Pre Reading Laplace Transform Properties 23 Derivation is as follows 1 In order

Pre reading laplace transform properties 23

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Pre-Reading: Laplace Transform Properties
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23 Derivation is as follows: { } ∫ 0 1 | | * + In order for the limit above to exist, we require that the function increase at a rate less than . This is also a basic property of the Laplace Transform; if increases faster than , then the function does not have a Laplace Transform. Since we have already assumed above that { } , as seen in the first line above, we are inherently assuming that the function does not increase faster than . Since this is the case, the limit can only be zero: { } [ ] As stated above, this property will be used to determine the effect of initial conditions. One can extend the theorem to any level of derivation: { } Integration in Time Domain Property 2∫ 3 This property is slightly more involved to derive, but otherwise provides duality to the differentiation property: 2∫ 3 ∫ 0∫ 1 0∫ 1 [ ]| 0 . / . /1
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24 Once again, we have assumed that exists, which inherently would mean that grows at a rate slower than . Thus, the limit by extension must approach zero. The second integral has equal limits, meaning its value will also be zero: . / 2∫ 3 [ ] Time Shifting / Translation in Time Domain Property { } This property is relatively straight-forward to derive using a simple change of variables. If we define a new variable , this gives . By extension, if , then , and if then . Lastly, we have . Using this change of variables: { } ∫ The value of the first integral, from to , is necessarily zero because for . As such: { } ∫ Frequency Shifting Property { } This property is the dual of the time shifting property; its derivation follows in much the same manner as above.
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25 Time Scale Change Property { } ( ) Valid for: Initial Value Theorem Assuming that does not contain any impulses, then: Property This theorem is one of two very useful properties of the Laplace Transform. Specifically, we can use this limit to determine the time-domain behavior of a circuit as time approaches zero (i.e., initial conditions). More importantly, we can do so directly using a frequency- domain expression, such as a transfer function, which is readily determined for a circuit. For example, consider a frequency-domain function, , for which we want to determine the time-domain initial conditions. Note that corresponds to an instant immediately after , as we assume for , and behavior at exactly is undefined. If we have the following: Then: Final Value Theorem If all the poles of lies in the left half of the complex (s) plane, then: Property This theorem is a very important one, as it will allow us to determine the steady-state response of a circuit. As noted above, one of the restrictions is on the poles of the function . We have not yet discussed the nature of poles and zeros, which will be covered in a later section. Let us do a simple example, and revisit the Final Value Theorem later.
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26 We are given the same function, , as before: For the above function, , let us assume that all poles are in the left half of the complex plane (and indeed, they are we will see how to determine this shortly). If this is the case, then we can apply the FVT to find the steady-state value of the function the value as time approaches infinity.
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