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Sahithyan's S2
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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

Section titled “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

Section titled “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

Section titled “Special functions”

Unit step function

Section titled “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

Section titled “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

Section titled “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

Section titled “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

Section titled “Properties”

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

Linearity

Section titled “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

Section titled “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

Section titled “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

Section titled “Time Scaling”
L{f(ta)}=aF(as)\mathcal{L}\left\{f\left(\frac{t}{a}\right)\right\} = aF\left(as\right)

Frequency Scaling

Section titled “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

Section titled “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

Section titled “Time shift”
L{f(t−T)}=e−sTF(s)\mathcal{L}\left\{f(t - T)\right\} = e^{-sT}F(s)

Frequency shift

Section titled “Frequency shift”
L−1{F(s+a)}=e−atf(t)\mathcal{L}^{-1}\{F(s+a)\} = e^{-at}f(t)

Theorems

Section titled “Theorems”

Initial Value Theorem

Section titled “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

Section titled “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

Section titled “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)}