Calculo de un circuito de doble polaridad de la forma:
$$f(t)=t , -2 \leq t \leq 2$$
Serie Trigonométrica de Fourier:
$$f(t)= \frac{a_{0}}{2} + \displaystyle\sum_{n=1}^\infty a_{n}\cos{n\omega_{0}t} + b_{n}\sin{n\omega_{0}t}$$
Serie Compleja de Fourier:
$$f(t)= C_{0} + \sum_{\substack{n=-\infty\\n\ne 0}}^\infty C_{n}e^{in\omega_{0}t}$$
$$T=4$$
$$\omega_{0}= \frac{2\pi}{T}=2\pi f=\frac{2\pi}{4}=\frac{\pi}{2}$$
$$f=\frac{1}{T}=\frac{\omega_{0}}{2\pi}=\frac{1}{4}=\cfrac{\cfrac{\pi}{2}}{\cfrac{2\pi}{1}}=\frac{\pi}{4\pi}=\frac{1}{4}$$
Coeficientes de Fourier para Serie Trigonometrica:
$$a_{0}= \frac{2}{T} \int_{d}^{d+T}f(t) \, dt$$
$$a_{n} = \frac{2}{T} \int_{d}^{d+T} f(t) \cos n \omega_{0}t\, dt$$
$$b_{n} = \frac{2}{T} \int_{d}^{d+t} f(t) \sin n \omega_{0}t\, dt$$
Iniciando el calculo:
$$a_{0} = \frac{2}{4} \int_{-2}^{2}t\,dt = \frac{1}{2} \bigg[\frac{t^2}{2}\bigg]_{-2}^{2} =
\frac{1}{2} \bigg[\frac{(-2)^2}{2} - \frac{2^2}{2}\bigg] =
\frac{1}{2} \big[0\big] = 0$$
$$ a_{n} = \frac{2}{4} \int_{-2}^{2}t \cos n \frac{\pi}{2} t\,dt$$
Tabla:
| $$+$$ |
$$t$$ |
$$\cos n \frac{\pi}{2} t$$ |
| $$-$$ |
$$1$$ |
$$\frac{\sin n \frac{\pi}{2} t }{n\frac{\pi}{2}}$$ |
| $$$$ |
$$0$$ |
$$\frac{1}{n\frac{\pi}{2}}\bigg[\frac{-\cos n \frac{\pi}{2}t}{n\frac{\pi}{2}}\bigg]$$ |
$$a_{n} = \frac{1}{2} \bigg[\frac{t\sin n \frac{\pi}{2} t }{n\frac{\pi}{2}}
+ \frac{1}{n\frac{\pi}{2}}\bigg[\frac{\cos n \frac{\pi}{2}t}{n\frac{\pi}{2}}\bigg]\bigg]_{-2}^{2}$$
$$ a_{n} = \frac{1}{2} \Bigg[\Bigg(\frac{2\sin n\frac{\pi}{2}2}{n\frac{\pi}{2}} + \frac{1}{n\frac{\pi}{2}}\bigg[\frac{\cos n \frac{\pi}{2}2}{n\frac{\pi}{2}}\bigg]\Bigg) - \Bigg(\frac{-2\sin n\frac{\pi}{2}-2}{n\frac{\pi}{2}} + \frac{1}{n\frac{\pi}{2}}\bigg[\frac{\cos n \frac{\pi}{2}-2}{n\frac{\pi}{2}}\bigg]\Bigg)\Bigg] $$
$$ a_{n} = \frac{1}{2} \Bigg[\Bigg(\frac{4\sin n\pi}{n\pi} + \frac{4 \cos n \pi}{n^2\pi^2}\Bigg) - \Bigg(\frac{-4\sin -n\pi}{n\pi} +\frac{4\cos -n\pi} {n^2\pi^2}\Bigg)\Bigg] $$
$$ a_{n} = \frac{1}{2} \Bigg[\Bigg(0 + \frac{4 \cos n \pi}{n^2\pi^2}\Bigg) - \Bigg(0 +\frac{4\cos -n\pi} {n^2\pi^2}\Bigg)\Bigg] $$
$$ a_{n} = \frac{1}{2} \Bigg[\frac{4 \cos n \pi}{n^2\pi^2} -\frac{4\cos -n\pi} {n^2\pi^2}\Bigg] = 0$$
$$ b_{n} = \frac{2}{4} \int_{-2}^{2}t \sin n \frac{\pi}{2} t\,dt$$
Tabla:
| $$+$$ |
$$t$$ |
$$\sin n \frac{\pi}{2} t$$ |
