.. admonition:: Work in Progress :class: warning This document is still under development and may change frequently. ========================== Time-Domain Magnetic Model ========================== .. toctree:: :maxdepth: 0 :caption: Time-Domain Magnetic Documentation :hidden: ../excitation_coil/excitation_coil_model materials/time_domain_magnetic_material This model uses the *quasi-static approximation* of Maxwell's equations where wave and displacement effects are neglected. This model is usually used to model low-frequency applications, where magnetic fields and eddy currents are dominating, such as in electric machines, transformers, and induction heating. The model here uses the so-called **A**-formulation (also known as *electric formulation*), where we are solving for the *magnetic vector potential* :math:`\mathbf{A}`, given by: .. math:: :label: eq:ElectricFormulation \operatorname{curl} \nu \operatorname{curl} \mathbf{A} + \sigma \frac{\partial \mathbf{A}}{\partial t} = \mathbf{J} + \nu \operatorname{curl} \mathbf{B}_r \quad, where: - :math:`\nu` is the rank-2 magnetic reluctivity tensor, - :math:`\sigma` is the rank-2 electrical conductivity tensor, - :math:`\mathbf{J}` is a prescribed divergence-free current density (:math:`\operatorname{div} \mathbf{J}= 0`), - :math:`\mathbf{B}_r` is the *remanent flux density*. Note that the *magnetic flux density* is related to the *magnetic vector potential* by: .. math:: \mathbf{B} = \operatorname{curl} \mathbf{A} \quad, The magnetic reluctivity is the inverse of the magnetic permeability tensor (:math:`\nu = \mu^{-1}`). The electric conductivity tensor :math:`\sigma` is usually a diagonal tensor, where the off-diagonal terms are zero. It describes how well a material conducts electric current. The *remanent flux density* is related to the *remanent magnetization* (:math:`\mathbf{M}_r`) by :math:`\mathbf{B}_r = \mu \mathbf{M}_r`. Eq. :eq:`eq:ElectricFormulation` is augmented by the *constitutive equations*: .. math:: \begin{alignat}{2} \mathbf{H} &= \nu \left( \mathbf{B} - \mathbf{B}_r \right) &\quad, \\ \mathbf{J} &= \sigma \mathbf{E} &\quad, \end{alignat} Note that in general the material properties: :math:`\nu`, :math:`\sigma`, and :math:`\mathbf{M}` can be nonlinear and dependent on other physical properties such as temperature. In Eq. :eq:`eq:ElectricFormulation`, the prescribed current density :math:`\mathbf{J}` on the right-hand side is set through *support models* such as the :doc:`Excitation Coil model `. Materials --------- The model :eq:`eq:ElectricFormulation` requires the provision of material properties, specifically the magnetic reluctivity (inverse permeability), the electrical conductivity and the magnetization. A material is created by specifying *methods* to compute each of those three *material properties*. For more details about materials, please see :doc:`Time-Domain Magnetic Material `. To simplify the setup for common materials, we provide a setup function for common materials, such as vacuum, copper, and iron using following functions. .. figure:: data/time_domain_magnetic_material_domain.png :width: 300 :alt: An example of showing a simulation with three materials: Air, Copper and Steel. :align: center An example of showing a simulation with three materials: Air, Copper and Steel. .. list-table:: Predefined time-domain magnetic materials :widths: 25 75 :header-rows: 1 * - Name - Description * - :doc:`Vacuum ` - Creates a vacuum material with .. math:: \begin{alignat}{2} \mu &= \mu_0 &\quad, \\ \sigma &= 0 &\quad, \\ \mathbf{M} &= 0 &\quad, \end{alignat} where :math:`\mu_0 \approx 4 \pi \times 10^{-7}` is the vacuum permeability. This material can be used for air or vacuum, but also for insulators like plastics, ceramics or glass. * - :doc:`Non-Magnetic ` - Creates a non magnetic material with vacuum permeability, specified electric conductivity and no magnetization. .. math:: \begin{alignat}{2} \mu &= \mu_0 &\quad, \\ \sigma &= \sigma_{\textrm{usr}} &\quad, \\ \mathbf{M} &= 0 &\quad, \end{alignat} Can be used to model non-magnetic but conductive materials, where :math:`\sigma_{\textrm{usr}}` is the electrical conductivity of the material specified by the user. This material can be used for metals like copper or aluminum, but also for conductive liquids like salt water. * - :doc:`Magnetic ` - Creates a magnetic material with non-vacuum permeability, specified electric conductivity and no magnetization. .. math:: \begin{alignat}{2} \mu &= \mu_{\text{usr}} &\quad, \\ \sigma &= \sigma_{\text{usr}} &\quad, \\ \mathbf{M} &= 0 &\quad, \end{alignat} Can be used to model magnetic materials like