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Schaltungstheorie brrrrrrr
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alexander
2026-01-25 01:43:59 +01:00
parent d56fe69e9d
commit 1c7b4decdb
3 changed files with 632 additions and 20 deletions
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src/cheatsheets/Schaltungstheorie.typ
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@@ -1,10 +1,17 @@
#import "../lib/common_rewrite.typ" : *
#import "@preview/mannot:0.3.1"
#import "@preview/zap:0.5.0"
#import "@preview/cetz:0.4.2"
#import "../lib/circuit.typ" : *
#import "@preview/unify:0.7.1": num,qty,numrange,qtyrange
#set math.mat(delim: "[")
#show math.equation.where(block: true): it => math.inline(it)
#set math.mat(delim: "[")
#show math.integral: it => math.limits(math.integral)
#show math.sum: it => math.limits(math.sum)
#let jVec(x) = $bold(underline(#x))$
#let jMat(x) = $bold(#x)$
#set page(
paper: "a4",
@@ -20,13 +27,14 @@
align: center,
columns: (1fr, 1fr, 1fr),
[#align(left, datetime.today().display("[day].[month].[year]"))],
[#align(center, counter(page).display("- 1 -"))],
[#align(center, counter(page).display("– 1 –"))],
[#align(right, image("../images/cc0.png", height: 5mm,))]
)
],
)
#let colorAllgemein = color.hsl(105.13deg, 92.13%, 75.1%)
#let colorEigenschaften = color.hsl(145.13deg, 92.13%, 75.1%)
#let colorEineTore = color.hsl(202.05deg, 92.13%, 75.1%)
#let colorZweiTore = color.hsl(235.9deg, 92.13%, 75.1%)
#let colorAnalyseVerfahren = color.hsl(280deg, 92.13%, 75.1%)
@@ -37,22 +45,151 @@
[Schaltungstheorie]
))
#columns(4, gutter: 2mm)[
#bgBlock(fill: colorAllgemein)[
#subHeading(fill: colorAllgemein)[Allgemeine]
*Konzentriertheitshypotese*\
$lambda >> d quad quad lambda = c/f quad quad c approx 3.10dot 10^8 m/s$
d: größe Bauteil Dimension
f: Maximal Frequenze
*Kirchhoff*
#grid(columns: (1fr, 1fr), row-gutter: 5mm, [
KCL: $sum_(k=1)^n i_k =0$ (Knotenregel)
], [
#cetz.canvas(length: 8mm, {
import cetz.draw: *
circle((0, 0),radius: 1)
line((angle: 0deg, radius: 0.2), (angle: 0deg, radius: 1.5), stroke: red)
line((angle: 120deg, radius: 1.5), (angle: 120deg, radius: 0.2), stroke: red)
line((angle: 240deg, radius: 1.5), (angle: 240deg, radius: 0.2), stroke: red)
set-style(mark: (end: ("straight")))
line((angle: 0deg, radius: 0.2), (angle: 0deg, radius: 0.8), stroke: red)
line((angle: 120deg, radius: 1.2), (angle: 120deg, radius: 0.4), stroke: red)
line((angle: 240deg, radius: 1.2), (angle: 240deg, radius: 0.4), stroke: red)
})
], [
KVL: $sum_(k=1)^n u_k =0$ (Maschenregel)
], [
#zap.circuit({
import zap: wire
import cetz.draw: *
registerAllCustom();
einTor("F1", (0, -1), (0, 1))
einTor("F2", (1.5, 0), (1.5, 1), flip: true)
einTor("F3", (1.5, 0), (1.5, -1), flip: true)
wire((0, -1), (1.5, -1))
wire((0, 1), (1.5, 1))
translate((0.75, 0))
