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Author SHA1 Message Date
alexander
83aa6764fe Merge branch 'main' of gitea.mintcalc.com:alexander/TUM-Formelsammlungen
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2026-02-02 14:46:23 +01:00
alexander
ad2c7f2919 started table 2026-02-02 14:45:59 +01:00
2 changed files with 303 additions and 193 deletions
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482
src/cheatsheets/Schaltungstheorie.typ
View File

@@ -1260,11 +1260,6 @@
)
]
#bgBlock(fill: colorAllgemein)[
#subHeading(fill: colorAllgemein)[Complex Zahlen]
]
// Complex AC
#bgBlock(fill: colorComplexAC)[
#subHeading(fill: colorComplexAC)[Komplex Wechselstrom Rechnnung]
@@ -1310,6 +1305,8 @@
$P_w = U_m^2 / 2R = (I_m^2 R)/2$
$P = 1/2 U I^* = 1/2 abs(U)^2 Y^* = 1/2 abs(I)^2 Z^*$
$U_"eff" = U_m/sqrt(2), I_"eff" = I_m / sqrt(2)$
]
@@ -1331,192 +1328,303 @@
#pagebreak()
#bgBlock(fill: colorZweiTore, width: 100%)[
#subHeading(fill: colorZweiTore)[Zwei-Tor-Übersichts]
#table(
fill: (x, y) => if calc.rem(y, 2) == 0 { tableFillHigh } else { tableFillLow },
columns: (auto, auto, auto, 1fr, 1fr, 1fr),
[*Name*],
[*Schaltbild*],
[*Ersatz-Schaltbild*],
[*Eigenschaften*],
[*Beschreibung*],
[*Knotenspannungs Analyse*],
[Nullor],
[],
[],
[],
[$A = mat(0, 0; 0, 0)$],
[],
[OpAmp \ lin],
[],
[],
[],
[],
[],
[OpAmp \ $U_"sat+"$],
[],
[],
[],
[],
[],
[OpAmp \ $U_"sat-"$],
[],
[],
[],
[],
[],
[VCVS],
[],
[],
[],
[$H' = mat(0, 0; mu, 0) quad A = mat(1/mu 0; 0, 0)$],
[],
[VCCS],
[],
[],
[],
[$G = mat(0, 0; g, 0) quad A = mat(0, -1/g; 0, 0)$],
[],
[CCVS],
[],
[],
[],
[$R = mat(0, 0, r, 0) quad A = mat(0, 0; 1/r, 0)$],
[],
[CCCS],
[],
[],
[],
[$H = mat(0, 0; beta, 0) quad A = mat(0, 0; 0, -1/beta)$],
[],
[Übertrager],
[],
[],
[],
[],
[],
[Gyrator],
[],
[],
[
- Antireziprok, Antisymetrisch
- Auch Positiv-Immitanz-Inverter
],
[$R = mat(0, -R_d; R_d, 0) quad G = mat(0, G_d; -G_d, 0) \ A = mat(0, R_d; 1/R_d, 0) quad A' = mat(0, -R_d; -1/R_d, 0)$],
[],
[NIK],
[],
[],
[],
[
- Akitv
- Antireziprok
- Symetrisch für $abs(k) = 1$
],
[$H = mat(0, -k; -k, 0) quad H' = mat(0, -1/k; -1/k, 0); A = mat(-k, 0; 0, 1/k) quad A'= mat(-1/k, 0; 0, k)$],
[T-Glied],
[],
[],
[],
[
],
[],
[$pi$-Glied],
[],
[],
[
]
)
]
// Tor Eigenschaften
#place(
bottom, float: true, scope: "parent",
bgBlock(fill: colorEigenschaften, width: 100%)[
#subHeading(fill: colorEigenschaften)[Tor Eigenschaften]
#bgBlock(fill: colorEigenschaften, width: 100%)[
#subHeading(fill: colorEigenschaften)[Tor Eigenschaften]
#table(
columns: (auto, auto, auto, auto),
inset: 2mm,
