Umsorttierung
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alexander
@@ -57,6 +57,8 @@
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#set text(8.5pt)
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#columns(4, gutter: 2mm)[
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// =============== Allgemein ===============
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// Allgemein
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#bgBlock(fill: colorAllgemein)[
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#subHeading(fill: colorAllgemein)[Allgemeine]
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@@ -180,11 +182,10 @@
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)
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]
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// Quell Wandlung
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// Kennline Addition
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#bgBlock(fill: colorEineTore)[
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#subHeading(fill: colorEineTore)[Ein-Tor]
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#subHeading(fill: colorEineTore)[Kennline Addition]
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/*
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#grid(
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columns: (auto, auto),
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column-gutter: 3mm,
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@@ -205,8 +206,7 @@
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$i_(cal(F),2) = i_(cal(F),1)$
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],
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)
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*Kennline Addition* \
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*/
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#grid(
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columns: (1fr, auto, 1fr),
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@@ -264,6 +264,47 @@
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)
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]
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// Dual Wandlung
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#bgBlock(fill: colorAllgemein)[
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#subHeading(fill: colorAllgemein)[Dual Wandlung]
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Stumpfe Ersetzung mit:
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#table(
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columns: (1fr, 1fr),
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fill: (x, y) => if calc.rem(y, 2) == 0 { tableFillHigh } else { tableFillLow },
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$ u --> R_d i^d $,
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$ i --> u^d / R_d $,
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$ Phi --> R_d q^d $,
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$ q --> Phi / R_d $,
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$ R --> G^d = R / R_d^2 $,
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$ G --> R^d = R_d^2 G $,
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$C --> L^d = R_d^2 C$,
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$L --> C^d = L / R_d^2$,
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$"KS" --> "LL"$, $"LL" -> "KS"$,
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$"Nullator" --> "Nullator"$, $"Norator" -> "Norator"$,
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$"Parallel" --> "Seriell"$,
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$"Seriell" --> "Parallel"$,
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table.cell(colspan: 2)[
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*Dualwandlung: Steurende & Ausgangs Größe*
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$
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"VCVS" &: u_"out" &= mu dot u_"in" &--> i_"out"^d = mu i_"in"^d\
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"VCCS" &: i_"out" &= g dot u_"in" &--> u_"out"^d = g R_d^2 dot i_"in"^d \
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"CCVS" &: u_"out" &= r dot i_"in" &--> i_"out"^d = r/R_d^2 dot u_"in"\
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"CCCS" &: i_"out" &= beta dot i_"in" &--> u_"out"^d = beta u_"in"^d\
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$
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],
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table.cell(colspan: 2)[$ (u, i) in cal(F) --> (u_d, i_d) in cal(F) = (R_d i, 1/R_d u) $]
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)
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]
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#colbreak()
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// Lineare Quelle
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#bgBlock(fill: colorEineTore)[
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#subHeading(fill: colorEineTore)[Lineare Quelle]
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@@ -409,7 +450,7 @@
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table.cell(colspan: 2, fill: tableFillLow)[
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#align(center, [*$i"-gesteuert" --> u"-gestuert"$*])
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$ G_i = 1/R_i quad quad quad I_0 = -U_0 1/R_i $
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#align(center, $G_i = 1/R_i quad quad quad I_0 = -U_0 1/R_i$)
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],
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)
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@@ -420,47 +461,7 @@
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*NICHT* die Pfeilrichtung ändern!
