FEM Teil hinzugefügt
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ENT4_FS.pdf
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ENT4_FS.tex
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ENT4_FS.tex
@ -416,12 +416,221 @@ hypertexnames=false % Zur korrekten Erstellung der Bookmarks
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\vec{U}_1^k = R_1 \cdot \vec{I}_1^k+\frac{d\vec{\psi}_1^k}{dt}+j\omega_k \cdot \vec{\psi}_1^k
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\end{equation}
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Ständerspannungsgleichung in Raumzeigerdarstellung
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\begin{equation}
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\vec{U}_1^S = R_1\cdot \vec{I}_1^S + \frac{d\vec{\psi}_1^S}{dt} = R_1\cdot \vec{I}_1^S + i_1\cdot \frac{d\vec{I}_1^S}{dt} + M\cdot \frac{d\vec{I}_2^S}{dt}
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\end{equation}
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Läuferspannungsgleichung
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\begin{equation}
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\vec{U}_2^k = R_2 \cdot \vec{I}_2^k+\frac{d\vec{\psi}_2^k}{dt}+j(\omega_k -\omega_L)\cdot \vec{\psi}_2^k
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\end{equation}
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..... nachher geht es weiter
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Läuferspannungsgleichung in Raumzeigerdarstellung
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\begin{equation}
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\vec{U}_2^L = R_2\cdot \vec{I}_2^L + \frac{d\vec{\psi}_2^L}{dt} = 0
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\end{equation}
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Läuferspannungsgleichung im Ständerkoordinatensystem
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\begin{equation}
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\vec{U}_2^S = R_2\cdot \frac{\vec{I}_\mu^S - \vec{I}_1^S}{1+\sigma_2} - j\omega_L\cdot M\cdot \vec{I}_\mu^S+M\frac{d\vec{I}_\mu^S}{dt}
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\end{equation}
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???
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\begin{equation}
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\vec{I}_1^k = I_\mu(1-j(\omega_L-\omega_K)\cdot T_2)+T_2\cdot \frac{dI_\mu}{dt}
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\end{equation}
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mit T\textsubscript{2}
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\begin{equation}
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T_2 = \frac{M\cdot (1+\sigma_2)}{R_2} = \frac{L_2}{R_2}
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\end{equation}
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Längskomponente (flussbildend): Feldbildung folgt mit Zeitkonstante T\textsubscript{2}
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\begin{equation}
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Re(\vec{I}_1^k) = I_{1d} = I_\mu + T_2 \frac{dI_\mu}{dt}
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\end{equation}
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Querkomponente (drehmomentbildend): Feldbildung folgt unverzögert
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\begin{equation}
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Im(\vec{I}_1^k) = I_{1q} = (\omega_K - \omega_L)\cdot T_2\cdot I_\mu
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\end{equation}
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Drehmoment\\
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ToDo: Herausfinden welche Formeln relevant sind
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\colorbox{yellow!60}{Numerische Feldberechnung}\\
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Magnetische Feldstärke = Magnetische Erregung
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\begin{equation}
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H = \frac{I}{l} [\frac{A}{m}]
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\end{equation}
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Maxwellsche Gleichungen in differentieller Form\\
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Durchflutungsgesetz
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\begin{equation}
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rot \vec{H} = \vec{S}
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\end{equation}
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Induktionsgesetz
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\begin{equation}
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rot \vec{E} = - \frac{\partial\vec{B}}{\partial t}
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\end{equation}
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Materialgesetz
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\begin{equation}
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\vec{B} = \vec{J} + \mu_0 \cdot \vec{H}
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\end{equation}
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Strömungsfeld für elektrische Leiter
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\begin{equation}
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\vec{S} = k \cdot \vec{E}
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\end{equation}
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Quellenfreiheit
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\begin{equation}
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div \vec{B} = 0
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\end{equation}
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Grundprinzip FEM
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\begin{itemize}
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\item Diskretisierung der Feldgebiete (mit Dreiecken 2D oder Tetraeder 3D)
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\item iterative Lösung
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\item magnetische Feldberechnung: magnetische Feldenergie unterschreitet vorgegeben Grenzwert
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\end{itemize}
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Da die Rotation für alle wirbelfreien Felder = 0 ist gilt:\\
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magnetische Flussdichte
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\begin{equation}
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\vec{B} = rot\vec{A}
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\end{equation}
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kartesische Koordinaten
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\begin{equation}
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rot \vec{A} = \frac{\partial \vec{A}}{\partial y}\cdot \vec{i} - \frac{\partial \vec{A}}{\partial x}\cdot \vec{j}
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\end{equation}
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Zylinderkoordinaten
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\begin{equation}
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rot \vec{A} = \frac{1}{r}\frac{\partial \vec{A}}{\partial y}\cdot \vec{e}_r - \frac{\partial \vec{A}}{\partial r}\cdot \vec{e}_\varphi
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\end{equation}
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Magnetische Vektorpotential/ magnetische Feldstärke\\
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kartesische Koordinaten
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\begin{equation}