align: horizon,
fill: (x, y) => if calc.rem(y, 2) == 1 { rgb("#c5c5c5") } else { white },
table.header([], [*Ein-Tor*], [*Zwei-Tor*], [*Reaktive Elemente*]),
[*passiv*\ (nimmt Energie auf)\ $not$aktiv],
[$forall (u,i) in cal(F): u dot i >= 0$],
[
$jMat(U)^T jMat(I) + jMat(I)^T jMat(U)$\
$forall vec(jVec(u),jVec(v)) in cal(F) : jVec(u)^T jVec(i) >=0$
],
[],
[*verlustlos*],
[
$forall (u,i) in cal(F): u dot i = 0$\
Kennline nur $u\/i$-Achsen
],
[$forall vec(jVec(u),jVec(v)) in cal(F) : jVec(u)^T jVec(i) = 0$],
[
$u\/q$-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],
[
Darstellbar: Matrix $+$ Aufpunkt\
$lambda_1 vec(jVec(u)_1, jVec(i)_1) + lambda_2 vec(jVec(u)_2, jVec(i)_2) in cal(F)$
],
[],
[*quellenfrei*],
[$(qty("0", "A"), qty("0", "V")) in cal(F)$],
[$(qty("0", "A"), qty("0", "V")) in cal(F)$],
[$(qty("0", "A"), qty("0", "V")) in cal(F)$],
[*streng linear*],
[linear UND quellenfrei],
[linear UND quellenfrei\ Darstellbar: Nur Matrix],
[],
[*ungepolt* \ (Punkt sym.)],
[$(u,i) in cal(F) <=> (-u, -i) in cal(F)\
g(u) = i, r(i) = u
$],
[
N/A
],
[],
[*symetrisch*\ $<=>$ Umkehrbar],
[N/A],
[
$jMat(A) = jMat(A')$\
$jMat(G) = jMat(P) jMat(G) jMat(P), space jMat(R) = jMat(P) jMat(R) jMat(P), quad jMat(P) = mat(0, 1; 1, 0) \
det(H) = 1, $
],
[],
[*Reziprok*],
[Immer Reziprok],
[
$cal(F)$ symetrisch $=> cal(F)$ reziprok
$jMat(U)^T jMat(I) - jMat(I)^T jMat(U) = 0 \
jMat(R)^T = jMat(R), quad jMat(G)^T = jMat(G) quad h_21 = -h_12 \ det(jMat(A)) = 1 quad det(jMat(A')) = 1 quad h'_21 = -h'_12$],
[],
[*$x$-gesteudert*], [Existiert $r(i) = u \/g(u) = i$], [Existiert die Matrix? siehe Tabelle],
[],
[Alle Beschreibung],
[Klar],
[$det(M) != 0$, Alle Eintrag $!= 0$]
)
]
#bgBlock(fill: colorZweiTore)[
#set text(size: 10pt)
#subHeading(fill: colorZweiTore)[Umrechnung Zweitormatrizen]
#table(
columns: (12mm, 1fr, 1fr, 1fr, 1fr, 1fr, 1fr),
align: center,
inset: (bottom: 4mm, top: 4mm),
gutter: 0.1mm,
fill: (x, y) => if x != 0 and calc.rem(x, 2) == 0 { rgb("#c5c5c5") } else { white },
table.cell(
inset: 0mm,
[
#cetz.canvas(length: 12mm,{
import cetz.draw : *
line((1,0), (0,1))
content((0.309, 0.25), "Out")
content((0.75, 0.75), "In")
})
]),
$bold(R)$,
$bold(G)$,
$bold(H)$,
$bold(H')$,
$bold(A)$,
$bold(A')$,
$bold(R)$,
$mat(r_11, r_12; r_21, r_22)$,
$jMat(G^(-1) =) 1/det(bold(G)) mat(g_22, -g_12; -g_21, g_11)$,
$1/h_22 mat(det(bold(H)), h_12; -h_21, 1)$,
$1/h'_11 mat(1, -h'_12; h'_21, det(bold(H')))$,
$1/a_21 mat(a_11, det(bold(A)); 1, a_22)$,
$1/a'_21 mat(a'_22, 1; det(bold(A')), a'_11)$,
$bold(G)$,
$jMat(R^(-1) =) 1/det(bold(R)) mat(r_22, -r_12; -r_21, r_11)$,
