Physical Field Theory¶

In this chapter we will describe physical field equations used in Agros2D.

Electrostatic¶

Electrostatic field can be described by Poisson partial differential equation

$\div \varepsilon\, \grad \varphi = \rho,$

where $\varepsilon$ is permittivity of the material, $\varphi$ is electrical scalar potential and $\rho$ is subdomain charge density. Electric field can be written as

$\vec{E} = - \grad \varphi$

and electric displacement is

$\vec{D} = \varepsilon \vec{E}.$

Maxwell stress tensor:

$\vec{S}_\mathrm{M} = \vec{E} \otimes \vec{D} - \frac{1}{2} \vec{E} \vec{D} \cdot \delta$

Boundary conditions¶

Dirichlet BC

Scalar potential $\varphi = f$ on the boundary is known.
Neumann BC

Surface charge density $D_\mathrm{n} = g$ on the boundary is known.

Boundary integrals¶

Charge:

$Q = \int_S D_\mathrm{n} \dif S = \int_S \varepsilon \frac{\partial \varphi}{\partial n} \dif S\,\,\,\mathrm{(C)}$

Subdomain integrals¶

Energy:

$W_\mathrm{e} = \int_V \frac{1}{2} \vec{E} \vec{D} \dif V\,\,\,\mathrm{(J)}$

Maxwell force:

$\vec{F}_\mathrm{M} = \int_S \vec{S}_\mathrm{M} \dif S = \int_V \div \vec{S}_\mathrm{M} \dif V\,\,\,\mathrm{(N)}$

Current Field¶

Current field can be described by Laplace partial differential equation

$\div \sigma\, \grad \varphi = 0,$

where $\sigma$ is electric conductivity of the material and $\varphi$ is electrical scalar potential. Electric field can be written as

$\vec{E} = - \grad \varphi$

and electric current density is

$\vec{J} = \sigma \vec{E}.$

Boundary conditions¶

Dirichlet BC

Scalar potential $\varphi = f$ on the boundary is known.
Neumann BC

Current density $J_\mathrm{n} = \sigma \frac{\partial \varphi}{\partial n} = g$ on the boundary is known.

Boundary integrals¶

Current:

$I = \int_S J_\mathrm{n} \dif S = \int_S \sigma \frac{\partial \varphi}{\partial n} \dif S\,\,\,\mathrm{(A)}$

General Magnetic Field¶

General magnetic field can be desribed by partial differential equation

$\curl \frac{1}{\mu}\, \left( \curl \vec{A} - \vec{B}_\mathrm{r} \right) + \sigma \vec{v} \times \curl \vec{A} + \sigma \frac{\partial \vec{A}}{\partial t} = \vec{J}_\mathrm{ext},$

where $\mu$ is permeability of the material, $\vec{A} = ( A_z\,\mathrm{or}\,A_{\varphi} )$ is component of the magnetic vector potential, $\vec{B}_\mathrm{r}$ is remanent flux density, $\vec{v}$ is velocity, $\sigma$ is electric conductivity and finally $J_\mathrm{ext} = ( \vec{J}_z\,\mathrm{or}\,J_{\varphi} )$ is component of source current density. Magnetic flux density is given by form

$\vec{B} = \curl \vec{A},$

magnetic field is

$\vec{H} = \frac{\vec{B}}{\mu},$

eddy current density is

$J_\mathrm{trans} = \sigma \frac{\partial \vec{A}}{\partial t},$

velocity current density is

$J_\mathrm{vel} = \sigma \vec{v} \times \vec{B} = \sigma \vec{v} \times \curl \vec{A},$

and total current density is

$J_\mathrm{tot} = J_\mathrm{ext} + J_\mathrm{trans} + J_\mathrm{vel}.$

Maxwell stress tensor:

$\mat{S}_\mathrm{M} = \vec{H} \otimes \vec{B} - \frac{1}{2} \vec{H} \vec{B} \cdot \mat{I}$

Boundary conditions¶

Dirichlet BC

Component of the magnetic vector potential $A = f$ on the boundary is known.
Neumann BC

Normal derivative of magnetic vector potential $\frac{\partial A}{\partial n} = g$ on the boundary is known.

Subdomain integrals¶

External current:

$I_\mathrm{ext} = \int_V J_\mathrm{ext} \dif V\,\,\,\mathrm{(A)}$

Eddy current:

$I_\mathrm{trans} = \int_V J_\mathrm{trans} \dif V\,\,\,\mathrm{(A)}$

Velocity current:

$I_\mathrm{vel} = \int_V J_\mathrm{vel} \dif V\,\,\,\mathrm{(A)}$

Total current:

$I_\mathrm{tot} = I_\mathrm{ext} + I_\mathrm{trans} + I_\mathrm{vel}$

Power losses:

$P_\mathrm{j} = \int_V \frac{J_\mathrm{tot}^2}{\sigma} \dif V\,\,\,\mathrm{(W)}$

Energy:

$W_\mathrm{m} = \int_V \frac{1}{2} \vec{H} \vec{B} \dif V\,\,\,\mathrm{(J)}$

Lorentz force:

