For the example above Case 1 Distinct Real Poles In the first and most basic

For the example above case 1 distinct real poles in

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to calculate, as mentioned above. For the example above: { } { } { } { } { } { } { } { } { } { } Case 1 Distinct Real Poles In the first (and most basic) case, we have a function in polynomial form. We assume that we can factor the denominator into distinct poles: We are assuming for this cases that ( ) all of the poles contain only real parts, and ( ) none of the poles are repeated (i.e., are all unique and not equal). In this case, the expansion takes the form: To determine the unknown coefficients , we can directly evaluate the following for each pole in the original function: [ ]| As an illustrative example, consider the following function , for which we want to determine the inverse Laplace transform.
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35 The partial fraction expansion, under the above rules, is as follows: The poles are given as , , and . We can directly evaluate the expression for above to determine the unknown coefficients. [ ]| | [ ]| | [ ]| | { } { } { } { } { } { } We calculate the inverse Laplace of , , and by examining the table of common Laplace Transforms and selecting the appropriate entry. Case 2 Repeated Real Poles It is often the case that we will have a function where all poles are real (as in Case 1), but one or more of the poles are repeated. For example, the following function has a single non-unique pole, , which is repeated times:
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36 We call the poles simple poles, while the pole is called a repeated real pole, as per the above. Naturally, the pole is only repeated if . In cases such as this, the partial fraction expansion must be changed slightly: [ ] [ ] In the above, the leftmost bracketed terms correspond to the expansion of the simple poles in the original function. The rightmost bracketed terms all correspond to the single repeated pole, . We have a new term and coefficient for each repetition of the pole, . If two different poles were repeated, we would include a set of such terms for each. Finally, in order to complete the procedure, we must also modify our method for calculating the unknown coefficients. First of all, the simple pole coefficients remain unchanged. For in the above: [ ]| However, for the coefficients of the repeated terms, we define the following: | | | | (…) | One can notice that we actually work backwards, calculating the coefficient for the highest power, first. The terms generated by these repeated real poles typically have similar inverse Laplace transforms.
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37 { } At this point, a second example would be useful in understanding the method. Consider the following function, , and calculate its expansion and inverse Laplace Transform. For the first pole, we have a simple pole, which is evaluated as in Case 1: | | Now, we start by finding and working backwards until reaching . | | | 0 1| | | 0 1| | This gives a final partial fraction expansion and inverse Laplace transform as follows: { } { } { } { }
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38 Case 3 Distinct Complex Poles Thus far, we have inherently assumed that all poles contain only real parts. But, it is possible to have complex poles which contain imaginary components. Such poles will only ever occur in complex conjugate pairs. As such, Case 3 covers instances where the poles are still all distinct, but contain a mix of pure real and pairs of complex conjugate poles. The
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