$mat(g_11, g_12; g_21, g_22)$,
$1/h_11 mat(1, -h_12; h_21, det(bold(H)))$,
$1/h'_22 mat(det(bold(H')), h'_12; -h'_21, 1)$,
$1/a_12 mat(a_22, -det(bold(A)); -1, a_11)$,
$1/a'_12 mat(a'_11, -1; -det(bold(A')), a'_22)$,
$bold(H)$,
$1/r_22 mat(det(bold(R)), r_12; -r_21, 1)$,
$1/g_11 mat(1, -g_12; g_21, det(bold(G)))$,
$mat(h_11, h_12; h_21, h_22)$,
$jMat(H')^(-1)= 1/det(bold(H')) mat(h'_22, -h'_12; -h'_21, h'_11)$,
$1/a_22 mat(a_12, det(bold(A)); -1, a_21)$,
$1/a'_11 mat(a'_12, 1; -det(bold(A')), a'_21)$,
$bold(H')$,
$1/r_11 mat(1, -r_12; r_21, det(bold(R)))$,
$1/g_22 mat(det(bold(G)), g_12; -g_21, 1)$,
$jMat(H^(-1))= 1/det(bold(H)) mat(h_22, -h_12; -h_21, h_11)$,
$mat(h'_11, h'_12; h'_21, h'_22)$,
$1/a_11 mat(a_21, -det(bold(A)); 1, a_12)$,
$1/a'_22 mat(a'_21, -1; det(bold(A')), a'_12)$,
$bold(A)$,
$1/r_21 mat(r_11, det(bold(R)); 1, r_22)$,
$1/g_21 mat(-g_22, -1; -det(bold(G)), -g_11)$,
$1/h_21 mat(-det(bold(H)), -h_11; -h_22, -1)$,
$1/h'_21 mat(1, h'_22; h'_11, det(bold(H')))$,
$mat(a_11, a_12; a_21, a_22)$,
$jMat(A'^(-1))= 1/det(bold(A')) mat(a'_22, a'_12; a'_21, a'_11)$,
$bold(A')$,
$1/r_12 mat(r_22, det(bold(R)); 1, r_11)$,
$1/g_12 mat(-g_11, -1; -det(bold(G)), -g_22)$,
$1/h_12 mat(1, h_11; h_22, det(bold(H)))$,
$1/h'_12 mat(-det(bold(H')), -h'_22; -h'_11, -1)$,
$jMat(A^(-1))= 1/det(bold(A)) mat(a_22, a_12; a_21, a_11)$,
$mat(a'_11, a'_12; a'_21, a'_22)$,
)
#table(
columns: (auto, auto, auto, auto),
inset: 2mm,
align: horizon,
fill: (x, y) => if calc.rem(y, 2) == 1 { rgb("#c5c5c5") } else { white },
columns: (12mm, 1fr, 1fr, 1fr, 1fr, 1fr, 1fr),
align: center,
inset: (bottom: 4mm, top: 4mm),
gutter: 0.1mm,
fill: (x, y) => if x != 0 and calc.rem(x, 2) == 0 { rgb("#c5c5c5") } else { white },
table.header([], [*Ein-Tor*], [*Zwei-Tor*], [*Reaktive Elemente*]),
[*passiv*\ (nimmt Energie auf)\ $not$aktiv],
[$forall (u,i) in cal(F): u dot i >= 0$],
[
$jMat(U)^T jMat(I) + jMat(I)^T jMat(U)$\
$forall vec(jVec(u),jVec(v)) in cal(F) : jVec(u)^T jVec(i) >=0$
],
[],
[*verlustlos*],
[
$forall (u,i) in cal(F): u dot i = 0$\
Kennline nur $u\/i$-Achsen
],
[$forall vec(jVec(u),jVec(v)) in cal(F) : jVec(u)^T jVec(i) = 0$],
[
$u\/q$-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],
[
Darstellbar: Matrix $+$ Aufpunkt\
$lambda_1 vec(jVec(u)_1, jVec(i)_1) + lambda_2 vec(jVec(u)_2, jVec(i)_2) in cal(F)$
],
[],
[*quellenfrei*],
[$(qty("0", "A"), qty("0", "V")) in cal(F)$],
[$(qty("0", "A"), qty("0", "V")) in cal(F)$],
[$(qty("0", "A"), qty("0", "V")) in cal(F)$],
[*streng linear*],
[linear UND quellenfrei],
