alexander/TUM-Formelsammlungen
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Merge branch 'main' of https://gitea.mintcalc.com/alexander/TUM-Formelsammlungen
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levi
2026-02-08 23:31:26 +01:00
parent cbd6fa82b7 612b991aed
commit 4da6d7ff0a
1 changed files with 299 additions and 26 deletions
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src/cheatsheets/Schaltungstheorie.typ
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@@ -205,8 +205,62 @@
$i_(cal(F),2) = i_(cal(F),1)$
],
)
*Kennline Addition* \
#grid(
columns: (1fr, auto, 1fr),
row-gutter: 2mm,
column-gutter: 3mm,
[Parallel-Schaltung], $-->$, [$i$-Richtung],
[Reihe-Schaltung], $-->$, [$u$-Richtung],
)
]
#bgBlock(fill: colorAllgemein)[
#subHeading(fill: colorAllgemein)[Spannungs Teiler/Strom Teiler]
#table(
columns: (1fr, 1fr),
fill: (x, y) => if calc.rem(x, 2) == 1 { tableFillLow } else { tableFillHigh },
[*Spannungsteiler*],
[*Stromteiler*],
zap.circuit({
import zap : *
node("N1", (0,1.5))
node("N2", (0,0))
node("N3", (0,-1.5))
resistor("R1", "N1", "N2", label: (content: $R_2$, distance: 0.1), scale: 0.7, fill: none)
resistor("R2", "N2", "N3", label: (content: $R_1$, distance: 0.1), scale: 0.7, fill: none)
joham.voltage((-0.4, 0), (-0.4, -1.5), $u_"out"$)
joham.voltage((0.8, 1.5), (0.8, -1.5), $u_"ges"$, anchor: "west")
wire((-0.6, 1.5),(0.8, 1.5))
wire((-0.6, -1.5),(0.8, -1.5))
wire((-0.6, 0),(0,0))
}),
zap.circuit({
import zap : *
node("N1", (0,0))
node("N2", (0,2))
resistor("R1", "N1", "N2", label: (content: $R_1$, distance: 0.1), scale: 0.7, fill: none, i: (content: $i_"out"$, anchor: "south", distance: 0.25, invert: true))
resistor("R2", (1,0), (1,2), label: (content: $R_2$, distance: 0.1), scale: 0.7, fill: none)
wire((-1, 0), "R2.in")
wire((-1, 2), "R2.out")
wire((-1, 2), "N2", i: (content: $i_"ges"$, distance: 0.1))
}),
$ u_"out" = R_1/(R_1 + R_2) $, $ i_"out" = R_2/(R_1 + R_2) i_"ges" $,
$ u_"out" = G_2/(G_1 + G_2) $, $ i_"out" = G_1/(G_1 + G_2) i_"ges" $
)
]
// Lineare Quelle
#bgBlock(fill: colorEineTore)[
@@ -440,6 +494,7 @@
$L --> C^d = L / R_d^2$,
$"KS" --> "LL"$, $"LL" -> "KS"$,
$"Nullator" --> "Nullator"$, $"Norator" -> "Norator"$,
$"Parallel" --> "Seriell"$,
$"Seriell" --> "Parallel"$,
@@ -447,20 +502,10 @@
table.cell(colspan: 2)[
*Dualwandlung: Steurende & Ausgangs Größe*
$
"VCVS": u_"out" = mu dot u_"in" &--> i_"out" R_d = mu dot i_"in" R_d \
"VCCS": u_"out" = g dot i_"in" &--> \
"CCVS": u_"out" = r dot i_"in" &--> \
"CCCS": i_"out" = beta dot i_"in" &--> \
$
],
table.cell(colspan: 2)[
*Dualwandlung: Nur Ausgangs Größe*
$
"VCVS": u_"out" = mu dot u_"in" &--> \
"VCCS": u_"out" = g dot i_"in" &--> \
"CCVS": u_"out" = r dot i_"in" &--> \
"CCCS": i_"out" = beta dot i_"in" &--> \
