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Started Newton Raphson
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alexander
2026-01-28 10:55:11 +01:00
parent 195b64517f
commit 52e2d52813
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src/cheatsheets/Schaltungstheorie.typ
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@@ -541,19 +541,48 @@
));
Klein-Signal: $u_"lin" = r_"lin" (i) = r'(i_"AP")i$
]
#colbreak()
#bgBlock(fill: colorZweiTore)[
#subHeading(fill: colorZweiTore)[Linearisierung (N-Tore)]
#bgBlock(fill: colorEineTore)[
#subHeading(fill: colorEineTore)[Linearisierung (Zweite-Tore)]
1. Arbeitspunk bestimmen $vec(jVec(u)_"AP", jVec(i)_"AP") hat(=) vec(jVec(x)_"AP", jVec(y)_"AP")$
1. Arbeitspunk bestimmen $vec(jVec(u)_"AP", jVec(i)_"AP")$
$f_1(x_1, x_2, ... x_n) &= y_1\
f_2(x_1, x_2, ... x_n) &= y_2\
&dots.v \
f_n (x_1, x_2, ... x_n) &= y_n
$
2. Ableitung
$bold(f)(jVec(x))=jVec(y)$
2. Ableitung: Jacobi-Matrix
$jMat(J) = mat(
#mannot.mark($delta y_1$, color: red)/(#mannot.mark($delta x_1$, color: red)), #mannot.mark($delta y_1$, color: red)/(#mannot.mark($delta x_2$, color: purple)), ..., #mannot.mark($delta y_1$, color: red)/(#mannot.mark($delta x_n$, color: blue));
#mannot.mark($delta y_2$, color: purple)/(#mannot.mark($delta x_1$, color: red)), #mannot.mark($delta y_2$, color: purple)/(#mannot.mark($delta x_2$, color: purple)), ..., #mannot.mark($delta y_2$, color: purple)/(#mannot.mark($delta x_n$, color: blue));
dots.v, dots.v, dots.down, dots.v;
#mannot.mark($delta y_n$, color: blue)/(#mannot.mark($delta x_1$, color: red)), #mannot.mark($delta y_n$, color: blue)/(#mannot.mark($delta x_2$, color: purple)), ..., #mannot.mark($delta y_n$, color: blue)/(#mannot.mark($delta x_n$, color: blue));
)$
*Großsignal Beschreibung:* \
$jVec(y)_"lin" = jMat(J)|_(jVec(x)_"AP") (jVec(x)_"lin" - jVec(x)_"AP") + jVec(y)_"AP"$
*Kleinsingal Beschreibung:* \
$jVec(y)_"lin" = jMat(J)|_(jVec(x)_"AP") jVec(x)_"lin"$
]
#bgBlock(fill: colorEineTore)[
#subHeading(fill: colorEineTore)[Newton-Raphson]
#subHeading(fill: colorEineTore)[Newton-Raphson (Eine-Tor)]
$x_(n+1) = x_n - f(x_n)/(f'(x_n))$
]
#bgBlock(fill: colorZweiTore)[
#subHeading(fill: colorZweiTore)[Newton-Raphson (Mehr-Tore)]
Nicht lineare Beschreibung in Nullraum/Impliziter Darstellung:
$f(jVec(x)) = jVec(0)$\
$jVec(x)_(n+1) = jVec(x)_n - (jMat(J)|_(jVec(x)_"AP"))^(-1) f(jVec(x))$
]
]
#pagebreak()