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]
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// Dual Wandlung
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#bgBlock(fill: colorAllgemein)[
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#subHeading(fill: colorAllgemein)[Dual Wandlung]
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Stumpfe Ersetzung mit:
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#table(
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columns: (1fr, 1fr),
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fill: (x, y) => if calc.rem(y, 2) == 0 { tableFillHigh } else { tableFillLow },
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$ u --> R_d i^d $,
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$ i --> u^d / R_d $,
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$ Phi --> R_d q^d $,
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$ q --> Phi / R_d $,
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$ R --> G^d = R / R_d^2 $,
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$ G --> R^d = R_d^2 G $,
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$C --> L^d = R_d^2 C$,
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$L --> C^d = L / R_d^2$,
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$"KS" --> "LL"$, $"LL" -> "KS"$,
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$"Nullator" --> "Nullator"$, $"Norator" -> "Norator"$,
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$"Parallel" --> "Seriell"$,
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$"Seriell" --> "Parallel"$,
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table.cell(colspan: 2)[
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*Dualwandlung: Steurende & Ausgangs Größe*
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$
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"VCVS" &: u_"out" &= mu dot u_"in" &--> i_"out"^d = mu i_"in"^d\
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"VCCS" &: i_"out" &= g dot u_"in" &--> u_"out"^d = g R_d^2 dot i_"in"^d \
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"CCVS" &: u_"out" &= r dot i_"in" &--> i_"out"^d = r/R_d^2 dot u_"in"\
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"CCCS" &: i_"out" &= beta dot i_"in" &--> u_"out"^d = beta u_"in"^d\
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$
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],
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table.cell(colspan: 2)[$ (u, i) in cal(F) --> (u_d, i_d) in cal(F) = (R_d i, 1/R_d u) $]
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)
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]
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#colbreak()
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// Linearsierung
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#bgBlock(fill: colorEineTore)[
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#subHeading(fill: colorEineTore)[Linearisierung (Ein-Tore)]
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@@ -469,7 +470,7 @@
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2. Ableitung $g_cal(F)(u)$/$r_cal(F)(i)$ bilden \ $g'_cal(F)(u)$/$r'_cal(F)(i)$
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#line(length: 100%)
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#line(length: 100%, stroke: (thickness: 0.2mm))
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*Stromgesteuert*
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@@ -514,7 +515,7 @@
