Motor-CAD: Eddy Current Modeling in Conductive Components

A. Magnetic Diffusion Equation

In Motor-CAD’s electromagnetic field formulation, eddy-current-carrying regions are governed by the magnetic diffusion equation: 

where: 

Symbol	Description
𝐀	Magnetic vector potential (B=∇×A𝐁=∇×𝐀)
𝜇	Magnetic permeability
𝜎	Electrical conductivity
𝐯	Local linear velocity (v=ω×r𝐯=𝝎×𝐫)
𝐁𝐫	Remanent flux density (non-zero for PMs, zero for unmagnetized metals)

• The motion-induced term  σ(v×∇×A)𝜎(𝐯×∇×𝐀) applies only to rotating components (e.g., the rotor). 
• For stationary conductive parts (e.g., the stator), this term is omitted.
  
  B. Induced Current Density and Eddy-Current Losses

From the time-varying magnetic vector potential, the induced current density is given by: 

J=σ(−∂A∂t)𝐉=𝜎(−∂𝐀∂t)

The corresponding eddy-current power loss density (power dissipated per unit volume) is: 

Motor-CAD’s transient electromagnetic solver enforces the zero-net-current condition for each conductive region: 

This ensures physical current continuity—particularly important for circumferentially segmented magnets, where localized eddy loops must sum to zero across the entire segment. 

 

C. 2D-to-3D Correction Factors 

Because Motor-CAD’s electromagnetic model is two-dimensional, axial effects are incorporated through an empirical 2D–3D correction factor, k2D3Dk2D3D. Three methods are available depending on the operating frequency and magnet dimensions: 

1. Ruoho Method — Low-Frequency Validity 

Used when the magnet width w is smaller than the skin depth  δ. 
Applicable for low-frequency operation. 

 

2. Wakao–Fujiwara Method — High-Frequency Validity 

Used when  w>δ, i.e., high-frequency conditions. 

Here, the skin depth is calculated as: 

3. Hybrid Method —Full-Frequency Range

Motor-CAD automatically transitions between Ruoho and Wakao–Fujiwara methods based on the ratio w/δ, ensuring accuracy across both low and high frequency regimes. 

Notes: 

• For V-shaped magnets, use the width of the magnet layer closest to the rotor surface (highest eddy-current loss region). 

• For U-shaped magnets, use the average magnet width. 

D. Sleeve and Banding Losses

Motor-CAD applies the same Ruoho-based correction to compute losses in rotor banding and stator sleeves: 

Component	Width (w)	Axial Length (L)
Rotor banding	Pole width	Magnet axial segment length
Stator sleeve	Pole width	k×Lstator+Lend overhang

where k is the sleeve axial length multiplier. 

References 

1. Ruoho et al., “Modeling Magnet Length in 2-D Finite-Element Analysis of Electric Machines,” IEEE Transactions on Magnetics, 2009. 

1. S. Wakao et al., “Useful Formulas of Analytical Investigation in Electromagnetic Field Computation (Part 6),” IEE Japan, 2005.