misc changes
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alexander
@@ -419,6 +419,54 @@
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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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$"Parallel" --> "Seriell"$,
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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" R_d = mu dot i_"in" R_d \
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"VCCS": u_"out" = g dot i_"in" &--> \
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"CCVS": u_"out" = r dot i_"in" &--> \
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"CCCS": i_"out" = beta dot i_"in" &--> \
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$
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],
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table.cell(colspan: 2)[
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*Dualwandlung: Nur Ausgangs Größe*
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$
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"VCVS": u_"out" = mu dot u_"in" &--> \
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"VCCS": u_"out" = g dot i_"in" &--> \
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"CCVS": u_"out" = r dot i_"in" &--> \
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"CCCS": i_"out" = beta dot i_"in" &--> \
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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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// Linearsierung
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#bgBlock(fill: colorEineTore)[
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@@ -517,8 +565,6 @@
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*Klein-Signal* $quad u_"lin" = r_"lin" (i) = r'(i_"AP")i$
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]
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#colbreak()
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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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@@ -594,6 +640,7 @@
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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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@@ -634,29 +681,82 @@
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// Tablauematrix
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#bgBlock(fill: colorAnalyseVerfahren)[
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#subHeading(fill: colorAnalyseVerfahren)[Tablauematrix]
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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: $jMat(A) jVec(i) = jVec(0)$\
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KVLs: $jMat(B) jVec(u) = jVec(0)$\
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Elementgleichungen: $jMat(N) jVec(u) + jMat(M) jVec(i) = jVec(e)$
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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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$ mat(
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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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) vec(jVec(u), jVec(i)) = vec(jVec(0), jVec(0), jVec(e)) $
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), tag: #<1>) vec(jVec(u)_b, jVec(i)_b) = vec(jVec(0), jVec(0), jVec(e))
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]
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// Machenstrom-/Knotenpotenzial-Analyse
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#bgBlock(fill: colorAnalyseVerfahren)[
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#subHeading(fill: colorAnalyseVerfahren)[Machenstrom-/Knotenpotenzial-Analyse]
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#mannot.annot(<1>, text($b - (n-1) space$, rgb("#00318b")), pos: left, dy: -2.8mm, dx: 1mm)
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#mannot.annot(<1>, text($n-1 space$, rgb("#00318b")), pos: left, dy: 0mm, dx: 1mm)
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#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: 2.8mm, dx: 1mm)
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#mannot.annot(<1>, text($2b$, rgb("#00318b")), pos: bottom, dy: -0.5mm)
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$
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#line(stroke: (thickness: 0.2mm), length: 100%)
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*Knotenspannungs-Analyse*
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KVL in Bildraum: $jVec(u)_b = jMat(A)^T jVec(u)_k$\
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KCLs in Nullraum: $jMat(A) jVec(i)_b = jVec(0)$\
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Elementgleichungen: $jMat(N) jVec(u)_b + jMat(M) jVec(i)_b = jVec(e)$
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$
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mannot.mark(mat(