| $$-$$ |
$$1$$ |
$$\frac{- \cos n \frac{\pi}{2} t }{n\frac{\pi}{2}}$$ |
| $$$$ |
$$0$$ |
$$-\frac{1}{n\frac{\pi}{2}}\bigg[\frac{\sin n \frac{\pi}{2}t}{n\frac{\pi}{2}}\bigg]$$ |
$$b_{n} = \frac{1}{2}\bigg[\frac{-2t \cos n \frac{\pi}{2}t}{n\pi} + \frac{4\sin n \frac{\pi}{2}t}{n^2 \pi^2}\bigg]_{-2}^{2}$$
$$b_{n} = \frac{1}{2}\Bigg[\Bigg(\frac{-2(2) \cos n \frac{\pi(2)}{2}}{n\pi} + \frac{4\sin n \frac{\pi(2)}{2}}{n^2 \pi^2}\Bigg) - \Bigg(\frac{-2(-2) \cos n \frac{\pi(-2)}{2}}{n\pi} + \frac{4\sin n \frac{\pi(-2)}{2}}{n^2 \pi^2}\Bigg)\Bigg]$$
$$b_{n} = \frac{1}{2}\Bigg[\Bigg(\frac{-4 \cos n \pi}{n\pi} + \frac{4\sin n \pi}{n^2 \pi^2}\Bigg) - \Bigg(\frac{4 \cos -n\pi }{n\pi} + \frac{4\sin -n\pi}{n^2 \pi^2}\Bigg)\Bigg]$$
$$b_{n} = \frac{1}{2}\Bigg[\Bigg(\frac{-4 \cos n \pi}{n\pi} +0\Bigg) - \Bigg(\frac{4 \cos -n\pi }{n\pi} + 0\Bigg)\Bigg]$$
$$b_{n} = \frac{1}{2}\Bigg[\frac{-4 \cos n \pi}{n\pi} - \frac{-4 \cos - n \pi}{n\pi}\Bigg]$$
$$b_{n} = \frac{1}{2}\Bigg[\frac{-4 (-1)^n)}{n\pi} - \frac{-4 (-1)^n}{n\pi}\Bigg]$$
$$b_{n} = \frac{1}{2}\Bigg[\frac{-8 (-1)^n}{n\pi}\Bigg]$$
$$b_{n} = \frac{-8 (-1)^n}{2n\pi} = \frac{-4 (-1)^n}{n\pi} = \frac{4 (-1)^{n+1}}{n\pi} $$
Coeficientes obtenidos:
$$
\begin{cases}
a_{0} = 0 \\
a_{n} = 0 \\
b_{n} = \frac{4 (-1)^{n+1}}{n\pi}
\end{cases}
$$
Por lo tanto, la serie es:
$$f(t)= 4\displaystyle\sum_{n=1}^\infty \frac{ (-1)^{n+1}}{n\pi}\sin{\frac{n\pi}{2}t}$$
Aproximación de orden 3:
$$f(t) = 4\bigg(\pi \sin \frac{\pi}{2}t - \frac{1}{2\pi} \sin \pi t + \frac{1}{3\pi} \sin \frac{3\pi}{2}t\bigg)$$
Ahora calcularemos los coeficientes para la serie Compleja:
$$f(t)= C_{0} + \sum_{\substack{n=-\infty\\n\ne 0}}^\infty C_{n}e^{in\omega_{0}t}$$
Los coeficientes:
$$C_{0} = \frac{1}{T} \int_{d}^{d+T} f(t) \, dt$$
$$C_{n} = \frac{1}{T} \int_{d}^{d+T} f(t)e^{-in \omega_{0} t} \, dt$$
Calculando:
$$C_{0} = \frac{1}{4} \int_{-2}^{2} t\,dt = \frac{1}{4}\Bigg[\frac{t^2}{2}\Bigg]_{-2}^{2} = 0$$
$$C_{n} = \frac{1}{4} \int_{-2}^{2} t e^{-\frac{i n \pi }{2} t} \, dt$$
Tabla:
| $$+$$ |
$$t$$ |
$$e^{-\frac{i n \pi }{2} t}$$ |
| $$-$$ |
$$1$$ |
$$-\frac{e^{-\frac{i n \pi }{2} t}}{\frac{i n \pi }{2}}$$ |
| $$$$ |
$$0$$ |
$$-\frac{e^{-\frac{i n \pi }{2} t}}{\frac{ n^2 \pi^2 }{4}}$$ |
