EGR 277

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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 Edition by Nilsson
• EGR 261 Chapters 12 - 17 in Electric Circuits,
  9th Edition by Nilsson
• Chapters 1, 2, 4 - 8 in Linear
  Signals Systems, 2nd Edition by Lathi

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Lecture 1 EGR 261 Signals Systems
Sequence of Electrical/Computer Engineering
Courses at TCC
Additional course offerings may be available
at the Tri-Cities Center
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Lecture 1 EGR 261 Signals Systems
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

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Lecture 1 EGR 261 Signals Systems
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.

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Lecture 1 EGR 261 Signals Systems
Sequence of Electrical/Computer Engineering
Courses at TCC

• 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

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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
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Lecture 1 EGR 261 Signals Systems
Notation F(s) L f(t) the Laplace transform
of f(t). f(t) L -1F(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.
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Lecture 1 EGR 261 Signals Systems
Definition
(one-sided Laplace transform) where s ? jw
complex frequency ? Res and w
Ims 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.

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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 ? gt 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.
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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).
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Lecture 1 EGR 261 Signals Systems
Example Find F(s) if f(t) cos(wot)u(t)
(Hint use Eulers Identity)
Example Find F(s) if f(t) sin(wot)u(t)
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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
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Lecture 1 EGR 261 Signals Systems
Laplace Transform Properties 1. Linearity
L af(t) aF(s)
Proof
L f1(t) f2(t) F1(s) F2(s)
2. Superposition
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)
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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 sa.
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)
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Lecture 1 EGR 261 Signals Systems
Laplace Transform Properties (continued) 4.
Time-Shifting
Note Be sure that all ts 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(4t - 4)u(t -
4)
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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
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Lecture 1 EGR 261 Signals Systems
Example Find L 4e-3tcos(4t - 6)u(t - 6)
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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)