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-jMat(A)^T, jMat(1), jMat(0);
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jMat(0), jMat(0), jMat(A);
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jMat(0), jMat(M), jMat(N)
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), tag: #<1>) vec(jVec(u)_k, jVec(u)_b, jVec(i)_b) = vec(jVec(0), jVec(0), jVec(e))
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#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: -2.8mm, dx: 1mm)
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#mannot.annot(<1>, text($n-1 space$, rgb("#00318b")), pos: left, dy: 0mm, dx: 1mm)
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#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: 2.8mm, dx: 1mm)
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#mannot.annot(<1>, text($2b + (n-1)$, rgb("#00318b")), pos: bottom, dy: -0.5mm)
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$
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#line(stroke: (thickness: 0.2mm), length: 100%)
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*Maschenstrom-Analyse*
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Nur für Planare Schaltungen
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KCL in Bildraum: $jVec(i)_b = jMat(B)^T jVec(i)_m$\
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KCLs 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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$
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mannot.mark(mat(
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jMat(B), jMat(0), jMat(0);
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jMat(0), jMat(1), -jMat(B)^T;
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jMat(M), jMat(N), jMat(0)
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), tag: #<1>) vec(jVec(u)_b, jVec(i)_b, jVec(i)_m) = 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)
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#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: 0mm, dx: 1mm)
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#mannot.annot(<1>, text($b space$, rgb("#00318b")), pos: left, dy: 2.8mm, dx: 1mm)
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#mannot.annot(<1>, text($3b - (n-1)$, rgb("#00318b")), pos: bottom, dy: -0.5mm)
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$
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#line(stroke: (thickness: 0.2mm), length: 100%)
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Nicht Lineare Gleichungen: $underline(f)'(jVec(u), jVec(i)) = 0$
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]
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// Reduziert Knotenpotenzial
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// Netwon Rephson
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#bgBlock(fill: colorAnalyseVerfahren)[
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#subHeading(fill: colorAnalyseVerfahren)[Reduzierte Knotenpotenzial-Analyse]
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#subHeading(fill: colorAnalyseVerfahren)[Netwen-Raphson]
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]
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// ZweiTor Beschreibungen
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@@ -885,15 +985,11 @@
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],
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// Linearsierung (N-Tore)
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#bgBlock(fill: colorZweiTore)[
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#subHeading(fill: colorZweiTore)[Linearisierung (N-Tore)]
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#bgBlock(fill: colorAnalyseVerfahren)[
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#subHeading(fill: colorAnalyseVerfahren)[Linearisierung (N-Tore)]
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1. Arbeitspunk bestimmen $vec(jVec(u)_"AP", jVec(i)_"AP") hat(=) vec(jVec(x)_"AP", jVec(y)_"AP")$
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$f_1(x_1, x_2, ... x_n) &= y_1\
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f_2(x_1, x_2, ... x_n) &= y_2\
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&dots.v \
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f_n (x_1, x_2, ... x_n) &= y_n$
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1. Arbeitspunk bestimmen/schätzen/ist gegeben \
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$vec(jVec(u)_"AP", jVec(i)_"AP") hat(=) vec(jVec(x)_"AP", jVec(y)_"AP")$
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$bold(f)(jVec(x))=jVec(y)$
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@@ -922,10 +1018,29 @@
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// Netwen-Raphson N-Tore
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#bgBlock(fill: colorZweiTore)[
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#subHeading(fill: colorZweiTore)[Newton-Raphson (N-Tore)]