"VCVS" &: u_"out" &= mu dot u_"in" &--> i_"out"^d = mu i_"in"^d\
"VCCS" &: i_"out" &= g dot u_"in" &--> u_"out"^d = g R_d^2 dot i_"in"^d \
"CCVS" &: u_"out" &= r dot i_"in" &--> i_"out"^d = r/R_d^2 dot u_"in"\
"CCCS" &: i_"out" &= beta dot i_"in" &--> u_"out"^d = beta u_"in"^d\
$
],
@@ -476,8 +521,6 @@
2. Ableitung $g_cal(F)(u)$/$r_cal(F)(i)$ bilden \ $g'_cal(F)(u)$/$r'_cal(F)(i)$
#colbreak()
#line(length: 100%)
*Stromgesteuert* $quad r(i) = u$
@@ -521,7 +564,7 @@
#linebreak()
*Klein-Signal* $quad i_"lin" = g_"lin" (u) = g'(u_"AP")u$
*Klein-Signal* $quad Delta i_"lin" = g_"lin" (Delta u) = g'(u_"AP") Delta u$
#line(length: 100%)
@@ -562,7 +605,7 @@
})),
);
#linebreak()
*Klein-Signal* $quad u_"lin" = r_"lin" (i) = r'(i_"AP")i$
*Klein-Signal* $quad Delta u_"lin" = r_"lin" (Delta i) = r'(i_"AP") Delta i$
]
// Graphen und Matrizen
@@ -984,6 +1027,26 @@
)
],
// KS/LL
#bgBlock(fill: colorZweiTore)[
#subHeading(fill: colorZweiTore)[Kurzschluss/Leerlauf Methode für Zweitore]
1. Was sind die Input-Größen/Output-Größen? \
$u"-gesteuert:" u = 0 &--> "Kurzschluss (KS)" \
i"-gesteuert:" i = 0 &--> "Leerlauf (LL)"$ \
2. Steuernde Größe 1 beschalten KS/LL \
$-->$ BEIDE gesteuerten Größe errechen \
(2. Spalte der Matrix)
3. Steuernde Größe 2 beschalten KS/LL \
$-->$ BEIDE gesteuerten Größe errechen \
(1. Spalte der Matrix)
4. Matrix/Gleichung aufstellen \
Addition der Beschreibung pro gesteuerter Größe \
(Superpositions Prinzip)
]
// Linearsierung (N-Tore)
#bgBlock(fill: colorAnalyseVerfahren)[
#subHeading(fill: colorAnalyseVerfahren)[Linearisierung (N-Tore)]
@@ -1406,30 +1469,227 @@
*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 $jVec(i)_q$)
#table(
columns: (1fr, 1fr),
fill: (x, y) => if calc.rem(y, 2) == 0 { tableFillHigh } else { tableFillLow },
grid(
columns: 3,
column-gutter: 5mm,
row-gutter: 3mm,
zap.circuit({
import zap : *
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)
resistor("G", (0,1.5), (1,1.5), scale: 0.4, fill: none, label: (content: $G$, distance: 0.6mm, anchor: "south"))
wire("V.in", (1,0))
}),
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"))
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))
})
resistor("G", (1,0), (1,1.5), n: "*-*", scale: 0.4, fill: none, label: (content: $G$, distance: 0.6mm))
wire("V.in", (1.5,0))
wire("V.out", (1.5,1.5))
}),
),
grid(
columns: 3,
column-gutter: 5mm,
row-gutter: 3mm,
zap.circuit({
import zap : *
vsource("V", (0,0), (0,1.5), scale: 0.4, fill: none)
joham.voltage((-0.3,1), (-0.3,0), $U_0$)
wire("V.in", (1,0))
wire("V.out", (1,1.5))
}),
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", (1,0), (1,1.5), n: "*-*", scale: 0.4, fill: none, label: (content: $G$, distance: 0.6mm))
resistor("-G", (2.3,1.5), (1,1.5), scale: 0.4, fill: none, label: (content: $-G$, distance: 0.6mm))