[linear UND quellenfrei\ Darstellbar: Nur Matrix],
[],
[*ungepolt* \ (Punkt sym.)],
[$(u,i) in cal(F) <=> (-u, -i) in cal(F)\
g(u) = i, r(i) = u
$],
[
N/A
],
[],
[*symetrisch*\ $<=>$ Umkehrbar],
[N/A],
[
$jMat(A) = jMat(A')$\
$jMat(G) = jMat(P) jMat(G) jMat(P), space jMat(R) = jMat(P) jMat(R) jMat(P), quad jMat(P) = mat(0, 1; 1, 0) \
det(H) = 1, $
],
[],
[*Reziprok*],
[Immer Reziprok],
[
$cal(F)$ symetrisch $=> cal(F)$ reziprok
$jMat(U)^T jMat(I) - jMat(I)^T jMat(U) = 0 \
jMat(R)^T = jMat(R), quad jMat(G)^T = jMat(G) quad h_21 = -h_12 \ det(jMat(A)) = 1 quad det(jMat(A')) = 1 quad h'_21 = -h'_12$],
[],
[*$x$-gesteudert*], [Existiert $r(i) = u \/g(u) = i$], [Existiert die Matrix? siehe Tabelle],
[],
[Alle Beschreibung],
[Klar],
[$det(M) != 0$, Alle Eintrag $!= 0$]
$bold(R) jVec(i) = jVec(u)$,
$bold(G) jVec(u) = jVec(i)$,
$bold(H) vec(i_1, u_2) = vec(u_1, i_2)$,
$bold(H') vec(u_1, i_2) = vec(i_1, u_2)$,
$bold(A) vec(u_2, -i_2) = vec(i_1, u_1)$,
$bold(A') vec(u_1, -i_1) = vec(i_2, u_2)$,
)
]
)
#place(bottom+left, scope: "parent", float: true)[
#bgBlock(fill: colorZweiTore)[
#set text(size: 10pt)
#subHeading(fill: colorZweiTore)[Umrechnung Zweitormatrizen]
#table(
columns: (12mm, 1fr, 1fr, 1fr, 1fr, 1fr, 1fr),
align: center,
inset: (bottom: 4mm, top: 4mm),
gutter: 0.1mm,
fill: (x, y) => if x != 0 and calc.rem(x, 2) == 0 { rgb("#c5c5c5") } else { white },
table.cell(
inset: 0mm,
[
#cetz.canvas(length: 12mm,{
import cetz.draw : *
line((1,0), (0,1))
content((0.309, 0.25), "Out")
content((0.75, 0.75), "In")
})
]),
$bold(R)$,
$bold(G)$,
$bold(H)$,
$bold(H')$,
$bold(A)$,
$bold(A')$,
$bold(R)$,
$mat(r_11, r_12; r_21, r_22)$,
$jMat(G^(-1) =) 1/det(bold(G)) mat(g_22, -g_12; -g_21, g_11)$,
$1/h_22 mat(det(bold(H)), h_12; -h_21, 1)$,
$1/h'_11 mat(1, -h'_12; h'_21, det(bold(H')))$,
$1/a_21 mat(a_11, det(bold(A)); 1, a_22)$,
$1/a'_21 mat(a'_22, 1; det(bold(A')), a'_11)$,
$bold(G)$,
$jMat(R^(-1) =) 1/det(bold(R)) mat(r_22, -r_12; -r_21, r_11)$,
$mat(g_11, g_12; g_21, g_22)$,
$1/h_11 mat(1, -h_12; h_21, det(bold(H)))$,
$1/h'_22 mat(det(bold(H')), h'_12; -h'_21, 1)$,
$1/a_12 mat(a_22, -det(bold(A)); -1, a_11)$,
$1/a'_12 mat(a'_11, -1; -det(bold(A')), a'_22)$,
$bold(H)$,
$1/r_22 mat(det(bold(R)), r_12; -r_21, 1)$,
$1/g_11 mat(1, -g_12; g_21, det(bold(G)))$,
$mat(h_11, h_12; h_21, h_22)$,
$jMat(H')^(-1)= 1/det(bold(H')) mat(h'_22, -h'_12; -h'_21, h'_11)$,
$1/a_22 mat(a_12, det(bold(A)); -1, a_21)$,
$1/a'_11 mat(a'_12, 1; -det(bold(A')), a'_21)$,
$bold(H')$,
$1/r_11 mat(1, -r_12; r_21, det(bold(R)))$,
$1/g_22 mat(det(bold(G)), g_12; -g_21, 1)$,