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*Klein-Signal* $quad Delta i_"lin" = g_"lin" (Delta u) = g'(u_"AP") Delta u$
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#line(length: 100%)
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#line(length: 100%, stroke: (thickness: 0.2mm))
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*Spannungsgesteuert*
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@@ -554,204 +555,14 @@
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);
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#linebreak()
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*Klein-Signal* $quad Delta u_"lin" = r_"lin" (Delta i) = r'(i_"AP") Delta i$
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]
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// Graphen und Matrizen
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#bgBlock(fill: colorAnalyseVerfahren)[
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#subHeading(fill: colorAnalyseVerfahren)[Graphen und Matrizen]
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$n:$ Knotenanzahle (mit Referenzknoten)
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$b:$ Zweiganzahle
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*Lineare Unabhänige KCL/KVLs*
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Für $2b$ unbekannte ($b$ Ströme + $b$ Spannungen)
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KCLs: $n-1$\
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KVLs: $b-(n-1)$
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#line(length: 100%, stroke: (thickness: 0.2mm))
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$bold(i_b)$ (oder $bold(i)$): Zweigstrom-Vektor \
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$bold(u_b)$ (oder $bold(u)$): Zweigspannungs-Vektor \
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$bold(i_m)$ : Maschenstrom-Vektor \
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#text(rgb(20%, 20%, 20%))[(Strom in einer viruellen Masche)] \
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$bold(u_k)$ : Kontenspannungs-Vektor \
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#text(rgb(20%, 20%, 20%))[(Spannung zwischen Referenzknoten und Knoten k)] \
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#line(length: 100%, stroke: (thickness: 0.2mm))
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Knotenzidenzmatrix $bold(A)$
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$bold(A) : bold(i_k) -> text("Knotenstrombilanz") = 0$ \
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$bold(A^T) : bold(u_b)-> bold(u_k)$
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$
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bold(A) = quad space space mannot.mark(
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mat(
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a_11, a_12, ..., a_(1m);
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a_21, a_22, ..., a_(2m);
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dots.v, dots.v, dots.down, dots.v;
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a_(n 1), a_(n 2), ..., a_(n m)
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), tag: #<1>
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)
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#mannot.annot(<1>, pos: left, text(rgb("#404296"))[#rotate(-90deg)[$<-$ Knoten \ ($n-1$)]], dx: 2mm)
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#mannot.annot(<1>, pos: bottom, text(rgb("#404296"))[Zweige ($b$) $->$], dy: -0.5mm)
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a in {-1, 0, 1}