$\vec{F} = \int_V J_\mathrm{tot} \times \vec{B} \dif V = \int_V J_\mathrm{tot} \times \curl \vec{A} \dif V\,\,\,\mathrm{(N)}$

Maxwell force:

$\vec{F}_\mathrm{M} = \oint_S \mat{S}_\mathrm{M} \dif S \,\,\,\mathrm{(N)}$

Torque (planar arrangement only):

$T_\mathrm{z} = \int_V \vec{r} \times \vec{F} \dif V\,\,\,\mathrm{(Nm)}$

Harmonic Magnetic Field¶

Harmonic magnetic field can be described by partial differential equation

$\curl \frac{1}{\mu}\, \left( \curl \vecfaz{A} - \vec{B}_\mathrm{r} \right) + \sigma \vec{v} \times \curl \vecfaz{A} + \mj \omega \sigma \vecfaz{A} = \vecfaz{J}_\mathrm{ext},$

where $\mu$ is permeability of the material, $\faz{A} = ( \faz{A}_z\,\mathrm{or}\,\faz{A}_{\varphi} )$ is component of the magnetic vector potential, $\omega = 2 \pi f$ is frequency, $\sigma$ is electric conductivity, $\vec{v}$ is velocity and finally $\faz{J}_\mathrm{ext} = ( \faz{J}_z\,\mathrm{or}\,\faz{J}_{\varphi} )$ is component of source current density. Magnetic flux density is given by form

$\vecfaz{B} = \curl \vecfaz{A},$

magnetic field is

$\vecfaz{H} = \frac{\vecfaz{B}}{\mu},$

eddy current density is

$\vecfaz{J}_\mathrm{trans} = \mj \omega \sigma \vecfaz{A},$

velocity current density is

$\vecfaz{J}_\mathrm{vel} = \sigma \vec{v} \times \vecfaz{B} = \sigma \vec{v} \times \curl \vecfaz{A},$

and total current density is

$\vecfaz{J}_\mathrm{tot} = \vecfaz{J}_\mathrm{ext} + \vecfaz{J}_\mathrm{trans} + \vecfaz{J}_\mathrm{vel}.$

Boundary conditions¶

Dirichlet BC

Component of the magnetic vector potential $\faz{A} = \faz{f}$ on the boundary is known.
Neumann BC

Normal derivative of magnetic vector potential $\frac{\partial \faz{A}}{\partial n} = \faz{g}$ on the boundary is known.

Subdomain integrals¶

External current:

$\faz{I}_\mathrm{ext} = \int_S \vecfaz{J}_\mathrm{ext} \dif S\,\,\,\mathrm{(A)}$

Eddy current:

$\faz{I}_\mathrm{trans} = \int_S \vecfaz{J}_\mathrm{trans} \dif S\,\,\,\mathrm{(A)}$

Velocity current:

$\faz{I}_\mathrm{vel} = \int_S \vecfaz{J}_\mathrm{vel} \dif S\,\,\,\mathrm{(A)}$

Total current:

$\faz{I}_\mathrm{tot} = \faz{I}_\mathrm{ext} + \faz{I}_\mathrm{trans} + \faz{I}_\mathrm{vel}$

Power losses:

$P = \int_V \frac{\left( \vecfaz{J}_\mathrm{tot} \cdot \vecfaz{J}_\mathrm{tot}^* \right)}{\sigma} \dif V\,\,\,\mathrm{(W)}$

Lorentz force:

$F_\mathrm{L} = \int_V \vecfaz{J}_\mathrm{tot} \times \vecfaz{B} \dif V\,\,\,\mathrm{(N)}$

Average energy:

$W_\mathrm{m} = \int_V \frac{1}{2} \vecfaz{H} \vecfaz{B} \dif V\,\,\,\mathrm{(N)}$

Heat Transfer¶

Heat transfer can be described by partial differential equation

$\div \lambda\, \grad T - \rho c_\mathrm{p} \frac{\partial T}{\partial t} = -w,$

where $\lambda$ is thermal conductivity, $T$ is temperature, $\rho$ is density, $c_\mathrm{p}$ is specific heat and finally $w$ is source of the inner heat (eddy current, chemical source, ...). Term with partial derivative is in steady-state analysis neglected. Thermal flux can be written as

$\vec{F} = \lambda\, \grad T$

and temperature gradient is

$\vec{G} = \grad T.$

Boundary conditions¶

Dirichlet BC

Temperature $T = f$ on the boundary is known.
Neumann BC

Thermal heat flux $q = - \lambda \frac{\partial T}{\partial n}$ on the boundary is known.
Mixed BC

Thermal heat flux due to convection into the environment $q = - \lambda \frac{\partial T}{\partial n} = \alpha \left( T - T_{\mathrm{ext}}\right)$ on the boundary is known.

Boundary integrals¶

Average temperature:

$T_\mathrm{avg} = \frac{1}{S} \int_S T \dif S\,\,\,\mathrm{(deg.)}$

Heat flux:

$F = \int_S \lambda \frac{\partial T}{\partial n} \dif S\,\,\,\mathrm{(W)}$

Subdomain integrals¶

Average temperature:

$T_\mathrm{avg} = \frac{1}{V} \int_V T \dif V\,\,\,\mathrm{(deg.)}$

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