Excitation Coils

Electric currents flowing through coils of conducting wire induce magnetic fields. Simcenter STAR-CCM+ allows you to include the effects of excitation coils in the magnetic field calculation.

Simcenter STAR-CCM+ calculates the electric current density produced by an excitation coil, $J_{coil}$ , and adds it as a source term in the magnetic vector potential equation (see Magnetic Vector Potential Models).

For the calculation of $J_{coil}$ , Simcenter STAR-CCM+ provides the Excitation Coil model, a basic approach that is generally suitable for magnetostatic cases, and the Finite Element Excitation Coil model, a more elaborate approach that is also suitable for eddy current simulations.

In both cases, the electric current density produced by an excitation coil can be written as:

Figure 1. EQUATION_DISPLAY

J_{c o i l} = I_{c o i l} τ

(4332)

where $I_{c o i l}$ is the electric current that flows through an individual coil turn and $τ$ is the coil unit electric current density (that is, the electric current density obtained when a current of 1A is circulating in a wire).

For a consistent problem definition, the function $τ$ must be divergence-free:

Figure 2. EQUATION_DISPLAY

\nabla\cdot τ = 0

(4333)

Although the Excitation Coil model allows you to define $τ$ using geometric properties, this model does not enforce Eqn. (4333) during the simulation. This approach can be sufficient in magnetostatic cases, but it is not suitable for eddy current simulations.

For eddy current simulations, the Finite Element Excitation Coil model automatically ensures that the compatibility requirement (Eqn. (4333)) is met, providing a dedicated solver that runs before the magnetic vector potential solution.

Finite Element Excitation Coil Model

To calculate $τ$ , Simcenter STAR-CCM+ solves the electric current conservation equation (Eqn. (4333)) directly. Eqn. (4333) can be written as:

Figure 3. EQUATION_DISPLAY

\nabla\cdot (- α \nabla ψ) = 0

(4334)

where $α$ is the coil conductivity and $ψ$ is a scalar potential. The conductivity $α$ is modified in an iterative process to ensure that the electric current density in the coil is homogeneous throughout a cross-section.

The boundary conditions at inflow, outflow, and insulating boundaries are:

Figure 4. EQUATION_DISPLAY

\begin{array}{l} \int_{S} τ \cdot d S = \mp s n_{t} & inflow, outflow \\ \int_{S} τ \cdot d S = 0 & insulating \end{array}

(4335)

where $n_{t}$ is the number of coil turns and $s = \pm 1$ is the specified local direction field factor that defines the direction of the electric current (Original = 1, Flipped = -1).

In 2D simulations, for a region of cross section $a$ , $τ$ simplifies to $τ = \frac{s n_{t}}{a} [0, 0, 1]$ .

Excitation Coil Model

In this approach, the unit current density $τ$ is specified by scaling a general vector field $f$ according to the flux through a specified cross section. For example, for cylindrical coils, $f$ can be defined as a constant vector profile [0, 1, 0] in a relevant cylindrical coordinate systems. $τ$ can then be written as:

Figure 5. EQUATION_DISPLAY

τ = \frac{n_{t}}{a_{0}} f

(4336)

where the scaling factor $a_{0}$ depends on the specified number of sections $n_{s}$ and reference area $a_{ref}$ through:

Figure 6. EQUATION_DISPLAY

a_{0} = \frac{1}{n_{s}} \int_{a_{ref}} | f \cdot d a |

(4337)

Reports

The magnetic flux linkage associated with a coil is defined as the line integral of the magnetic vector potential $A$ around the contour of the coil:

Figure 7. EQUATION_DISPLAY

ψ = \int_{A} B \cdot d a = \int_{L} A \cdot d l

(4338)

By writing the line element $d l$ as:

Figure 8. EQUATION_DISPLAY

d l = τ d V

(4339)

Eqn. (4338) can be written in terms of $a_{0}$ and $n_{t}$ as:

Figure 9. EQUATION_DISPLAY

ψ = \int_{V} A \cdot τ dV = \int_{V} ρ_{ψ} dV

(4340)

where $ρ_{ψ}$ is called the specific magnetic flux linkage:

Figure 10. EQUATION_DISPLAY

ρ_{ψ} = A \cdot τ

(4341)

The time derivative of the specific magnetic flux linkage defines the specific electromotive force induced by the coil:

Figure 11. EQUATION_DISPLAY

ρ_{ε} = \frac{d ρ_{ψ}}{d t}

(4342)

The electrical resistance of a conducting coil can be calculated as:

Figure 12. EQUATION_DISPLAY

R = \int_{V} \frac{1}{σ p_{c}} {| τ |}^{2} ⅆ V

(4343)

where $σ$ is the electrical conductivity and $p_{c}$ is the   percentage of the coil cross section that is covered by conducting material.