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$
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#linebreak()
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$-1 &: "In Knoten rein" \
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1 &: "Aus Knoten raus"$
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#line(length: 100%, stroke: (thickness: 0.2mm))
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Mascheninsidenz Matrix $bold(B)$\
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$bold(B) : bold(u_b) -> text("Zweigspannungsbilanz") = 0$ \
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$bold(B^T) : bold(i_m) -> i_b$
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$
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bold(B) = quad space space mannot.mark(mat(
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b_11, b_12, ..., b_(1m);
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b_21, b_22, ..., b_(2m);
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dots.v, dots.v, dots.down, dots.v;
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b_(n 1), b_(n 2), ..., b_(n m)
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), tag: #<1>)
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#mannot.annot(<1>, pos:left, text(rgb("#404296"))[#rotate(-90deg)[$<-$ Maschen \ $b-(n-1)$]], dx: 4mm)
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#mannot.annot(<1>, pos:bottom, text(rgb("#404296"))[Zweige ($b$) $->$], dy: -0.5mm)
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b in {-1, 0, 1}
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$
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#linebreak()
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$-1 &: "Gegen Maschenrichtung" \
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1 &: "In Maschenrichtung"$
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#line(length: 100%, stroke: (thickness: 0.2mm))
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#colbreak()
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*KCL und KVL* \
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KCL in Nullraum: $bold(A) bold(i_b) = bold(0)$ \
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KVL in Bildraum: $bold(A^T) bold(u_k) = bold(u_b)$
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KVL in Nullraum: $bold(B) bold(u_b) = bold(0)$ \
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KCL in Bildraum: $bold(B^T) bold(i_m) = bold(i_b)$ \
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#line(length: 100%, stroke: (thickness: 0.2mm))
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*Tellegen'sche Satz* \
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$bold(A B^T) = bold(B^T A) = 0$ \
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Prüfen oben ein AP stimmt:
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$bold(u_b^T i_b) = 0$
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]
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// Baumkonzept
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#bgBlock(fill: colorAnalyseVerfahren)[
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#subHeading(fill: colorAnalyseVerfahren)[Baumkonzept]
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1. Baum einzeichnen (Keine Schleifen!) \
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Muss alle Knoten umfassen
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2. $n-1$ KCLs: \
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Superknoten mit NUR einer Baumkante \
