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0 +w 192 544 192 496 0 +w 192 496 192 400 0 +w 192 400 160 288 0 +w 288 288 272 208 0 +w 528 288 528 208 0 +w 544 400 608 400 0 +207 608 288 624 192 4 A +207 608 400 624 512 4 B +g 192 400 128 400 0 0 +r 528 288 608 288 0 1500 +r 464 400 544 400 0 6000 +r 544 496 192 496 0 12000 diff --git "a/inl\303\244mning_1.pdf" "b/inl\303\244mning_1/inl\303\244mning_1.pdf" similarity index 100% rename from "inl\303\244mning_1.pdf" rename to "inl\303\244mning_1/inl\303\244mning_1.pdf" diff --git "a/inl\303\244mning_1/main.tex" "b/inl\303\244mning_1/main.tex" new file mode 100644 index 0000000..92acf71 --- /dev/null +++ "b/inl\303\244mning_1/main.tex" @@ -0,0 +1,328 @@ +\documentclass{article} +\usepackage[utf8]{inputenc} +\usepackage{circuitikz} +\usepackage{pdfpages} +\usepackage[left=3cm, right=3cm]{geometry} +\usepackage{amsmath} +\usepackage{hyperref} + +\title{Inlämningsuppgift 1-168} +\author{Nils Forssén, nilfo359, Y2.b} +\date{17 September 2022} + +\begin{document} +\maketitle +\newpage + +\section{Uppgift} +\includegraphics[clip, trim=0cm 15cm 0cm 0cm, width=1.0\textwidth]{nilfo359_part_1__id_0168.pdf} + +\section{Omskrivning av krets} + +Vi ritar om ovan krets med följande förenklingar: +\begin{itemize} + \item Resistanser kopplade parallellt med en ideell spänningskälla elimineras dÃ¥ spänningskällan garanterar given spänning över resistansen. Strömmen genom kopplingen blir oförändrad. + \item Resistanser kopplade i serie med en ideell strömkälla elimineras dÃ¥ strömkällan garanter en ström genom resistansen. Spänningen över kopplingen blir oförändrad. +\end{itemize} +Vi inför även beteckningar $V_1$, $V_2$, $V_3$ och $V_4$ motsvarande potentialen i respektive nod (Nod 1, 2, 3 och 4) som vi kommer betrakta när vi senare tillämpar \emph{nodanalys}. Detta ger oss följande krets som är ekvivalent med ovan given enport. +\begin{center} +\begin{circuitikz}[american] +\draw (0,-4) to [short, *-*] (0,-2) +(0,-2) to [R, l=$R_5$] (8,-2) + to [short, -*] (8,-4) + to [R, l=$R_{10}$] (12, -4) + to [short, *-o] (13, -4)node[right] {A}; +\draw (8, -4) to [R, l=$R_9$, *-*] (8, -8) + to [short, *-*] (8, -8) + to [short, *-o] (13, -8)node[right] {B}; +\draw (12, -4) to [american current source, l=$I_1$, *-*] (12, -8); +\draw (8, -4) to [R, l=$R_6$, *-*] (4, -4) node[above]{$V_2$} +to [R, l=$R_2$, *-*] (0, -4) node[left]{$V_3$} +to [R, l=$R_1$, *-*] (0, -8); +\draw (2, -8) to [battery, l=$E_1$, *-*] (0, -8); +\draw (2, -8) to [R, l=$R_3$, *-*] (4, -8) +to [american current source, l=$I_2$, *-*] (8, -8); +\draw (0, -8) to [short, *-*] (0, -10); +\draw (8, -10) to [battery, l=$E_2$, *-*] (0, -10); +\draw (8, -10) to [short, *-*] (8, -8); +\draw (4, -4) to [R,l=$R_4$, *-*] (4, -8); + +\draw (4, -8) to [short, -o] (5, -9) node[right]{$V_1$}; +\draw (8, -4) to [short, *-o] (9, -5) node[right]{$V_4$}; +\end{circuitikz} +\end{center} + +\section{Nodanalys} +För att slutligen ta fram en Norton-eller-Theveninekvivalent till den givna kretsen väljer jag att först använda metoden \emph{nodanalys} för att ta fram spänningensskillnaden mellan $A$ och $B$ (tomgÃ¥ngsspänningen av enporten). Enligt nodanalys metodiken identifierar vi en ensam spänningskälla, $E_2$ och sätter denna som en supernod. Vi väljer sedan att använda den negativa grenen av spänningskällan som referensnod och jordar den. Vidare sÃ¥ ansätter vi en spänningspolaritet över varje resistans $R_i$ vilket ger upphov till en ström genom respektive resistans betecknat med en enkel pil pÃ¥ dess gren. DÃ¥ fÃ¥s följande nät. + +\begin{center} +\begin{circuitikz}[american] +\draw (0,-4) to [short, *-*] (0,-2) +(0,-2) to [R, l=$R_5$, v_<=$ $] (8,-2) + to [short, -*, i_<=$ $] (8,-4) + to [R, l=$R_{10}$, v_>=$ $] (12, -4) + to [short, *-o] (13, -4)node[right] {A}; +\draw (8, -4) to [R, l=$R_9$, *-*, v_<=$ $, i_<=$ $] (8, -8) + to [short, *-*] (8, -8) + to [short, *-o] (13, -8)node[right] {B}; +\draw (12, -4) to [american current source, l=$I_1$, *-*] (12, -8); +\draw (8, -4) to [R, l=$R_6$, *-*, v_>=$ $, i_>=$ $] (4, -4) node[above]{$V_2$} +to [R, l=$R_2$, *-*, v_<=$ $, i_<=$ $] (0, -4) node[left]{$V_3$} +to [R, l=$R_1$, *-*, v_>=$ $, i_>=$ $] (0, -8); +\draw (2, -8) to [battery, l=$E_1$, *-*] (0, -8); +\draw (2, -8) to [R, l=$R_3$, *-*, v_>=$ $, i_>=$ $] (4, -8) +to [american current source, l=$I_2$, *-*] (8, -8); +\draw (0, -8) to [short, *-*] (0, -10); +\draw (8, -10) to [battery, l=$E_2$, *-*] (0, -10); +\draw (8, -10) to [short, *-*] (8, -8); +\draw (4, -4) to [R,l=$R_4$, *-*, v_>=$ $, i_>=$ $] (4, -8); + +\draw (4, -8) to [short, -o] (5, -9) node[right]{$V_1$};; +\draw (8, -4) to [short, *-o] (9, -5) node[right]{$V_4$}; +\draw (0, -10) node[ground]{}; +\end{circuitikz} +\end{center} + +Om vi nu tillämpar Kirchoffs strömlag (summan av alla inkommande strömmar i varje nod skall vara 0) pÃ¥ nod ett till fyra sÃ¥ fÃ¥r vi följande ekvationer. +\begin{equation} +\text{KCL i nod 1:\quad} -I_2 + \cfrac{(0 + E_1) - V_1 }{R_3} + \cfrac{V_2 - V_1}{R_4} = 0 +\end{equation} +\begin{equation} + \text{KCL i nod 2:\quad} -\cfrac{V_2 - V_1}{R_4} + \cfrac{V_3 - V_2}{R_2} + \cfrac{V_4 - V_2}{R_6} = 0 +\end{equation} +\begin{equation} + \text{KCL i nod 3:\quad} -\cfrac{V_3}{R_1} - \cfrac{V_3 - V_2}{R_2} + \cfrac{V_4 - V_3}{R_5} = 0 +\end{equation} +\begin{equation} + \text{KCL i nod 4:\quad} -\cfrac{V_4 - V_2}{R_6} + \cfrac{E_2 - V_4}{R_9} - \cfrac{V_4 - V_3}{R_5} = 0 +\end{equation} +Dessa fyra ekvationer räcker för att lösa fram potentialen i var och en av de sökta noderna. Vi ställer upp ekvationssystemetet i matrisform. +\begin{align*} + \begin{bmatrix} + -\left(\cfrac{1}{R_3} + \cfrac{1}{R_4}\right) & \cfrac{1}{R_4} & 