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zb__{Ujg5x7I$dL<-anBzD2ildar5%{_H}Ny1Dx_hFT6qsu+IDazFsr@H~?n`VbGWR z29ZoyAc08dn>cN4@b=qa4vtLeIdh=2)IXPMZ#RDTUGJbze}FM46wDbjL;?Z*s~wUe zul8T9@2YR;Yv|B)IQP)vC@$y86T}IsFqM+83s~u~l%LFF}ph^R}@eP$c z46=EU&jWGb68#JYvbI)RTdVEsqi4bk4<}MmxQU5u1|u|vh6r)K%Hnwt$REr#}XQ+Kz?M>cJ2ENjXXfzXLc zVGF>UzPK_S%mEm;!DXS~q7F`|p}_$3#ymwZV44upx$YS#M~}kp-E>~wJL2NtzWcys z{Tcc&8p%V4`Y&Gehtr?QM2m{}DJlL)n=+B64oyd|u2*eP>&!X}X+aPKLohbZ7P5q5 mmN=Xnj-esXL5L6&jQ"), fill: blue, stroke: blue) - line((2.5, 0), (2.5, -0.8), mark: (end: ">"), fill: blue, stroke: blue) content((0.5, 0.5), [1]) - content((2.5, -1), [Median = 4]) }) ``` diff --git a/src/bildverarbeitung_1.typ b/src/bildverarbeitung_1.typ index 6877d6b..5bc609c 100644 --- a/src/bildverarbeitung_1.typ +++ b/src/bildverarbeitung_1.typ @@ -74,3 +74,41 @@ cv::circle(img, cv::Point(10, 10), 20, cv::Scalar(0, 0, 128), 30); [cv::Scalar], [Farbe in BRG], [30], [Linienbreite (-1 für geffülter Kreis], ) + +=== Bild einlesen +#table(columns: 1fr, [```cpp +std::string filename = "mond.png"; +cv::Mat img = cv::imread(filename, cv::IMREAD_ANYCOLOR); +```]) + +=== Bild zu einem Graubild konvertieren +#table(columns: 1fr, [```cpp +cv::Mat img_grayray; +cv::cvtColor(img, img_gray, cv::COLOR_BGR2GRAY); +```]) + +=== Bild von Grau zu Farb +#table(columns: 1fr, [```cpp +cv::Mat img_color; +cv::cvtColor(img_gray, imgNotes, cv::COLOR_GRAY2BGR); +```]) + +=== Maximale und Minimale Helligkeit finden +#table(columns: 1fr, [```cpp +double minGray, maxGray; +cv::Point minLoc, maxLoc; +cv::minMaxLoc(imgGray, &minGray, &maxGray, &minLoc, &maxLoc); + +// auf x oder y zugreifen: +std::cout << maxLoc.x << std::endl; +std::cout << maxLoc.y << std::endl; +```]) + +=== Spalten/Reihen umfärben +#table(columns: 1fr, [```cpp +imgGray.col(maxLoc.x).setTo(cv::Scalar(32)); +imgGray.row(maxLoc.y).setTo(cv::Scalar(32)); +```]) + + + diff --git a/src/elektronik_2.typ b/src/elektronik_2.typ index 38ed6a6..1e04a32 100644 --- a/src/elektronik_2.typ +++ b/src/elektronik_2.typ @@ -1,10 +1,22 @@ #import "@preview/zap:0.4.0" == Elektronik 2 -=== Feldeffekttransistor +=== Vierquadrantenkennlinienfeld +#image("../img/elektronik_2/Kombiniertes_Kennlinienfeld_Transistor_2.svg") + +#table(columns: (120pt, 1fr), +[Ausgangskennlinienfeld \ (oben rechts)], +[Sobald $U_"CE"$ einen kleinen Schwellenwert überschreitet, bleibt der Strom $I_C$ fast konstant, egal wie weit man die Spannung erhöht. Der Transistor wirkt hier wie eine stromgesteuerte Stromquelle.], +[Stromsteuerkennlinie \ (oben links)], [#grid(columns: (100pt, 1fr), +[$ B eq frac(I_c, I_B) $], [Ein