Circuit Theory/Transform Tables

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Circuit Theory

[edit] Laplace Transform Appendix

The Laplace Transform is defined as such:

$F(s) = \mathcal{L} \left\{f(t)\right\} =\int_{0^-}^\infty e^{-st} f(t)\,dt.$

The result of the transform of a time-domain function f(t) is F(s).

Time Domain	Laplace Domain
$x(t)=\frac{1}{2\pi j} \int_{\sigma-j\infty}^{\sigma+j\infty} X(s)e^{st}ds$	$X(s)=\int_{-\infty}^\infty x(t)e^{-st}dt$
$δ(t)$	$1$
$δ(t - a)$	$e - a s$
$u (t)$	$\frac{1}{s}$
$u (t - a)$	$\frac{e^{-as}}{s}$
$t u (t)$	$\frac{1}{s^2}$
$t n u (t)$	$\frac{n!}{s^{n+1}}$
$\frac{1}{\sqrt{\pi t}}u(t)$	$\frac{1}{\sqrt{s}}$
$e a t u (t)$	$\frac{1}{s-a}$
$t n e a t u (t)$	$\frac{n!}{(s-a)^{n+1}}$
$cos(ω t) u (t)$	$\frac{s}{s^2+\omega^2}$
$sin(ω t) u (t)$	$\frac{\omega}{s^2+\omega^2}$
$cosh(ω t) u (t)$	$\frac{s}{s^2-\omega^2}$
$sinh(ω t) u (t)$	$\frac{\omega}{s^2-\omega^2}$
$e a t cos(ω t) u (t)$	$\frac{s-a}{(s-a)^2+\omega^2}$
$e a t sin(ω t) u (t)$	$\frac{\omega}{(s-a)^2+\omega^2}$
$\frac{1}{2\omega^3}(\sin \omega t-\omega t \cos \omega t)$	$\frac{1}{(s^2+\omega^2)^2}$
$\frac{t}{2\omega} \sin \omega t$	$\frac{s}{(s^2+\omega^2)^2}$
$\frac{1}{2\omega}(\sin \omega t+\omega t \cos \omega t)$	$\frac{s^2}{(s^2+\omega^2)^2}$

Property	Definition
Linearity	$\mathcal{L}\left\{a f(t) + b g(t) \right\} = a F(s) + b G(s)$
Differentiation	$\mathcal{L}\{f'\} = s \mathcal{L}\{f\} - f(0^-)$ $\mathcal{L}\{f''\} = s^2 \mathcal{L}\{f\} - s f(0^-) - f'(0^-)$ $\mathcal{L}\left\{ f^{(n)} \right\} = s^n \mathcal{L}\{f\} - s^{n - 1} f(0^-) - \cdots - f^{(n - 1)}(0^-)$
Frequency Division	$\mathcal{L}\{ t f(t)\} = -F'(s)$ $\mathcal{L}\{ t^{n} f(t)\} = (-1)^{n} F^{(n)}(s)$
Frequency Integration	$\mathcal{L}\left\{ \frac{f(t)}{t} \right\} = \int_s^\infty F(\sigma)\, d\sigma$
Time Integration	$\mathcal{L}\left\{ \int_0^t f(\tau)\, d\tau \right\} = \mathcal{L}\left\{ u(t) * f(t)\right\} = {1 \over s} F(s)$
Scaling	$\mathcal{L} \left\{ f(at) \right\} = {1 \over a} F \left ( {s \over a} \right )$
Initial value theorem	$f(0^+)=\lim_{s\to \infty}{sF(s)}$
Final value theorem	$f(\infty)=\lim_{s\to 0}{sF(s)}$
Frequency Shifts	$\mathcal{L}\left\{ e^{at} f(t) \right\} = F(s - a)$ $\mathcal{L}^{-1} \left\{ F(s - a) \right\} = e^{at} f(t)$
Time Shifts	$\mathcal{L}\left\{ f(t - a) u(t - a) \right\} = e^{-as} F(s)$ $\mathcal{L}^{-1} \left\{ e^{-as} F(s) \right\} = f(t - a) u(t - a)$
Convolution Theorem	$\mathcal{L}\{f(t) * g(t)\} = F(s) G(s)$

Where:

$f(t) = \mathcal{L}^{-1} \{ F(s) \}$

$g(t) = \mathcal{L}^{-1} \{ G(s) \}$

s = σ + j ω

The Fourier Transform is defined as such:

$F(j\omega) = \mathcal{F} \left\{f(t) \right\} = \int_{-\infty}^\infty f(t) e^{-j\omega t}dt$

The result of the transform of a time-domain function f(t) is F(jω).

Time Domain	Fourier Domain
$x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega)e^{j \omega t}d \omega$	$X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t$
$1$	$2πδ(ω)$
$- 0.5 + u (t)$	$\frac{1}{j \omega}$
$δ(t)$	$1$
$δ(t - c)$	$e - j ω c$
$u (t)$	$\pi \delta(\omega)+\frac{1}{j \omega}$
$e - b t u (t)$	$\frac{1}{j \omega+b}$
$cosω 0 t$	$π[δ(ω + ω 0) + δ(ω - ω 0)]$
$cos(ω 0 t + θ)$	$π[e - j θ δ(ω + ω 0) + e j θ δ(ω - ω 0)]$
$sinω 0 t$	$j π[δ(ω + ω 0) - δ(ω - ω 0)]$
$sin(ω 0 t + θ)$	$j π[e - j θ δ(ω + ω 0) - e j θ δ(ω - ω 0)]$
$rect(\frac{t}{\tau})$	$\tau sinc \frac{\tau \omega}{2 \pi}$
$\tau sinc \frac{\tau t}{2 \pi}$	$2π p τ (ω)$
$(1-\frac{2 \|t\|}{\tau})p_\tau (t)$	$\frac{\tau}{2} sinc^2 \frac{\tau \omega}{4 \pi}$
$\frac{\tau}{2} sinc^2 ( \frac{\tau t}{4 \pi} )$	$2 \pi (1-\frac{2\|\omega\|}{\tau})p_\tau (\omega)$