#import "../lib/common_rewrite.typ" : * #import "@preview/mannot:0.3.1" #import "@preview/zap:0.5.0" #show math.equation.where(block: true): it => math.inline(it) #set page( paper: "a4", margin: ( bottom: 10mm, top: 5mm, left: 5mm, right: 5mm ), flipped:true, footer: context [ #grid( align: center, columns: (1fr, 1fr, 1fr), [#align(left, datetime.today().display("[day].[month].[year]"))], [#align(center, counter(page).display("- 1 -"))], [#align(right, image("../images/cc0.png", height: 5mm,))] ) ], ) #let colorAllgemein = color.hsl(105.13deg, 92.13%, 75.1%) #let colorEineTore = color.hsl(202.05deg, 92.13%, 75.1%) #let colorZweiTore = color.hsl(235.9deg, 92.13%, 75.1%) #let colorAnalyseVerfahren = color.hsl(280deg, 92.13%, 75.1%) #let colorComplexAC = color.hsl(356.92deg, 92.13%, 75.1%) #let colorMathe = color.hsl(34.87deg, 92.13%, 75.1%) #place(top+center, scope: "parent", float: true, heading( [Schaltungstheorie] )) #columns(4, gutter: 2mm)[ #bgBlock(fill: colorEineTore)[ #subHeading(fill: colorEineTore)[Quelle Wandlung] #zap.circuit({ import zap: * set-style(scale: (x: 0.75, y:0.75), fill: none) resistor("R1", (-2, 0), (0, 0)) vsource("V1", (-2, 0), (-2, -2)) wire((-2, -2), (0, -2)) node("n1", (0, 0), label: "1") node("n2", (0, -2), label: "2") }) ] #bgBlock(fill: colorAnalyseVerfahren)[ #subHeading(fill: colorAnalyseVerfahren)[Graphen und Matrizen] $bold(i_b)$ (oder $bold(i)$): Zweigstrom-Vektor \ $bold(u_b)$ (oder $bold(u)$): Zweigspannungs-Vektor \ $bold(i_m)$ : Maschenstrom-Vektor \ #text(rgb(20%, 20%, 20%))[(Strom in einer viruellen Masche)] \ $bold(u_k)$ : Kontenspannungs-Vektor \ #text(rgb(20%, 20%, 20%))[(Spannung zwischen Referenzknoten und Knoten k)] \ #line(length: 100%, stroke: (thickness: 0.2mm)) Knotenzidenzmatrix $bold(A)$ $bold(A) : bold(i_k) -> text("Knotenstrombianz") = 0$ \ $bold(A^T) : bold(u_b)-> bold(u_k)$ $ bold(A) = quad mannot.mark(mat( a_11, a_12, ..., a_(1m); a_21, a_22, ..., a_(2m); dots.v, dots.v, dots.down, dots.v; a_(n 1), a_(n 2), ..., a_(n m) ), tag: #<1>) #mannot.annot(<1>, pos:left, text(rgb("#404296"))[#rotate(-90deg)[$<-$ Knoten]], dx: 5mm) #mannot.annot(<1>, pos:bottom, text(rgb("#404296"))[Zweige $->$], dy: -0.5mm) a in {-1, 0, 1} $ #line(length: 100%, stroke: (thickness: 0.2mm)) Mascheninsidenz Matrix $bold(B)$\ $bold(B) : bold(u_b) -> text("Zweigspannungsbilanz") = 0$ \ $bold(B^T) : bold(i_m) -> i_b$ $ bold(B) = quad mannot.mark(mat( b_11, b_12, ..., b_(1m); b_21, b_22, ..., b_(2m); dots.v, dots.v, dots.down, dots.v; b_(n 1), b_(n 2), ..., b_(n m) ), tag: #<1>) #mannot.annot(<1>, pos:left, text(rgb("#404296"))[#rotate(-90deg)[$<-$ Maschen]], dx: 6mm) #mannot.annot(<1>, pos:bottom, text(rgb("#404296"))[Zweige $->$], dy: -0.5mm) b in {-1, 0, 1} $ #line(length: 100%, stroke: (thickness: 0.2mm)) *KCL