kleiner Strom an der Basis steuert also linear einen viel größeren Strom am Kollektor.])], +[Eingangskennlinie \ (unten links)], +[Da die Basis-Emitter-Strecke physikalisch eine Diode ist, fließt erst ab einer Schwellenspannung (bei Silizium ca. 0,6V bis 0,7V) ein nennenswerter Strom. Danach steigt der Strom exponentiell an.], +[Spannungsrückwirkung \ (unten rechts)], +[Im Idealfall sollte die Ausgangsspannung keinen Einfluss auf den Eingang haben. In der Realität sieht man eine minimale Verschiebung der Kennlinien (der Early-Effekt spielt hier eine Rolle), was als "Rückwirkung" bezeichnet wird.], +) + === Logikschaltungen === Sinussignale -=== Sinussignale === Passive Zweipole === Aktive Zweipole (Versorgungen) === Passive Vierpole diff --git a/src/lineare_algebra_2.typ b/src/lineare_algebra_2.typ index e69de29..3e04bfe 100644 --- a/src/lineare_algebra_2.typ +++ b/src/lineare_algebra_2.typ @@ -0,0 +1,156 @@ +== Matrix +Matrix wird mit *Zeile* X *Spalten* beschrieben. Das ist eine $2 crossmark 3 "-Matrix:" A eq mat(delim: "[", 2, 1, 3; 7, 5, 2)$ + +=== Addition und Subtraktion +#grid(columns: (1fr, 1fr), gutter: 40pt, [ +$ A plus B :eq mat(delim: "[", +A^1_1 plus B^1_1, A^1_2 plus B^1_2, dots.h, A^1_n plus B^1_n; +A^2_1 plus B^2_1, A^2_2 plus B^2_2, dots.h, A^2_n plus B^2_n; +dots.v, dots.v, dots.down, dots.v; +A^m_1 plus B^m_1, A^m_2 plus B^m_2, dots.h, A^m_n plus B^m_n; +) $ +], [ +$ A minus B :eq mat(delim: "[", +A^1_1 minus B^1_1, A^1_2 minus B^1_2, dots.h, A^1_n minus B^1_n; +A^2_1 minus B^2_1, A^2_2 minus B^2_2, dots.h, A^2_n minus B^2_n; +dots.v, dots.v, dots.down, dots.v; +A^m_1 minus B^m_1, A^m_2 minus B^m_2, dots.h, A^m_n minus B^m_n; +) $ +]) + +=== Multiplikation +$ a dot A :eq mat(delim: "[", +a dot A^1_1, a dot A^1_2, dots.h, a dot A^1_n; +a dot A^2_1, a dot A^2_2, dots.h, a dot A^2_n; +dots.v, dots.v, dots.down, dots.v; +a dot A^m_1, a dot A^m_2, dots.h, a dot A^m_n; +) $ + +=== Transposition +#grid(columns: (1fr, 1fr), gutter: 40pt, [ +$ A^T :eq mat(delim: "[", +A^1_1, A^2_1, dots.h, A^m_1; +A^1_2, A^2_2, dots.h, A^m_2; +dots.v, dots.v, dots.down, dots.v; +A^1_n, A^2_n, dots.h, A^m_n; +) $ +], [ +Bsp.: +$ mat(delim: "[", +2, -1 +)^T eq +mat(delim: "[", +2; -1 +) $ +$ mat(delim: "[", +a, b, c; d, e, f +)^T eq +mat(delim: "[", +a, d; b, e; c, f +) $ +]) + +=== Matrix-Multiplikation +$ M eq A dot B $ + +#image("../img/linalg_2/2026-02-23-211625_hyprshot.png") +// $ +// #let hi(c, body) = box(fill: c, inset: 2pt, radius: 2pt)[#body] +// +// mat(delim: "[", +// #box(fill: yellow, inset: 2pt, radius: 2pt)[1], +// #box(fill: green, inset: 