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$jMat(A) = mat(jMat(1)_(n-1), jMat(A)_e)$ \
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3. $b - (n-1)$ KVLs: \
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Maschen mit NUR einer NICHT Baumkante \
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$jMat(B) = mat(jMat(B)_t, jMat(1)_(b-(n-1)))$
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*Nur bei Baumkonzept:* $B_t = - A_e^T$
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]
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|
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// Tablauematrix
|
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#bgBlock(fill: colorAnalyseVerfahren)[
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#subHeading(fill: colorAnalyseVerfahren)[Allgemeine Analyse Verfahren]
|
||||
|
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*Tableaugleichung*
|
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Alle Element Gleichungen in Nullraum + KVLs/KCLs in eine Matrix
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KCLs in Nullraum: $jMat(A) jVec(i)_b = jVec(0)$\
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KVLs in Nullraum: $jMat(B) jVec(u)_b = jVec(0)$\
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Elementgleichungen: $jMat(N) jVec(u)_b + jMat(M) jVec(i)_b = jVec(e)$
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$ mannot.mark(mat(
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jMat(B), jMat(0);
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jMat(0), jMat(A);
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jMat(M), jMat(N)
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), tag: #<1>) vec(jVec(u)_b, jVec(i)_b) = vec(jVec(0), jVec(0), jVec(e))
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#mannot.annot(<1>, text($b - (n-1) space$, rgb("#00318b")), pos: left, dy: -2.8mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($n-1 space$, rgb("#00318b")), pos: left, dy: 0mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: 2.8mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($2b$, rgb("#00318b")), pos: bottom, dy: -0.5mm)
|
||||
$
|
||||
|
||||
#line(stroke: (thickness: 0.2mm), length: 100%)
|
||||
|
||||
*Knotenspannungs-Analyse*
|
||||
|
||||
KVL in Bildraum: $jVec(u)_b = jMat(A)^T jVec(u)_k$\
|
||||
KCLs in Nullraum: $jMat(A) jVec(i)_b = jVec(0)$\
|
||||
Elementgleichungen: $jMat(N) jVec(u)_b + jMat(M) jVec(i)_b = jVec(e)$
|
||||
|
||||
$
|
||||
mannot.mark(mat(
|
||||
-jMat(A)^T, jMat(1), jMat(0);
|
||||
jMat(0), jMat(0), jMat(A);
|
||||
jMat(0), jMat(M), jMat(N)
|
||||
), tag: #<1>) vec(jVec(u)_k, jVec(u)_b, jVec(i)_b) = vec(jVec(0), jVec(0), jVec(e))
|
||||
|
||||
#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: -2.8mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($n-1 space$, rgb("#00318b")), pos: left, dy: 0mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: 2.8mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($2b + (n-1)$, rgb("#00318b")), pos: bottom, dy: -0.5mm)
|
||||
$
|
||||
|
||||
#line(stroke: (thickness: 0.2mm), length: 100%)
|
||||
|
||||
*Maschenstrom-Analyse*
|
||||
|
||||
Nur für Planare Schaltungen
|
||||
|
||||
KCL in Bildraum: $jVec(i)_b = jMat(B)^T jVec(i)_m$\
|
||||
KCLs in Nullraum: $jMat(B) jVec(u)_b = jVec(0)$\
|
||||
Elementgleichungen: $jMat(N) jVec(u)_b + jMat(M) jVec(i)_b = jVec(e)$
|
||||
|
||||
$
|
||||
mannot.mark(mat(
|
||||
jMat(B), jMat(0), jMat(0);
|
||||
jMat(0), jMat(1), -jMat(B)^T;
|
||||
jMat(M), jMat(N), jMat(0)
|
||||