wire("V.in", (2.3,0))
wire("V.out", "G.out")
}),
),
grid(
columns: 3,
column-gutter: 5mm,
row-gutter: 3mm,
zap.circuit({
import zap : *
vsource("V", (0,0), (0,1.5), scale: 0.4, fill: none)
joham.voltage((-0.3,1), (-0.3,0), $U_0$)
wire("V.in", (1,0))
wire("V.out", (1,1.5))
}),
align(center+horizon, $==>$),
zap.circuit({
import zap : *
joham.gyrator("G", (0,0), scale: 0.4, constant: $G_d$, invert: true)
joham.ground("GND", "G.12")
node("A", "G.12")
node("B", (-0.75,0.67), label: (content: $alpha$, distance: 0.1))
isource("V", "G.12", "G.11", scale: 0.4, fill: none, i: (content: $G_d U_0 space$, invert: true, distance: 0.3, anchor: "east"))
}),
),
[*Übertrager/NIK* siehe Zweitor-Tabelle],
scale(90%, grid(
columns: (auto, 0mm, 1fr),
column-gutter: 5mm,
row-gutter: 3mm,
[
*VCVS + 1 Knoten* \
#zap.circuit({
import zap : vsource, node, disource, dvsource, wire
import cetz.draw : line, rect, mark, content
node("A", (0,0), fill: false)
node("B", (0,1), fill: false)
node("C", (0.8,0), fill: false)
node("D", (0.8,1), fill: false)
joham.voltage((0, 1), (0, 0), $u_"in"$, anchor: "west")
dvsource("S", (0.8,0), (0.8,1), fill: none, scale: 0.4)
joham.voltage((1.1, 1), (1.1, 0), $u_"out"$, anchor: "west")
})
$u_"out" = mu u_"in"$
],
align(center+horizon, $==>$),
[
#zap.circuit({
import zap : vsource, node, disource, dvsource, wire
import cetz.draw : line, rect, mark, content
joham.gyrator("G0", (3,0), scale: 0.4, invert: true, constant: $G_d$)
disource("S", "G0.11", "G0.12", scale: 0.4, fill: none, i: (content: $i_"out"$, distance: 0.3, anchor: "west", invert: true))
node("N4", "G0.11", label: (content: $beta$, distance: 0.1),)
node("N3", "G0.12",)
joham.ground("GND1", "N3")
node("A", (1, 0.67), fill: false)
node("B", (1, -0.67), fill: false)
joham.voltage((1, 0.7), (1, -0.7), $u_"in"$, anchor: "west")
})
$i_"out" = mu G_d u_"in"$
],
)),
scale(90%, grid(
columns: (auto, 0mm, 1fr),
column-gutter: 5mm,
row-gutter: 3mm,
[
*CCCS + 1 Knoten* \
#zap.circuit({
import zap : vsource, node, disource, dvsource, wire
import cetz.draw : line, rect, mark, content
node("A", (0,0), fill: false)
node("B", (0,1), fill: false)
node("C", (0.8,0), fill: false)
node("D", (0.8,1), fill: false)
wire((0,0), (0,1), i: (content: $i_"in"$, invert: true, anchor: "east", distance: 0.1), size: 0.2)
disource("S", (0.8,0), (0.8,1), fill: none, scale: 0.4)
line((0.8,0.15), (0.8,0.1), mark: (end: ">", scale: 0.2, fill: black))
content((0.9, 0.15), $i_"out"$, anchor: "west")
})
$i_"out" = beta i_"in"$
],
align(center+horizon, $==>$),
[
#zap.circuit({
import zap : vsource, node, disource, dvsource, wire
import cetz.draw : line, rect, mark, content
joham.gyrator("G1", (0,0), scale: 0.4, constant: $G_d$)
node("A", (1.8,-0.67))
node("B", (1.8,0.67))
