diff --git a/src/cheatsheets/Schaltungstheorie.typ b/src/cheatsheets/Schaltungstheorie.typ index 720899e..e2d3db3 100644 --- a/src/cheatsheets/Schaltungstheorie.typ +++ b/src/cheatsheets/Schaltungstheorie.typ @@ -920,11 +920,6 @@ ) ] - #bgBlock(fill: colorAllgemein)[ - #subHeading(fill: colorAllgemein)[Complex Zahlen] - - ] - // Complex AC #bgBlock(fill: colorComplexAC)[ #subHeading(fill: colorComplexAC)[Komplex Wechselstrom Rechnnung] @@ -970,6 +965,8 @@ $P_w = U_m^2 / 2R = (I_m^2 R)/2$ + $P = 1/2 U I^* = 1/2 abs(U)^2 Y^* = 1/2 abs(I)^2 Z^*$ + $U_"eff" = U_m/sqrt(2), I_"eff" = I_m / sqrt(2)$ ] @@ -991,192 +988,303 @@ #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 }, - - [], - $bold(R) jVec(i) = jVec(u)$, - $bold(G) jVec(u) = jVec(i)$, - $bold(H) vec(i_1, u_2) = vec(u_1, i_2)$, - $bold(H') vec(u_1, i_2) = vec(i_1, u_2)$, - $bold(A) vec(u_2, -i_2) = vec(i_1, u_1)$, - $bold(A') vec(u_1, -i_1) = vec(i_2, u_2)$, - - ) - ] ] diff --git a/src/lib/common_rewrite.typ b/src/lib/common_rewrite.typ index a501c8a..51e10d3 100644 --- a/src/lib/common_rewrite.typ +++ b/src/lib/common_rewrite.typ @@ -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,