$$C_{n} = \frac{1}{4} \Bigg[-\frac{te^{-\frac{i n \pi }{2} t}}{\frac{i n \pi }{2}} + \frac{e^{-\frac{i n \pi }{2} t}}{\frac{ n^2 \pi^2 }{4}}\Bigg]_{-2}^{2}$$
$$C_{n} = \frac{1}{4} \Bigg[\Bigg(-\frac{(2)e^{-\frac{i n \pi }{2} (2)}}{\frac{i n \pi }{2}} + \frac{e^{-\frac{i n \pi }{2} (2)}}{\frac{ n^2 \pi^2 }{4}}\Bigg) - \Bigg(-\frac{(-2)e^{-\frac{i n \pi }{2} (-2)}}{\frac{i n \pi }{2}} + \frac{e^{-\frac{i n \pi }{2} (-2)}}{\frac{ n^2 \pi^2 }{4}}\Bigg)\Bigg]$$
$$C_{n} = \frac{1}{4} \Bigg[\Bigg(-\frac{4e^{-i n \pi}}{i n \pi} + \frac{4e^{-i n \pi}}{n^2 \pi^2}\Bigg) -
\Bigg(\frac{4e^{i n \pi}}{i n \pi}
+ \frac{4e^{i n \pi}}{ n^2 \pi^2}\Bigg)\Bigg]$$
$$C_{n} = \frac{1}{4} \Bigg[-\frac{4e^{-i n \pi}}{i n \pi} + \frac{4e^{-i n \pi}}{n^2 \pi^2} - \frac{4e^{i n \pi}}{i n \pi} - \frac{4e^{i n \pi}}{ n^2 \pi^2}\Bigg]$$
$$C_{n} = \frac{1}{4} \Bigg[-\frac{4(-1)^n}{i n \pi} + \frac{4(-1)^n}{n^2 \pi^2} - \frac{4(-1)^n}{i n \pi} - \frac{4(-1)^n}{ n^2 \pi^2}\Bigg]$$
$$C_{n} = \frac{1}{4} \Bigg[-\frac{4(-1)^n}{i n \pi} - \frac{4(-1)^n}{i n \pi} \Bigg]$$
$$C_{n} = \frac{1}{4} \Bigg[- \frac{8(-1)^n}{i n \pi} \Bigg]$$
$$C_{n} = - \frac{8(-1)^n}{4i n \pi} = \frac{-2(-1)^n}{in\pi} = \frac{2(-1)^{n+1}}{in\pi}$$
Por lo tanto, la serie compleja es:
$$f(t)= 2\sum_{\substack{n=-\infty\\n\ne 0}}^\infty \frac{(-1)^{n+1}}{in\pi} \cdot e^{i n \frac{\pi}{2} t }$$
Calculo de Potencia de Orden 2:
$$P= {\lvert C_{0} \rvert }^2 + 2 \sum_{n=1}^2 {\lvert C_{n} \rvert }^2$$
$$P= {\lvert C_{0} \rvert }^2 + 2{\lvert C_{1} \rvert }^2 + 2{\lvert C_{2} \rvert }^2$$
$$C_{n} = \frac{2(-1)^{n+1}}{in\pi}$$
Tabla:
| $$n$$ |
$$C_{n}$$ |
$$\lvert C_{n} \rvert$$
$$Amplitud$$
|
$${\lvert C_{n} \rvert}^2$$
$$Potencia$$
|
$$\sphericalangle C_{n}$$
$$Fase$$
|
| $$0$$ |
$$0$$ |
$$0$$ |
$$0$$ |
$$2\pi$$ |
| $$1$$ |
$$\frac{-2i}{\pi}$$ |
$$\frac{2}{\pi}$$ |
$$\frac{4}{{\pi}^2}$$ |
$$\frac{3 \pi }{2}$$ |
| $$2$$ |
$$\frac{i}{\pi}$$ |
$$\frac{1}{\pi}$$ |
$$\frac{1}{{\pi}^2}$$ |
$$\frac{\pi}{2}$$ |
| $$3$$ |
$$\frac{-2i}{3\pi}$$ |
$$\frac{2}{3\pi}$$ |
$$\frac{4}{9{\pi}^2}$$ |
$$\frac{3\pi}{2}$$ |
Calculo de la Potencia:
$$ P \approx 2\Bigg( \frac{4}{{\pi}^2} \Bigg) + 2\Bigg( \frac{1}{{\pi}^2}\Bigg) \approx \frac{1}{{\pi}^2} \Bigg( 8 +2 \Bigg)$$
$$ \approx 1.0132 [W] (Watts)$$
Espectro de Potencia & Fase con 3 armonicos:
$$n \omega_{0} = \frac{n}{2 \pi} f_{0} [Hz]$$