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Nicht lineare Beschreibung in Nullraum/Impliziter Darstellung:
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$f(jVec(x)) = jVec(0)$\
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Nicht lineare Beschreibung in Nullraum/Implizit Darstellung:
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$jVec(f)'(jVec(x)) = jVec(0)$\
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$jVec(x)_(n+1) = jVec(x)_n - (jMat(J)|_(jVec(x)_"AP"))^(-1) f(jVec(x))$
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0. Jakobi Matrix $jMat(J)(x^((k)))$ aufstellen (ableiten)
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Schrit $n+1$ Berechnen:
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1. Jacobi Matrix $jMat(J)(x^((n)))$ ausrechnen
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2. Linearisiertes Gleichungs System aufstellen: \
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$M_"lin"^((n)) jVec(u)^((n)) + N_"lin"^((n)) jVec(i)^((n))$ aus $jMat(J)(x^((n)))$ rauslesen
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$jVec(e)_"lin" = jMat(J)(x^((n))) dot x^((n)) - jVec(f)(x^((n)))$
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$n$te-Linearisierte Elementgleichungen: \
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$M_"lin"^((n)) jVec(u)^((n)) + N_"lin"^((n)) jVec(i)^((n)) = jVec(e)^((n))_"lin"$
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Tablaue: $jMat(T)^((n)) jVec(x)^((n + 1)) = jVec(e)^((n))$ \
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3. Gleichungsystem lösen: $jVec(x)^((n+1))$
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4. Fehler $epsilon$ berechnen
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]
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// Reaktive Elemeten
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@@ -940,12 +1055,16 @@
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row-gutter: 10mm,
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column-gutter: 2mm,
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[
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*Kapazitiv*
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$[i(t)] = unit("A")$\
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$[q(t)] = unit("A s") = unit("C")$\
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],
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grid.vline(stroke: 0.75pt),
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[],
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[
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*Induktiv*
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$[u(t)] = unit("V")$ \
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$[Phi(t)] = unit("V s") = unit("W b")$ \
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],
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@@ -965,7 +1084,11 @@
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],
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)
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$W(t_1, t_2) = integral_(t_1)^(t_2) P(tau) d tau = integral_(t_1)^(t_2) u(tau) i(tau) d tau$
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#linebreak()
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$P(t) = u(t) i(t)$
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$W(t_1, t_2) = integral_(t_1)^(t_2) P(tau) d tau$
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$W(t_1, t_2) > 0$: Nimmt Energie auf\
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$W(t_1, t_2) = 0$: Verlustlos\
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@@ -981,13 +1104,17 @@
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[*Kapazitiv*],
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grid.vline(stroke: 0.75pt),
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[],
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[*Induktivität*],
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[*Induktiv*],
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[
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$q = c(u) \ u = chi(q)$\
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$W(t_1, t_2), integral_(q_1 = q(t_1))^(q_2 = q(t_2)) chi(q) d q$
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],
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[],
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[
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$Phi = l(i) \ i = lambda(Phi)$
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$Phi = l(i) \ i = lambda(Phi)$ \
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$W(t_1, t_2), integral_(Phi_1 = Phi(t_1))^(Phi_2 = Phi(t_2)) lambda(Phi) d Phi$
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],
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[$u,q$ stetig und beschränkt],
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@@ -1080,7 +1207,7 @@