circle((0, 0), radius: 4mm, stroke: blue)
translate((0, 1mm))
mark((angle: 180deg, radius: 4mm), 90deg, symbol: "straight", stroke: blue, scale: 0.75)
translate((0, -2mm))
mark((angle: 0deg, radius: 4mm), 270deg, symbol: "straight", stroke: blue, scale: 0.75)
})
])
]
#bgBlock(fill: colorEineTore)[
#subHeading(fill: colorEineTore)[Quelle Wandlung]
#zap.circuit({
import zap: *
set-style(scale: (x: 0.75, y:0.75), fill: none)
resistor("R1", (-2, 0), (0, 0))
vsource("V1", (-2, 0), (-2, -2))
wire((-2, -2), (0, -2))
node("n1", (0, 0), label: "1")
node("n2", (0, -2), label: "2")
})
#grid(
columns: (auto, auto),
column-gutter: 3mm,
zap.circuit({
import zap: *
registerAllCustom();
einTor("F1", (0,0), (2,0), i: (content: $i_(cal(F),1)$, anchor: "south-east", label-distance: -10pt),
u: $u_1$
)
einTor("F1", (0,0), (2,0), i: (content: $i_(cal(F),2)$, anchor: "west", label-distance: -10pt, invert: true))
}),
[
Tor-Bedinung:\
$i_(cal(F),2) = i_(cal(F),1)$
],
)
]
#colbreak()
#bgBlock(fill: colorEineTore)[
#subHeading(fill: colorEineTore)[Quelle Wandlung]
#grid(
columns: (1fr, 1fr),
align: center,
scale(x: 65%, y: 65%,
zap.circuit({
import zap : *
import cetz.draw : line, content
vsource("U0", (0, 1.5), (0, -1.5), fill: none)
node("n2", (3.5,1.5), fill: false)
node("n3", (3.5,-1.5), fill: false)
resistor("R0", "U0.in", "n2", fill: none, label: (content: $R_i$, anchor: "south", distance: 0.1))
wire("U0.out", "n3")
wire((2.49,1.5), "n2", i: (content: $i$, anchor: "south", invert: true))
set-style(mark: (end: ">", fill: black))
line((3.5, 1.2), (3.5, -1.2), stroke: 0.5pt)
content((3.9, 0), $u$)
line((0.7, 0.5), (0.7, -0.5), stroke: 0.5pt)
content((1.7, 0), $U_0 = I_0 / G_i$)
})),
scale(x: 65%, y: 65%,
zap.circuit({
import zap : *
import cetz.draw : line, content
isource("I0", (0, 1.5), (0, -1.5), fill: none)
node("n0", (2,1.5))
node("n1", (2,-1.5))
node("n2", (3.5,1.5), fill: false)
node("n3", (3.5,-1.5), fill: false)
resistor("g0", "n1", "n0", fill: none, label: (content: $G_i$, anchor: "south", distance: 0.2))
wire("I0.in", "n0", "n2")
wire("I0.out", "n1", "n3")
wire("n0", "n2", i: (content: $i$, anchor: "south", invert: true))
wire((0,0.6), "I0.in", i: (content: $I_0 = U_0 / R_i$, anchor: "east", invert: false, distance: 0.2))
set-style(mark: (end: ">", fill: black))
line((3.5, 1.2), (3.5, -1.2), stroke: 0.5pt)
content((3.9, 0), $u$)
})),
[
$U_0 = I_0 R_i$
$R_i=1/G_i$
$r(i) = R_i i + U_0$
],
[
$I_0 = U_0 G_i$
$G_i=1/R_i$
$g(u) = G_i u + I_0$
]
);
]
// Graphen und Matrizen
#bgBlock(fill: colorAnalyseVerfahren)[
#subHeading(fill: colorAnalyseVerfahren)[Graphen und Matrizen]
@@ -126,6 +263,7 @@
$bold(u_b^T i_b) = 0$
]
// Baumkonzept
#bgBlock(fill: colorAnalyseVerfahren)[
#subHeading(fill: colorAnalyseVerfahren)[Baumkonzept]
KCLs: $n-1$\
@@ -134,6 +272,7 @@
Baum einzeichnen (Keine Schleifen!)