), tag: #<1>) vec(jVec(u)_b, jVec(i)_b, jVec(i)_m) = vec(jVec(0), jVec(0), jVec(e))
|
||||
|
||||
#mannot.annot(<1>, text($b - (n-1) space$, rgb("#00318b")), pos: left, dy: -2.8mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: 0mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: 2.8mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($3b - (n-1)$, rgb("#00318b")), pos: bottom, dy: -0.5mm)
|
||||
$
|
||||
|
||||
#line(stroke: (thickness: 0.2mm), length: 100%)
|
||||
|
||||
Nicht Lineare Gleichungen: $underline(f)'(jVec(u), jVec(i)) = 0$
|
||||
]
|
||||
|
||||
// Netwon Rephson
|
||||
#bgBlock(fill: colorAnalyseVerfahren)[
|
||||
#subHeading(fill: colorAnalyseVerfahren)[Netwen-Raphson]
|
||||
|
||||
|
||||
*Newton-Raphson*
|
||||
$x_(n+1) = x_n - f(x_n)/(f'(x_n))$
|
||||
]
|
||||
|
||||
// =============== Zwei Tore ===============
|
||||
// ZweiTor Beschreibungen
|
||||
#bgBlock(fill: colorZweiTore)[
|
||||
#subHeading(fill: colorZweiTore)[Zwei-Tor Beschreibungen]
|
||||
@@ -1026,6 +837,196 @@
|
||||
(Superpositions Prinzip)
|
||||
]
|
||||
|
||||
#colbreak()
|
||||
// Graphen und Matrizen
|
||||
#bgBlock(fill: colorAnalyseVerfahren)[
|
||||
#subHeading(fill: colorAnalyseVerfahren)[Graphen und Matrizen]
|
||||
|
||||
$n:$ Knotenanzahle (mit Referenzknoten)
|
||||
|
||||
$b:$ Zweiganzahle
|
||||
|
||||
*Lineare Unabhänige KCL/KVLs*
|
||||
|
||||
Für $2b$ unbekannte ($b$ Ströme + $b$ Spannungen)
|
||||
|
||||
KCLs: $n-1$\
|
||||
KVLs: $b-(n-1)$
|
||||
|
||||
#line(length: 100%, stroke: (thickness: 0.2mm))
|
||||
|
||||
$bold(i_b)$ (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 $bold(A)$
|
||||
|
||||
$bold(A) : bold(i_k) -> text("Knotenstrombilanz") = 0$ \
|
||||
$bold(A^T) : bold(u_b)-> bold(u_k)$
|
||||
$
|
||||
bold(A) = quad space space 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 \ ($n-1$)]], dx: 2mm)
|
||||
#mannot.annot(<1>, pos: bottom, text(rgb("#404296"))[Zweige ($b$) $->$], dy: -0.5mm)
|
||||
a in {-1, 0, 1}
|
||||
$
|
||||
#linebreak()
|
||||
$-1 &: "In Knoten rein" \
|
||||
1 &: "Aus Knoten raus"$
|
||||
|
||||
#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 space space 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 \ $b-(n-1)$]], dx: 4mm)
|
||||
#mannot.annot(<1>, pos:bottom, text(rgb("#404296"))[Zweige ($b$) $->$], dy: -0.5mm)
|
||||
|
||||
b in {-1, 0, 1}
|
||||
$
|
||||
|
||||
#linebreak()
|
||||
$-1 &: "Gegen Maschenrichtung" \
|
||||
1 &: "In Maschenrichtung"$
|
||||
|
||||
#line(length: 100%, stroke: (thickness: 0.2mm))
|
||||
|
||||
#colbreak()
|
||||
*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: $bold(B^T) bold(i_m) = bold(i_b)$ \
|
||||
|
||||
#line(length: 100%, stroke: (thickness: 0.2mm))
|
||||
|
||||
|
||||
*Tellegen'sche Satz* \
|
||||
$bold(A B^T) = bold(B^T A) = 0$ \
|
||||
|
||||
Prüfen oben ein AP stimmt:
|
||||
$bold(u_b^T i_b) = 0$
|
||||
]
|
||||
|
||||
// Baumkonzept
|
||||
#bgBlock(fill: colorAnalyseVerfahren)[
|
||||
#subHeading(fill: colorAnalyseVerfahren)[Baumkonzept]
|
||||
|
||||
|
||||
1. Baum einzeichnen (Keine Schleifen!) \
|
||||
Muss alle Knoten umfassen
|
||||
|
||||
2. $n-1$ KCLs: \
|
||||
Superknoten mit NUR einer Baumkante \
|
||||
$jMat(A) = mat(jMat(1)_(n-1), jMat(A)_e)$ \
|
||||
|
||||
|
||||
3. $b - (n-1)$ KVLs: \
|
||||
Maschen mit NUR einer NICHT Baumkante \
|
||||
$jMat(B) = mat(jMat(B)_t, jMat(1)_(b-(n-1)))$
|
||||
|
||||
|
||||
*Nur bei Baumkonzept:* $B_t = - A_e^T$
|
||||
|
||||
]
|
||||
|
||||
// Tablauematrix
|
||||
#bgBlock(fill: colorAnalyseVerfahren)[
|
||||
#subHeading(fill: colorAnalyseVerfahren)[Allgemeine Analyse Verfahren]