0 & 0 \\[6pt] + \cfrac{1}{R_4} & -\left(\cfrac{1}{R_4} + \cfrac{1}{R_2} + \cfrac{1}{R_6}\right) & \cfrac{1}{R_2} & \cfrac{1}{R_6} \\[6pt] + 0 & \cfrac{1}{R_2} & -\left(\cfrac{1}{R_1} + \cfrac{1}{R_2} + \cfrac{1}{R_5}\right) & \cfrac{1}{R_5} \\[6pt] + 0 & \cfrac{1}{R_6} & \cfrac{1}{R_5} & -\left(\cfrac{1}{R_6} + \cfrac{1}{R_9} + \cfrac{1}{R_5}\right) \\[6pt] + \end{bmatrix} + \begin{bmatrix} + V_1 \\[6pt] + V_2 \\[6pt] + V_3 \\[6pt] + V_4 \\[6pt] + \end{bmatrix} + = + \begin{bmatrix} I_2 - E_1\cfrac{1}{R_3} \\[6pt] 0 \\[6pt] 0 \\[6pt] I_1 - E_2\cfrac{1}{R_9} \end{bmatrix} +\end{align*} +Med de värden för resistanser, strömmar och potentialer som ges i uppgiften (notera att resistanser har enhet Ohm ($\Omega$), strömmar har enhet Ampere (A) och spänningar/potentialer har enhet Volt (V): +\begin{align*} + \begin{bmatrix} + -\left(\cfrac{1}{8000} + \cfrac{1}{7500}\right) & \cfrac{1}{7500} & 0 & 0 \\[6pt] + \cfrac{1}{7500} & -\left(\cfrac{1}{7500} + \cfrac{1}{7500} + \cfrac{1}{2000}\right) & \cfrac{1}{7500} & \cfrac{1}{2000} \\[6pt] + 0 & \cfrac{1}{7500} & -\left(\cfrac{1}{2500} + \cfrac{1}{7500} + \cfrac{1}{8000}\right) & \cfrac{1}{8000} \\[6pt] + 0 & \cfrac{1}{2000} & \cfrac{1}{8000} & -\left(\cfrac{1}{2000} + \cfrac{1}{3500} + \cfrac{1}{8000}\right) \\[6pt] + \end{bmatrix} + \begin{bmatrix} + V_1 \\[6pt] + V_2 \\[6pt] + V_3 \\[6pt] + V_4 \\[6pt] + \end{bmatrix} + \\= + \begin{bmatrix} \cfrac{2}{1000} - \cfrac{3}{8000} \\[6pt] 0 \\[6pt] 0 \\[6pt] \cfrac{1}{1000} - \cfrac{9}{3500} \end{bmatrix} +\end{align*} +Vi löser ekvationsystemet med matlab och fÃ¥r dÃ¥ avrundat till 3 värdesiffror följande potentialer mätt i Volt. +\begin{align*} + \begin{bmatrix} + V_1 \\[6pt] + V_2 \\[6pt] + V_3 \\[6pt] + V_4 \\[6pt] + \end{bmatrix} + \approx + \begin{bmatrix} + -6.16 \\[6pt] + 0.256 \\[6pt] + 0.417 \\[6pt] + 1.92 \\[6pt] + \end{bmatrix} +\end{align*} +Eftersom vi slutligen söker spänningsskillnaden mellan punkt A och B (tomgÃ¥ngsspänningen av enporten) sÃ¥ behöver vi ta reda pÃ¥ potentialen i punkt A. Punkt B ser vi direkt har potentialen, $V_B$, 9 Volt relativt jord. Spänningsskillnaden $U_{10}$ mellan punkt A och Nod 4 fÃ¥s enkelt av ohms lag över resistor $R_{10}$. +\begin{equation} +U_{10} = R_{10}I_1 = 1.5 V +\end{equation} +DÃ¥ vi vet riktningen pÃ¥ strömen fÃ¥r vi att potentialen i punkt A, $V_A$, blir potentialen i Nod 4, $V_4$, - spänningsskillnaden $U_{10}$. Spänningsskillnaden, kretsens tomgÃ¥ngsspänningen $U_0$, mellan A och B fÃ¥s dÃ¥ till. +\begin{equation} + V_A = V_4 - U_{10} = 0.423 V +\end{equation} +\begin{equation} + V_B = 9 V +\end{equation} +\begin{equation} + U_0 = V_A - V_B = -8.58 V +\end{equation} +Notera att vi ansätter riktiningen pÃ¥ $U_0$ som: +\begin{center} + \begin{circuitikz}[american] +\draw[draw=white] (0, 