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Inngangssignalene til systemet er oftest analoge, og en digitalisering av disse før signalbehandlingen er derfor nødvendig. + +For Ã¥ ungÃ¥ at signalene i omgjøringen fra analog til digital skal fÃ¥ alvorlige aliasfeil, ønsker vi Ã¥ designe et anti-alias-filter som sett i \autoref{fig:problem}. + +\begin{figure}[!htbp] + \centering + \includegraphics[width=0.7\textwidth]{figurer/D7Problem.pdf} + \caption{Blokkdiagram for systemet med anti-alias-filteret.} + \label{fig:problem} +\end{figure} + +Hovedprinsippet til anti-aliasfilteret er Ã¥ begrense frekvensen pÃ¥ signalet $v_1$ som kommer inn til analog-til-digital-konvertereren, med andre ord et lavpass-filter (se rød strek i \autoref{fig:baandstopp}). Dersom vi tar en punktprøve med frekvens $f_s$, sier punktprøvingsteoremet at den høyeste frekvensen som skal slippe gjennom til utgangen $v_2$, mÃ¥ være $B = \frac{1}{2} f_s $. + +Det er ønskelig at signalet under $B = \frac{1}{2} f_s $ er sÃ¥ uendret som mulig, at knekkfrekvensen (se \autoref{sec:prinsipielllosning}) ikke blir lavere enn $f_0 = 0.75 \cdot B$, og at amplituderesponsen til systemet har en dempning pÃ¥ minst $-10$ dB ved frekvensen $B = \frac{1}{2} f_s $. + +\section{Prinsipiell løsning} +\label{sec:prinsipielllosning} + +Det Ã¥ desgine et lavpass-filter som er likt teorien er praktisk talt umulig Ã¥ gjennomføre. Derfor designer man et filter som har en veldig bratt endring i dempning av frekvenser rundt den ønskede knekkfrekvensen, se \autoref{fig:baandstopp}. +Vi definerer knekkfrekvensen der amplituden har sunket med $3$ dB eller at amplituden har blitt $ \frac{1}{\sqrt{2}} $ av den opprinnelige amplituden. + +\begin{figure}[!htbp] + \centering + \includegraphics[width=\textwidth]{figurer/Baandstopp.pdf} + \caption{Teoretisk lavpass-filter i rød og praktisk utførbart lavpass-filter i blÃ¥tt.} + \label{fig:baandstopp} +\end{figure} + +Vi ønsker Ã¥ ha et filter som demper signalet vi sender inn pÃ¥ $v_1 $ veldig mye over frekvensen $B = \frac{1}{2} f_s $. + +Det finnes flere typer filter-topologier som løser problemet, men den som har den flateste amplituderesponsen er et Butterworth-filter \cite{butterworth}. +For Ã¥ oppnÃ¥ knekkfrekvensen pÃ¥ $-3$ dB trenger vi ikke Ã¥ gjøre noe annet enn Ã¥ velge at knekkfrekvensen $f_0 = 0.75 \cdot B = \frac{3}{8} f_s$. Da gir Butterworth en dempning pÃ¥ $-3$ dB. +For Ã¥ oppnÃ¥ brattheten vi ønsker pÃ¥ filteret for Ã¥ oppnÃ¥ en dempning pÃ¥ $-10$ dB ved $B$ mÃ¥ vi velge en stor nok orden $n$ for filteret slik at filteret blir tilsrekkelig bratt. + +Det gjøres ved Ã¥ bruke formelen som er gitt i (\ref{eq:filtergradFor}). + +\begin{align} + n = \frac{1}{2} \frac{\ln \left(A^{-2} - 1\right)}{\ln \left( \frac{f}{f_0}\right)} \label{eq:filtergradFor} +\end{align} + +Der $A$ er amplituden (ikke i dB), $f$ er frekvensen vi ønsker en gitt dempning pÃ¥, $f_0 $ er ønsket knekkfrekvens og $n$ er graden til filteret. Dette tallet mÃ¥ rundes opp til nærmeste heltall, bÃ¥de fordi vi ikke kan lage et halvt filter og fordi dersom vi runder ned vil vi kunne bomme pÃ¥ mengde dempning ved ønsket frekvens. + +Dersom vi setter inn $f = B = \frac{1}{2} f_s$ og $f_0 = f_0 = \frac{3}{8} f_s$ fÃ¥r vi (\ref{eq:filtergrad}). + +\begin{align} + n &= \frac{1}{2} \frac{\ln \left(A^{-2} - 1\right)}{\ln \left( \frac{\frac{1}{2} f_s}{\frac{3}{8} f_s}\right)} \nonumber\\ + n &= \frac{1}{2} \frac{\ln \left(A^{-2} - 1\right)}{\ln \left( \frac{4}{3}\right)}\label{eq:filtergrad} +\end{align} + +For Ã¥ lage selve filterene bruker vi en sÃ¥kalt Sallan-Key topologi \cite{sallan-key} for et lavpassfilter, se \autoref{circ:sallen-key-start}. Dette er et andreordens-filter og disse kan seriekobles for Ã¥ legge til flere ordener. Dersom det er to i serie vil vi kunne ha opp til et fjerdeordens-filter, tre sÃ¥ fÃ¥r vi sjetteordens o.s.v. + +\begin{figure}[!htpb] + \centering + \begin{circuitikz} + \draw + (0,0) node[op amp,yscale=-1](opamp){} + + (opamp.+) to [short] ++(-1,0) coordinate(C2) + to [C, l_=$C_2$, *-] ++(0,-2) + to ++(0,0) node[ground]{} + + (C2) to [R, l_=$R$] ++(-2,0) coordinate(R2) + to [short, *-] ++(0,1) coordinate(C1) + to [C, l=$C_1 $] (C1-|opamp.out) + to [short, -*] (opamp.out) + + (R2) to [R, l_=$R$, -o] ++(-2,0) node[left] {$v_1$} + + (opamp.-) to [short] ++(0,-1) coordinate(fb) + to [short] (fb-|opamp.out) + to [short] (opamp.out) + to [short, -o] ++(1,0) node[right] {$v_2$} + + ; + \end{circuitikz} + \caption{Kretstopologi for et andregrads lavpass-filter i Sallen-Key topologi.} + \label{circ:sallen-key-start} +\end{figure} + + +For Ã¥ regne ut de forskjellige komponentverdiene har vi noen formler. Siden vi bruker operasjonsforsterkere (opamp) er det ganske standard Ã¥ bruke motstandsverdier mellom $1k\Omega$ og $100k\Omega$. Motstandene kan da velges fritt innenfor disse grenser. + +$\omega_0 $ er knekkfrekvensen i radianer og gitt i (\ref{eq:knekkfreq}). +\begin{align} + \omega_0 = 2\pi \cdot f_0 \label{eq:knekkfreq} +\end{align} + +To tidskonstanter $\tau_1$ og $\tau_2$ i (\ref{eq:tidskonstanter}). + +\begin{align} + \tau_1 = \frac{1}{\omega_0 \zeta} \quad \text{og} \quad \tau_2 = \frac{1}{\omega_0^2\tau_1} \label{eq:tidskonstanter} +\end{align} + +Da er kondensatorverdiene gitt som i (\ref{eq:kondiser}). + +\begin{align} + C_1 = \frac{\tau_1}{R} \quad \text{og} \quad C_2 = \frac{\tau_2}{R} \label{eq:kondiser} +\end{align} + +Verdiene for $\zeta$ kan regnes ut eller sÃ¥ er de gitt for et normal Butterworth-filter for orden 1 til 6 i \autoref{tab:sallenKeyLosning}. Her kommer $\zeta$-verdiene i par som gjelder for et sett med et Sallen-Key-filter. + + +\begin{table}[!htpb] +\centering +\caption{$\zeta$-verdier for forskjellige ordener $n$ for et butterwurth-filter med et Allan-Key-filter topologi.} +\label{tab:sallenKeyLosning} +\begin{tabular}{|l|l|l|l|} +\hline \hline + & \multicolumn{3}{l|}{Polpar \textit{i}} \\ \hline +$n$ & $1$ & $2$ & $3$ \\ \hline +\hline +$1$ & $1$ & & \\ \hline +$2$ & $0.70711$ & & \\ \hline +$3$ & $1$ & $0.5$ & \\ \hline +$4$ & $0.92388$ & $0.38268$ & \\ \hline +$5$ & $1$ & $0.80902$ & $0.30902$ \\ \hline +$6$ & $0.96593$ & $0.70711$ & $0.25882$ \\ \hline +\hline +\end{tabular}% +\end{table} + + +\section{Realisering og test} +\label{sec:realisering} +\subsection{Realisering} + +Analog til digital konvertereren har en punktprøvehastighet pÃ¥ $f_s = 9.9$kHz. De andre viktige frekvensene blir da. + +\begin{align*} + f_s &= 9.9\text{kHz} \\ + B &= 4.85\text{kHz} \\ + f_0 &= 3.71\text{kHz} +\end{align*} + +Siden vi skal ha $10$ dB dempning mÃ¥ vi gjøre det om til ikke dB for at den skal funke med (\ref{eq:filtergrad}). + +\begin{align} + A[dB] &= -10 \nonumber \\ + A &= 10^{\frac{A[dB]}{20}} = 10^{\frac{-10}{20}} \approx 0.3162 \label{eq:amplitude} +\end{align} + +Da blir orden fra (\ref{eq:filtergrad}). +\begin{align} + n = \frac{1}{2} \frac{\ln \left(0.3162^{-2} - 1\right)}{\ln \left( \frac{4}{3}\right)} \approx 3.81 \xRightarrow{\text{Rundes opp}} 4 \label{eq:orden} +\end{align} + +Kretstopologien for filteret blir som i \autoref{circ:nestenFerdig}. + +\begin{figure}[!htpb] + \centering + \begin{circuitikz}[scale=0.92, every node/.style={transform shape}] + \draw + (0,0) node[op amp,yscale=-1](opamp){} + + (opamp.+) to [short] ++(-1,0) coordinate(C2) + to [C, l_=$C_2$, *-] ++(0,-2) + to ++(0,0) node[ground]{} + + (C2) to [R, l_=$R$] ++(-2,0) coordinate(R2) + to [short, *-] ++(0,1) coordinate(C1) + to [C, l=$C_1 $] (C1-|opamp.out) + to [short, -*] (opamp.out) + + (R2) to [R, l_=$R$, -o] ++(-2,0) node[left] {$v_1$} + + (opamp.-) to [short] ++(0,-1) coordinate(fb) + to [short] (fb-|opamp.out) + to [short] (opamp.out) + to [short, -] ++(1,0) coordinate(end1) + + ++(6.19,0) node[op amp,yscale=-1](opampB){} + + (opampB.+) to [short] ++(-1,0) coordinate(C4) + to [C, l_=$C_4$, *-] ++(0,-2) + to ++(0,0) node[ground]{} + + (C4) to [R, l_=$R$] ++(-2,0) coordinate(R4) + to [short, *-] ++(0,1) coordinate(C3) + to [C, l=$C_3 $] (C3-|opampB.out) + to [short, -*] (opampB.out) + + (R4) to [R, l_=$R$, -] ++(-2,0) coordinate(start2) + + (opampB.-) to [short] ++(0,-1) coordinate(fb) + to [short] (fb-|opampB.out) + to [short] (opampB.out) + to [short, -o] ++(1,0) node[right] {$v_2$} + + (end1) to [short] (start2) + + ; + \end{circuitikz} + \caption{4. ordens lavpass-filter uten komponentverdier.} + \label{circ:nestenFerdig} +\end{figure} + +Velger $R = 1k\Omega$, og siden dette er et 4. ordens filter, har den fra \autoref{tab:sallenKeyLosning} + +\begin{align*} + \zeta_1 = 0.92388 \quad \text{og} \quad \zeta_2 = 0.38268 +\end{align*} + +Knekkfrekvensen $ \omega_0 = 2\pi \cdot 3.71\text{kHz} = 7.42\pi \cdot 10^3 \text{ rad/s}$ +Da blir tidskonstantene for kondensatorene som i (\ref{eq:tid1}) og (\ref{eq:tid2}). +\begin{align} + \tau_1 = \frac{1}{\omega_0 \zeta_1} \approx 46\mu\text{s} \quad &\text{og} \quad \tau_2 = \frac{1}{\omega_0^2\tau_1} \approx 40\mu\text{s} \label{eq:tid1}\\ + \tau_3 = \frac{1}{\omega_0 \zeta_2} \approx 112\mu\text{s} \quad &\text{og} \quad \tau_4 = \frac{1}{\omega_0^2\tau_3} \approx 16\mu\text{s} \label{eq:tid2} +\end{align} + +Og dermed blir kondensatorene som under. +\begin{align*} + C_1 = \frac{\tau_1}{R} \approx 46 \text{nF} \quad \text{og} \quad C_2 = \frac{\tau_2}{R} \approx 40 \text{nF} \\ + C_3 = \frac{\tau_3}{R} \approx 112 \text{nF} \quad \text{og} \quad C_4 = \frac{\tau_4}{R} \approx 16 \text{nF} +\end{align*} + +Justerer verdiene til standardverdier sett i (\ref{eq:kond1}) og (\ref{eq:kond2}). + +\begin{align} + C_1 = 50 \text{nF} = 100 \text{nF} + 100 \text{nF} \quad &\text{og} \quad C_2 = 40 \text{nF} = 10 \text{nF}|| 10 \text{nF}|| 10 \text{nF}|| 10 \text{nF} \label{eq:kond1} \\ + C_3 = 110 \text{nF} = 100 \text{nF} || 10\text{nF} \quad &\text{og} \quad C_4 = 15 \text{nF} = (10\text{nF} + 10\text{nF}) || 10 \text{nF} \label{eq:kond2} +\end{align} + + +Den fullførte kretsen er som i \autoref{circ:ferdig}. + +\begin{figure}[!htpb] + \centering + \begin{circuitikz}[scale=0.92, every node/.style={transform shape}] + \draw + (0,0) node[op amp,yscale=-1](opamp){} + + (opamp.