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Sinusoids

Period: T

Time necessary to go through one cycle

Frequency:f = 1/T

Cycles per second

Radian frequency: w = 2pf

Amplitude: VM

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Phase

-8

-6

-4

-2

0

2

4

6

8

0 0.01 0.02 0.03 0.04 0.05

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Leading and Lagging Phase

x1(t) leadsx2(t) by q-

x2(t) lagsx1(t) by q-

qw tXtx M cos)( 11

w tXtx M cos)( 22

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Complex Numbers

x is the real part

y is the imaginarypart

zis the magnitude

q is the phase

q

x

y

real

axis

imaginary

axis

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More Complex Numbers

Polar Coordinates: A =zq

Rectangular Coordinates: A =x +jy

qcoszx qsinzy

22yxz

x

y1tanq

There is a good chance that your calculator

will convert from rectangular to polar and from

polar to rectangular.

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Arithmetic With Complex

Numbers We need to be able to perform computation with

complex numbers.

Addition Subtraction

Multiplication

Division

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Addition

Addition is most easily performed in rectangular

coordinates:

A =x +jy

B =z+jw

A + B = (x + z) +j(y + w)

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Subtraction

Subtraction is most easily performed in

rectangular coordinates:

A =x +jy

B =z+jw

A - B = (x - z) +j(y - w)

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Subtraction

Real

Axis

Imaginary

Axis

AB

A - B

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Multiplication

Multiplication is most easily performed in polar

coordinates:

A =AMq

B =BM

AB = (AMBM) (q )

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Multiplication

Real

Axis

Imaginary

Axis

A

B

A B

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Division

Division is most easily performed in polar

coordinates:

A =AMq

B =BM

A / B = (AM/BM) (q )

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Division

Real

Axis

Imaginary

Axis

A

B

A / B

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Phasors

A phasor is a complex number that represents the

magnitude and phase of a sinusoid:

qw tXM cos

q MXX

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Complex Exponentials

A complex exponential is the mathematical tool

needed to obtain phasor of a sinusoidal function. A complex exponential is

ejwt = cos wt+j sin wt

A complex numberA =zq can be represented

A =zq =z ejq =zcos q +j zsin q

We represent a real-valued sinusoid as the realpart of a complex exponential.

Complex exponentials provide the link betweentime functions and phasors.

Complex exponentials make solving for ACsteady state an algebraic problem

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Complex Exponentials (contd)

Aejwt=z ejqejwt=z ej(wt+q

z ej(wt+q =zcos (wt+q +j zsin (wt+q

Re[Aejwt] =zcos (wt+q

What do you get when you multipleA withejwt

for the real part?

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Sinusoids, Complex Exponentials,

and Phasors Sinusoid:

zcos (wt+q Re[Aejwt]

Complex exponential:

Aejwt=z ej(wt+q, Az ejq,

Phasor for the above sinusoid:

V =zq