]
// Machenstrom-/Knotenpotenzial-Analyse
#bgBlock(fill: colorAnalyseVerfahren)[
#subHeading(fill: colorAnalyseVerfahren)[Machenstrom-/Knotenpotenzial-Analyse]
]
@@ -141,29 +280,399 @@
#bgBlock(fill: colorAnalyseVerfahren)[
#subHeading(fill: colorAnalyseVerfahren)[Reduzierte Knotenpotenzial-Analyse]
]
#bgBlock(fill: colorZweiTore)[
#subHeading(fill: colorZweiTore)[Zweitor Verschaltung]
#grid(
columns: (auto, auto),
gutter: 5mm,
[
Parallel-Parallel \
$jMat(G)_"ges" = jMat(G_1) + jMat(G_2)$
],
[
#zap.circuit({
import zap : *
registerAllCustom();
zweiTor("G1", (0,-6mm), $jMat(G_1)$)
zweiTor("G2", (0,6mm), $jMat(G_2)$)
zwire("G1.in1", (-8mm, -3.5mm), (-8mm, 8.5mm), "G2.in1")
zwire("G1.in0", (-10mm, -8.5mm), (-10mm, 3.5mm), "G2.in0")
node("n0", (13mm, 8.5mm), fill: false)
node("n1", (13mm, -8.5mm), fill: false)
node("n00", (10mm, 8.5mm))
node("n11", (8mm, -8.5mm))
wire("n0", "n00")
wire("n1", "n11")
zwire("G1.out1", (10mm, -3.5mm), (10mm, 8.5mm), "G2.out1")
zwire("G1.out0", (8mm, -8.5mm), (8mm, 3.5mm), "G2.out0")
node("n2", (-13mm, 8.5mm), fill: false)
node("n3", (-13mm, -8.5mm), fill: false)
node("n22", (-8mm, 8.5mm))
node("n33", (-10mm, -8.5mm))
wire("n2", "n22")
wire("n3", "n33")
})
],
[
Seriel-Seriel \
$jMat(G)_"ges" = jMat(G_1) + jMat(G_2)$
],
[
#zap.circuit({
import zap : *
registerAllCustom();
zweiTor("G1", (0,-6mm), $jMat(R_1)$)
zweiTor("G2", (0,6mm), $jMat(R_2)$)
node("n0", (13mm, 8.5mm), fill: false)
node("n1", (13mm, -8.5mm), fill: false)
node("n2", (-13mm, 8.5mm), fill: false)
node("n3", (-13mm, -8.5mm), fill: false)
zwire("G1.in1", (-8mm, -3.5mm), (-8mm, 3.5mm), "G2.in0")
zwire("G1.out1", (8mm, -3.5mm), (8mm, 3.5mm), "G2.out0")
zwire("G1.in0", "n3")
zwire("G1.out0", "n1")
zwire("G2.out1", "n0")
zwire("G2.in1", "n2")
})
],
[
Seriel-Parallel \
$jMat(H)_"ges" = jMat(H_1) + jMat(H_2)$
],
[
#zap.circuit({
import zap : *
registerAllCustom();
zweiTor("G1", (0,-6mm), $jMat(H_1)$)
zweiTor("G2", (0,6mm), $jMat(H_2)$)
node("n0", (13mm, 8.5mm), fill: false)
node("n1", (13mm, -8.5mm), fill: false)
node("n2", (-13mm, 8.5mm), fill: false)
node("n3", (-13mm, -8.5mm), fill: false)
zwire("G1.in1", (-8mm, -3.5mm), (-8mm, 3.5mm), "G2.in0")
zwire("G1.in0", "n3")
zwire("G2.in1", "n2")
zwire("G1.out1", (10mm, -3.5mm), (10mm, 8.5mm), "G2.out1")
zwire("G1.out0", (8mm, -8.5mm), (8mm, 3.5mm), "G2.out0")
node("n00", (10mm, 8.5mm))
node("n11", (8mm, -8.5mm))
wire("n0", "n00")
wire("n1", "n11")
})
],
[
Parallel-Seriel \
$jMat(H')_"ges" = jMat(H'_1) + jMat(H'_2)$
],
[
#zap.circuit({
import zap : *
registerAllCustom();
zweiTor("G1", (0,-6mm), $jMat(H'_1)$)
zweiTor("G2", (0,6mm), $jMat(H'_2)$)
node("n0", (13mm, 8.5mm), fill: false)
