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Common Constants in Electrical Power Systems Engineering

Speed of electric current (in 12-gauge copper wire) ≈ v ≈ 300 m/μs

Vacuum permittivity (electric constant) = ε₀ =8.85418782 × 10^-12 m^-3 kg^-1 s⁴ A²

WIL: Making Quick References

Learning 1:
Quick references are sometimes also referred to as reference cards, reference sheets, crib sheets, and cheat sheets.

Learning 2:
Quick references used more in the early stages. If information is used often enough, the reference eventually gets to the mind.

Learning 3:
Having physical quick reference sheets helps with memorization using the method of loci.

Learning 4:
Yes search engines exist, committing some information to memory still helps though because of 1) current low bandwidth between the information found from search engines and information we need as well as 2) ongoing limited development, adoption, and accessibility of “ultra-high bandwidth brain-machine interfaces to connect humans and computers”(NEURALINK)

Common Laplace Transforms

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

Common laplace transforms

$\text{a is a constant}$

$L \left\{ a \right\} =$ $\frac{a}{s}$

$L \left\{ t\right\} =$ $\frac{1}{s^2}$

$L \left\{ t^n \right\} =$ $\frac{n!}{s^{n+1}}$

$L \left\{ e^{-at}\right\} =$ $\frac{1}{s+a}$

$L \left\{ te^{-at}\right\} =$ $\frac{1}{(s+a)^2}$

$L \left\{ 1 - e^{-at}\right\} =$ $\frac{a}{s(s+a)}$

$L \left\{ sin(\omega t)\right\} =$ $\frac{\omega}{s^2 + \omega^2}$

$L \left\{ cos(\omega t)\right\} =$ $\frac{s}{s^2 + \omega^2}$

$L \left\{ 1 - cos(\omega t)\right\} =$ $\frac{\omega^2}{s(s^2 + \omega^2)}$

$L \left\{ \omega t sin(\omega t) \right\} =$ $\frac{2 \omega^2 s}{(s^2 + \omega^2)^2}$

$L \left\{ sin(\omega t) - \omega t cos(\omega t) \right\} =$ $\frac{2 \omega^3}{(s^2 + \omega^2)^2}$

$L \left\{ sin(\omega t + \phi) \right\} =$ $\frac{sin(\phi) + \omega cos(\phi) }{s^2 + \omega^2}$

$L \left\{ e^{-at} sin(\omega t) \right\} =$ $\frac{\omega}{(s + a)^2 + \omega^2}$

$L \left\{ e^{-at} cos(\omega t) \right\} =$ $\frac{s+a}{(s + a)^2 + \omega^2}$

$L \left\{ e^{-at} (cos(\omega t) - \frac{a}{\omega}sin(\omega t )) \right\} =$ $\frac{s}{(s + a)^2 + \omega^2}$

$L \left\{ e^{-at} + \frac{a}{\omega}sin(\omega t ) - cos(\omega t) \right\} =$ $\frac{a^2 + \omega^2}{(s + a)(s^2 + \omega^2)}$

$L \left\{ sinh(\beta t \right\} =$ $\frac{\beta}{s^2- \beta^2}$

$L \left\{e^{-at} sinh(\beta t \right\} =$ $\frac{\beta}{(s+a)^2- \beta^2}$

$L \left\{cosh(\beta t) \right\} =$ $\frac{s}{s^2 - \beta^2}$

$L \left\{e^{-at} sinh(\beta t \right\} =$ $\frac{s+a}{(s+a)^2- \beta^2}$

$L \left\{e^{-at} f(t) \right\} =$ $F(s+a)$

$L \left\{tf(t) \right\} =$ $-F(s)$

$L \left\{ x \right\} =$ $\bar{x}$

$L \left\{ \dot{x} \right\} =$ $s \bar{x} - x(0)$

$L \left\{ \ddot{x} \right\} =$ $s^2 \bar{x} - sx(0) - \dot{x}(0)$

$\text{Unit Step Function:}$

$L \left\{ u(t-a) \right\} = s^{-1}e^{-as}$

$L \left\{ f(t-a)u(t-a) \right\} = F(s)e^{-sa}$

$\text{Dirac Delta Function:}$

$L \left\{ \delta (t) \right\} =$ $1$

$L \left\{ \delta (t-a) \right\} =$ $e^{-as}$

$\text{Convolution Integral:}$

$\int_{0}^{t} f(z) g(t-z) dz = F(s)G(s)$

$\int_{0}^{t} f(t - z)g(z) dz = F(s)G(s)$