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| -rw-r--r-- | exercise02/exercise02.tex | 348 | ||||
| -rw-r--r-- | main/exercise02.tex | 5 |
2 files changed, 345 insertions, 8 deletions
diff --git a/exercise02/exercise02.tex b/exercise02/exercise02.tex index 580ac67..db2fa8c 100644 --- a/exercise02/exercise02.tex +++ b/exercise02/exercise02.tex @@ -107,11 +107,13 @@ \task \begin{equation*} x(t) = \begin{cases} - \SI{-0.2}{V}, &\quad \text{ if} \; \left(\SI{-0.5}{s} + n \cdot \SI{2}{s}\right) \leq t < \left(\SI{0.5}{s} + n \cdot \SI{2}{s}\right) \\ - \SI{-0.2}{V}, &\quad \text{ if} \; \left(\SI{0.5}{s} + n \cdot \SI{2}{s}\right) \leq t < \left(\SI{1.5}{s} + n \cdot \SI{2}{s}\right) \\ + \SI{0.8}{V}, &\quad \text{ if} \; \left(-\frac{T_0}{4} + n \cdot T_0\right) \leq t < \left(\frac{T_0}{4} + n \cdot T_0\right) \\ + \SI{-0.2}{V}, &\quad \text{ if} \; \left(\frac{T_0}{4} + n \cdot T_0\right) \leq t < \left(\frac{3T_0}{4} + n \cdot T_0\right) \\ \end{cases} \qquad \forall \; n \in \mathbb{Z} \end{equation*} + with $T_0 = \SI{2}{s}$ + \task \begin{itemize} \item Period: $T_0 = \SI{2}{s}$ @@ -119,6 +121,207 @@ \item Base angular frequency: $\omega_0 = \SI{3.14}{s^{-1}}$ \end{itemize} + \task + \begin{equation*} + \begin{split} + a_n &= \frac{2}{T_0} \left( \int\limits_{-\frac{T_0}{4}}^{\frac{T_0}{4}} \SI{0.8}{V} \cdot \cos\left(n\frac{2\pi}{T_0}t\right) \, \mathrm{d} t - \int\limits_{\frac{T_0}{4}}^{\frac{3T_0}{4}} \SI{0.2}{V} \cdot \cos\left(n\frac{2\pi}{T_0}t\right) \, \mathrm{d} t \right) \\ + &= \frac{2}{T_0} \frac{T_0}{2 \pi n} \left( \SI{0.8}{V} \cdot \left[\sin\left(n\frac{2\pi}{T_0}t\right)\right]_{-\frac{T_0}{4}}^{\frac{T_0}{4}} - \SI{0.2}{V} \cdot \left[\sin\left(n\frac{2\pi}{T_0}t\right)\right]_{\frac{T_0}{4}}^{\frac{3T_0}{4}} \right) \\ + &= \frac{1}{\pi n} \left( + \SI{0.8}{V} \cdot \left( + \sin\left(\frac{\pi}{2}n\right) + - \underbrace{\sin\left(-\frac{\pi}{2}n\right)}_{= \sin\left(\frac{\pi}{2}n\right)} + \right) + - \SI{0.2}{V} \cdot \left( + \underbrace{\sin\left(\frac{3\pi}{2}n\right)}_{= \sin\left(\frac{\pi}{2}n\right)} + - \sin\left(\frac{\pi}{2}n\right) + \right) + \right) \\ + &= \frac{2}{\pi n} \left( \SI{0.8}{V} \cdot \sin\left(\frac{\pi}{2}n\right) - \SI{0.2}{V} \cdot \sin\left(\frac{\pi}{2}n\right) \right) \\ + &= \frac{\SI{2,0}{V}}{\pi n} \cdot \sin\left(\frac{\pi}{2}n\right) \\ + &= \begin{cases} + \frac{\SI{2,0}{V}}{\pi n} (-1)^{\frac{n+3}{2}} &\quad \; \forall n \text{ odd}, \\ + 0 &\quad \; \forall n \text{ even}. + \end{cases} + \end{split} + \end{equation*} + + $a_0$ (DC bias) needs special treatment. + \begin{equation*} + \begin{split} + a_0 &= \frac{1}{T_0} \left( \int\limits_{-\frac{T_0}{4}}^{\frac{T_0}{4}} \SI{0.8}{V} \, \mathrm{d} t - \int\limits_{\frac{T_0}{4}}^{\frac{3T_0}{4}} \SI{0.2}{V} \, \mathrm{d} t \right) \\ + &= \frac{1}{T_0} \left( \SI{0.8}{V} \cdot \left[t\right]_{-\frac{T_0}{4}}^{\frac{T_0}{4}} - \SI{0.2}{V} \cdot \left[t\right]_{\frac{T_0}{4}}^{\frac{3T_0}{4}} \right) \\ + &= \frac{1}{T_0} \left( \SI{0.8}{V} \cdot \left(\frac{T_0}{4} - \left(-\frac{T_0}{4}\right)\right) - \SI{0.2}{V} \cdot \left(\frac{3T_0}{4} - \frac{T_0}{4}\right) \right) \\ + &= \frac{1}{T_0} \left( \SI{0.8}{V} \cdot \frac{T_0}{2} - \SI{0.2}{V} \cdot \frac{T_0}{2} \right) \\ + &= \SI{0.3}{V} + \end{split} + \end{equation*} + + \begin{equation*} + \begin{split} + b_n &= \frac{2}{T_0} \left( \int\limits_{-\frac{T_0}{4}}^{\frac{T_0}{4}} \SI{0.8}{V} \cdot \sin\left(n\frac{2\pi}{T_0}t\right) \, \mathrm{d} t - \int\limits_{\frac{T_0}{4}}^{\frac{3T_0}{4}} \SI{0.2}{V} \cdot \sin\left(n\frac{2\pi}{T_0}t\right) \, \mathrm{d} t \right) \\ + &= \frac{2}{T_0} \frac{T_0}{2 \pi n} \left( \SI{0.8}{V} \cdot \left[-\cos\left(n\frac{2\pi}{T_0}t\right)\right]_{-\frac{T_0}{4}}^{\frac{T_0}{4}} - \SI{0.2}{V} \cdot \left[-\cos\left(n\frac{2\pi}{T_0}t\right)\right]_{\frac{T_0}{4}}^{\frac{3T_0}{4}} \right) \\ + &= \frac{1}{\pi n} \left( + \SI{0.8}{V} \cdot \left( + -\cos\left(\frac{\pi}{2}n\right) + + \underbrace{\cos\left(-\frac{\pi}{2}n\right)}_{= \cos\left(\frac{\pi}{2}n\right)} + \right) + - \SI{0.2}{V} \cdot \left( + -\underbrace{\cos\left(\frac{3\pi}{2}n\right)}_{= \cos\left(\frac{\pi}{2}n\right)} + + \cos\left(\frac{\pi}{2}n\right) + \right) + \right) \\ + &= 0 + \end{split} + \end{equation*} + + \task + $\underline{c}_n$: + \begin{equation*} + \begin{split} + b_n &= \frac{1}{T_0} \left( \int\limits_{-\frac{T_0}{4}}^{\frac{T_0}{4}} \SI{0.8}{V} \cdot e^{-j n\frac{2\pi}{T_0}t} \, \mathrm{d} t - \int\limits_{\frac{T_0}{4}}^{\frac{3T_0}{4}} \SI{0.2}{V} \cdot e^{-j n\frac{2\pi}{T_0}t} \, \mathrm{d} t \right) \\ + &= \frac{1}{T_0} \frac{T_0}{2 \pi n (-j)} \left( + \SI{0.8}{V} \cdot \left[e^{-j n\frac{2\pi}{T_0}t}\right]_{-\frac{T_0}{4}}^{\frac{T_0}{4}} + - \SI{0.2}{V} \cdot \left[e^{-j n\frac{2\pi}{T_0}t}\right]_{\frac{T_0}{4}}^{\frac{3T_0}{4}} + \right) \\ + &= j \frac{1}{2 \pi n} \left( + \SI{0.8}{V} \left( + e^{-j n\frac{\pi}{2}} + - e^{+j n\frac{\pi}{2}} + \right) + - \SI{0.2}{V} \left( + e^{-j n\frac{3 \pi}{2}} + - e^{-j n\frac{\pi}{2}} + \right) + \right) \\ + &= \begin{cases} + \frac{\SI{1,0}{V}}{\pi n} e^{j\frac{n+3}{2}\pi} &\quad \; \forall n \text{ odd}, \\ + 0 &\quad \; \forall n \text{ even}. + \end{cases} + \end{split} + \end{equation*} + Following has been used: + \begin{equation*} + \begin{split} + \left(e^{-j n\frac{\pi}{2}} - e^{+j n\frac{\pi}{2}}\right) &= \begin{cases} + -j &\quad \forall n = 4 k + 1, k \in \mathbb{Z}, \\ + +j &\quad \forall n = 4 k + 3, k \in \mathbb{Z}, \\ + 0 &\quad \forall n = 2 k, k \in \mathbb{Z}. + \end{cases} \\ + \left(e^{-j n\frac{3 \pi}{2}} - e^{-j n\frac{\pi}{2}}\right) &= \begin{cases} + +j &\quad \forall n = 4 k + 1, k \in \mathbb{Z}, \\ + -j &\quad \forall n = 4 k + 3, k \in \mathbb{Z}, \\ + 0 &\quad \forall n = 2 k, k \in \mathbb{Z}. + \end{cases} + \end{split} + \end{equation*} + + \task + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$n$}, + ylabel={$\left|\underline{c}_n\right| in \si{V}$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-5.5, + xmax=5.5, + xtick={-5, -4, ..., 5}, + ymin=0, + ymax=0.45, + ytick={0, 0.1, ..., 0.4} + ] + \addplot[red, thick] coordinates {(-5,0) (-5,0.064)}; + \addplot[red, thick] coordinates {(-3,0) (-3,0.106)}; + \addplot[red, thick] coordinates {(-1,0) (-1,0.318)}; + \addplot[red, thick] coordinates {(0,0) (0,0.3)}; + \addplot[red, thick] coordinates {(1,0) (1,0.318)}; + \addplot[red, thick] coordinates {(3,0) (3,0.106)}; + \addplot[red, thick] coordinates {(5,0) (5,0.064)}; + + \addplot[red, thick, only marks, mark=o] coordinates {(-5,0.064)}; + \addplot[red, thick, only marks, mark=o] coordinates {(-4,0)}; + \addplot[red, thick, only marks, mark=o] coordinates {(-3,0.106)}; + \addplot[red, thick, only marks, mark=o] coordinates {(-2,0)}; + \addplot[red, thick, only marks, mark=o] coordinates {(-1,0.318)}; + \addplot[red, thick, only marks, mark=o] coordinates {(0,0.3)}; + \addplot[red, thick, only marks, mark=o] coordinates {(1,0.318)}; + \addplot[red, thick, only marks, mark=o] coordinates {(2,0)}; + \addplot[red, thick, only marks, mark=o] coordinates {(3,0.106)}; + \addplot[red, thick, only marks, mark=o] coordinates {(4,0)}; + \addplot[red, thick, only marks, mark=o] coordinates {(5,0.064)}; + \end{axis} + \end{tikzpicture} + \end{figure} + + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega \text{ in } \si{s^{-1}}$}, + ylabel={$\left|\underline{H}\left(j \omega\right)\right|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-5.5, + xmax=5.5, + ymin=-3.5, + ymax=3.5, + ytick={-3.14159, -1.5708, 1.5708, 3.14159}, + yticklabels={$-\pi\hspace{0.30cm}$, $-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi\hspace{0.10cm}$}, + ] + \addplot[red, thick] coordinates {(-5,0) (-5,0)}; + \addplot[red, thick] coordinates {(-3,0) (-3,-3.14159)}; + \addplot[red, thick] coordinates {(-1,0) (-1,0)}; + \addplot[red, thick] coordinates {(0,0) (0,0)}; + \addplot[red, thick] coordinates {(1,0) (1,0)}; + \addplot[red, thick] coordinates {(3,0) (3,3.14159)}; + \addplot[red, thick] coordinates {(5,0) (5,0)}; + + \addplot[red, thick, only marks, mark=o] coordinates {(-5,0)}; + \addplot[red, thick, only marks, mark=o] coordinates {(-4,0)}; + \addplot[red, thick, only marks, mark=o] coordinates {(-3,-3.14159)}; + \addplot[red, thick, only marks, mark=o] coordinates {(-2,0)}; + \addplot[red, thick, only marks, mark=o] coordinates {(-1,0)}; + \addplot[red, thick, only marks, mark=o] coordinates {(0,0)}; + \addplot[red, thick, only marks, mark=o] coordinates {(1,0)}; + \addplot[red, thick, only marks, mark=o] coordinates {(2,0)}; + \addplot[red, thick, only marks, mark=o] coordinates {(3,3.14159)}; + \addplot[red, thick, only marks, mark=o] coordinates {(4,0)}; + \addplot[red, thick, only marks, mark=o] coordinates {(5,0)}; + \end{axis} + \end{tikzpicture} + \end{figure} + %TODO \end{tasks} \end{solution} @@ -238,7 +441,7 @@ height={0.25\textheight}, width=0.9\linewidth, scale only axis, - xlabel={$\omega \text{ in } \si{Hz}$}, + xlabel={$\omega \text{ in } \si{s^{-1}}$}, ylabel={$\left|\underline{X}\left(j \omega\right)\right| \text{ in } \si{V/Hz}$}, %grid style={line width=.6pt, color=lightgray}, %grid=both, @@ -278,7 +481,7 @@ height={0.25\textheight}, width=0.9\linewidth, scale only axis, - xlabel={$\omega \text{ in } \si{Hz}$}, + xlabel={$\omega \text{ in } \si{s^{-1}}$}, ylabel={$\arg\left(\underline{X}\left(j \omega\right)\right) \text{ in } \si{\degree}$}, %grid style={line width=.6pt, color=lightgray}, %grid=both, @@ -343,7 +546,64 @@ \end{question} \begin{solution} - %TODO + \begin{tasks} + \task + Network analysis: + \begin{figure}[H] + \centering + \begin{circuitikz} + \draw (0, 0) to[L, l=$L$, v=$u_L(t)$, i=$i_C(t)$, o-] ++(2,0) to[short, i=$i_o(t)$, *-o] ++(2,0); + \draw (2, 0) to[C, l=$C$, i=$i_C(t)$, -*] ++(0,-2); + \draw (0, -2) to[short, o-o] ++(4,0); + + \draw (0, 0) to[open, v=$u_i(t)$] (0, -2); + \draw (4, 0) to[open, v^=$u_o(t)$] (4, -2); + \end{circuitikz} + \end{figure} + \begin{itemize} + \item Kirchhoff's current law at the only point in the circuit: + \begin{equation*} + i_L(t) = i_C(t) + i_o(t) + \end{equation*} + \item No load at the output: $i_o(t) = 0$ + \item Kirchhoff's coltage law at the only closed loop in the circuit: + \begin{equation*} + u_i(t) = u_L(t) + u_o(t) + \end{equation*} + \item Differential equation for capacitors: + \begin{equation*} + i_C(t) = C \frac{\mathrm{d} u_o(t)}{\mathrm{d} t} + \end{equation*} + \item Differential equation for inductors: + \begin{equation*} + u_L(t) = L \frac{\mathrm{d} u_L(t)}{\mathrm{d} t} = L C \frac{\mathrm{d}^2 u_o(t)}{\mathrm{d} t^2} + \end{equation*} + \end{itemize} + Finally, + \begin{equation*} + u_i(t) = L C \frac{\mathrm{d}^2 u_o(t)}{\mathrm{d} t^2} + u_o(t) + \end{equation*} + + \task + \begin{equation*} + \begin{split} + u_i(t) &= L C \frac{\mathrm{d}^2 u_o(t)}{\mathrm{d} t^2} + u_o(t) \\ + \underline{U}_i \left(j \omega\right) &= \left( \left(j \omega\right)^2 L C + 1 \right) \underline{U}_o \left(j \omega\right) + \end{split} + \end{equation*} + \begin{equation*} + \begin{split} + \underline{H} \left(j \omega\right) &= \frac{\underline{U}_o \left(j \omega\right)}{\underline{U}_i \left(j \omega\right)} \\ + &= \frac{1}{\left(j \omega\right)^2 L C + 1} + \end{split} + \end{equation*} + + \task + Yes, this is a real circuit which can be implemented with electronic components. Real components do not have knowledge of the future. + + \task + Second order low pass filter + \end{tasks} \end{solution} \begin{question}[subtitle={Amplitude and Phase Response}] @@ -430,6 +690,40 @@ \end{split} \end{equation*} + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega \text{ in } \si{s^{-1}}$}, + ylabel={$\left|\underline{H}\left(j \omega\right)\right|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-21e3, + xmax=21e3, + ymin=0, + ymax=1.2, + ytick={0, 0.5, 1} + ] + \addplot[blue, thick, smooth, domain=-20e3:20e3, samples=100] plot (\x, {sqrt( (4.7e-5 * \x)^2 / ((4.7e-5 * \x)^2 + 1) )}); + \end{axis} + \end{tikzpicture} + \end{figure} + \task \begin{equation*} \begin{split} @@ -446,6 +740,48 @@ \end{split} \end{equation*} + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega \text{ in } \si{s^{-1}}$}, + ylabel={$\left|\underline{H}\left(j \omega\right)\right|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-21e3, + xmax=21e3, + %ymin=-3.5, + %ymax=3.5, + %ytick={-3.14159, -1.5708, 1.5708, 3.14159}, + %yticklabels={$-\pi\hspace{0.30cm}$, $-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi\hspace{0.10cm}$}, + %yticklabels={$\SI{-180}{\degree}$, $\SI{-90}{\degree}$, $\SI{90}{\degree}$, $\SI{180}{\degree}$}, + ymin=-1.7, + ymax=1.7, + ytick={-1.5708, 1.5708}, + yticklabels={$\SI{-90}{\degree}$, $\SI{90}{\degree}$}, + ] + %\addplot[blue, thick, smooth, domain=-20e3:20e3, samples=100] plot (\x, {(2*pi/360) * atan2(((4.7e-05*\x)/((4.7e-05*\x)^2+1)), ((4.7e-05)^2/((4.7e-05*\x)^2+1)))}); + \addplot[blue, thick, smooth, domain=1:20e3, samples=50] plot (\x, {(2*pi/360) * atan(1/((4.7e-5)*\x))}); + \addplot[blue, thick, smooth, domain=-20e3:-1, samples=50] plot (\x, {(2*pi/360) * atan(1/((4.7e-5)*\x))}); + \end{axis} + \end{tikzpicture} + \end{figure} + \task \begin{equation*} \underline{U}_i\left(j \omega\right) = \mathcal{F}\left\{u_i(t)\right\} = \SI{2}{V} \pi \left(\delta\left(\omega - 2 \pi \cdot \SI{2.5}{kHz}\right) + \delta\left(\omega + 2 \pi \cdot \SI{2.5}{kHz}\right)\right) @@ -463,7 +799,5 @@ u_o(t) = \mathcal{F}^{-1}\left\{\underline{U}_i\left(j \omega\right)\right\} = \SI{1.19}{V} \cdot \cos\left(2 \pi \cdot \SI{2.5}{kHz} \cdot t - \SI{53.6}{\degree}\right) \end{equation*} The signal has been attenuated and phase-shifted. - - %TODO \end{tasks} \end{solution} diff --git a/main/exercise02.tex b/main/exercise02.tex index 05a4d0d..9591b77 100644 --- a/main/exercise02.tex +++ b/main/exercise02.tex @@ -43,7 +43,10 @@ \begin{appendix} -%\include{appendix/crlb} +\chapter{Solutions} + +\printsolutions +\clearpage \end{appendix} |