node("n1", (13mm, -8.5mm), fill: false)
node("n2", (-13mm, 8.5mm), fill: false)
node("n3", (-13mm, -8.5mm), fill: false)
zwire("G1.out1", (8mm, -3.5mm), (8mm, 3.5mm), "G2.out0")
zwire("G1.out0", "n1")
zwire("G2.out1", "n0")
zwire("G1.in1", (-8mm, -3.5mm), (-8mm, 8.5mm), "G2.in1")
zwire("G1.in0", (-10mm, -8.5mm), (-10mm, 3.5mm), "G2.in0")
node("n22", (-8mm, 8.5mm))
node("n33", (-10mm, -8.5mm))
wire("n2", "n22")
wire("n3", "n33")
})
],
[
Kette normal \
$jMat(A)_"ges" = jMat(A_1) dot jMat(A_2)$
],
[
#zap.circuit({
import zap : *
registerAllCustom();
zweiTor("A2", (6mm, 0), $jMat(A_2)$, height: 8mm, width: 8mm)
zweiTor("A1", (-6mm, 0), $jMat(A_1)$, height: 8mm, width: 8mm)
node("n0", (-13mm, 2mm), fill: false)
node("n1", (-13mm, -2mm), fill: false)
node("n2", (13mm, 2mm), fill: false)
node("n3", (13mm, -2mm), fill: false)
wire((-13mm, 2mm), "A1.in1")
wire((-13mm, -2mm), "A1.in0")
wire((13mm, 2mm), "A2.out1")
wire((13mm, -2mm), "A2.out0")
wire("A2.in1", "A1.out1")
wire("A2.in0", "A1.out0")
})
],
[
Kette invers \
$jMat(A')_"ges" = jMat(A'_2) dot jMat(A'_1)$
],
[
#zap.circuit({
import zap : *
registerAllCustom();
zweiTor("A2", (6mm, 0), $jMat(A'_2)$, height: 8mm, width: 8mm)
zweiTor("A1", (-6mm, 0), $jMat(A'_1)$, height: 8mm, width: 8mm)
node("n0", (-13mm, 2mm), fill: false)
node("n1", (-13mm, -2mm), fill: false)
node("n2", (13mm, 2mm), fill: false)
node("n3", (13mm, -2mm), fill: false)
wire((-13mm, 2mm), "A1.in1")
wire((-13mm, -2mm), "A1.in0")
wire((13mm, 2mm), "A2.out1")
wire((13mm, -2mm), "A2.out0")
wire("A2.in1", "A1.out1")
wire("A2.in0", "A1.out0")
})
]
)
],
#bgBlock(fill: colorEineTore)[
#subHeading(fill: colorEineTore)[Linearisierung (Ein-Tore)]
1. Arbeitspunkt bestimmen \ $"AP" =(u_"AP", i_"AP")$
2. Ableitung $g_cal(F)(u)$/$r_cal(F)(i)$ bilden \ $g'_cal(F)(u)$/$r'_cal(F)(i)$
*Stromgesteuert $r(i) = u$*
Groß-Signal:
$i_"lin" = g_"lin" (u) = g'_cal(F)(u_"AP")(u-u_"AP") + i_"AP" = \
g'_cal(F)(u_"AP")u - g'_cal(F)(u_"AP")u_"AP" + i_"AP"
$
$I_0 = i_"AP" - g'_cal(F)(u_"AP")u_"AP"$
$g_0 = g'_cal(F)(u_"AP")u$
#block(
height: 20mm,
scale(x: 75%, y: 75%,
zap.circuit({
import zap : *
import cetz.draw : line, content
isource("I0", (0, 1.5), (0, -1.5), fill: none)
node("n0", (2,1.5))
node("n1", (2,-1.5))
node("n2", (3.5,1.5), fill: false)
node("n3", (3.5,-1.5), fill: false)
resistor("g0", "n1", "n0", fill: none, label: (content: $g_0$, anchor: "south", distance: 0.2))
wire("I0.in", "n0", "n2")
wire("I0.out", "n1", "n3")
wire("n0", "n2", i: (content: $i_"lin"$, anchor: "south", invert: true))
wire((0,0.6), "I0.in", i: (content: $I_0$, anchor: "east", invert: true))
set-style(mark: (end: ">", fill: black))