disource("S", "A", "B", scale: 0.4, fill: none, i: (content: $i_"out"$, distance: 0.2, label-distance: -0.4, anchor: "east", invert: true))
node("N1", "G1.21", label: (content: $alpha$, distance: 0.1),)
node("N2", "G1.22",)
joham.ground("GND1", "N2")
joham.voltage("G1.21", "G1.22", $u_"in"$)
})
$i_"out" = beta G_d u_in$
],
)),
table.cell(
align(center+horizon, grid(
columns: (auto, 0mm, auto),
column-gutter: 5mm,
row-gutter: 3mm,
[
*CCVS + 2 Knoten* \
#zap.circuit({
import zap : vsource, node, disource, dvsource, wire
import cetz.draw : line, rect, mark, content
node("A", (0,0), fill: false)
node("B", (0,1), fill: false)
node("C", (0.8,0), fill: false)
node("D", (0.8,1), fill: false)
wire((0,0), (0,1), i: (content: $i_"in"$, invert: true, anchor: "east", distance: 0.1), size: 0.2)
dvsource("S", (0.8,0), (0.8,1), fill: none, scale: 0.4)
joham.voltage((1.1, 1), (1.1, 0), $u_"out"$, anchor: "west")
})
$u_"out" = r i_"in"$
],
align(center+horizon, $==>$),
[
#zap.circuit({
import zap : vsource, node, disource, dvsource, wire
import cetz.draw : line, rect, mark, content
joham.gyrator("G1", (0,0), scale: 0.4, constant: $G_d$)
joham.gyrator("G0", (3,0), scale: 0.4, invert: true, constant: $G_d$)
disource("S", "G0.11", "G0.12", scale: 0.4, fill: none, i: (content: $i_"out"$, distance: 0.3, anchor: "west", invert: true))
node("N1", "G1.21", label: (content: $alpha$, distance: 0.1),)
node("N4", "G0.11", label: (content: $beta$, distance: 0.1),)
node("N2", "G1.22",)
node("N3", "G0.12",)
joham.ground("GND1", "N3")
joham.ground("GND1", "N2")
joham.voltage("G1.21", "G1.22", $u_"in"$)
})
$i_"out" = r G_d^2 u_"in"$
],
)),
colspan: 2
)
)
#table(
columns: (1fr, 1fr),
fill: (x, y) => if calc.rem(y, 2) == 0 { tableFillHigh } else { tableFillLow },
@@ -2101,7 +2361,7 @@
disource("S", (0.8,0), (0.8,1), fill: none, scale: 0.4)
line((0.8,0.10), (0.8,0.09), mark: (end: ">", scale: 0.4, fill: black))
content((0.9, 0.15), $i_"out" = g i_"in"$, anchor: "west")
content((0.9, 0.15), $i_"out" = g u_"in"$, anchor: "west")
})], [], [
- NICHT Verlustlos
@@ -2490,7 +2750,6 @@
inset: (bottom: 4mm, top: 4mm),
gutter: 0.1mm,
fill: (x, y) => if x != 0 and calc.rem(x, 2) == 0 { rgb("#c5c5c5") } else { white },
[],
$bold(R) jVec(i) = jVec(u)$,
$bold(G) jVec(u) = jVec(i)$,
@@ -2498,5 +2757,19 @@
$bold(H') vec(u_1, i_2) = vec(i_1, u_2)$,
$bold(A) vec(u_2, -i_2) = vec(i_1, u_1)$,
$bold(A') vec(u_1, -i_1) = vec(i_2, u_2)$,
[],
[Stromge-steuert],
[Spannung-gesteuert],
[Hybrid-beschreibung],
[Inverse-Hybrid],
[Ketten-Beschreibung],
[Inverse-Ketten],
[],
$jVec(r)vec(i_1, i_2) = vec(u_1, u_2)$,
$jVec(g)vec(u_1, u_2) = vec(i_1, i_2)$,
$jVec(h) vec(i_1, u_2) = vec(u_1, i_2)$,
$jVec(h') vec(u_1, i_2) = vec(i_1, u_2)$,
$jVec(a) vec(u_2, -i_2) = vec(i_1, u_1)$,
$jVec(a') vec(u_1, -i_1) = vec(i_2, u_2)$,
)
]