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