line((3.5, 1.2), (3.5, -1.2), stroke: 0.5pt)
content((3.9, 0), $u$)
})
));
#linebreak()
Klein-Signal:$i_"lin" = g_"lin" (u) = g'(u_"AP")u$
*Spannungsgesteuert $g(u) = i$*
$u_"lin" = r_"lin" (i) = r'(i_"AP")(i-i_"AP") + u_"AP"$
$U_0 = u_"AP" - r'(i_"AP") i_"AP"$
#block(
height: 25mm,
scale(x: 75%, y: 75%,
zap.circuit({
import zap : *
import cetz.draw : line, content
vsource("U0", (0, 1.5), (0, -1.5), fill: none)
node("n2", (3.5,1.5), fill: false)
node("n3", (3.5,-1.5), fill: false)
resistor("R0", "U0.in", "n2", fill: none, label: (content: $r_0$, anchor: "south", distance: 0.1))
wire("U0.out", "n3")
wire((2.49,1.5), "n2", i: (content: $i$, anchor: "south", invert: true))
set-style(mark: (end: ">", fill: black))
line((3.5, 1.2), (3.5, -1.2), stroke: 0.5pt)
content((3.9, 0), $u_"lin"$)
line((0.7, 0.5), (0.7, -0.5), stroke: 0.5pt)
content((1.2, 0), $U_0$)
})
));
Klein-Signal: $u_"lin" = r_"lin" (i) = r'(i_"AP")i$
]
#bgBlock(fill: colorEineTore)[
#subHeading(fill: colorEineTore)[Linearisierung (Zweite-Tore)]
1. Arbeitspunk bestimmen $vec(jVec(u)_"AP", jVec(i)_"AP")$
2. Ableitung
]
#bgBlock(fill: colorEineTore)[
#subHeading(fill: colorEineTore)[Newton-Raphson]
]
]
#pagebreak()
// Tor Eigenschaften
#place(
bottom, float: true, scope: "parent",
bgBlock(fill: colorEigenschaften, width: 100%)[
#subHeading(fill: colorEigenschaften)[Tor Eigenschaften]
#table(
columns: (auto, auto, auto, auto),
inset: 2mm,
align: horizon,
table.header([], [*Ein-Tor*], [*Zwei-Tor*], [*Complex AC*]),
[*passiv*\ ($not$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*],
[$forall (u,i) in cal(F): u dot i = 0$],
[$forall vec(jVec(u),jVec(v)) in cal(F) : jVec(u)^T jVec(i) = 0$],
[],
[*linear*],
[Kennline ist Gerade],
[Darstellbar: Matrix $+$ Aufpunkt],
[],
[*quellenfrei*],
[$(qty("0", "A"), qty("0", "V")) in cal(F)$],
[],
[],
[*streng linear*],
[linear UND quellenfrei],
[linear UND quellenfrei\ Darstellbar: Nur Matrix],
[],
[*ungepolt* \ (Punkt sym.)],
[$(u,i) in cal(F) <=> (-u, -i) in cal(F)$],
[
],
[],
[*symetrisch*\ $<=>$ Umkehrbar \ (Achsen sym.) ],
[$(u,i) in cal(F) <=> (u, -i) in cal(F)$],
[
$jMat(A) = jMat(A')$\
$jMat(G) = jMat(P) jMat(G) jMat(P), space jMat(R) = jMat(P) jMat(R) jMat(P), quad jMat(P) = mat(0, 1; 1, 0)$
],
[],
[*Reziprok*],
[Immer Reziprok],
[
$cal(F)$ symetrisch $=> cal(F)$ reziprok
$jMat(U)^T jMat(I) - jMat(I)^T jMat(U) = 0 \
jMat(R)^T = jMat(R), quad jMat(G)^T = jMat(G) quad h_21 = -h_12 \ det(jMat(A)) = 1 quad det(jMat(A')) = 1 quad h'_21 = -h'_12$],
[],
[*Strom-gst.*],
[$r(i) = u$],
[],
[],
[*Spannung-gst.*],
[$g(u) = i$]
)
]
)
#place(bottom+left, scope: "parent", float: true)[