+) to [short] ++(-1,0) coordinate(C2) + to [C, l_=$40 \text{nF}$, *-] ++(0,-2) + to ++(0,0) node[ground]{} + + (C2) to [R, l_=$1k\Omega$] ++(-2,0) coordinate(R2) + to [short, *-] ++(0,1) coordinate(C1) + to [C, l=$50 \text{nF} $] (C1-|opamp.out) + to [short, -*] (opamp.out) + + (R2) to [R, l_=$1k\Omega$, -o] ++(-2,0) node[left] {$v_1$} + + (opamp.-) to [short] ++(0,-1) coordinate(fb) + to [short] (fb-|opamp.out) + to [short] (opamp.out) + to [short, -] ++(1,0) coordinate(end1) + + ++(6.19,0) node[op amp,yscale=-1](opampB){} + + (opampB.+) to [short] ++(-1,0) coordinate(C4) + to [C, l_=$15 \text{nF}$, *-] ++(0,-2) + to ++(0,0) node[ground]{} + + (C4) to [R, l_=$1k\Omega$] ++(-2,0) coordinate(R4) + to [short, *-] ++(0,1) coordinate(C3) + to [C, l=$110 \text{nF} $] (C3-|opampB.out) + to [short, -*] (opampB.out) + + (R4) to [R, l_=$1k\Omega$, -] ++(-2,0) coordinate(start2) + + (opampB.-) to [short] ++(0,-1) coordinate(fb) + to [short] (fb-|opampB.out) + to [short] (opampB.out) + to [short, -o] ++(1,0) node[right] {$v_2$} + + (end1) to [short] (start2) + + ; + \end{circuitikz} + \caption{4. ordens lavpass-filter uten komponentverdier.} + \label{circ:ferdig} +\end{figure} + +\subsection{Test} + +For Ã¥ teste ble en Analog Discovery brukt, med network analyser. Se \autoref{fig:knekk} og \autoref{fig:-10}. + +\begin{figure}[!htbp] + \centering + \includegraphics[width=\textwidth]{figurer/Knekkfrekvens.png} + \caption{Amplitude ved ønsket knekkfrekvens.} + \label{fig:knekk} +\end{figure} + +\begin{figure}[!htbp] + \centering + \includegraphics[width=\textwidth]{figurer/-10dB.png} + \caption{Frekvens ved $10$dB dempning.} + \label{fig:-10} +\end{figure} + +Som vi ser, ser vi at ved frekvensen $f_k = 3.71$kHz er amplituden $-2.9$dB. Vi ser ogsÃ¥ at frekvensen ved $10$dB dempning er lik $B = \frac{1}{2} f_s = 4.85$kHz. Begge disse er innenfor kravene som er gitt for anti-alias-filteret. + +\section{Konklusjon} +\label{sec:konklusjon} + +Ut i fra spesifikasjonene som ble oppgitt i problemstillingen, er filteret som er designet velfungerende. Det kunne vært gjort mer nøyaktig ved Ã¥ bruke riktigere komponeneter som er næremere i verdi. +Ellers fungerte kretsen meget bra. + + +%Bibliografi: Legg til flere elementer ved Ã¥ legge til flere \bibitem:-------- +\phantomsection +\addcontentsline{toc}{section}{Referanser} +\begin{thebibliography}{99} + +\bibitem{butterworth} + Wikipedia contributors. (2019, September 27). \textit{Butterworth filter}. In Wikipedia, The Free Encyclopedia. Retrieved 10:22, October 8, 2019, from \url{https://en.wikipedia.org/w/index.php?title=Butterworth_filter&oldid=918135860} + +\bibitem{sallan-key} + Wikipedia contributors. (2019, August 6). \textit{Sallen–Key topology}. In Wikipedia, The Free Encyclopedia. Retrieved 11:21, October 8, 2019, from \url{https://en.wikipedia.org/w/index.php?title=Sallen%E2%80%93Key_topology&oldid=909548354} + + +\end{thebibliography}{} + +\clearpage +\appendix +%Tillegg. 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