

dvCi 1

1

dt

dvCi 2

2

21

21

21

biaidt

dvCb

dt

dvCa

dt

bvavdC


6

ODE

Linear time-invariant (LTI) systems are usually modeled usingconstant coefficient, linear,

ordinary differential (or integral) equations

Time-invariant

Linear system (no square etc. of derivatives)

+v(t)

R

L

i(t)

Ci

Cdt

diR

dt

idL

dt

dv

dt

diLdi

CRiv

t

1

1

2

2

+v(t)

R

L

i(t)

+v(t)

Ri(t)

C

iRdt

diLv c

c

t

vCdt

dvRv

diC

iRv

OR , 1


7

Same math: LTI systems from other fields

MF

Ffriction=D·voutput:input : ,

,

vFDvdt

dvMF

dt

dvaaMF

M F

Fspring= - k·x

k

output:input : ,

2

2

xFxkdt

xdMF

xkaMF

M F

k

D output:input :

2

2

xF

xkdt

dxD

dt

xdMF

vDxkaMF

auto suspension system


8

Satellite control, riding a bicycle, process control, nuclear system control, …

LTI systems from other fields (cont’d)

A1: area 1

R2R1

A2: area 2

h1

h2

u(t)

2

2

1

2122

1

2111 )(

R

h

R

hh

dt

dhA

R

hhtu

dt

dhA

inputOutput

(OR “state variables”)

Rh

)(1

)(

tuhRdt

dhA

R

htu

dt

dhA


9

LTI systems from other fields (cont’d) (Arms race)

X: arms production of country 1 Y: arms production of country 2

nYmXdt

dY

bXaYdt

dX

)(2

2

anXmanbdt

dXnb

dt

Xd

YbXdt

dX

a

1

competitioncurrent strength


10

LTI systems from other fields (cont’d) (Oligopoly/duopoly) – economics, game theory

Two companies in a market, producing same product

produced) more when declines price ,( ,)(

demand)-(supply volumeproductionfor priceMarket : )(

babaxxp

xxp

product producing ofcost 2,company ,)(

product producing ofcost 1,company ,)(

2company by production of Volume :

1company by production of Volume :

222222

111111

2

1

xcxbxC

xcxbxC

x

x

222212

2

111212

1

11211

)(2

)(1

)()(1 :Profit1

cxbbxaxaxP

cxbbxaxaxP

xCxxpxP

21222

12111

2

2

bbaxaxmdt

dx

bbaxaxmdt

dx

If change in production is proportional to marginal change in profit


11

So…

RLC circuits are

…linear time-invariant systems

…described by linear ordinary differential equations (ODE)

…which are used to model many other systems

in different sciences and engineering fields

In fact, they’re mathematically same as many other

physical/social/economic etc. systems (analogous)

Hence, the math and the techniques we learn here apply to many

other problems in other domains


12

Response of LTI Systems

- Response to complex exponentials

- AC circuit analysis


13

How do we handle these diff. equations? There are various ways to approach this problem

We will start with Steady state response … to complex exponentials

)(... 01 tfxadt

dxa

dt

xda

n

n

n

0... 01 tt

nt

n

n xadt

dxa

dt

xda

There are two components of the solution

(Homogenous equation) xt : Complementary solution, Natural response (systems own nature), Transient response (usually dies away)

)(... 01 tfxadt

dxa

dt

xda s

sns

n

n xs : Particular solution, Steady state response

)()(0)()(

...)(

01 tftfxxadt

xxda

dt

xxda st

stn

stn

n


14

The complex exponential

Everything starts with the simple but important fact that;

Corollaries: If we assume complex exponential input, then

all currents/voltages in an RLC circuit are also complex exponentials

Differential equations become algebraic (polynomial) equations

tsts

esdt

ed

Derivative of a complex exponential is itself, multiplied by a constant

ion)superposit ,(linearity generalin , ,

,ts

i

st

ieXiv

Xeiv

Complex exponentials are eigen functions of LTI systems

Linear systems – vector spaces – basis expansion

Laplace domain – exp(st), Fourier domain – exp(jw)


15

The complex exponential (cont’d) From diff. equations to polynomial equations

0

,

:form in the be tohas

:Assume

2

2

22

2

2

AkXsDXMXs

ekXsDXMXseA

eAFF

ekXsDXMXsekXesDXeMXsF

eXx

xkdt

dxD

dt

xdMF

stst

st

stststst

st

Note:

Complex exp. input Complex exp. output

(works both ways)


