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									Lecture #13       EGR 272 – Circuit Theory II
Read: Chapters 12 and 13 in Electric Circuits, 6th 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:           Courses Using Laplace Transforms:
• Solving differential equations          • Circuit Theory II
• Analyzing circuits in the s-domain      • Electronics
• Transfer functions                      • Linear Systems
• Frequency response                      • Control Theory
• Applications in many courses            • Discrete Time Systems (z-transforms)
                                          • Communications
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, calculator usage will
be restricted on the test covering this material by giving a 2-part test as follows:
Part 1: (No calculators allowed) Covers finding Laplace transforms and inverse
Laplace transforms
Part 2: (Calculators allowed) Circuit applications using Laplace transforms          1
Lecture #13       EGR 272 – Circuit Theory II

Notation:
F(s) = L {f(t)} = the Laplace transform of f(t).
f(t) = L -1{F(s)} = the inverse Laplace transform of f(t).

Uniqueness: Every f(t) has a unique F(s) and every F(s) has a unique f(t).


                                   L
  time domain           f(t)                     F(s)              s-domain
                                                             (or frequency domain)
                                   L -1



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 #13         EGR 272 – Circuit Theory II

Definition:
          
 F(s)    
          0
            f(t)e-st dt   (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:
                  jw

                                              Note: The s-plane is sometimes
                          s-plane             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 #13       EGR 272 – Circuit Theory II

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.




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Lecture #13       EGR 272 – Circuit Theory II
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 #13      EGR 272 – Circuit Theory II
Example: Find F(s) if f(t) = cos(wot)u(t)      Euler's Identities
           (Hint: use Euler’s Identity)        e  jx = cos(x)  jsin(x)
                                                        e +jx + e-jx
                                               cos(x) =
                                                              2
                                                        e +jx - e-jx
                                               sin(x) =
                                                             2j




Example: Find F(s) if f(t) = sin(wot)u(t)




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Lecture #13 EGR 272 – Circuit Theory II
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
1     Linearity                  L {af(t)} = aF(s)
2     Superposition              L {f1(t) + f2(t) } = F1(s) + F2(s)
3     Modulation                 L {e-atf(t)} = F(s + a)
4     Time-Shifting              L {f(t - )u(t - )} = e-sF(s)
5     Scaling                    L f(at)  1 F  s 
                                                  
                                               a a

6     Real Differentiation           d       
                                 L    f(t)   sF(s) - f(0)
                                      dt     
                                     t          1
7     Real Integration           L     f(t)dt   F(s)
                                     0          s
                                                  d
8     Complex Differentiation    L   tf(t)   F(s)
                                                  ds
                                               
                                      f(t) 
9     Complex Integration        L            F(s)ds
                                      t  s
                                 L
10    Convolution                    {f(t) * g(t)} = F(s)·G(s)                  7
Lecture #13     EGR 272 – Circuit Theory II
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)




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Lecture #13      EGR 272 – Circuit Theory II
Laplace Transform Properties: (continued)            This means that if you know F(s)
                                                     for any f(t), then the result of
3. Modulation: L {e-atf(t)} = F(s + a)               multiplying f(t) by e-at is that you
   Proof:                                            replace each s in F(s) by s+a.




   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 #13     EGR 272 – Circuit Theory II
Laplace Transform Properties: (continued)
                                                        Note: Be sure that all t’s
4. Time-Shifting:     L {f(t - )u(t - )} = e-sF(s)   are in the (t - ) form when
                                                        using this property.
  Example: Find L {4e-2(t - 3) u(t - 3)}




  Example: Find L {10e-2(t - 4)sin(4[t - 4])u(t - 4)}




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Lecture #13      EGR 272 – Circuit Theory II
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 #13      EGR 272 – Circuit Theory II
Example: Find L {4e-3tcos(4[t - 6])u(t - 6)}




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Lecture #13     EGR 272 – Circuit Theory II
Laplace Transform Properties: (continued) In other words, the result of replacing

5. Scaling: L f(at)  F  
                        1 s               each (t) in a function with (at) is that
                                        each s in the function is replaced by s/a
                        a 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)




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