NETWORKS 2: 090920201

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CHAPTER 14

• NETWORKS 2 0909202-01
• 16 November 2005 Lecture 7
• ROWAN UNIVERSITY
• College of Engineering
• Dr Peter Mark Jansson, PP PE
• DEPARTMENT OF ELECTRICAL COMPUTER ENGINEERING
• Autumn Semester 2005 Quarter Two

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Chapter 14 key concepts

• Todays learning objectives
• Review HW problems section 11.4 (1,24)
• information society
• review the Laplace L transform method
• applicability, benefits, properties
• impulse function, time delay, freq. shift
• inverse Laplace transform L-1
• solution of diff. eqns. with Laplace
• transfer function

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Information Society Electrical Communication
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Figure 14.3-1 The transform method.
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Laplace applicability

• first we learned methods that worked for
• dc circuits
• sinusoidal sources
• now we consider a method that works for sources
  of many more forms
• step functions (dc and other), and/or
• exponentials (dc and other), and/or
• sinusoidal sources
• the Laplace transform applies for all signals
  that are zero when t lt 0.
• i.e, where Lf(t) f(t) 0 when t lt 0

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Benefits of Laplace

• all functions of time f(t) that are physically
  possible always have a Laplace transform
• the Laplace transform is unique for every unique
  f(t) function
• the Laplace transform is linear
• the Laplace has an inverse transform
• all calculations can become algebra for solution
  in the frequency - F(s) - domain
• Laplace tables simplify our calculations
• clear applicability in the information age

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example 1

• Find the Laplace transform L of the following
  f(t) e-2t sin t

Write LC1 for equation above
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impulse and time delay

• an impulse is a pulse of infinite amplitude for
  an infinitesimal time whose area is finite
• the Laplace transform L of an impulse function
  is 1

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impulse and time delay

• the Laplace of a time delay ?
• f(t- ?) is L f(t- ?)u(t- ? ) e-s? F(s)
• Where F(s) L u(t) 1/s

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example 2

• Find the Laplace transform of the following
  product of a step function and time delay f(t)
  Au(t- ?)

If the time delay was 5 seconds and amplitude of
the step function was 3, write the Laplace as
LC2
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frequency shift property

• the Laplace of e-at f(t) yields another very
  helpful formula
• L e-at f(t) F (s a)
• An example is where f(t) sin ?t
• Since L sin ?t ? /(s2 ?2 )
• L e-at sin ?t ? /((sa)2 ?2 )

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example 3

• Find the Laplace transform of the following
  product of an exponential function and time
  delay
• g(t) e-4t u(t-3)

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example 3
First from the time delay property we find
And from the frequency shift property we find
Simplify this expression and write the Laplace as
LC3
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Learning Check 4

• using the linearity property and the frequency
  shift property find
• L 6u(t) 5e-8tu(t)

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Inverse Laplace Transform

• critical to the LaPlace method is the existence
  of an inverse transform.
• it is remarkably straight-forward
• f(t) L -1 F(s)
• If F(s) 8/s 4/(s5) 12/(s216)
• what is f(t)?

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Inverse Laplace Transform

• it is remarkably straight-forward
• f(t) L -1 F(s)
• L -1 8/s 4/(s5) 12/(s216)
• L -18/s L 14/(s5) L -1 12/(s216)
• 8 4e-5t LC5

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Do you remember Partial Fraction Expansion?
To find A, multiply through by its denominator
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Partial Fraction Expansion explained
Choose a value of s so the denominator goes to
zero (in our case 1), consequently A 2
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Partial Fraction Expansion explained

• Solve for B
• Multiply all terms by Bs denominator
• Choose a value of s so that Bs denominator goes
  to zero,
• Solve for B LC6

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Another useful relationship explained transforms
with complex poles
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Lets solve an example of an inverse transform
with complex poles
f(t) L -1 F(s)
Any idea how we can get it in this form?
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Lets go its quite easy really
Now can you see it in this form?
What is a? what is c? what is ?? Solve for d as
LC7
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And now the final step - the inverse transform
f(t) L -1 F(s)
Solve for f(t) as LC8
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Solutions of circuit problems with diff eqs via
Laplace
General method

• Identify the circuit variables
• Write the diff eqs and i.d. initial conditions
• Obtain Laplace Transform for all terms
• Solve for unknown variables in s-domain
• Obtain the inverse Laplace for solution in the
  time domain

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example 4
v1 -
i1(t) 7 e6t A for t gt 0, i(0) 0
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midterm examination review

• 30 of final Net II grade
• covers Chapters 10 and 11
• through section 11.8
• study helps Lectures 1-7 sample problems and
  learning checks
• homework problems in assn. 1 - 3

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midterm examination review - continued

• key concepts
• history of electric power (10.2, 11.2)
• sinusoidal sources (10.3 10.7)
• developing/interpreting the equations phase
  angle
• complex forcing functions
• phasors (frequency time domains)
• complex numbers (apps. B C)
• rectangular, exponential and polar notation
• trigonometric identities
• impedance and admittance (10.8)
• KCL, KVL, etc. (10.9 10.11)
• phasor diagrams (10.12)

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midterm examination review - continued

• key concepts
• instantaneous power (11.3)
• real (average) power (11.3, 11.5)
• effective value of periodic waveforms (11.4)
• root-mean-square
• complex power (11.5)
• complex, apparent, real, and imaginary power
• more detail on frequency and time domain
• power factor (11.6)
• leading and lagging power factor, pf phase angle
• maximum power transfer (11.8)

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midterm examination review sample problems

• key concepts
• LaPlace Transform
• Why Use LaPlace?
• How to use LaPlace