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#subHeading(fill: colorComplexAC)[Komplex Wechselstrom Rechnnung]
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Im Eingeschwungenem Zustand
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$u(t) =U_m "Re"{e^(j omega t + phi)}$
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$u(t) = "Re"{U_m e^(j omega t + phi)}$
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$u(t) = U_m cos(omega t + alpha)$ \
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$i(t) = I_m cos(omega t + beta)$
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@@ -1239,7 +1366,16 @@
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#ComplexNumbersSection(i: $j$)
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*Trigonometire*
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#grid(
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columns: (auto, auto),
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column-gutter: 2mm,
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row-gutter: 3mm,
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$cos(x) = cos(-x)$, $sin(-x) = -sin(x)$, $cos(x)^2 + sin(x)^2 = 1$, $cos(x) = sin(x + pi/2)$, $sin(x) = cos(x - pi/2)$, $$,
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grid.cell(colspan: 2, $cos(x + y) = cos(x)cos(y) - sin(x)sin(y)$),
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grid.cell(colspan: 2, $sin(x + y) = sin(x)cos(y) + cos(x)sin(y)$)
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)
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]
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// SinTable
|
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@@ -1260,11 +1396,40 @@
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#columns(2)[
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#bgBlock(fill: colorAnalyseVerfahren)[
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#subHeading(fill: colorAnalyseVerfahren)[Knotenpotenzial-Analyse Komponetent]
|
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#subHeading(fill: colorAnalyseVerfahren)[Reduzierte Knotenpotenzial-Analyse Komponetent]
|
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#import mannot: *
|
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|
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#let ImageHeight = 2.5cm
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|
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Reduzierte Knotenpotenzial-Analyse: $jMat(G_k) = jVec(u)_k = jVec(i)_q$
|
||||
|
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*Cramersche Regel:* $u_(k i) = (det jMat(G)_(k i))/(det jMat(G)_k)$ ($jMat(G)_(k i)$ entshet aus $G_k$ durch ersetzen der $i$-ten Splate mit $jVec(i)_q$)
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|
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#table(
|
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grid(
|
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columns: 3,
|
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zap.circuit({
|
||||
import zap : *
|
||||
|
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vsource("V", (0,0), (0,1.5), scale: 0.4, fill: none)
|
||||
joham.voltage((-0.3,1), (-0.3,0), $U_0$)
|
||||
resistor("G", (0,1.5), (1,1.5), scale: 0.4, fill: none)
|
||||
wire("V.in", (1,0))
|
||||
}),
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||||
align(center+horizon, $==>$),
|
||||
zap.circuit({
|
||||
import zap : *
|
||||
|
||||
isource("V", (0,0), (0,1.5), scale: 0.4, fill: none, i: (content: $G U_0 space $, invert: false, distance: 0.2, anchor: "east"))
|
||||
|
||||
resistor("G", (0,1.5), (1,1.5), scale: 0.4, fill: none)
|
||||
wire("V.in", (1,0))
|
||||
})
|
||||
),
|
||||
|
||||
|
||||
)
|
||||
|
||||
#table(
|
||||
columns: (1fr, 1fr),
|
||||
fill: (x, y) => if calc.rem(y, 2) == 0 { tableFillHigh } else { tableFillLow },
|
||||
@@ -1725,7 +1890,7 @@
|
||||
|
||||
// Zwei-Tor Tabelle
|
||||
|
||||
#grid(columns: (2fr, 1fr))[
|
||||
#grid(columns: (1fr))[
|
||||
#bgBlock(fill: colorZweiTore, width: 100%)[
|
||||
#subHeading(fill: colorZweiTore)[Zwei-Tor-Übersichts]
|
||||
|
||||
@@ -1748,7 +1913,7 @@
|
||||
|
||||
joham.voltage((to: "Op.plus", rel: (-0.25, 0)), (to: "Op.minus", rel: (-0.25, 0)), $u_"in"$)
|
||||
|
||||
joham.voltage((to: "Op.out", rel: (0.25, 0)), (to: "Op.out", rel: (0.25, -0.7)), $u_"out"$, anchor: "west")
|
||||
joham.voltage((to: "Op.out", rel: (0.25, 0)), (to: "Op.out", rel: (0.25, -0.7)), $u_"out"$, anchor: "west")
|
||||
})
|
||||
|
||||
#table(
|
||||
@@ -1914,7 +2079,25 @@
|
||||
$ i_2 &= - ü i_1 &quad i_1 &= - 1/ü i_2 \
|
||||
u_2 &= 1/ü u_1 &quad u_1 &= ü u_2
|
||||
$
|
||||
], [],
|
||||
], [
|
||||
#align(horizon+center, zap.circuit({
|
||||
import zap : *
|
||||
import cetz.draw : line, rect, mark, content
|
||||
|
||||
joham.gyrator("G1", (0,0), scale: 0.4, constant: $G_d$)
|
||||
joham.gyrator("G2", (1.6,0), scale: 0.4, constant: $ü G_d$)
|
