CHEM-E4106 - Electrochemistry D, Lecture, 8.1.2024-20.2.2024
Kurssiasetusten perusteella kurssi on päättynyt 21.02.2024 Etsi kursseja: CHEM-E4106
Kirja
Suorituksen vaatimukset
11. Appendices
11.4. Laplace transforms
Definition: | \( \mathcal{L}\{f(t)\}=F(s)=\int_0^{\infty}{e^{-st}}f(t)dt \) |
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Derivation: | \( \mathcal{L}\{f'(t)\}=sF(s)-f(0) \) |
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Integration |
\( \mathcal{L}\left\{\int_{0}^{t}{f(u)du}\right\}=\frac{1}{s}F(s) \) \( \mathcal{L}\left\{ \int_{0}^{t}\int_{0}^{u}{f(v)dvdu} \right\}=\frac{1}{s^2}F(s) \) |
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Linearity: | \( \mathcal{L}\{Af(t)+Bg(t)\}=AF(s)+BG(s) \) |
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Transfer in Laplace domain: | \( \mathcal{L}\{e^{at}f(t)\}=F(s-a) \) |
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Transfer in time domain: | \( \mathcal{L}\{f(t-a)H_s(t-a)\}=e^{-as}F(s), a>0 \) \( H_s(t-a)=\begin{cases}0,t<\\1, t>a\end{cases} \) Heaviside step function |
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Convolution: | \( \mathcal{L}\left\{\int_{0}^{t}{f(t-u)g(u)du}\right\}=F(s)G(s) \) |
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Heaviside theory: | \( \mathcal{L}\left\{\sum\limits_{n=1}^m{\frac{p(a_n)}{q'(a_n)}e^{a_nt}}\right\}=\frac{P(s)}{Q(s)} \), where roots of Q(s) are: a1, a2, ..., an. |
Most common transforms in electrochemistry
f(t) |
F(s) |
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\(1\) | \(\displaystyle \frac{1}{s} \) |
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\( t \) |
\(\displaystyle \frac{1}{s^2} \) |
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\(\displaystyle \frac{t^{n-1}}{(n-1)!} \) |
\(\displaystyle \frac{1}{s^n} \) | ||
\(\displaystyle\frac{1}{\sqrt{\pi t}}\) |
\( \displaystyle\frac{1}{\sqrt{s}} \) |
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\(\displaystyle2\sqrt{\frac{t}{\pi}}\) |
\(\displaystyle \frac{1}{s\sqrt{s}} \) |
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\(\displaystyle e^{-at} \) |
\(\displaystyle \frac{1}{s+a} \) |
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\( \sin(\omega t) \) |
\(\displaystyle \frac{\omega}{s^2+\omega^2} \) |
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\( \cos(\omega t) \) |
\(\displaystyle \frac{s}{s^2+\omega^2} \) |
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\( \displaystyle\frac{1}{\sqrt{\pi t}}-ae^{a^2t}\text{erfc}(a\sqrt{t}) \) |
\(\displaystyle \frac{1}{\sqrt{s}+a} \) |
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\( \displaystyle\frac{1}{\sqrt{\pi t}}+ae^{a^2t}\text{erf}(a\sqrt{t}) \) |
\(\displaystyle \frac{\sqrt{s}}{s-a^2} \) |
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\( \displaystyle e^{a^2t}\text{erf}({a\sqrt{t}}) \) |
\(\displaystyle \frac{a}{\sqrt{s}(s-a^2)} \) |
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\( \displaystyle1-e^{a^2t}\text{erfc}({a\sqrt{t}}) \) |
\(\displaystyle \frac{a}{s(\sqrt{s}+a)} \) |
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\( \displaystyle e^{a^2t}\text{erfc}({a\sqrt{t}}) \) |
\(\displaystyle \frac{1}{\sqrt{s}(\sqrt{s}+a)} \) |
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\( \displaystyle\frac{a}{2\sqrt{\pi t^3}}\exp\left(-\frac{a^2}{4t}\right) \) |
\(\displaystyle e^{-a\sqrt{s}},a \geq0 \) |
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\(\displaystyle \frac{1}{\sqrt{\pi t}} \text{exp}\left(-\frac{a^2}{4t}\right) \) |
\(\displaystyle \frac{1}{\sqrt{s}}e^{-a\sqrt{s}}, a \geq0 \) | ||
\( \displaystyle\text{erfc}\left(\frac{a}{2\sqrt{t}}\right) \) |
\(\displaystyle \frac{1}{s}e^{-a\sqrt{s}}, a \geq0 \) |
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\(\displaystyle 2\sqrt{\frac{t}{\pi}}\exp\left(-\frac{a^2}{4t}\right)-a\text{ erfc}\left(\frac{a}{2\sqrt{t}}\right) \) |
\(\displaystyle \frac{1}{s\sqrt{s}}e^{-a\sqrt{s}}, a \geq0 \) |
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\(\displaystyle \frac{1}{\sqrt{\pi t}}\exp\left(-\frac{a^2}{4t}\right)-be^{ab+b^2t}\text{erfc}\left(\frac{a}{2\sqrt{t}}+b\sqrt{t}\right) \) |
\(\displaystyle \frac{e^{-a\sqrt{s}}}{\sqrt{s}+b}, a \geq0 \) |