Lecture #1 EGR 261 – Signals & Systems

Lecture #1 EGR 261 – Signals & Systems Welcome to EGR 261 Signals & Systems • Syllabus • Homework • Web page • Office hours EGR 260: Chapters 1 - 8 in Electric Circuits, 9th Editionby Nilsson EGR 261: Chapters 12 - 17 in Electric Circuits, 9th Editionby Nilsson Chapters 1, 2, 4 - 8 in Linear Signals & Systems, 2nd Edition by Lathi

Lecture #1 EGR 261 – Signals & Systems Sequence of Electrical/Computer Engineering Courses at TCC EGR 110 Engineering Graphics EGR 125 (4 cr) C++ Programming for Engineers EGR 260 (3 cr) Circuit Analysis ODU equiv: ECE 201 Offered: F, Sp * EGR 267 (3 cr) EGR Analysis Tools ODU equiv: ECE 200 Offered: F, ?? EGR 270 (4 cr) Fund. Of Computer EGR ODU equiv: ECE 241 Offered: Sp, Su * MTH 279 (4 cr) Differential Equations EGR 261 (3 cr) Signals & Systems ODU equiv: ECE 202 Offered: F, Sp * EGR 262 (2 cr) Fund. Circuits Lab ODU equiv: ECE 287 Offered: Sp, Su * * Additional course offerings may be available at the Tri-Cities Center

Lecture #1 EGR 261 – Signals & Systems 3 Changes to Electrical/Computer Engineering Courses at TCC • Due to recent changes at ODU, TCC will make the following changes to the sequence of electrical/computer engineering courses: • EGR 260-261 will be replaced by EGR 271-272 • EGR 267 will no longer be offered • No changes to EGR 262 or EGR 270 • The changes will be phased in as follows: • Fall 2013: First time EGR 271 will be offered • Last time EGR 261 will be offered • Spring 2014: First time EGR 272 will be offered • See the chart on the following page for additional scheduling information

Lecture #1 EGR 261 – Signals & Systems 4 Content differences between EGR 260-261 and EGR 271-272 • The new course sequence at ODU actually is a return to the format that they used several years ago and is similar to the format used by many universities. • How is EGR 271 different from EGR 260? • EGR 271 will be more manageable as it will cover less material (Ch. 1-6 in Nilsson instead of Ch. 1-8 covered in EGR 260). • MATLAB solutions for problems will be added to EGR 271. • How is EGR 272 different from EGR 261? • EGR 272 will cover Ch. 7-10, 12-15 in Nilsson instead of Ch. 12-17 covered in EGR 261 + additional material from a second textbook in EGR 261). • AC circuit analysis will be added to EGR 272 (Ch. 9-10 in Nilsson). • MATLAB solutions for problems will be added to EGR 272. • Material on Fourier Series, Fourier transforms, convolution, and properties of linear signals and systems will be moved to a junior-level course at ODU.

Lecture #1 EGR 261 – Signals & Systems 5 Sequence of Electrical/Computer Engineering Courses at TCC EGR 271 (3 cr) Circuit Theory I ODU equiv: ECE 201 Offered: F, Sp, Su MTH 279 (4 cr) Differential Equations EGR 125 (4 cr) Into to Engineering Methods (C++) EGR 272 (3 cr) Circuit Theory II ODU equiv: ECE 202 Offered: F, Sp EGR 262 (2 cr) Fund. Circuits Lab ODU equiv: ECE 287 Offered: F, Sp, Su EGR 270 (4 cr) Fund. Of Computer EGR ODU equiv: ECE 241 Offered: F, Sp, Su Notes: Classes available at the Virginia Beach Campus, the Chesapeake Campus, and the Tri-Cities Center EGR 271-272 transfers to Virginia Tech as ECE 2004 EGR 270 transfers to Virginia Tech as ECE 2504 EGR 262 does not transfer to Virginia Tech

Lecture #1 EGR 261 – Signals & Systems Read: Ch. 12, Sect. 1-9 in Electric Circuits, 9th Edition by Nilsson Handout: Laplace Transform Properties and Common Laplace Transform Pairs Laplace Transforms – an extremely important topic in EE! Key Uses of Laplace Transforms: • Solving differential equations • Analyzing circuits in the s-domain • Transfer functions • Frequency response • Applications in many courses Testing: Calculators have become increasingly powerful in recent years and can often be used to find Laplace transforms and inverse Laplace transforms. However, it is also easy to make mistakes with the calculators and if the student is not familiar with the material, the mistakes might easily go undetected. As a result, no calculators will be allowed on Test #1 in this class. They will be allowed on all other tests and on the final exam. Courses Using Laplace Transforms: • Signals and Systems • Electronics • Control Theory • Discrete Time Systems (z-transforms) • Communications • Others No calculators allowed on Test #1