0) node[right]{A} to [short, v_>=$U_0$, o-o] (0,-2) node[right]{B}; +\draw (-1, 0) to [short, -o] (0,0) ; +\draw (-1, -2) to [short, -o] (0,-2) ; +\end{circuitikz} +\end{center} + +\section{Inre resistans} +Med mÃ¥let att ta fram en Norton-eller-Theveninekvivalens till den givna kretsen studerar vi den inre resistansen av kretsen. För att ta fram den inre resistansen sÃ¥ bortser vi ifrÃ¥n alla spännings och strömkällor. Dessa byts i originalkretsen ut mot en koppling och ett avbrott respektive. Detta motiveras med att en ideell spänningskälla har i regel 0 resistans (som en vanlig koppling) och en ideell strömkälla har i regel oändlig resistans (som ett avbrott). Det nya nätet med endast resistanser blir dÃ¥. +\begin{center} +\begin{circuitikz}[american] +\draw (0,-4) to [short, *-*] (0,-2) +(0,-2) to [R, l=$R_5$] (8,-2) + to [short, -*] (8,-4) + to [R, l=$R_{10}$] (12, -4) + to [short, *-o] (13, -4)node[right] {A}; +\draw (8, -4) to [R, l=$R_9$, *-*] (8, -8) + to [short, *-*] (8, -8) + to [short, *-o] (13, -8)node[right] {B}; +\draw (8, -4) to [R, l=$R_6$, *-*] (4, -4) +to [R, l=$R_2$, *-*] (0, -4) +to [R, l=$R_1$, *-*] (0, -8); +\draw (2, -8) to [short, *-*] (0, -8); +\draw (2, -8) to [R, l=$R_3$, *-*] (4, -8); +\draw (0, -8) to [short, *-*] (0, -10); +\draw (8, -10) to [short, *-*] (0, -10); +\draw (8, -10) to [short, *-*] (8, -8); +\draw (4, -4) to [R,l=$R_4$, *-*] (4, -8); +\end{circuitikz} +\end{center} +Vi fortsätter med att förenkla det nya resistansnätet med vetskapen att resistanser i serie och parallellkopplingar kan kombineras till en summaresistans enligt formler nedan. Vi inför motsvarande beteckningar. +\begin{align*} + \text{För $R_i$ i serie med $R_j$:\quad} R_{i|j} = R_i + R_j \\ + \text{För $R_i$ parallellt med $R_j$:\quad} R_{i//j} = \cfrac{R_iR_j}{R_i + R_j}' +\end{align*} +Med ovan notation kan resistansnätet förenklas ekvivalent enligt följande. +\begin{center} +\begin{circuitikz}[american] +\draw (0,-4) to [short, *-*] (0,-2) +(0,-2) to [R, l=$R_5$] (8,-2) + to [short, -*] (8,-4) + to [R, l=$R_{10}$] (12, -4) + to [short, *-o] (13, -4)node[right] {A}; +\draw (8, -4) to [R, l=$R_9$, *-*] (8, -8) + to [short, *-*] (8, -8) + to [short, *-o] (13, -8)node[right] {B}; +\draw (8, -4) to [R, l=$R_6$, *-*] (4, -4) +to [R, l=$R_2$, *-*] (0, -4) +to [R, l=$R_1$, *-*] (0, -8); +\draw (0, -8) to (4, -8); +\draw (8, -8) to [short, *-*] (0, -8); +\draw (4, -4) to [R, l=$R_{3|4}$, *-*] (4, -8); +\end{circuitikz} +\end{center} +Nedan utnyttjar jag Y-$\Delta$-transformation (som boken definierar) för att fortsätta förenkla nätet. +\begin{center} +\begin{circuitikz}[american] +\draw (0,-4) to [short, *-*] (0,-2) +(0,-2) to [R, l=$R_5$] (8,-2) + to [short, -*] (8,-4) + to [R, l=$R_{10}$] (12, -4) + to [short, *-o] (13, -4)node[right] {A}; +\draw (8, -4) to [R, l=$R_9$, *-*] (8, -8) + to [short, *-*] (8, -8) + to [short, *-o] (13, -8)node[right] {B}; +\draw (8, -4) to [R, l=$R_{\Delta 1}$, *-*] (4, -8) +to [R, l=$R_{\Delta 2}$, *-*] (0, -4) +to [R, l=$R_{\Delta 3}$, *-*] (8, -4); +\draw (0, -8) to (4, -8); +\draw (8, -8) to [short, *-*] (0, -8); +\draw (0, -4) to [R, l=$R_{1}$, *-*] (0, -8); +\end{circuitikz} +\end{center} +\begin{align*} + \text{Där\quad} R_{\Delta 1} = R_{3|4}R_6\left(\cfrac{1}{R_{3|4}} + \cfrac{1}{R_{6}} + \cfrac{1}{R_{2}}\right) \\ + R_{\Delta 2} = R_{3|4}R_2\left(\cfrac{1}{R_{3|4}} + \cfrac{1}{R_{6}} + \cfrac{1}{R_{2}}\right) \\ + R_{\Delta 3} = R_{6}R_2\left(\cfrac{1}{R_{3|4}} + \cfrac{1}{R_{6}} + \cfrac{1}{R_{2}}\right) +\end{align*} +\begin{center} +\begin{circuitikz}[american] +\draw (8,-4) + to [R, l=$R_{10}$] (12, -4) + to [short, *-o] (13, -4)node[right] {A}; +\draw (8, -4) to [R, l=$R_{\Delta 1//9}$, *-*] (8, -8) + to [short, *-*] (8, -8) + to [short, *-o] (13, -8)node[right] {B}; +\draw (0, -4) to [R, l=$R_{\Delta 3 // 5}$, *-*] (8, -4); +\draw (0, -8) to (4, -8); +\draw (8, -8) to [short, *-*] (0, -8); +\draw (0, -4) to [R, l=$R_{\Delta 2 // 1}$, *-*] (0, -8); +\end{circuitikz} +\begin{circuitikz}[american] +\draw (8,-4) + to [R, l=$R_{10}$] (12, -4) + to [short, *-o] (13, -4)node[right] {A}; +\draw (8, -4) to [R, l=$R_{\Delta 1//9}$, *-*] (8, -8) + to [short, *-*] (8, -8) + to [short, *-o] (13, -8)node[right] {B}; +\draw (0, -4) to [short, *-*] (8, -4); +\draw (0, -8) to (4, -8); +\draw (8, -8) to [short, *-*] (0, -8); +\draw (0, -4) to [R, l=$R_{(\Delta 2 // 1)|(\Delta 3 // 5)}$, *-*] (0, -8); +\end{circuitikz} +\begin{circuitikz}[american] +\draw (8,-4) + to [R, l=$R_{10}$] (12, -4) + to [short, *-o] (13, -4)node[right] {A}; +\draw (8, -4) to [R, l=$R_{(\Delta 1//9) // ((\Delta 2 // 1)|(\Delta 3 // 5))}$, *-*] (8, -8) + to [short, *-*] (8, -8) + to [short, *-o] (13, -8)node[right] {B}; +\end{circuitikz} \\ +\begin{circuitikz}[american] +\draw (8,-4) + to [short, *-o] (13, -4)node[right] {A}; +\draw (8, -4) to [R, l=$R_{((\Delta 1//9) // ((\Delta 2 // 1)|(\Delta 3 // 5))) |10}$, *-*] (8, -8) + to [short, *-*] (8, -8) + to [short, *-o] (13, -8)node[right] {B}; +\end{circuitikz} +\end{center} +Slutligen fÃ¥r vi bara en resistans som är ekvivalent med resistansnätet vi började med. Denna resistans representerar den inre resistansen av hela kretsen. Vi kan räkna ut den med hjälp av formlerna ovan och insättning av resistansvärdena frÃ¥n originalkretsen. Beräkningen med alla resistansvärden införda utförs i matlab och är avrundat till 3 värdesiffror. +\begin{center} +\begin{align*} + R_{3|4} &= R_3 + R_4 && \\ + R_{\Delta 1} &= R_{3|4}R_6\left(\cfrac{1}{R_{3|4}} + \cfrac{1}{R_{6}} + \cfrac{1}{R_{2}}\right) && \\ + %&= \cfrac{15500*2000}{\cfrac{1}{15500} + \cfrac{1}{2000} + \cfrac{1}{7500}} = 