Balder W. Holst

Laplace Transformation

Se regneregler.

Overføringsfunktion

$$\text{impuls} \rightarrow \text{respons}$$

$$ H(s) = \mathcal{L}{h(t)} = \int_{-\infty}^{\infty} h(t) e^{-st} , dt, \s s = \sigma + j\omega$$

$h(t)$: Impulse response in the time domain. $s$: Input signal

$$H(s) = \frac{\text{output}(s)}{\text{input}(s)}$$

Poles

$$H(s) = \frac{\beta}{\alpha} \Rightarrow \begin{cases} \beta = 0 &\Rightarrow \text{nulpunkter} \ \alpha = 0 &\Rightarrow \text{poles} \end{cases} $$ $\alpha$: den karakteristikle ligning

Pol = Singularitet i frekvensdomæne

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If poles are on the imaginary axis the impulse response will result in a constant oscillation at the frequency.

Table

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Inverse Laplace Transformation

$$\text{respons} \rightarrow \text{impuls}$$

Try to use tables instead of this equation.

$$ f(t) = \frac{1}{2\pi j} \int_{\sigma_{c}-j\infty} TODO $$

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