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AC Signals

Week 3 - Lecture 1 Mark Bocko

Topics:Brief mathematical review

Afewsimplederiva/ves SimpleharmonicoscillatorNaturalbasee

Complexnumbers Eulersformula

Phasors1

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Limits and Derivatives

2

x

f(x)

x0+x

f(x0+x)

x0

f(x0)

Slope = f(x0+x) - f(x0)x0+x x0

f(x0+x) - f(x0)x

=

Make x 0

limx0

f(x0+ x) f(x

0)

x

d

dxf(x)

x0

f'(x0)

tangent

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tangent x0 x

f(x)

x0+x

f(x0+x)

f(x0)

x0+x

f(x0+x)

f(x0)

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A few simple derivatives we will need

4

d

dx x= 1

d

dx

ax = a

d

dxx

n= nx

n1

d

dxf(x) g(x)

= f(x) g'(x) + f '(x) g(x)

d

dxf(g(x))

= f '(g(x)) g'(x)

d

dxsin(x) = cos(x)

d

dxcos(x) = sin(x)

Product rule:

d

dx e

x=

e

x

Chain rule:

d

dxx sin(x)

= x cos(x) + 1 sin(x)

d

dxsin(x2 )

= cos(x

2 ) 2xd

dxe

ax= ae

ax

d

dxConst = 0

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Whats so special about e?

5

d

dxe

x= e

x The only exponential function (ax) where the slope ofthe function equals the value of the function at everypoint.

Exploding rabbit population: Lets say that there is a colony ofrabbits and that each pair of rabbits has 2 offspring per year.

How does the population grow over time? (Assume that we start withjust two rabbits and that rabbits are immortal and never perish theyjust keep making more rabbits.)

Let y(t) = number of rabbits at time t, and y(t=0) = 2, (initial condition)

d

dt

y = 1 ythen (1 because it takes 2 rabbits to produce 2 more rabbits)

y(t) = Aetsolution is

e=2.71828182845904523536028747135266249775724709369995

and since y(t=0) = 2 A = 2

y(t) = 2et 010,000

20,000

30,000

40,000

50,000

0 5 10

years

rabbits

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sin(t)

d

dtsin(t) = cos(t)

cos(t)

d

dtcos(t) =

sin(t)

- sin(t)

d

dt sin(t)( ) = cos(t)

- cos(t)d

dt cos(t)( ) = sin(t)

Derivatives of sin, cos

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vmax

-vmax

t

xmax

-xmax

t

Simple Harmonic Oscillator

7

F

mk

A

x = xmax v = 0

B

C

x = -xmax v = 0D

E

x(t) = xmaxsin(t)B

D

v(t) = vmaxcos(t)A

B

C

D

E

CA E

x = 0 v = vmax

x = 0 v = - vmax

x = 0 v = vmax

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8

mk

Simple Harmonic Oscillator

d

dtx = v

d

dtv = a

d

dt

dx

dt

d2x

dt2= a

, velocity

, accelerationF = ma

Newtons 2nd Law

F = -kx

Hookes Law

m

d2x

dt2=

kx

x

d2x

dt2=

k

mx

d2x

dt2=

2x

2

k

mlet , so

Can we find a function that satisfies this differential equation?

x = x0sin t( )

d

dtx = x

0 cos t( )

d2

dt2x = x

0

2sin t( )

x(t) = x0sin t( )

k

mwhereso

It works!

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Complex Numbers

9

The need for imaginary numbers

x2 4 = 0 x2 = 4 , so x = +2 and x = -2 are solutions

How about: x2 + 4 = 0

x2 = -4 = (-1) (4) so x = -1 4 x = 2 -1

j = -1

j1 = j j2 = -1 j3 = -j j4 = 1 j5 = j

Complex Number: z = x + j y

realpart

imaginarypart

Re(z) = x

Im(z) = y

x = 2j

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10

Complex Number Arithmetic

Addition: (a + jb) + (c + jd) = (a+c) + j(b+d)

Multiplication: (a + jb) x (c + jd) = (ac-bd) + j(bc+ad)

Complex Conjugate: (a + jb) (a jb)

cc z = a + jb z* = a jb

|z|2 = z x z* = a2 + b2

Graphing Complex Numbers:

r

a

b

real

imagz = (a + jb)

b

a

r2= a

2+ b

2, r = a

2+ b

2

tan =b

a

, = tan1 b

a

sin =b

r, b = r sin

cos=

a

r, a

=

rcos

r = |z|

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11

Complex Exponential Function

sinx = x

x

3

3!+ x

5

5!

x

7

7!+ ...

cosx = 1 x

2

2 !+

x4

4!

x6

6!+ ...

ex

= 1 +x

1 !

+x

2

2 !

+x

3

3!

+x

4

4!

+x

5

5!

+ ...

ej

= 1 +j

1 !+

(j)2

2 !+

(j)3

3!+

(j)4

4!+

(j)5

5!+ ...

ej

= 1 2

2 !+

4

4! ...

+ j

1 !

3

3!+

5

5! ...

ej

= cos + jsin Eulers Formula

n! = n(n-1)(n-2)1

Taylor Series ExpansionshMps:en.wikipedia.orgwikiTaylor_series

ej

Consider

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A remarkable equation!

ej = cos + jsin

rej

= rcos + jr sin

ej

+ 1 = 0or equivalently

real

imag

b

a

r

rcos

rsin

ej0

= cos 0 + jsin 0 = 1

ej

2= cos

2+ jsin

2= j

ej3

2= cos

3

2+ jsin

3

2= j

ej

= cos + jsin =1

ej2

= cos2 + jsin2 = 1

real

imag

ej0

ej/2

ej

ej3/2ej21

j

-1 0

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Complex Arithmetic revisited

13

Simplifications using complex exponential notation:

r1e

j1 r

2e

j2= r

1r2e

j 1+

2( )Multiplication:

r1e

j1

r2e

j2

=

r1

r2

ej

1

2( )Division:

rej

2

= r2

Magnitude:

ej

+ ej

2= cos

ej

ej

2= jsin

rej

= r

UsefulIdentities

ejt

= cost + jsint

Complex phasors and sinusoidal oscillations

Circular motion in the complex plane