und KVL* \ KCL in Nullraum: $ bold(A) bold(i_b) = bold(0)$ \ KVL in Bildraum: $ bold(A^T) bold(u_k) = bold(u_b)$ KVL in Nullraum: $bold(B) bold(u_b) = bold(0)$ \ KCL in Bildraum: $bold(B^T) bold(i_m) = bold(i_b)$ \ *Tellegen'sche Satz* \ $bold(A B^T) = bold(B^T A) = 0$ \ $bold(u_b^T i_b) = 0$ ] #bgBlock(fill: colorAnalyseVerfahren)[ #subHeading(fill: colorAnalyseVerfahren)[Baumkonzept] ] #bgBlock(fill: colorAnalyseVerfahren)[ #subHeading(fill: colorAnalyseVerfahren)[Machenstrom-/Knotenpotenzial-Analyse] ] #bgBlock(fill: colorAnalyseVerfahren)[ #subHeading(fill: colorAnalyseVerfahren)[Reduzierte Knotenpotenzial-Analyse] ] ] #pagebreak() #place(bottom+left, scope: "parent", float: true)[ #bgBlock(fill: colorZweiTore)[ #subHeading(fill: colorZweiTore)[Umrechnung Zweitormatrizen] #show table.cell: it => pad(), #table( columns: (auto, 1fr, 1fr, 1fr, 1fr, 1fr, 1fr), align: center, gutter: 0.1mm, [In $->$], $bold(R)$, $bold(G)$, $bold(H)$, $bold(H')$, $bold(A)$, $bold(A')$, $bold(R)$, $mat(r_11, r_12; r_21, r_22)$, $1/det(bold(G)) mat(g_22, -g_12; -g_21, g_11)$, $1/h_22 mat(det(bold(H)), h_12; -h_21, 1)$, $1/h'_11 mat(1, -h'_12; h'_21, det(bold(H')))$, $1/a_21 mat(a_11, det(bold(A)); 1, a_22)$, $1/a'_21 mat(a'_22, 1; det(bold(A')), a'_11)$, $bold(G)$, $1/det(bold(R)) mat(r_22, -r_12; -r_21, r_11)$, $mat(g_11, g_12; g_21, g_22)$, $1/h_11 mat(1, -h_12; h_21, det(bold(H)))$, $1/h'_22 mat(det(bold(H')), h'_12; -h'_21, 1)$, $1/a_12 mat(a_22, -det(bold(A)); -1, a_11)$, $1/a'_12 mat(a'_11, -1; -det(bold(A')), a'_22)$, $bold(H)$, $1/r_22 mat(det(bold(R)), r_12; -r_21, 1)$, $1/g_11 mat(1, -g_12; g_21, det(bold(G)))$, $mat(h_11, h_12; h_21, h_22)$, $1/det(bold(H')) mat(h'_22, -h'_12; -h'_21, h'_11)$, $1/a_22 mat(a_12, det(bold(A)); -1, a_21)$, $1/a'_11 mat(a'_12, 1; -det(bold(A')), a'_21)$, $bold(H')$, $1/r_11 mat(1, -r_12; r_21, det(bold(R)))$, $1/g_22 mat(det(bold(G)), g_12; -g_21, 1)$, $1/det(bold(H)) mat(h_22, -h_12; -h_21, h_11)$, $mat(h'_11, h'_12; h'_21, h'_22)$, $1/a_11 mat(a_21, -det(bold(A)); 1, a_12)$, $1/a'_22 mat(a'_21, -1; det(bold(A')), a'_12)$, $bold(A)$, $1/r_21 mat(r_11, det(bold(R)); 1, r_22)$, $1/g_21 mat(-g_22, -1; -det(bold(G)), -g_11)$, $1/h_21 mat(-det(bold(H)), -h_11; -h_22, -1)$, $1/h'_21 mat(1, h'_22; h'_11, det(bold(H')))$, $mat(a_11, a_12; a_21, a_22)$, $1/det(bold(A')) mat(a'_22, a'_12; a'_21, a'_11)$, $bold(A')$, $1/r_12 mat(r_22, det(bold(R)); 1, r_11)$, $1/g_12 mat(-g_11, -1; -det(bold(G)), -g_22)$, $1/h_12 mat(1, h_11; h_22, det(bold(H)))$, $1/h'_12 mat(-det(bold(H')), -h'_22; -h'_11, -1)$, $1/det(bold(A)) mat(a_22, a_12; a_21, a_11)$, $mat(a'_11, a'_12; a'_21, a'_22)$, ) ] ] #place(bottom+left, scope: "parent", float: true)[ #bgBlock(fill: colorAllgemein, [ #subHeading(fill: colorAllgemein, [Sin-Table]) #sinTable ]) ]