LAPLACE TRANSFORMS

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LAPLACE TRANSFORMS
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INTRODUCTION
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The Laplace Transformation
Time Domain
Frequency Domain
Laplace Transform
Differential equations
Algebraic equations
Input excitation e(t) Output response r(t)
Input excitation E(s) Output response R(s)
Inverse Laplace Transform
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THE LAPLACE TRANSFORM
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THE INVERSE LAPLACE TRANSFORM
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Functional Laplace Transform Pairs
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Operational Laplace Transform Pairs
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Inverse Laplace Transform

• The inverse Laplace transform is usually more
  difficult than a simple table conversion.

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Partial Fraction Expansion

• If we can break the right-hand side of the
  equation into a sum of terms and each term is in
  a table of Laplace transforms, we can get the
  inverse transform of the equation (partial
  fraction expansion).

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Repeated Roots

• In general, there will be a term on the
  right-hand side for each root of the polynomial
  in the denominator of the left-hand side.
  Multiple roots for factors such as (s2)n will
  have a term for each power of the factor from 1
  to n.

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Complex Roots

• Complex roots are common, and they always occur
  in conjugate pairs. The two constants in the
  numerator of the complex conjugate terms are also
  complex conjugates.

where K is the complex conjugate of K.
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Solution of Partial Fraction Expansion

• The solution of each distinct (non-multiple)
  root, real or complex uses a two step process.
• The first step in evaluating the constant is to
  multiply both sides of the equation by the factor
  in the denominator of the constant you wish to
  find.
• The second step is to replace s on both sides of
  the equation by the root of the factor by which
  you multiplied in step 1

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The partial fraction expansion is
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• The inverse Laplace transform is found from the
  functional table pairs to be

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Repeated Roots

• Any unrepeated roots are found as before.
• The constants of the repeated roots (s-a)m are
  found by first breaking the quotient into a
  partial fraction expansion with descending powers
  from m to 0

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• The constants are found using one of the
  following

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The partial fraction expansion yields
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The inverse Laplace transform derived from the
functional table pairs yields
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A Second Method for Repeated Roots
Equating like terms
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Thus
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Another Method for Repeated Roots
As before, we can solve for K2 in the usual
manner.
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Unrepeated Complex Roots

• Unrepeated complex roots are solved similar to
  the process for unrepeated real roots. That is
  you multiply by one of the denominator terms in
  the partial fraction and solve for the
  appropriate constant.
• Once you have found one of the constants, the
  other constant is simply the complex conjugate.

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Complex Unrepeated Roots
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Case 1 Functions with repeated linear roots

• Consider the following example
• F(s) should be decomposed for Partial Fraction
  Expansion as follows

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• Using the residue method

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• so
• and f(t) -6e-t (6 12t)e-2t u(t)

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Case 2 Functions with complex roots

• If a function F(s) has a complex pole (i.e., a
  complex root in the denominator), it can be
  handled in two ways
• 1) By keeping the complex roots in the form of a
  quadratic
• 2) By finding the complex roots and using
  complex numbers to evaluate the coefficients

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• Example Both methods will be illustrated using
  the following example.
• Note that the quadratic terms has complex roots.

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Method 1 Quadratic factors in F(s)

• F(s) should be decomposed for Partial Fraction
  Expansion as follows

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A) Find A, B, and C by hand (for the quadratic
factor method)

• Combining the terms on the right with a common
  denominator and then equating numerators yields

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• so
• now manipulating the quadratic term into the form
  for decaying cosine and sine terms

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• so
• The two sinusoidal terms may be combined if
  desired using the following identity

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• so

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Method 2 Complex roots in F(s)

• Note that the roots of are
• so

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A) Find A, B, and C by hand (for the complex root
method)

• F(s) should be decomposed for Partial Fraction
  Expansion as follows

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• The inverse transform of the two terms with
  complex roots will yield a single time-domain
  term of the form
• Using the Residue Theorem

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• So,
• This can be broken up into separate sine and
  cosine terms using

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• so