2pt, radius: 2pt)[2],; +// #box(fill: aqua, inset: 2pt, radius: 2pt)[3], +// #box(fill: orange, inset: 2pt, radius: 2pt)[4] +// ) +// dot +// mat(delim: "[", +// #box(fill: red, inset: 2pt, radius: 2pt)[5], +// #box(fill: purple, inset: 2pt, radius: 2pt)[6],; +// #box(fill: blue, inset: 2pt, radius: 2pt)[7], +// #box(fill: lime, inset: 2pt, radius: 2pt)[8] +// ) +// eq +// mat(delim: "[", +// #hi(yellow, 1) dot #hi(orange, 5) plus #hi(green, 2) dot #hi(blue, 7), +// #hi(yellow, 1) dot #hi(orange, 5) plus #hi(green, 2) dot #hi(blue, 7), +// ) +// $ + +- $mat(delim: "[", +1, 2; 3, 4 +) dot mat(delim: "[", +5; 6 +) eq +mat(delim: "[", +1 dot 5 plus 2 dot 6; 3 dot 5 plus 4 dot 6 +) eq mat(delim: "[", +17; 39 +) $ + +- $mat(delim: "[", +1, 2; 3, 4 +) dot mat(delim: "[", +5, 6 +) eq +mat(delim: "[", +1 dot 5 plus 3 dot 6, 2 dot 5 plus 4 dot 6 +) eq mat(delim: "[", +23, 34 +) $ + +- $mat(delim: "[", +1, 2 +) dot mat(delim: "[", +3; 4 +) eq +mat(delim: "[", +1 dot 3 plus 2 dot 4 +) eq mat(delim: "[", +11 +) $ + +- $mat(delim: "[", +1; 2 +) dot mat(delim: "[", +3, 4 +) eq +mat(delim: "[", +1 dot 3, 1 dot 4; 2 dot 3, 2 dot 4 +) eq mat(delim: "[", +3, 4; 6, 8 +) $ + +=== Symmetrische Matrix +ist: #box(stroke: 1pt + red, inset: (x: 1em, y: 0.5em), [$A^T eq A$]) dann ist die Matrix Symmetrische. \ +z.b.: $mat(delim: "[", +1, 2; 2, 3 +)$ + +=== Schiefsymmetrische Matrix +ist: #box(stroke: 1pt + red, inset: (x: 1em, y: 0.5em), [$A^T eq minus A$]) dann ist die Matrix Symmetrische. \ +z.b.: $mat(delim: "[", +1, -2; 2, 0 +)$ +oder +$mat(delim: "[", +0, -1, 2; 1, 0, -3; -2, 3, 0 +)$ + +=== Nullmatrix +$0 eq mat(delim: "[", +0, 0, dots.h, 0; +0, 0, dots.h, 0; +dots.v, dots.v, dots.down, dots.v; +0, 0, dots.h, 0; +)$ + +=== Einheitsmatrix +$bb(1) eq mat(delim: "[", +1, 0, dots.h, 0; +0, 1, dots.h, 0; +dots.v, dots.v, dots.down, dots.v; +0, 0, dots.h, 1; +)$ + + diff --git a/src/physik_2.typ b/src/physik_2.typ index dfc021d..8e9e3af 100644 --- a/src/physik_2.typ +++ b/src/physik_2.typ @@ -87,7 +87,126 @@ $ frac(gamma_E, gamma_B) eq frac(1, epsilon_0 dot mu_0) eq c^2 $ $gamma_E eq 1$ \ $gamma_B eq 10^(-7) frac(N, A^2)$ +=== E-Feld +$bold(F)_E eq Q dot E$ #h(20pt) $[E] eq frac(N, C)$ +=== B-Feld +$bold(F)_B eq Q dot v crossmark B$ #h(20pt) $[B] eq frac(N, A dot m) eq T "(Tesla)"$ + + + + + + + + + + + + + + + + + + + + + + + + + + + +#pagebreak() +#pagebreak() +#pagebreak() == Schwingungen und Wellen +Periodendauer $T [s]$ \ +Frequenz $f eq frac(1, T)$ #h(20pt) $[f] eq "Hz"$ + + +=== Ort-Zeit Funktion +$x(t) eq A dot cos(omega t plus delta)$ \ +// Phase $omega t dot delta$ \ +// Phasenwinkel $delta$ Wert von $delta$ bei $t eq 0$ \ +// Kreisfrequenz $omega$ \ +// $x(t) eq x(t plus T)$ wir setzen $delta eq 0$ \ +// $A cos(omega t) &eq A dot cos(omega (t plus T)) \ +// &eq A dot cos(omega t plus omega T)$ \ +// da $cos 2 pi$ periodisch $omega T eq 2 pi$ + +// $omega eq frac(2 pi, T) eq frac(2 pi, frac(1, f)) eq 2 pi f$ \ +// $[omega] eq frac(1, s)$ +$omega eq frac(2 pi, T) eq 2 pi f$ #h(20pt) $[omega] eq frac(1, s)$ + +Geschwindigkeit: \ +// $V_x &eq frac(d x (t), d t) \ +// &eq - A omega sin(omega t plus delta)$ +$V_x eq - A omega sin(omega t plus delta)$ + +Beschleunigung: \ +// $a_x &eq frac(d v_x, d t) eq frac(d^2 x(t), d t^2) \ +// &eq - A omega^2 cos(omega t plus delta) \ +// &eq minus omega^2 dot x(t)$ +$a_x &eq - A omega^2 cos(omega t plus delta) \ +&eq minus omega^2 dot x(t)$ + +// Amplitude von:\ +// $v_x : A omega$ \ +// $a_x : A omega^2$ \ + + + +=== Newtonssches Gesetz (Bewegungsgleichung) +// $m dot a eq sum F$ \ +// $m dot a_x eq minus k dot x arrow a_x eq minus frac(k, m) dot x$ \ +// oder \ +// $m dot a_x eq minus omega^2 dot x dot m$ \ +// gleichsetzen der gleichungen: \ +// $minus k dot x eq minus m dot omega^2 dot x$ \ +// $k eq m dot omega^2$ \ +// $omega eq sqrt(frac(k, m))$ \ +// mit: \ +// $omega eq frac(s pi, T)$ $arrow$ $T eq 2 pi sqrt(frac(m,k))$ \ +$omega eq sqrt(frac(k, m))$ \ + +Dehnung der Feder: \ +$k dot Delta l eq m dot g$ + +// = pp S 21 +// $F eq minus k dot x$ \ +// $E_"pot" eq minus integral_0^x F d s eq minus integral_0^x (minus k s ) d s eq frac(1, 2) k x^2$ $(c eq 0)$ +// +// $E_"pot" (x eq 0) eq 0$ Bei Gleichgewichtslage \ +// $E_"pot" (x = A) eq frac(1, 2) k A^2)$ Maximeirt ... +// $E_"ges" &eq E_"pot" plus E_"kin" eq frac(1, 2) k x^2 plus frac(1, 2) m v^2 \ +// &eq frac(1, 2) k (A cos(omega t plus delta))^2 plus frac(1, 2) m (minus A omega sin(omega t plus delta))^2 \ +// &eq frac(1, 2) k A^2 cos^2(omega t plus delta) plus frac(1, 2) m A^2 omega^2 sin^2(omega t plus delta) \ +// &eq frac(1, 2) k A^2 underbrace((cos^2(omega t plus delta) plus sin^2omega t plus delta), eq 1) \ +// &eq frac(1, 2) k A^2 $ + +=== Gleichgewichtslage +$E_"pot" (x) eq 0$ Bei Gleichgewichtslage \ + + +// = pp S 27 +// Federkraft von m: \ +// $F eq minus k y$ \ +// Gewichtskraft von m: \ +// $F eq m dot x$ \ +// Newtown 2: \ +// $m dot a_y eq minus k y plus m dot g$ \ +// Einführung neue Koordinate: \ +// $y^' eq y minus y_0$ +// +// +// $sum f eq -(y^' plus y_0) dot k plus m dot g$ \ +// mit $k y_0 eq m dot g$: +// $sum F eq - y^' dot k$ +// +// $ y^'(t) eq A dot cos(omega t plus delta) $ +// masse m schwingt um Gleichgewichtslage $y_0$