|
||||
|
||||
*Tableaugleichung*
|
||||
|
||||
Alle Element Gleichungen in Nullraum + KVLs/KCLs in eine Matrix
|
||||
|
||||
KCLs in Nullraum: $jMat(A) jVec(i)_b = jVec(0)$\
|
||||
KVLs in Nullraum: $jMat(B) jVec(u)_b = jVec(0)$\
|
||||
Elementgleichungen: $jMat(N) jVec(u)_b + jMat(M) jVec(i)_b = jVec(e)$
|
||||
|
||||
$ mannot.mark(mat(
|
||||
jMat(B), jMat(0);
|
||||
jMat(0), jMat(A);
|
||||
jMat(M), jMat(N)
|
||||
), tag: #<1>) vec(jVec(u)_b, jVec(i)_b) = vec(jVec(0), jVec(0), jVec(e))
|
||||
|
||||
#mannot.annot(<1>, text($b - (n-1) space$, rgb("#00318b")), pos: left, dy: -2.8mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($n-1 space$, rgb("#00318b")), pos: left, dy: 0mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: 2.8mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($2b$, rgb("#00318b")), pos: bottom, dy: -0.5mm)
|
||||
$
|
||||
|
||||
#line(stroke: (thickness: 0.2mm), length: 100%)
|
||||
|
||||
*Knotenspannungs-Analyse*
|
||||
|
||||
KVL in Bildraum: $jVec(u)_b = jMat(A)^T jVec(u)_k$\
|
||||
KCLs in Nullraum: $jMat(A) jVec(i)_b = jVec(0)$\
|
||||
Elementgleichungen: $jMat(N) jVec(u)_b + jMat(M) jVec(i)_b = jVec(e)$
|
||||
|
||||
$
|
||||
mannot.mark(mat(
|
||||
-jMat(A)^T, jMat(1), jMat(0);
|
||||
jMat(0), jMat(0), jMat(A);
|
||||
jMat(0), jMat(M), jMat(N)
|
||||
), tag: #<1>) vec(jVec(u)_k, jVec(u)_b, jVec(i)_b) = vec(jVec(0), jVec(0), jVec(e))
|
||||
|
||||
#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: -2.8mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($n-1 space$, rgb("#00318b")), pos: left, dy: 0mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: 2.8mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($2b + (n-1)$, rgb("#00318b")), pos: bottom, dy: -0.5mm)
|
||||
$
|
||||
|
||||
#line(stroke: (thickness: 0.2mm), length: 100%)
|
||||
|
||||
*Maschenstrom-Analyse*
|
||||
|
||||
Nur für Planare Schaltungen
|
||||
|
||||
KCL in Bildraum: $jVec(i)_b = jMat(B)^T jVec(i)_m$\
|
||||
KCLs in Nullraum: $jMat(B) jVec(u)_b = jVec(0)$\
|
||||
Elementgleichungen: $jMat(N) jVec(u)_b + jMat(M) jVec(i)_b = jVec(e)$
|
||||
|
||||
$
|
||||
mannot.mark(mat(
|
||||
jMat(B), jMat(0), jMat(0);
|
||||
jMat(0), jMat(1), -jMat(B)^T;
|
||||
jMat(M), jMat(N), jMat(0)
|
||||
), tag: #<1>) vec(jVec(u)_b, jVec(i)_b, jVec(i)_m) = vec(jVec(0), jVec(0), jVec(e))
|
||||
|
||||
#mannot.annot(<1>, text($b - (n-1) space$, rgb("#00318b")), pos: left, dy: -2.8mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: 0mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: 2.8mm, dx: 1mm)
|
||||
#mannot.annot(<1>, text($3b - (n-1)$, rgb("#00318b")), pos: bottom, dy: -0.5mm)
|
||||
$
|
||||
|
||||
#line(stroke: (thickness: 0.2mm), length: 100%)
|
||||
|
||||
Nicht Lineare Gleichungen: $underline(f)'(jVec(u), jVec(i)) = 0$
|
||||
]
|
||||
|
||||
// Linearsierung (N-Tore)
|
||||
#bgBlock(fill: colorAnalyseVerfahren)[
|
||||
#subHeading(fill: colorAnalyseVerfahren)[Linearisierung (N-Tore)]
|
||||
@@ -1051,15 +1052,9 @@
|
||||
$jVec(y)_"lin" = jMat(J)|_(jVec(x)_"AP") jVec(x)_"lin"$
|
||||
]
|
||||
|
||||
// Newton-Raphson
|
||||
#bgBlock(fill: colorEineTore)[
|
||||
#subHeading(fill: colorEineTore)[Newton-Raphson (Eine-Tor)]
|
||||
$x_(n+1) = x_n - f(x_n)/(f'(x_n))$
|
||||
]
|
||||
|
||||
// Netwen-Raphson N-Tore
|
||||
#bgBlock(fill: colorZweiTore)[
|
||||
#subHeading(fill: colorZweiTore)[Newton-Raphson (N-Tore)]
|
||||
#bgBlock(fill: colorAnalyseVerfahren)[
|
||||
#subHeading(fill: colorAnalyseVerfahren)[Newton-Raphson (N-Tore)]
|
||||
Nicht lineare Beschreibung in Nullraum/Implizit Darstellung:
|
||||
$jVec(f)'(jVec(x)) = jVec(0)$\