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\vec{H} = -grad_{\varphi m} = -(\frac{\partial_{\varphi m}}{\partial x}\cdot \vec{i} + \frac{\partial_ {\varphi m}}{\partial y} \cdot \vec{j})
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\end{equation}
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Zylinderkoordinaten
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\begin{equation}
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\vec{H} = -grad_{\varphi m} = -(\frac{\partial_{\varphi m}}{\partial r}\cdot \vec{e}_r + \frac{1}{r}\frac{\partial_ {\varphi m}}{\partial \varphi} \cdot \vec{e}_\varphi)
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\end{equation}
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Magnetische Energiedichte je Längeneinheit
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\begin{equation}
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\frac{dW_{mag}}{l} = \frac{1}{2} \mu H^2 dA
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\end{equation}
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Betrag der magnetischen Feldstärke
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\begin{equation}
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H^2 = (\frac{\partial_{\varphi m}}{\partial x})^2 + (\frac{\partial_{\varphi m}}{\partial y})
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\end{equation}
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partielle Ableitungen des Skalarprodukts
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\begin{equation}
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\frac{\partial_{\varphi m}}{\partial x} = \frac{1}{2A}[(y_2-y_3)\varphi_{m1}+(y_3-y_1)\varphi_{m2}+(y_1-y_2)\varphi_{m3}]
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\end{equation}
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\begin{equation}
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\frac{\partial_{\varphi m}}{\partial y} = \frac{1}{2A}[(x_3-x_2)\varphi_{m1}+(x_1-x_3)\varphi_{m2}+(x_2-x_1)\varphi_{m3}]
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\end{equation}
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Minimum magnetische Feldenergie
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\begin{equation}
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\frac{\partial W_{mag}/l}{\partial \varphi_m} = 0
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\end{equation}
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Mathematisches Konzept der FEM
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"starke Formulierung"
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\begin{equation}
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Res = rot \vec{H} - \vec{S} = \frac{1}{\mu} rot rot \vec{A}-\vec{S}
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\end{equation}
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"schwache Formulierung " ????\\
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Flussdichte
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\begin{equation}
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B = J + \mu_0 H = \mu_0 \mu_r H
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\end{equation}
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aus den gemessenen Kennliniepunkten Geradengleichung
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\begin{equation}
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\frac{1}{\mu_r -1} = a^* + b^* \cdot H
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\end{equation}
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Permeabilität
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\begin{equation}
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\mu_r = f(H) =1+ \frac{1}{a^*+b^*\cdot H}
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\end{equation}
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induzierte Spannung
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\begin{equation}
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|u_{ind}|= w \cdot \frac{d\phi}{dt} = \omega \cdot w \cdot \phi
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\end{equation}
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magnetische Spannung
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\begin{equation}
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V_m = H_\delta \cdot \delta
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\end{equation}
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Strombelag
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\begin{equation}
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A =
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\end{equation}
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Grundwelle
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\begin{equation}
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B_1(\theta) = \mu_0 \frac{2\omega}{\pi\delta}\cdot cos(\theta)\cdot i(t)
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\end{equation}
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Zonungsfaktor
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\begin{equation}
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\xi_{Z,1} = \frac{|\vec{U}_{res}|}{|\vec{U}_1|+|\vec{U}_2|+|\vec{U}_3|}
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\end{equation}
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Sehnungsfaktor
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\begin{equation}
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\xi_{S,1} = sin(\frac{\tau_\omega}{\tau_p})
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\end{equation}
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Wicklungsfaktor
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\begin{equation}
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\xi_1 = \xi_{Z,1} \cdot \xi_{S,1}
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\end{equation}
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Wirksame Windungszahl
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\begin{equation}
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w_1 = N \cdot \xi_1
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\end{equation}
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Grundstrombelag
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\begin{equation}
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a_p(x,t) = A_p \cdot cos(px-\omega_1 t -\varphi_1)
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\end{equation}
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Amplitude der Grundwelle
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\begin{equation}
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A_p = \frac{3}{\pi}\cdot A = \frac{3\cdot N_1 \xi_p \cdot }{\pi \cdot R}
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\end{equation}
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Magnetische Spanung über dem Luftspalt
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\begin{equation}
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V(x,t) = \frac{1}{p} \cdot A_p \cdot R \cdot sin(px-\omega_1 t - \varphi_1)
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\end{equation}
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Amplitude B-Feld Grundwelle
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\begin{equation}
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B_p = \frac{\mu_0}{\delta^{''}} \frac{3\cdot N_1 \xi_p}{p\cdot \pi} \cdot \sqrt{2} \cdot I_\mu
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\end{equation}
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\end{multicols*}
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\begin{multicols*}{2}
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