Transverse Magnetic Modes

For magnetic fields that lie on a 2D domain, the magnetic vector potential has only one component perpendicular to the 2D domain. This configuration is known as Transverse Magnetic (TM).

Transverse Magnetic Potential Equations

Simcenter STAR-CCM+ calculate the transverse magnetic potential using the finite volume approach. When the transverse magnetic potential has harmonic time dependence, Simcenter STAR-CCM+ solves for the complex phasor representation of the transverse magnetic potential.

Transverse Magnetic Potential

For a magnetic field that lies on the x-y plane, Simcenter STAR-CCM+ calculates the transverse magnetic potential $A_{z}$ from:

Figure 1. EQUATION_DISPLAY

- \int_{A}^{} \frac{1}{μ} \nabla A_{z} \cdot d a + \int_{V}^{} σ \frac{\partial A_{z}}{\partial t} d V = \int_{V}^{} J_{z, e x} d V

(4319)

which is obtained by integrating Eqn. (4243) over the cell domain. Eqn. (4319) is spatially discretized and solved using the finite volume method.

In steady and quasi-unsteady applications, the transient term, $\partial A / \partial t$ , vanishes.

Harmonic Balance FV Transverse Magnetic Potential

For single harmonic time dependence, Simcenter STAR-CCM+ calculates the complex transverse magnetic potential

{\hat{A}}_{z}

from:

Figure 2. EQUATION_DISPLAY

\begin{array}{l} \begin{matrix} - \oint_{A} (\frac{1}{μ} \nabla A_{z}^{'}) \cdot d a - ω_{0} \int_{V} (σ^{"} A_{z}^{'} + σ^{'} A_{z}^{"}) d V = \int_{V} J_{z, e x}^{'} \end{matrix} d V \\ \begin{matrix} - \oint_{A} (\frac{1}{μ} \nabla A_{z}^{"}) \cdot d a + ω_{0} \int_{V} (σ^{'} A_{z}^{'} - σ^{"} A_{z}^{'}) d V = \int_{V} J_{z, e x}^{"} \end{matrix} d V \end{array}

(4320)

which is obtained by integrating Eqn. (4269) over the cell domain. Eqn. (4320) is discretized and solved using the finite volume method.

Current Conservation

In transient simulations, you can choose whether Simcenter STAR-CCM+ calculates the transverse magnetic potential while imposing the conservation of electric current within the region. When conserving the electric current, Simcenter STAR-CCM+ scales the total electric current density in Eqn. (4319), or Eqn. (4320), such that:

Figure 3. EQUATION_DISPLAY

\int_{A}^{} J_{z} d a = 0

(4321)

where the integral is over the surface that bounds the whole region. The total electric current density in Eqn. (4320) is generally complex, ${\hat{J}}_{z}$ .

Source Terms

For transverse magnetic modes, the electric current density $J_{ex}$ (see Eqn. (4311)) has only one component, $J_{z, ex}$ , which is normal to the 2D domain. For single harmonic time dependence (Eqn. (4320)), ${\hat{J}}_{z, ex}$ is a complex quantity.

Boundary and Interface Conditions

At the domain boundaries, the solution must satisfy either Dirichlet boundary conditions, which define the magnetic vector potential $A_{z}$ normal to the boundary, or Neumann boundary conditions, which define the electric current sheet $J_{S, z}$ normal to the boundary. In the harmonic case (Eqn. (4320)), the prescribed quantities are complex.

In many applications, such as electrical machines, the cross-sectional field analysis can be reduced to either an odd number or an even number of poles by using anti-periodic and periodic interfaces, respectively. The magnetic potential has opposite sign at each side of an anti-periodic interface:

Figure 4. EQUATION_DISPLAY

\begin{matrix} A_{z} (r, θ) = {- A}_{z} (r, θ + \frac{2 (k - 1) π}{p}) & k = 1, 2, 3... \end{matrix}

(4322)

and the same sign at each side of a periodic interface:

Figure 5. EQUATION_DISPLAY

\begin{matrix} A_{z} (r, θ) = A_{z} (r, θ + \frac{2 k π}{p}) & k = 1, 2, 3... \end{matrix}

(4323)

where $p$ is the even or odd number of pole pairs and $r$ , $θ$ are polar coordinates.