|
||||
|
||||
@@ -1084,8 +1079,9 @@
|
||||
|
||||
4. Fehler $epsilon$ berechnen
|
||||
]
|
||||
|
||||
|
||||
// Reaktive Elemeten
|
||||
#colbreak()
|
||||
#bgBlock(fill: colorComplexAC)[
|
||||
#subHeading(fill: colorComplexAC)[Reaktive Element]
|
||||
|
||||
@@ -1161,6 +1157,7 @@
|
||||
Das gilt *NUR* zweit Abhänige Darstellung, *NICHT* für Frequenz Abhänige Darstellung
|
||||
]
|
||||
|
||||
#colbreak()
|
||||
#bgBlock(fill: colorComplexAC)[
|
||||
#subHeading(fill: colorComplexAC)[Reaktive Bauelemente]
|
||||
#grid(
|
||||
@@ -1256,46 +1253,7 @@
|
||||
|
||||
]
|
||||
|
||||
// Reaktive Dual Wandlung
|
||||
#bgBlock(fill: colorComplexAC)[
|
||||
#subHeading(fill: colorComplexAC)[Reaktive Dualwandlung]
|
||||
|
||||
#grid(
|
||||
columns: (1fr, 1fr),
|
||||
row-gutter: 4mm,
|
||||
$u --> R_d i^d$, $i --> u^d/R_d$,
|
||||
$q --> Phi^d / R_d$, $Phi --> q^d R_d$,
|
||||
)
|
||||
]
|
||||
|
||||
// Complex AC
|
||||
#bgBlock(fill: colorComplexAC)[
|
||||
#subHeading(fill: colorComplexAC)[Komplex Wechselstrom Rechnnung]
|
||||
*Nur für Lineare und Eingeschwungenene Schaltungen*
|
||||
|
||||
$u(t) = "Re"{U_m e^(j omega t + phi)}$
|
||||
|
||||
$u(t) = U_m cos(omega t + alpha)$ \
|
||||
$i(t) = I_m cos(omega t + beta)$
|
||||
|
||||
$(d u)/(d t) = A_m$
|
||||
|
||||
Kreisfrequenz: $omega = 2 pi f$ \
|
||||
Amplitude: $A_m$ \
|
||||
Phaseverschieben: $alpha$
|
||||
|
||||
*Ohm:* $u(t) = R I_m cos(omega t + beta) = omega A_m cos(omega)$)
|
||||
|
||||
*Serienschaltung*\
|
||||
$u_1(t) = U_1 cos(omega t)$\
|
||||
$u_2(t) = U_2 cos(omega t + phi)$
|
||||
|
||||
$u(t)_"ges" = u_1(t) + u_2(t) = \
|
||||
U_"ges"^2 = U_1^2 + 2 U_1 U_2 + U_2^2 \
|
||||
tan(phi) = (U_2 sin(phi))/(U_1 + U_2 cos(phi))$
|
||||
]
|
||||
|
||||
|
||||
// Komplexe Beziehung
|
||||
#bgBlock(fill: colorComplexAC)[
|
||||
#subHeading(fill: colorComplexAC)[Komplex Beziehungen]
|
||||
#let size = 1.4
|
||||
@@ -1367,6 +1325,33 @@
|
||||
])
|
||||
]
|
||||
|
||||
// Complex AC
|
||||
#bgBlock(fill: colorComplexAC)[
|
||||
#subHeading(fill: colorComplexAC)[Komplex Wechselstrom Rechnnung]
|
||||
*Nur für Lineare und Eingeschwungenene Schaltungen*
|
||||
|
||||
$u(t) = "Re"{U_m e^(j omega t + phi)}$
|
||||
|
||||
$u(t) = U_m cos(omega t + alpha)$ \
|
||||
$i(t) = I_m cos(omega t + beta)$
|
||||
|
||||
$(d u)/(d t) = A_m$
|
||||
|
||||
Kreisfrequenz: $omega = 2 pi f$ \
|
||||
Amplitude: $A_m$ \
|
||||
Phaseverschieben: $alpha$
|
||||
|
||||
*Ohm:* $u(t) = R I_m cos(omega t + beta) = omega A_m cos(omega)$)
|
||||
|
||||
*Serienschaltung*\
|
||||
$u_1(t) = U_1 cos(omega t)$\
|
||||
$u_2(t) = U_2 cos(omega t + phi)$
|
||||
|
||||
$u(t)_"ges" = u_1(t) + u_2(t) = \
|
||||
U_"ges"^2 = U_1^2 + 2 U_1 U_2 + U_2^2 \
|
||||
tan(phi) = (U_2 sin(phi))/(U_1 + U_2 cos(phi))$
|
||||
]
|
||||
|
||||
// AC Components
|
||||
#bgBlock(fill: colorComplexAC)[
|
||||
#subHeading(fill: colorComplexAC)[Komplexe Komponent]
|
||||
@@ -1403,6 +1388,7 @@
|
||||
)
|
||||
]
|
||||
|
||||
#colbreak()
|
||||
#bgBlock(fill: colorComplexAC)[
|
||||
#subHeading(fill: colorComplexAC)[*Levi's Lustig Leistung*]
|
||||
$P=P_W + j P_B$\
|
||||
@@ -1426,6 +1412,7 @@
|
||||
$U_"eff" = U_m/sqrt(2), I_"eff" = I_m / sqrt(2)$
|
||||
]
|
||||
|
||||
#colbreak()
|
||||
// Komplexe Zahlen
|
||||
#bgBlock(fill: colorAllgemein)[
|
||||
#subHeading(fill: colorAllgemein)[Komplexe Zahlen]
|
||||
@@ -1444,6 +1431,7 @@
|
||||
)
|
||||
]
|
||||
|
||||
#colbreak()
|
||||
// SinTable
|
||||
#bgBlock(fill: colorAllgemein, [
|
||||
#subHeading(fill: colorAllgemein, [Sin-Table])
|
||||
|
||||
Reference in New Issue
Block a user