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Sahithyan's S2
Sahithyan's S2 — Theory of Electricity

Laplace Transform

A mathematical operation that transforms a function of time f(t)f(t), into a function of a complex variable, denoted as F(s)F(s). Mathematically, the Laplace Transform of a function f(t)f(t) is defined as:

F(s)=L{f(t)}=∫0∞e−stf(t) dtF(s) = \mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, \text{d}t

Here:

  • tt represents time
  • s=σ+jωs = \sigma + j\omega, where σ,ω∈R\sigma, \omega \in \mathbb{R}
  • e−ste^{-st} is the exponential decay factor that weights the function f(t)f(t).

Can only be applied to casual functions. Converts a differential equation (time domain) into a linear complex algebraic equation (frequency domain). After Laplace transformation is applied, the resulting functions are capitalized by convention.

Casual function

A function of time f(t)f(t) satisfying:

f(t)=0fort<0f(t) = 0 \quad \text{for} \quad t < 0

Inverse Laplace transform

If L{f(t)}=F(s)\mathcal{L}\{f(t)\} = F(s), then the inverse Laplace Transform is given by:

f(t)=L−1{F(s)}=12πj∫σ−j∞σ+j∞estF(s) dsf(t) = \mathcal{L}^{-1}\{F(s)\} = \frac{1}{2\pi j} \int_{\sigma - j\infty}^{\sigma + j\infty} e^{st} F(s) \, \text{d}s

Special functions

Unit step function

u(t)={0,t<01,t>0u(t) = \begin{cases} 0, & t<0 \\ 1, & t>0 \end{cases}

Laplace transform of the unit step function is:

L{u(t)}=1s\mathcal{L}\{u(t)\} = \frac{1}{s}

Unit impulse function

δ(t)={0,t≠0∞,t=0\delta(t) = \begin{cases} 0, & t\neq 0 \\ \infty, & t=0 \end{cases}

Laplace transform of the unit impulse function is:

L{δ(t)}=1\mathcal{L}\{\delta(t)\} = 1

Area under the curve is 11.

∫−∞∞δ(t) dt=1\int_{-\infty}^{\infty} \delta(t)\, \text{d}t = 1

And it has a special property:

∫−∞∞f(t−k) δ(t) dt=f(k)\int_{-\infty}^{\infty} f(t-k)\,\delta(t)\, \text{d}t = f(k)

Unit ramp function

r(t)={0,t<0t,t>0r(t) = \begin{cases} 0, & t<0 \\ t, & t>0 \end{cases}

Laplace transform of the unit ramp function is:

L{r(t)}=1s2\mathcal{L}\{r(t)\} = \frac{1}{s^2}

Circuit transformation

Both AC and DC voltage sources are transformed according to the Laplace transform table. Resistors are included as is.

An inductor LL is converted to LsLs. To cater for initial current, a voltage source Li(0+)L i(0^+) is added aiding the voltage.

A capacitor CC is converted to 1Cs\frac{1}{Cs}. To cater for initial voltage, a voltage source v(0+)s\frac{v(0^+)}{s} is added opposing the voltage.

Properties

Suppose f(t)f(t) has a Laplace transform F(s)F(s) for the below definitions.

Linearity

aa and bb are constants.

L{af(t)+bg(t)}=aL{f(t)}+bL{g(t)}\mathcal{L}\{a f(t) + b g(t)\} = a \mathcal{L}\{f(t)\} + b \mathcal{L}\{g(t)\}

Differentiation

L{f′(t)}=sF(s)−f(0+)\mathcal{L}\{f'(t)\} = sF(s) - f(0^+) L{f′′(t)}=s2F(s)−sf(0+)−f′(0+)\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0^+) - f'(0^+) L{f(n)(t)}=snF(s)−∑k=1n−1skf(0+)−∑k=1n−1f(k)(0+)\mathcal{L}\{f^{(n)}(t)\} = s^nF(s) - \sum_{k=1}^{n-1} {s^{k} f(0^+)} - \sum_{k=1}^{n-1} f^{(k)}(0^+)

Integration

L{∫0tf(t) dt}=F(s)s\mathcal{L}\left\{\int_{0}^{t} f(t) \, \text{d}t\right\} = \frac{F(s)}{s}

Time Scaling

L{f(ta)}=aF(as)\mathcal{L}\left\{f\left(\frac{t}{a}\right)\right\} = aF\left(as\right)

Frequency Scaling

L−1{F(sa)}=af(at)\mathcal{L}^{-1}\left\{F\left(\frac{s}{a}\right)\right\} = af\left(at\right)

Multiplication by t

L{tnf(t)}=(−1)ndnF(s)dsn\mathcal{L}\left\{t^n f(t)\right\} = (-1)^n \frac{\text{d}^n F(s)}{\text{d}s^n}

Time shift

L{f(t−T)}=e−sTF(s)\mathcal{L}\left\{f(t - T)\right\} = e^{-sT}F(s)

Frequency shift

L−1{F(s+a)}=e−atf(t)\mathcal{L}^{-1}\{F(s+a)\} = e^{-at}f(t)

Theorems

Initial Value Theorem

f(0+)=lim⁡t→0+f(t)=lim⁡s→∞sF(s)f(0^+) = \lim_{t \to 0+} f(t) = \lim_{s \to \infty} sF(s)

Final Value Theorem

f(∞)=lim⁡t→∞f(t)=lim⁡s→0sF(s)f(\infty) = \lim_{t \to \infty} f(t)= \lim_{s \to 0} sF(s)

Laplace transform table

Function NameFunctionLaplace Transform
Unit Impulseδ(t)\delta(t)11
Unit Stepu(t)u(t)1s\frac{1}{s}
Polynomialtnt^nn!sn+1\frac{n!}{s^{n+1}}
Exponentiale−ate^{-at}1(s+a)\frac{1}{(s + a)}
Sine Wavesin⁡ωt\sin \omega tω(s2+ω2)\frac{\omega}{(s^2 + \omega^2)}
Cosine Wavecos⁡ωt\cos \omega ts(s2+ω2)\frac{s}{(s^2 + \omega^2)}
Damped Sine Wavee−atsin⁡ωte^{-at} \sin \omega tω(s+a)2+ω2\frac{\omega}{(s + a)^2 + \omega^2}
Damped Cosine Wavee−atcos⁡ωte^{-at} \cos \omega t(s+a)(s+a)2+ω2\frac{(s + a)}{(s + a)^2 + \omega^2}
Sinh Wavesinh⁡at\sinh ata(s2−a2)\frac{a}{(s^2 - a^2)}
Cosh Wavecosh⁡at\cosh ats(s2−a2)\frac{s}{(s^2 - a^2)}
Damped Sinh Wavee−btsinh⁡ate^{-bt} \sinh ata(s+b)2−a2\frac{a}{(s + b)^2 - a^2}
Damped Cosh Wavee−btcosh⁡ate^{-bt} \cosh ats+b(s+b)2−a2\frac{s + b}{(s + b)^2 - a^2}
When a≠ba \neq be−at−e−btb−a\frac{e^{-at} - e^{-bt}}{b - a}1(s+a)(s+b)\frac{1}{(s + a)(s + b)}
When a≠ba \neq ba.e−at−b.e−bta−b\frac{a.e^{-at} - b.e^{-bt}}{a - b}s(s+a)(s+b)\frac{s}{(s + a)(s + b)}