#bgBlock(fill: colorZweiTore)[
#set text(size: 10pt)
#subHeading(fill: colorZweiTore)[Umrechnung Zweitormatrizen]
#table(
columns: (auto, 1fr, 1fr, 1fr, 1fr, 1fr, 1fr),
columns: (12mm, 1fr, 1fr, 1fr, 1fr, 1fr, 1fr),
align: center,
inset: (bottom: 4mm, top: 4mm),
gutter: 0.1mm,
[In $->$], $bold(R)$, $bold(G)$, $bold(H)$, $bold(H')$, $bold(A)$, $bold(A')$,
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")
})
]),
$bold(R)$,
$bold(G)$,
$bold(H)$,
$bold(H')$,
$bold(A)$,
$bold(A')$,
$bold(R)$,
$mat(r_11, r_12; r_21, r_22)$,
$1/det(bold(G)) mat(g_22, -g_12; -g_21, g_11)$,
$jMat(G^(-1) =) 1/det(bold(G)) mat(g_22, -g_12; -g_21, g_11)$,
$1/h_22 mat(det(bold(H)), h_12; -h_21, 1)$,
$1/h'_11 mat(1, -h'_12; h'_21, det(bold(H')))$,
$1/a_21 mat(a_11, det(bold(A)); 1, a_22)$,
$1/a'_21 mat(a'_22, 1; det(bold(A')), a'_11)$,
$bold(G)$,
$1/det(bold(R)) mat(r_22, -r_12; -r_21, r_11)$,
$jMat(R^(-1) =) 1/det(bold(R)) mat(r_22, -r_12; -r_21, r_11)$,
$mat(g_11, g_12; g_21, g_22)$,
$1/h_11 mat(1, -h_12; h_21, det(bold(H)))$,
$1/h'_22 mat(det(bold(H')), h'_12; -h'_21, 1)$,
@@ -174,14 +683,14 @@
$1/r_22 mat(det(bold(R)), r_12; -r_21, 1)$,
$1/g_11 mat(1, -g_12; g_21, det(bold(G)))$,
$mat(h_11, h_12; h_21, h_22)$,
$1/det(bold(H')) mat(h'_22, -h'_12; -h'_21, h'_11)$,
$jMat(H')^(-1)= 1/det(bold(H')) mat(h'_22, -h'_12; -h'_21, h'_11)$,
$1/a_22 mat(a_12, det(bold(A)); -1, a_21)$,
$1/a'_11 mat(a'_12, 1; -det(bold(A')), a'_21)$,
$bold(H')$,
$1/r_11 mat(1, -r_12; r_21, det(bold(R)))$,
$1/g_22 mat(det(bold(G)), g_12; -g_21, 1)$,
$1/det(bold(H)) mat(h_22, -h_12; -h_21, h_11)$,
$jMat(H^(-1))= 1/det(bold(H)) mat(h_22, -h_12; -h_21, h_11)$,
$mat(h'_11, h'_12; h'_21, h'_22)$,
$1/a_11 mat(a_21, -det(bold(A)); 1, a_12)$,
$1/a'_22 mat(a'_21, -1; det(bold(A')), a'_12)$,
@@ -192,16 +701,33 @@
$1/h_21 mat(-det(bold(H)), -h_11; -h_22, -1)$,
$1/h'_21 mat(1, h'_22; h'_11, det(bold(H')))$,
$mat(a_11, a_12; a_21, a_22)$,
$1/det(bold(A')) mat(a'_22, a'_12; a'_21, a'_11)$,
$jMat(A'^(-1))= 1/det(bold(A')) mat(a'_22, a'_12; a'_21, a'_11)$,
$bold(A')$,
$1/r_12 mat(r_22, det(bold(R)); 1, r_11)$,
$1/g_12 mat(-g_11, -1; -det(bold(G)), -g_22)$,
$1/h_12 mat(1, h_11; h_22, det(bold(H)))$,
$1/h'_12 mat(-det(bold(H')), -h'_22; -h'_11, -1)$,
$1/det(bold(A)) mat(a_22, a_12; a_21, a_11)$,
$jMat(A^(-1))= 1/det(bold(A)) mat(a_22, a_12; a_21, a_11)$,
$mat(a'_11, a'_12; a'_21, a'_22)$,
)
#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 },
[],
$bold(R) jVec(i) = jVec(u)$,
$bold(G) jVec(u) = jVec(i)$,
$bold(H) vec(i_1, u_2) = vec(u_1, i_2)$,
$bold(H') vec(u_1, i_2) = vec(i_1, u_2)$,
$bold(A) vec(u_2, -i_2) = vec(i_1, u_1)$,
$bold(A') vec(u_1, -i_1) = vec(i_2, u_2)$,
)
]
]