Electromagnetics for CAE Engineers
Maxwell's Equations to Motor Simulation

Electromagnetic CAE encompasses an enormous range of applications: high-voltage insulation design in power transformers, eddy-current losses in motor laminations, antenna radiation patterns, and precise torque prediction in permanent magnet motors for EVs. The unifying framework is Maxwell's four equations, which we'll build up from first principles before tackling the FEM discretization and practical modeling considerations.

1. Basic Electromagnetic Quantities

1.1 Material Parameters

2. Maxwell's Equations

2.1 Maxwell's Equations (Differential Form)

2.2 Integral Form (Physical Intuition)

2.3 Constitutive Relations for Linear Isotropic Media

2.4 Wave Equation from Maxwell's Equations

Combining Faraday and Ampère-Maxwell in a lossless medium:

This is the wave equation — electromagnetic waves travel at speed $c$ in the medium.

3. Electrostatic Analysis

3.1 Governing Equation

For static charges with no time variation, Faraday's law gives $\nabla\times\mathbf{E} = 0$, allowing us to define a scalar electric potential $\phi$:

Combined with Gauss's law $\nabla\cdot\mathbf{D} = \rho_e$:

3.2 Electric Field Singularities at Sharp Edges

The electric field near a sharp conducting edge (corner angle $\alpha$) varies as $r^{\pi/\alpha - 1}$, where r is the distance from the edge. For a right-angle corner (α = π/2): $E \sim r^1$ (finite). For a sharper notch (α < π/2): field intensifies. For a reentrant corner (α > π): field singularity — this is why all high-voltage electrodes use rounded edges (large radius of curvature).

3.3 Dielectric Breakdown Strength

4. Magnetostatic Analysis

4.1 Governing Equations

From $\nabla\cdot\mathbf{B} = 0$, define the magnetic vector potential $\mathbf{A}$:

Substituting into Ampère's law $\nabla\times\mathbf{H} = \mathbf{J}$:

For linear isotropic media (constant μ), using the Coulomb gauge ($\nabla\cdot\mathbf{A} = 0$):

4.2 Nonlinear B-H Curve

For ferromagnetic materials, $\mathbf{B} = \mu(H)\mathbf{H}$ is nonlinear. The permeability $\mu_r$ drops significantly at magnetic saturation:

• Silicon steel (M270-35A): peak μ_r ≈ 6000 at low H; saturates to μ_r ≈ 1 above ~1.8 T
• The FEM solver uses Newton-Raphson iteration with the tangent permeability $\mu_T = dB/dH$
• Hysteresis (different loading/unloading B-H curves) can be ignored for magnetostatics but matters for core loss calculations

4.3 Magnetic Force Calculation

Two methods are commonly used in FEM to compute force/torque:

Maxwell Stress Tensor:

Virtual Work (Coulomb's method):

5. Eddy Current Analysis and Skin Effect

5.1 Skin Depth

where $\rho = 1/\sigma$ is resistivity, $\omega = 2\pi f$ is angular frequency, and $\mu = \mu_r\mu_0$.

5.2 Eddy Current Problem Formulation

The time-harmonic (phasor) form of the eddy current problem (quasi-static, neglect displacement current):

where $\hat{\mathbf{A}}$ is the complex amplitude of the magnetic vector potential and $\hat{\mathbf{J}}_s$ is the source current density. The $j\omega\sigma$ term couples the magnetic and electric fields and causes the skin effect.

5.3 Eddy Current Losses (Core Loss)

The time-averaged eddy current power loss per unit volume:

This is why motor cores are laminated — thin sheets insulated from each other reduce eddy current loops. The Steinmetz equation captures total core loss including hysteresis:

6. Electromagnetic Waves and High-Frequency Analysis

When the structure size is comparable to the wavelength $\lambda = c/f$, wave effects dominate. The boundary between "quasi-static" and "wave" regimes:

6.1 Helmholtz Wave Equation

For time-harmonic fields ($e^{j\omega t}$ convention):

In lossy media (conductor): $k = \omega\sqrt{\mu(\varepsilon - j\sigma/\omega)}$ — complex wavenumber where the imaginary part describes attenuation.

6.2 Scattering Parameters (S-parameters)

For microwave circuits and antennas, the response is characterized by scattering parameters:

Return loss $= -20\log_{10}|S_{11}|$ dB. For antenna design: |S₁₁| < -10 dB (VSWR < 2) is the common bandwidth criterion.

7. FEM Electromagnetic Analysis

7.1 Edge Elements (Nédélec / Whitney Elements)

For a tetrahedron with nodes 1,2,3,4 and edges 12, 13, ..., 34, the edge element basis functions are:

where $\lambda_i$ are the nodal basis functions (barycentric coordinates). These functions have constant tangential components along edge ij and zero tangential component along all other edges — exactly the right behavior for tangential continuity.

7.2 Gauge Conditions

The magnetic vector potential $\mathbf{A}$ is not unique — adding $\nabla\psi$ for any scalar $\psi$ leaves $\mathbf{B} = \nabla\times\mathbf{A}$ unchanged. A gauge condition must fix this:

• Coulomb gauge: $\nabla\cdot\mathbf{A} = 0$ — natural choice for magnetostatics
• Lorenz gauge: $\nabla\cdot\mathbf{A} + \mu\varepsilon\dot{\phi} = 0$ — natural for wave problems
• Tree-cotree gauge: Used in FEM to eliminate null space of the curl operator

7.3 Absorbing Boundary Conditions for Wave Problems

Electromagnetic wave simulations require absorbing boundary conditions to prevent artificial reflections at the computational domain boundary:

• First-order ABC (Silver-Müller): $\hat{\mathbf{n}}\times\mathbf{H} + \sqrt{\varepsilon/\mu}\,\hat{\mathbf{n}}\times(\hat{\mathbf{n}}\times\mathbf{E}) = 0$
• PML (Perfectly Matched Layer): A fictitious lossy medium that absorbs outgoing waves without reflection — the industry standard for FDTD and FEM wave simulations

8. Motor and Actuator CAE

8.1 Torque Calculation in FEM

Torque computed by integrating the Maxwell stress tensor over a cylindrical surface in the air gap:

where $B_r$ and $B_\theta$ are the radial and tangential flux density components. This integral is performed on a cylindrical surface between rotor and stator — results are insensitive to the exact position of this surface if it's in the air gap.

8.2 Cogging Torque

Cogging torque is the pulsating torque due to the interaction of permanent magnet flux with stator slot geometry — it exists even with no current in the windings. It's a key NVH (noise/vibration) concern in EV drivetrains. In FEM:

• Compute torque vs. rotor angle with zero stator current
• Period = 360° / LCM(number of poles, number of slots)
• Reduction strategies: skewing (rotor or stator), optimizing slot opening, fractional slot/pole combinations

8.3 Key Outputs of Motor EM Simulation

Related Interactive Tools

• Skin Depth Calculator — Compute δ vs. frequency for any material
• RC/RL Circuit Transient Response — Time constant visualization
• Transformer Basic Design Calculator