$jMat(H^(-1))= 1/det(bold(H)) mat(h_22, -h_12; -h_21, h_11)$,
$mat(h'_11, h'_12; h'_21, h'_22)$,
$1/a_11 mat(a_21, -det(bold(A)); 1, a_12)$,
$1/a'_22 mat(a'_21, -1; det(bold(A')), a'_12)$,
$bold(A)$,
$1/r_21 mat(r_11, det(bold(R)); 1, r_22)$,
$1/g_21 mat(-g_22, -1; -det(bold(G)), -g_11)$,
$1/h_21 mat(-det(bold(H)), -h_11; -h_22, -1)$,
$1/h'_21 mat(1, h'_22; h'_11, det(bold(H')))$,
$mat(a_11, a_12; a_21, a_22)$,
$jMat(A'^(-1))= 1/det(bold(A')) mat(a'_22, a'_12; a'_21, a'_11)$,
$bold(A')$,
$1/r_12 mat(r_22, det(bold(R)); 1, r_11)$,
$1/g_12 mat(-g_11, -1; -det(bold(G)), -g_22)$,
$1/h_12 mat(1, h_11; h_22, det(bold(H)))$,
$1/h'_12 mat(-det(bold(H')), -h'_22; -h'_11, -1)$,
$jMat(A^(-1))= 1/det(bold(A)) mat(a_22, a_12; a_21, a_11)$,
$mat(a'_11, a'_12; a'_21, a'_22)$,
)
#table(
columns: (12mm, 1fr, 1fr, 1fr, 1fr, 1fr, 1fr),
align: center,
inset: (bottom: 4mm, top: 4mm),
gutter: 0.1mm,
fill: (x, y) => if x != 0 and calc.rem(x, 2) == 0 { rgb("#c5c5c5") } else { white },
[],
$bold(R) jVec(i) = jVec(u)$,
$bold(G) jVec(u) = jVec(i)$,
$bold(H) vec(i_1, u_2) = vec(u_1, i_2)$,
$bold(H') vec(u_1, i_2) = vec(i_1, u_2)$,
$bold(A) vec(u_2, -i_2) = vec(i_1, u_1)$,
$bold(A') vec(u_1, -i_1) = vec(i_2, u_2)$,
)
]
]

14
src/lib/common_rewrite.typ
View File

@@ -1,4 +1,4 @@
#let bgBlock(body, fill: color, width: 100%) = block(body, fill:fill.lighten(80%), width: width, inset: (bottom: 2mm))
#let bgBlock(body, fill: color, width: 100%) = block(body, fill:fill.lighten(80%), width: width, inset: (bottom: 2mm, left: 2mm, right: 2mm,))
#let SeperatorLine = line(length: 100%, stroke: (paint: black, thickness: 0.3mm))
#let MathAlignLeft(e) = {
@@ -6,7 +6,7 @@
}
#let subHeading(body, fill: color) = {
box(
move(dx: -2mm, dy: 0mm, box(
align(
top+center,
text(
@@ -17,10 +17,10 @@
)
),
fill: fill,
width: 100%,
width: 100% + 4mm,
inset: 1mm,
height: auto
)
))
}
#let MathAlignLeft(e) = {
@@ -58,11 +58,13 @@
#let ComplexNumbersSection(i: $i$) = [
$1/#i = #i^(-1) = -#i quad quad #i^2=-1 quad quad sqrt(#i) = 1/sqrt(2) + 1/sqrt(2)#i$
$z in CC = a + b #i quad quad quad z = r dot e^(phi #i)$ \
$z in CC = a + b #i quad quad quad z = r dot e^(#i phi)$ \
$z_0 + z_1 = (a_0 + a_1) + (b_0 + b_1) #i$\
$z_0 dot z_1 = (a_1 a_2 - b_1 b_2) + #i (a_1b_2 + a_2 b_1) = r_0 r_1 e^(#i (phi_0 + phi_1))$\
$z^x = r^x dot e^(phi #i dot x) quad x in RR$ \
$z_0/z_1 = r_0/r_1 e^(#i (phi_0 - phi_1)) quad quad quad$
$z_0/z_1 = r_0/r_1 e^(#i (phi_0 - phi_1)) quad quad quad$
$z^* = a - #i b = r e^(-#i phi)$
$r = abs(z) quad phi = cases(
+ arccos(a/r) space : space b >= 0,