Definition
The transform is useful for converting differentiation and integration in the time domain into much easier multiplication and division in the Laplace domain
Here are some general transforms (you can look up the rest of these)
The formal definition is
x(t) | x(s) |
---|---|
x’(t) | s |
x’’(t) | |
t | |
Use a table for this stuff |
Properties
- Time delay (shift)
- Linearity
- Differentiation
- Integration
- Time exponential
- A shift in time exponential weighting in the s domain does the same thing in time
- Convolution
When in the LaPlace domain you do a lot of Partial Fraction Decomposition.
Theorems
- IVT:
- FVT: holds only when exists
We use the LaPlace transforms to create Transfer Functions
You can get the impulse response of a system by doing the reverse LaPlace transform of a transfer function