4.44*e^{10} \Omega && \\ + R_{\Delta 2} &= R_{3|4}R_2\left(\cfrac{1}{R_{3|4}} + \cfrac{1}{R_{6}} + \cfrac{1}{R_{2}}\right) && \\ + %&=\cfrac{15500*7500}{\cfrac{1}{15500} + \cfrac{1}{2000} + \cfrac{1}{7500}} = 1.67*e^{11} \Omega && \\ + R_{\Delta 3} &= R_{6}R_2\left(\cfrac{1}{R_{3|4}} + \cfrac{1}{R_{6}} + \cfrac{1}{R_{2}}\right) && \\ + %&=\cfrac{2000*7500}{\cfrac{1}{15500} + \cfrac{1}{2000} + \cfrac{1}{7500}} = 2.15*e^{10} \Omega && \\ \\ + R_{\left(\Delta 2 // 1\right)|\left(\Delta 3 // 5\right)} =& \cfrac{R_{\Delta 2}R_1}{R_{\Delta 2} + R_1} + \cfrac{R_{\Delta 3}R_5}{R_{\Delta 3} + R_5} && \\ \\ + R_{\Delta 1//9} =& \cfrac{R_{\Delta 1}R_9}{R_{\Delta 1} + R_9} && \\ \\ + R_{((\Delta 1//9) // ((\Delta 2 // 1)|(\Delta 3 // 5))) |10} =& \cfrac{R_{(\Delta 2 // 1)|(\Delta 3 // 5)}R_{\Delta 1//9}}{R_{(\Delta 2 // 1)|(\Delta 3 // 5)} + R_{\Delta 1//9}} + R_{10} && \\ \\ + \approx& 3.60 k\Omega +\end{align*} +\end{center} +Den inre resistansen av originalkretsen är därmed ungefär $3.60k\Omega$. +\section{Thévenin ekvivalent krets} +Nu när vi har studerat den givna kretsen och kommit fram till att kretsen har en inre resistans av ungefär $3.60k\Omega$ och en tomgÃ¥ngsspänning pÃ¥ ungefär $-8.58V$ sÃ¥ kan vi ersätta hela kretsen med en ekvivalent krets enligt Thévenins teorem. +\begin{center} +\begin{circuitikz}[american] +\draw (4, 0) node[right] {A} + to [R, l=$3.60k\Omega$, o-*] (0, 0); +\draw (0, -4) to [battery, l=$8.58V$, *-*] (0,0); +\draw (0, -4) to [short, *-o] (4, -4) node[right] {B}; +\draw[draw=white] (4, 0) to [short, v_<=$ $] (4, -4); +\end{circuitikz} +\end{center} +Notera riktningen pÃ¥ spänningskällan i enlighet med teckenbytet. Liknande skulle vi kunna räkna ut kortslutningsströmmen med hjälp av ohms lag pÃ¥ tomgÃ¥ngsspänningen och den inre resistansen. PÃ¥ sÃ¥ sätt kan man istället teckna en Norton ekvivalent krets, men den delen utelämnas dÃ¥ uppgiften endast sökte en ersättningskrets.\\ \\ + +Uppgiften avklarad. \\ \\ + +Som bilagor hittar ni skärmbilder frÃ¥n ett webbaserat simuleringsprogram jag är bekväm med som bekräftar mina uträkningar. +\newpage +\section{Bilagor} +Simuleringsprogram: \url{https://www.falstad.com/circuit/} \\ +\textbf{Originalkrets, tomgÃ¥ngsspänning och kortslutningsström:} +\begin{center} + \includegraphics[scale=0.8]{original.png} +\end{center} +\textbf{Thévenin ekvivalent krets, tomgÃ¥ngsspänning och kortslutningsström:} +\begin{center} + \includegraphics[scale=0.8]{ekvivalent.png} +\end{center} +\end{document} diff --git "a/inl\303\244mning_1/nilfo359_part_1__id_0168.pdf" "b/inl\303\244mning_1/nilfo359_part_1__id_0168.pdf" new file mode 100644 index 0000000000000000000000000000000000000000..8dd8b4f09daa5703d6400833eead7c4c9e74ec79 GIT binary patch literal 22869 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