Voltage Forced Coils

This validation case describes a study performed in SimScale to demonstrate a method for modeling 3D voltage forced coils. The validation is based on the formulation studied by P.J. Leonard and D. Rodger\(^1\) in their research, “Voltage Forced Coils for 3D Finite Element Electromagnetic Models.”
In many electromagnetic devices, the magnetic field is generated by a coil or set of coils. When the terminal voltage is specified, rather than a known current, the current becomes an unknown variable that must be solved.

Modeling Assumptions

For this validation, the coil is treated as an ideal stranded coil. It is assumed that:

The current is uniform across the cross-section of each wire (negligible skin and proximity effects).
The wires are packed sufficiently close that the current density can be modeled as homogeneous across the total coil cross-section.

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Validation Scenario

The study evaluates the formulation in two stages:

Free Space: The coil is modeled in isolation to establish baseline behavior.
Conducting Environment: The experiment is repeated with conducting aluminum plates placed above and below the coil to solve a classic eddy current problem.
A circular coil of rectangular cross-section was subjected to a 20V step voltage input using the SimScale Stranded Coil option. The results verify the accuracy of the voltage-forced formulation against established analytical and experimental benchmarks.

Geometry

Two geometric variations were analyzed for this validation: a ‘Coil-with-Plates’ setup, where the coil is centered between two metallic sheets, and an ‘In-Air’ setup, which includes only the coil geometry.

Coil domain — Figure 1: Computational domain representing the coil and the plates.

An air domain is created around the assembly in order to run the simulation. The resulting geometry is the following:

Figure 2: Final geometry used in the present validation case

Analysis Type and Mesh

Analysis Type: Electromagnetics

Model: Time-Transient Magnetostatics

Mesh and Element Types: The meshes from this validation case were created in SimScale with the Standard meshing algorithm.

Find below an overview of the meshes used in this validation study:

Mesh (With Plate)	Mesh Type	Nodes	Element Type
Coarse Mesh	Standard	125364	3D tetrahedral
Moderate Mesh	Standard	213707	3D tetrahedral
Fine Mesh	Standard	545421	3D tetrahedral

Table 1: Standard mesh metrics for the “With Plate” case. The meshes consist exclusively of tetrahedral elements

Mesh (In Air)	Mesh Type	Nodes	Element Type
Coarse Mesh	Standard	75131	3D tetrahedral
Moderate Mesh	Standard	129543	3D tetrahedral
Fine Mesh	Standard	327021	3D tetrahedral

Table 2: Standard mesh metrics for the “In Air” case. The meshes consist exclusively of tetrahedral elements

Figure 3 shows a cross section of how the finest mesh captures the surfaces of the assembly:

Mesh_Cross section — Figure 3: Cross section of the Finest mesh appearance, with 545421 nodes.

Simulation Setup

Material:

Air: flow region
- Material behavior: Soft magnetic
- \((σ)\) Electric conductivity: 0 \(S/m\)
- Magnetic permeability type: Constant
- \((μ_r)\) Relative magnetic permeability: 1
- Core losses: None
Coil
- Material behavior: Soft magnetic
- \((σ)\) Electric conductivity: 3.1e7 \(S/m\)
  - The electric conductivity is calculated based on a resistance of 12.4 ohms
- Magnetic permeability type: Constant
- \((μ_r)\) Relative magnetic permeability: 1
- Core losses: None
Plate
- Material behavior: Soft magnetic
- \((σ)\) Electric conductivity: 3.33e7 \(S/m\)
- Core losses: None

Coils:

Two coils with the same setup are present in this validation case.

Since a quarter model is used, the setup involves an open coil with the following settings:

Topology: Closed
Coil type: Stranded
Number of turns: 700
Wire diameter: 0.001215 \(m\)
Additional resistance: 0 ohms
Excitation: Voltage
\(U(t)\) Voltage: 20 \(V\)

Reference Solution

The reference publication\(^1\) presents experimental data for the current on the coil.

Result Comparison

A mesh sensitivity study was performed with a set of three meshes (Table 1 and 2), focusing on the current observed on the coil. Figures 4 and 5 show how the current evolves as more nodes are added to the mesh:

Figure 4: Mesh sensitivity study for the ‘with plate’ case, showing current transient results for the coarse, moderate, and fine meshes.

Figure 5: Mesh sensitivity study for the ‘in air’ case, showing current transient results for the coarse, moderate, and fine meshes.

The sensitivity study shows great stability between the meshes, despite the great increase in mesh density, indicating mesh-independent results.

Figure 6 presents a comparison between the results obtained in SimScale and the experimental data reported by P.J. Leonard and D. Rodger\(^1\). The simulation demonstrates high fidelity and strong correlation with the experimental benchmarks across both tested environments:

In-Air Scenario: The baseline magnetic behavior aligns closely with the theoretical expectations for an isolated coil.
With-Plate Scenario: The model accurately captures the influence of the conducting plates, reflecting the damping effects of the induced eddy currents.

The data shows a consistent trend, confirming that the SimScale Stranded Coil formulation effectively replicates the physical behavior of voltage-forced electromagnetic systems. This high degree of agreement validates the modeling approach for use in more complex, non-linear, or transient industrial applications.