||||
joham.ground("G", (0.8, -0.67))
|
||||
|
||||
node("N1", "G1.12", fill: false, label: (content: $beta$, distance: 0.1))
|
||||
node("N1", "G1.11", fill: false, label: (content: $alpha$, distance: 0.1))
|
||||
node("N1", "G2.22", fill: false, label: (content: $delta$, distance: 0.1))
|
||||
node("N1", "G2.21", fill: false, label: (content: $gamma$, distance: 0.1))
|
||||
|
||||
node("N1", (0.8, 0.67), label: (content: $epsilon$, distance: 0.1))
|
||||
node("N1", (0.8, -0.67))
|
||||
|
||||
}))
|
||||
],
|
||||
[
|
||||
- Verlustlos
|
||||
- Reziprok
|
||||
@@ -1948,8 +2131,11 @@
|
||||
content((-0.7, -0.8), text([Output $cal(F)^d$], fill: rgb("#8b2000")))
|
||||
|
||||
content((0.8, -0.8), text([Input $cal(F)$], fill: rgb("#00318b")))
|
||||
|
||||
})],
|
||||
})
|
||||
$ u_1 = - R_d i_2 &quad i_1 = 1/R_d u_2 \
|
||||
u_2 = R_d i_1 &quad u_2 = - 1/R_d u_1
|
||||
$
|
||||
],
|
||||
[],
|
||||
[
|
||||
- Verlustlos
|
||||
@@ -1958,14 +2144,77 @@
|
||||
|
||||
Der Pfeil zeigt AUF die NORMAL Eintor
|
||||
],
|
||||
[$ R = mat(0, -R_d; R_d, 0) &quad G = mat(0, G_d; -G_d, 0) \ A = mat(0, R_d; 1/R_d, 0) &quad A' = mat(0, -R_d; -1/R_d, 0)
|
||||
[$ R = mat(0, -R_d; R_d, 0) &quad G = mat(0, G_d; -G_d, 0) \
|
||||
A = mat(0, R_d; 1/R_d, 0) &quad A' = mat(0, -R_d; -1/R_d, 0) \
|
||||
R_d = 1/G_d
|
||||
|
||||
$],
|
||||
|
||||
[NIK],
|
||||
[#zap.circuit({
|
||||
joham.zweitor("NIK", (0,0), label: "NIK")
|
||||
})],
|
||||
[],
|
||||
[#grid(
|
||||
columns: (auto, auto),
|
||||
column-gutter: -3mm,
|
||||
align(center+horizon, scale(75%, zap.circuit({
|
||||
import zap : *
|
||||
import cetz.draw : line, rect, mark, content
|
||||
|
||||
node("A", (-2, 0), fill: false)
|
||||
node("B", (2, 0), fill: false)
|
||||
node("C", (-2, 1.5), fill: false)
|
||||
node("D", (2, 1.5), fill: false)
|
||||
|
||||
wire("A", "B")
|
||||
|
||||
joham.norator("Q1", (-1.5, 2), (1.5, 2), scale: 0.5)
|
||||
node("N1", (-1.5,1.5))
|
||||
node("N2", (0,1.5))
|
||||
node("N3", (1.5,1.5))
|
||||
resistor("R1", "N1", "N2", scale: 0.5, fill: none, label: (content: $R$, anchor: "south", distance: 1mm))
|
||||
resistor("R2", "N2", "N3", scale: 0.5, fill: none, label: (content: $R$, anchor: "south", distance: 1mm))
|
||||
wire("N1", "Q1.in")
|
||||
wire("N3", "Q1.out")
|
||||
wire("N1", "C", i: (content: $i_1$, invert: true))
|
||||
wire("N3", "D", i: (content: $i_2$, invert: true))
|
||||
joham.nullator("Q0", "N2", n:"*-*", (0,0), scale: 0.5)
|
||||
|
||||
joham.voltage("C", "A", $u_1$)
|
||||
joham.voltage("D", "B", $u_2$)
|
||||
|
||||
content((0,-0.4), $k = +1$, anchor: "south")
|
||||
}))),
|
||||
align(center+horizon, scale(75%, zap.circuit({
|
||||
import zap : *
|
||||
import cetz.draw : line, rect, mark, content
|
||||
|
||||
node("A", (-2, 0), fill: false)
|
||||
node("B", (2, 0), fill: false)
|
||||
node("C", (-2, 1.5), fill: false)
|
||||
node("D", (2, 1.5), fill: false)
|
||||
|
||||
wire("A", "B")
|
||||
|
||||
joham.nullator("Q1", (-1.5, 2), (1.5, 2), scale: 0.5)
|
||||
node("N1", (-1.5,1.5))
|
||||
node("N2", (0,1.5))
|
||||
node("N3", (1.5,1.5))
|
||||
resistor("R1", "N1", "N2", scale: 0.5, fill: none, label: (content: $R$, anchor: "south", distance: 1mm))
|
||||
resistor("R2", "N2", "N3", scale: 0.5, fill: none, label: (content: $R$, anchor: "south", distance: 1mm))
|
||||
wire("N1", "Q1.in")
|
||||
wire("N3", "Q1.out")
|
||||
wire("N1", "C", i: (content: $i_1$, invert: true))
|
||||
wire("N3", "D", i: (content: $i_2$, invert: true))
|
||||
joham.norator("Q0", "N2", n:"*-*", (0,0), scale: 0.5)
|
||||
|
||||
joham.voltage("C", "A", $u_1$)
|
||||
joham.voltage("D", "B", $u_2$)
|
||||
|
||||
content((0,-0.4), $k = -1$, anchor: "south")
|
||||
}))))
|
||||
|
||||
],
|
||||
[
|
||||
- Aktiv
|
||||
- Antireziprok
|
||||
@@ -1973,7 +2222,7 @@
|
||||
],
|
||||
[$ H = mat(0, -k; -k, 0) quad H' = mat(0, -1/k; -1/k, 0) \ A = mat(-k, 0; 0, 1/k) quad A'= mat(-1/k, 0; 0, k) $]
|
||||
)
|
||||
]
|
||||
]
|
||||
]
|
||||
|
||||
// Knoten Spannungs Analyse
|
||||
@@ -2005,12 +2254,17 @@
|
||||
Kennline nur $u\/i$-Achsen
|
||||
],
|
||||
[$forall vec(jVec(u), jVec(v)) in cal(F) : jVec(u)^T jVec(i) = 0$],
|
||||
grid(columns: (auto, auto),
|
||||
column-gutter: 5mm,
|
||||
[
|
||||
$u\/q$-Plot: Wenn keine Schleifen \
|
||||
$i\/Phi$-Plot: Wenn keine Schleifen \
|
||||
$i\/Phi$-Plot: Wenn keine Schleifen
|
||||
], [
|
||||
$u\/i$-Plot: Wenn Auf Achse \
|
||||
$Phi\/q$-Plot: Wenn auf Achse \
|
||||
],
|
||||
], [
|
||||
|
||||
]),
|
||||
|
||||
[*linear*],
|
||||
[Kennline ist Gerade],
|
||||
|
||||
@@ -69,7 +69,11 @@
|
||||
$z^* = a - #i b = r e^(-#i phi)$
|
||||
|
||||
Konjungiert Erweitern:\
|
||||
$(a + b #i)/(c + d #i) = ((a + b #i)(c - d #i))/(c^2 + d² )$
|
||||
#grid(columns: (1fr, 1fr),
|
||||
$(a + b #i)/(c + d #i) = ((a + b #i)(c - d #i))/(c^2 + d² )$,
|
||||
$1/(a + j b) = (a - j b)/(a^2 + b^2)$
|
||||
)
|
||||
|
||||
|
||||
$r = abs(z) quad phi = cases(
|
||||
+ arccos(a/r) space : space a >= 0,
|
||||
|
||||
Reference in New Issue
Block a user