16

Complex exponentials with (s=j) Let’s assume our inputs are in the form:

Why? Because …

and, cos(t+) is AC circuits (I.e. very important practical case)

tjtjjtj eXeeAeA

i

tji

ieA Or, more generally(linearity, superposition)

Special case of for jws

eX ts

tj

ii

tjjtjj

tj

ieX

eeA

eeA

tA

tjtAeA

22)cos(

OR , )sin()cos(

)(

Phasor

Can obtain AC response by summing complex exp. responses

(shortcut: use Re{.} operation)


17

Element characteristics for ejt input

R:

C:

L:

tjtjjtj eXeeAeAtv )( AssumePhasor

tjeR

XiRiv ,

Only amplitude change (attenuated)

v=R·i still holds.

iCj

vCji

v

vCjXejCdt

dvCi tj

1 ,

1

Same signal, multiplied by jC

Like v=R·i . Ratio is a constant, independent of time.Can solve like a DC (constant) circuit.

iLjvLji

v

vLj

eLj

XdeX

Li

dt

diLv tjt j

,

11


18

Impedance

For complex exponential v(t), i(t), with frequency

Capacitor characteristics C:

Inductor characteristics L:

and are just like resistance values

(constant ratio), but are complex values

We call this ratio of complex v to i, ( ), “Impedance” (Z)

iCj

v 1

iLjv

Cj1

Lj

i

v

i

v


19

Impedance (cont’d)

R: Resistive part X: Reactance

|Z|: Magnitude (how it attenuates the input – change on input amplitude)

z: Phase (change on input phase)

)()( zjeZjXRZ

NOTE: A LTI system changes only TWO things in a complex exponential (or a sinusoid):

- Amplitude (multiplied by magnitude of transfer function, |H()|)

- Phase (shifted by phase of transfer function, H)

Multiply magnitudes, add phase angles

Impedance can be purely real (Resistance), purely imaginary (Capacitance or Inductance), or a mix (mixed circuits).


20

A closer look at C, L impedances C:

L:

)1

(1

Cj

CjZ

Negative Reactance: “Capacitive reactance”

)( LjZ Positive Reactance: “Inductive reactance”

)2

(1

je

CZ

)2

(

jeLZ

Magnitude inversely proportional to .

Pass high frequencies, stop low (in: v, output: i)

Open circuit for DC, short circuit for high freq.

Magnitude proportional to .

Pass low frequencies, stop high (in: v, output: i)

Short circuit for DC, open circuit for high freq.

Phase shifted by –900 (voltage 900 behind current)

2

jej

)2

(

jej

Phase shifted by 900 (voltage 900 ahead of current)

v = Z·i


21

AC circuit analysis

For a given input AC voltage (or current):

Find corresponding complex signal V (or I), such that:

Replace all C with and all L with ,

Solve like a DC circuit (reductions, equivalent Z, loop currents, node voltages, superposition, etc),

Find complex output I (or V)

Convert back to real world by

)cos()( OR )cos()( 00 tAtitAtv

tjj eeAtVtvtV 0)( )()(Re

Cj1

Lj

)(Re)( OR )(Re)( tVtvtIti

Phasor


22

AC circuit analysis - phasorsA small simplification:

Phasor carries the amplitude and phase information LTI system (i.e. RLC circuit) changes

only the amplitude and phase

tjj eeAtVtAtv 00 )( )cos()(Phasor

For single AC analysis, we can use only the phasors of voltages and currents

Replace voltage (or current) sources with corresponding phasors

When multiplying/dividing with Z,multiply/divide magnitudes and amplitudes, add/subtract phase angles

Convert output phasor back to a sinusoid

jeAtA 00 )cos(

)cos( 00 tAeA j

)(00)(00 , ZZ j

jjj e

Z

A

Z

eAeAZeAZ


23

Simple AC circuits

Series RC:

Series RL:

+v(t)

Ri(t)

C v

RCj

Rv

CjRZ

vi

CjRZ

11

11

1

1

Cj1

+v(t)

Ri(t)

L Lj v

RL

j

Rv

LjRZ

vi

LjRZ

1

11

Transfer Function


24

Simple AC circuits

Series RLC:

Parallel RLC:

+v(t)

R

L

i(t)

CCj

1

Ljv

RCRL

j

Rv

CjLjRZ

vi

CjLjRZ

11

11

1

1

+v(t) R L

i(t)

C Cj1

Lj

C

L

vL

CjR

vCjLjRZ

vi

CjLjR

Z

1111

111

Transfer Function

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Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker [email protected] Signals and...

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