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Introduction toMath Methods in Electrical Engineering

An EE100 Lecture

Dr. Bob Berinatobob.berinato@dynetics.com

www.dynetics.com

Fall 2014

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• Good Engineers are Good Problem Solvers …

and the best problem solvers have a large toolbox of math methods and techniques

• Several tools we will consider today Complex Number Notation and Applications Properties of Curves Derivatives Integrals

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EE (and Math) at Work: Synthetic Aperture Radar (SAR)

• Radar can measure range (round-trip time delay) and Doppler shift• A moving radar can use the data it collects to create an “image”• Math methods are central to understanding and engineering SAR systems

Complex Math and Fourier Transforms are essential tools

www.sandia.gov/radar/imageryku.html

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What are the 5 most important “numbers” in mathematics?

2 integers: 0, 1

2 irrationals: , e

1 complex:

Could these possibly be related in one equation?

1j

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Leonhard Euler – An Amazing Individual!

01je

1 ,sincos jje j

a

jb

1

cos

sin

Euler’s Formula: expresses sines and cosines in complex number notation

1

2

2 6

1

k k

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Application: The Phasor

• Consider the following complex signal, called a phasor

• This is the complex notation for a sinusoid with phase and amplitude A

• Note that at time t = 0, the phasor has value Aexp(j)

• Then the phase increases (and repeats every 2 radians, i.e., every integer multiple of 1/f0)

• Grapically, this is shown as arotating vector of length A that makes a complete revolution every 1/f0 seconds

tfjAtz o2exp

zRe

zIm

... ,1 ,00f

t

A

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Application: Signal Notation

• Real Signal Notation:

• Complex Signal Notation:

ttftats o 2cos

ttftjatsttfjta

tfjtjtatfjtmtz

oo

oo

2sin2exp

2expexp2exp

Magnitude Phase

Center FrequencyComplex Envelope,

or Amplitude Magnitude Phase

Re[z(t)] = s(t), the “real” signal

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The Fourier Series and Fourier Transform

• The Fourier Series is a representation of a periodic signal (repeats over and over again with period T) as the weighted sum of phasors at discrete frequencies

• Similarly, the (inverse) Fourier Transform is a representation of a signal as the weighted sum (integral) of phasors at a continuum of frequencies

dtftjtstsfS

dfftjfSfSts

2exp

2exp1

T

kftfjCts k

k

kkk

, 2exp

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What Do You See?

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Some Things You Might See

• Maximum value at t = 0, where v(0) = 1• Zero-crossings at t = 1, 2, 3, …• Many local maxima and minima

There are actually an infinite number

Positive and negative values The area under the curve has both

positive and negative contributors• The curve is “even”, i.e., symmetric

about t = 0

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The Name of This Curve

t

tttv

sinsinc)(

• This function appears in many contexts in EE The output of an ideal low-pass filter The diffraction pattern from a rectangular optical or

microwave aperture The spectrum of a radar pulse Sampling theory Fourier series and Fourier transforms

• Note that this curve has a “name” because it is important in many applications – don’t be intimidated by fancy names of curves Other examples of interesting curves with names are “sin”,

“cos”, “log”, “bessel” functions, “legendre” polynomials, …

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Finding the Maxima and Minima of a Curve

• A very common problem in many fields is to find the maxima and minima of a function

• The solution relates to the idea of the slope of a curve• For a line y = mx + b, the slope is equal to m (and is constant

for all x)

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12

Δ xx

xyxy

x

ym

x

y

2x1x

1xy

2xy

x

y

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The Derivative

• For a general curve, the “derivative” of the function gives the slope – this is the first major part of Calculus The derivative is a function of x, i.e., it changes as x

changes It can be thought of as the slope of the tangent line

x

xyxxy

x

y

dx

dyxx Δ

Δlim

Δlim

0Δ0Δ

x

y

0x x0x

x

y

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So Where are the Maxima and Minima?

• The maxima and minima occur where the derivative equals zero!

• If the slope decreases as x increases, then it is a maximum• If the slope increases as x increases, then it is a minimum

x

y

0x x0x

x

0 as 0 xy

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The Maxima and Minima of v(t) = sinc(t)?

zero crossings of dv/dt yield maxima and minima

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How Did We Find The Value of sinc(0)?

t

tttv

sinsinc)(

• At t = 0, the value of sinc(0) is 0/0 – This is undefined!• The derivative comes to the rescue• L’Hopital’s Rule: When the function equals 0/0, the value is

obtained by taking the derivative of the numerator and denominator

• From calculus we learn

• Thus

k

dt

ktdtaa

dt

atd , cos

sin

1

0 cos0sinc

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• Suppose we consider a simple pulse function

• The area under this curve is clearly equal to 1• But what if we have a curve like v(t) = sinc(t)?• This motivates the “integral” of a function – this is the second

major part of Calculus• Concept: Partition the t-axis into a large number of

approximately rectangular slices• For the pulse example above, the t-axis could be divided into

slices that are each 1-sec long Each slice has an area of 10-6

The sum of the 106 slices within the pulse is equal to 1

The Area Under a Curve

elsewhere ,0

,1)( 2

12

1 tttv

(t)

t½ –½

1

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• Let’s find the integral of a function between t = a and t = b• It can be approximated by the 46 slices shown below

• In the limit, we write

The Integral

t

tv

a b

45

0 4646Area

k

kabav

ab

b

a

N

kN

dttvN

kabav

N

ab1

0

limArea

green – positivered – negative

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The Area Under the Sinc Curve

t

tttv

sinsinc)(

• Interestingly enough, with an infinite number of positive and negative contributors to the area under the curve, the total area is 1

12

12sinc2sincArea

0

dttdtt

sinc(t) is an “even” function so the total area is twice the area of the positive-time area

It isn’t easy to derive this so don’t worry about how this is found

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A Great New Learning Opportunity

• Join the MOOC revolution … Massive Open Online Courses A great way to add to your toolbox and grow your problem-solving skills

• www.coursera.org, www.edx.org, and www.udacity.com offer a wide array of math, science, and engineering courses

• These are the same courses being taught at major universities, and require students to do homework and take exams

• In the last few years, I’ve taken the following Coursera courses Cryptography I from Stanford Game Theory from Stanford Intro to Philosophy from University of Edinburgh Leading Strategic Innovation in Organizations from Vanderbilt Computational Neuroscience from University of Washington Introduction to Functional Analysis from Ecole Centrale Paris Future of (Mostly) Higher Education from Duke University

• UAH hosted its 1st MOOC this year, Intro to Chemical Engineering by Dr. Chittur

• The courses can be quite challenging, but real learning occurs!

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Summary

• Electrical Engineering is full of challenging mathematics• The best EE students (and professionals) are those who do well in math• Important concepts for you to master during your undergraduate program

include Calculus – Limits, Derivatives, Integrals Differential Equations – Ordinary & Partial Linear Algebra and Linear Operators Complex Analysis (where “j ” lives) Fourier Series, Fourier Transforms, and Laplace Transforms Probability and Statistics

• Don’t just memorize formulas, know how to derive them and what they mean!!!

Dirac: “I understand what an equation means if I have a way of figuring out the characteristics of its solutions without actually solving it.”

• Learn to Love Math – It Can Be Contagious

What are you doing with your 2,000,000 minutes???What are you doing with your 2,000,000 minutes???