ECEN3714 Network Analysis Lecture #1 11 January 2016 Dr. George Scheets n Review Appendix B (Complex Numbers) n Read 12.1 – 12.2 n Ungraded Homework Problems: 12.1, 3, & 4

ECEN3714 Network Analysis Lecture #2 13 January 2016 Dr. George Scheets n Read 12.3 – 12.4 n Ungraded Homework Problems 12.5, 8, & 10 n Labs commence Wednesday, 20 January n Quiz #1, Friday, 22 January

ECEN3714 Network Analysis Lecture #3 15 January 2016 Dr. George Scheets n Read 12.5 n Ungraded Homework Problems 12.13, 14, & 15 n Labs commence Wednesday, 20 January n Quiz #1, Friday, 22 January

OSI IEEE n January General Meeting n 5:30 - 6:30 pm, Wednesday, 20 January n ES201b n Reps from Georgia-Pacific will present n +3 pts extra credit & dinner will be served n All are invited

You can't always trust fancy programs.

Grading n In Class: Quizzes, Tests, Final Exam Open Book & Open Notes WARNING! Study for them like they’re closed book! n Ungraded Homework: Assigned most every class Not collected Solutions Provided Payoff: Tests & Quizzes

Why work the ungraded Homework problems? n An Analogy: Linear Systems vs. Soccer n Reading text = Reading a book about Soccer n Looking at the problem solutions = watching a scrimmage n Working the problems = practicing or playing in a scrimmage n Quiz = Exhibition Game or Scrimmage n Test = Big Game

To succeed in this class... n Show some self-discipline!! Important!! For every hour of class put in 1-2 hours of your own effort. n PROFESSOR'S LAMENT If you put in the time You should do fine. If you don't, You likely won't.

Cheating n Don’t do it! If caught, expect to get an ‘F!’ for the course. n My idol: Judge Isaac Parker U.S. Court: Western District of Arkansas a.k.a. “Hanging Judge Parker” a.k.a. “Hanging Judge Parker”

Labs n Start at Scheduled Time on Week #2 n But NOT in scheduled place n First 2 Wednesday Labs in EN 510 n First 2 Friday Labs in EN 019

5 Hertz Square Wave... 1 volt peak, 2 volts peak-to-peak, 0 mean

Generating a Square Wave vp 5 Hz 1/3 vp 15 Hz

Generating a Square Wave /5 vp 25 Hz Hz + 15 Hz

Generating a Square Wave /7 vp 35 Hz Hz + 15 Hz + 25 Hz

Generating a Square Wave Hz + 15 Hz + 25 Hz + 35 Hz cos2*pi*5t - (1/3)cos2*pi*15t + (1/5)cos2*pi*25t - (1/7)cos2*pi*35t)

Generating a Square Wave... 5 cycle per second square wave generated using first 50 cosines, Absolute Bandwidth = 495 Hertz

Generating a Square Wave... 5 cycle per second square wave generated using first 100 cosines, Absolute Bandwidth = 995 Hertz

Sines & Cosines n Can be used to construct any time domain waveform n Periodic x(t) = ∑ [ a i cos(2πf i t) + b i sin(2πf i t) ] n Cosines & sines are 90 degrees apart cos(2πft) + j sin(2πft) n Phasor e jπft = cos(2πft) + j sin(2πft) n cos(2πft) = Real {e jπft } n sin(2πft) = Imaginary {e jπft } n Wikipedia Example Wikipedia Example Wikipedia Example

Phasor Projection n Projection on Real Axis = Cosine n Projection on Imaginary Axis = Sine Snapshot after 1 phasor revolution

Complex Numbers Rectangular & Polar Coordinates Easiest to use... Addition (x+y)Rectangular Subtraction (x-y)Rectangular Multiplication (x*y)Rectangular or Polar Division (x/y)Polar 3 ways to represent a complex number Ex) 9 + j9 = 81 / 45o = 81ejπ/4

Last Time… n Two complex numbers x = 7 + j4 = / o = 8.062e j0.1652π y = 2 – j4 = 4.472/ o = 4.472e -j0.3524π y = 2 – j4 = 4.472/ o = 4.472e -j0.3524π

Pierre-Simon Marquis de Laplace n Born 1749 n Died 1827 n French Mathematician & Astronomer n Previously, you've had y(t) = function{ x(t) } u Solved in time domain (derivatives?, integrals?) n In 1785, Laplace noticed it's frequently easier to solve these via x(t) → X(s) →Y(s) → y(t) transform massage transform

Time Bounds n None Specified? Assume 0 - < t < ∞ = 0 < t < ∞ u (Default bounds for this class) n Assume time function = 0 where not specified u Example: x(t) = 7t; t > 3 Assume x(t) = 0 when t 3 Assume x(t) = 0 when t < 3

Laplace Transform F(s) = f(t) e -st dt 0-0- ∞ "s" is a complex number = σ + jω Fourier Transform is similar σ = 0 Lower Bound = -∞

Correlation n Provides a measure of how "alike" x(t) and y(t) are n If integral evaluates positive u x(t1) and y(t1) tend to be doing same thing t1 an arbitrary time u if x(t1) is positive, y(t1) tends to be positive u if x(t1) is negative, y(t1) tends to be negative x(t) y(t) dt

Correlation n If integral evaluates negative u x(t1) and y(t1) tend to be doing the opposite n If evaluates = 0 u x(t) & y(t) are not related (uncorrelated) no predictability x(t) y(t) dt

Laplace Transform of f(t) = e -2t F(s) = e -2t e -st dt 0-0- ∞

Laplace Transform of e -2t F(0) = e -2t e -0t dt 0-0- ∞ Evaluated at s = 0.

Laplace Transform of e -2t t e -0t = u(t) t e -2t n This evaluates to F(0) = 1/2

Laplace Transform of e -2t F(2) = e -2t e -2t dt 0-0- ∞ F(s) = e -2t e -st dt 0-0- ∞

Laplace Transform of e -2t t e -2t n Product is e -4t, which has area F(2) = 1/4. t e -st evaluated at s = 2 Ideally, these two waveforms would have the highest + correlation. Laplace Transform is an imperfect correlator.

Normalized Energy e -2t e -st dt ∞ 0-0- e -st dt ∞ 0-0- NE(s) =

Normalized Energy Plot s NE(s) = s 0.5 /(s+2) Peak is at s = 2.

Correlation & Laplace Transform n Somewhat similar x(t) y(t) dt

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015, 2008, 2005 by Pearson Education, Inc. All rights reserved. TABLE 12.1 An Abbreviated List of Laplace Transform Pairs

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015, 2008, 2005 by Pearson Education, Inc. All rights reserved. TABLE 12.2 An Abbreviated List of Operational Transforms