iron, cobalt or nickel when non-linear effects are not important where :math:`\mu_{\text{usr}}` is the magnetic permeability and :math:`\sigma_{\text{usr}}` is the electrical conductivity of the material specified by the user. * - :doc:`Non-Linear Magnetic ` - Creates a non-linear magnetic material with a constant electrical conductivity. .. math:: \begin{alignat}{2} \mu &= \mu(|\mathbf{B}|)_{\text{usr}} &\quad, \\ \sigma &= \sigma_{\text{usr}} &\quad, \\ \mathbf{M} &= 0 &\quad, \end{alignat} Can be used to model magnetic materials like iron or steel when non-linear effects are important, where :math:`\mu(|\mathbf{B}|)_{\text{usr}}` is the non-linear magnetic permeability of the material and :math:`\sigma_{\text{usr}}` is the electrical conductivity of the material specified by the user. In general a bh curve needs to be provided to define the non-linear magnetic properties. * - :doc:`Magnet ` - Creates a linear magnet .. math:: \begin{alignat}{2} \mu &= \mu_{\text{usr}} &\quad, \\ \sigma &= \sigma_{\text{usr}} &\quad, \\ \mathbf{M} &= \mathbf{M}_{\text{usr}} &\quad, \end{alignat} Can be used to model permanent magnets like neodymium or ferrite magnets where :math:`\mu_{\text{usr}}` is the magnetic permeability of the material and :math:`\sigma_{\text{usr}}` is the electric conductivity of the material and :math:`\mathbf{M}_{\text{usr}}` is the constant magnetization of the material specified by the user. Anisotropic materials can be defined by specifying the permeability tensor and the conductivity tensor, see :doc:`Time-Domain Magnetic Material ` for more details. More complex materials like hysteric, laminated and superconducting materials are supported in their respective modules. Conditions ---------- Following conditions are available for the *time-domain magnetic model*: .. list-table:: List of supported conditions :widths: 25 25 50 :header-rows: 1 * - Name - Type - Description * - :doc:`Tangential Magnetic Flux ` - Boundary Condition - As boundary conditions, we support the *tangential magnetic flux* given by: .. math:: \mathbf{n} \cdot \mathbf{B} = 0 \quad \text{on} \quad \Gamma_0, on the boundary :math:`\Gamma_0`. * - :doc:`Normal Magnetic Field ` - Boundary Condition - The *normal magnetic field* given by: .. math:: \left. \mathbf{H} \times \mathbf{n} \right|_{\Gamma_0} = 0 \quad, * - :doc:`Tangential Magnetic Field ` - Boundary Condition - The *tangential magnetic field* given by: .. math:: \left. \mathbf{H} \times \mathbf{n} \right|_{\Gamma_0} = \mathbf{H}_{\text{usr}} \quad, where :math:`\mathbf{H}_{\text{usr}}` is the *user-defined* external magnetic field imposed on the boundary :math:`\Gamma_0`. Reports ------- Following reports are available for the *time-domain magnetic model* for visualization: .. list-table:: List of reports :widths: 25 25 50 :header-rows: 1 * - Name - Type - Description * - :doc:`Magnetic Force Report ` - Scalar Field - Returns the magnetic force density on an object * - :doc:`Magnetic Torque Report ` - Scalar Field - Returns the magnetic force density on an object Coefficient Functions --------------------- Following functions are available for the *time-domain magnetic model* for visualization or querying: .. list-table:: List of functions :widths: 25 25 50 :header-rows: 1 * - Name - Type - Description * - Relative Magnetic Permeability - Scalar Field - The relative permeability is defined as: .. math:: \mu_r = \frac{1}{3 \mu_0} \frac{1}{\operatorname{tr}\left( \bar{\bar{\nu}} \right)}, where :math:`\operatorname{tr}` is the trace operator and :math:`\mu_0` is the vacuum permeability. * - Magnetic Flux Density - Vector Field - The magnetic flux density is computed as: .. math:: \mathbf{B} = \mathbf{\nabla} \times \mathbf{A}. * - Magnetic Field - Vector Field - The magnetic field is given by: .. math:: \mathbf{H} = \bar{\bar{\nu}} \mathbf{B} - \mathbf{H}_c, where :math:`\mathbf{H}_c` is the coercive force. * - Magnetic Energy Density - Scalar Field - The magnetic energy density is defined as: .. math:: \epsilon = \int_{0}^{|\mathbf{B}|} \mathbf{H} \cdot \mathbf{B} \, \mathrm{d}B. * - Electromechanical Stress Tensor - Tensor Field - The stress tensor is expressed as: .. math:: \bar{\bar{\sigma}}_{\mathrm{em}} = \mathbf{H} \otimes \mathbf{B} - \frac{1}{2} \left( \mathbf{H} \cdot \mathbf{B} \right) \mathbb{I}, where :math:`\mathbb{I}` is the :math:`3 \times 3` identity matrix. * - Ohmic Heating - Scalar Field - The ohmic heating is computed as: .. math:: P_\Omega = \mathbf{J} \cdot \mathbf{E}. * - Lorentz Force Density - Vector Field - The Lorentz force density is given by: .. math:: \mathbf{F}_L = \mathbf{J} \times \mathbf{B}.