18 Valid If were instead a complex number the transform result is the same but

18 valid if were instead a complex number the

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18 [ ] Valid: If were instead a complex number, the transform result is the same, but with a slightly more complex ROC: Valid: Sinusoidal Function The last function we will examine directly will be the basic sinusoidal function. This function is of particular importance; we will use it (and the Laplace Transform) to construct the fundamentals behind the Bode Plot. The Laplace Transform of the basic sine function is slightly more complex than the previous examples, but it starts in the same manner we first apply the mathematical definition. 𝐴
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19 { } ∫ To complete the integration, we can use Euler‟s Formula to convert the sine function into two exponential functions: [ ] [ ] [ ] [ ]| [ ( ) ( )] [ ] ( Valid for ) [ ] Valid for This result will be used again later when constructing Bode Plots. One can also see that the process involved with calculating the transform is becoming involved. As such, for most real-world problems we will instead use a pre-calculated table of common Laplace Transforms. This will allow us to solve most common problems directly, without having to explicitly calculate the above integral. However, this table on its own is not sufficient to construct many of the complex transforms that we will encounter. To assist in this endeavour, the following section will also introduce some common properties of the Laplace Transform which, when used along with the table of common transforms, can complete most simple and complex problems.
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20 To avoid having to derive each Laplace Transform from first principles, we can make use of two tables to quickly construct the Laplace Transform for most common functions. This first table provides a list of several elementary functions and their transforms. It will be available on any test or exam that requires you to directly calculate a Laplace Transform. We can later use a table of properties of the Laplace Transform to combine these elementary functions into more complex transforms, as well. Function, Laplace Transform, , Unit Impulse (Delta Dirac) , Unit Step Function Ramp Function Power Function Periodic Unit Impulse ( ) Exponential Decay Delayed Impulse Pre-Reading: Table of Laplace Transforms
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21 Sine Function Cosine Function Hyperbolic Sine Function Hyperbolic Cosine Function ( ) ( ) [ ] General Quadratic Lag
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22 There are also a number of useful properties of the Laplace Transform which we can use to combine entries from the previous table into more complex transforms. We will start with the most basic property, linearity. This property actually allows two related transformations multiplication by a constant and addition / subtraction. Multiplication by a Constant Property { } { } This property can be seen as an extension of the properties of integration: { } ∫ { } Addition and Subtraction Property { } { } { } Again, this property is an extension of the integral which forms the basis of Laplace: { } ∫ ( ) Differentiation in Time Domain Property { } This property will be important to us later when we use the Laplace Transform to determine the effect of initial conditions on a circuit. It is also occasionally used to construct some complex transforms.
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