L L-1 Lecture #1 EGR 261 – Signals & Systems Notation: F(s) = L {f(t)} = the Laplace transform of f(t). f(t) = L-1{F(s)} = the inverse Laplace transform of F(s). Uniqueness: Every f(t) has a unique F(s) and every F(s) has a unique f(t). Note: Transferring to the s-domain when using Laplace transforms is similar to transferring to the phasor domain for AC circuit analysis.

jw s-plane  Lecture #1 EGR 261 – Signals & Systems Definition: (one-sided Laplace transform) where s =  + jw = complex frequency  = Re[s] and w = Im[s] sometimes complex frequency values are displayed on the s-plane as follows: Note: The s-plane is sometimes used to plot the roots of systems, determine system stability, and more. It is used routinely in later courses, such as Control Theory.

Lecture #1 EGR 261 – Signals & Systems Convergence: A negative exponent (real part) is required within the integral definition of the Laplace Transform for it to converge, so Laplace Transforms are often defined over a specific range (such as for  > 0). Convergence will discussed in the first couple of examples in this course to illustrate the point, but will not be stressed afterwards as convergence is not typically a problem in circuits problems. Determining Laplace Transforms - Laplace transforms can be found by: 1) Definition - use the integral definition of the Laplace transform 2) Tables - tables of Laplace transforms are common in engineering and math texts 3) Using properties of Laplace transforms - if the Laplace transforms of a few basic functions are known, properties of Laplace transforms can be used to find the Laplace transforms of more complex functions.

Lecture #1 EGR 261 – Signals & Systems Example: If f(t) = u(t), find F(s) using the definition of the Laplace transform. List the range over which the transform is defined (converges). Example: If f(t) = e-at u(t),find F(s) using the definition of the Laplace transform. List the range over which the transform is defined (converges).

Lecture #1 EGR 261 – Signals & Systems Example: Find F(s) if f(t) = cos(wot)u(t) (Hint: use Euler’s Identity) Example: Find F(s) if f(t) = sin(wot)u(t)

Lecture #1 EGR 261 – Signals & Systems Laplace Transform Properties Laplace transforms of complicated functions may be found by using known transforms of simple functions and then by applying properties in order to see the effect on the Laplace transform due to some modification to the time function. Ten properties will be discussed as shown below (also see handout). Table of Laplace Transform Properties

Lecture #1 EGR 261 – Signals & Systems Laplace Transform Properties: 1. Linearity: L{af(t)} = aF(s) Proof: 2. Superposition: L{f1(t) + f2(t) } = F1(s) + F2(s) Example: Use the results of the last two examples plus the two properties above to find F(s) if f(t) = 25(1 – e-3t )u(t)

Lecture #1 EGR 261 – Signals & Systems Laplace Transform Properties: (continued) 3. Modulation: This means that if you know F(s) for any f(t), then the result of multiplying f(t) by e-at is that you replace each s in F(s) by s+a. L {e-atf(t)} = F(s + a) Proof: Example: Find V(s) if v(t) = 10e-2t cos(3t)u(t) Example: Find I(s) if i(t) = 4e-20t sin(7t)u(t)

Lecture #1 EGR 261 – Signals & Systems Laplace Transform Properties: (continued) 4. Time-Shifting: Note: Be sure that all t’s are in the (t -) form when using this property. L {f(t - )u(t - )} = e-sF(s) Example: Find L {4e-2(t - 3) u(t - 3)} Example: Find L {10e-2(t - 4)sin(4[t - 4])u(t - 4)}

Lecture #1 EGR 261 – Signals & Systems Example: Find F(s) if f(t) = 4e-3t u(t - 5) using 2 approaches: A) By applying modulation and then time-shifting B) By applying time-shifting and then modulation

Lecture #1 EGR 261 – Signals & Systems Example: Find L {4e-3tcos(4[t - 6])u(t - 6)}

Lecture #1 EGR 261 – Signals & Systems Laplace Transform Properties: (continued) 5. Scaling: In other words, the result of replacing each (t) in a function with (at) is that each s in the function is replaced by s/a and the function is also divided by a. Note: This is not a commonly used property. Example: Find F(s) if f(t) = 12cos(3t)u(t)

Lecture #1 EGR 261 – Signals & Systems