Magnetic field analysis in electrical machines (e.g., motors, generators, transformers) is critical for optimizing performance, efficiency, and thermal management. Below is a structured approach to perform magnetic field analysis in MATLAB, leveraging its built-in PDE Toolbox and Finite Element Analysis (FEA) capabilities. 1. Problem Setup Key Equations For magnetostatic problems, Maxwell’s equations reduce to: ∇ ⋅ B = 0 and ∇ × H = J ∇⋅B=0and∇×H=J where: B = μ H B=μH (magnetic flux density), μ = μ 0 μ r μ=μ 0 μ r (permeability), J J is the current density. Typical Applications Motor/generator stator-rotor flux distribution. Inductance calculation. Core losses (hysteresis/eddy currents). Force/torque computation. 2. MATLAB Workflow Step 1: Define Geometry Use MATLAB’s PDE Toolbox to model the machine cross-section (2D analysis is common for simplicity). matlab Copy % Example: Create a simple motor geometry (stator + rotor + airgap) model = createpde(‘electromagnetic’,’magnetostatic’); % Define circles for stator, rotor, and airgap stator = [3,4,0,0,0.1,0.1]’; % Outer radius = 0.1 m rotor = [3,4,0,0,0.05,0.05]’; % Inner radius = 0.05 m airgap = [3,4,0,0,0.055,0.055]’; % Airgap radius = 0.055 m gd = [stator, rotor, airgap]; ns = char(‘stator’,’rotor’,’airgap’); g = decsg(gd, ‘stator – rotor – airgap’, ns’); geometryFromEdges(model, g); Step 2: Assign Material Properties Define μ r μ r (relative permeability) for each region: matlab Copy % Stator/Rotor (Silicon Steel, μr = 2000) electromagneticProperties(model, ‘RelativePermeability’, 2000, ‘Region’, ‘stator’); electromagneticProperties(model, ‘RelativePermeability’, 2000, ‘Region’, ‘rotor’); % Airgap (μr = 1) electromagneticProperties(model, ‘RelativePermeability’, 1, ‘Region’, ‘airgap’); Step 3: Define Current Sources Assign current density J J to windings: matlab Copy % Define a coil with current density J = 1e6 A/m² electromagneticSource(model, ‘CurrentDensity’, 1e6, ‘Face’, [coil_faces]); Step 4: Boundary Conditions Dirichlet BC: Set magnetic potential A z = 0 A z =0 at outer boundaries. Periodic BC: For symmetry in rotating machines. matlab Copy % Apply Dirichlet BC on outer stator boundary electromagneticBC(model, ‘Edge’, [outer_edges], ‘MagneticPotential’, 0); Step 5: Generate Mesh matlab Copy generateMesh(model, ‘Hmax’, 0.005); % Hmax controls mesh fineness pdeplot(model) Step 6: Solve Magnetostatic Problem matlab Copy result = solve(model); Step 7: Post-Processing Visualize magnetic flux density ( B B): matlab Copy Bx = result.MagneticFluxDensity.x; By = result.MagneticFluxDensity.y; B = sqrt(Bx.^2 + By.^2); pdeplot(model, ‘XYData’, B, ‘FlowData’, [Bx, By], ‘ColorMap’, ‘jet’); title(‘Magnetic Flux Density (B)’); 3. Torque Calculation For motors, compute torque using the Maxwell Stress Tensor: matlab Copy % Integrate B_tangential * B_normal over the rotor surface [theta, Br, Btheta] = calculateFluxTangential(result, rotor_boundary); mu0 = 4*pi*1e-7; torque = (1/mu0) * trapz(theta, Br .* Btheta) * rotor_radius^2; 4. Example: Permanent Magnet Motor matlab Copy % Define PM material (μr = 1.05, coercivity Hc = -1e6 A/m) electromagneticProperties(model, ‘RelativePermeability’, 1.05, … ‘Magnetization’, [0, -1e6], ‘Face’, pm_faces); 5. Advanced Features Transient Analysis: Use solvepde with time-stepping for eddy currents. Parameter Sweep: Optimize geometry/materials using parfor. Export to Simulink: Couple FEA results with system-level simulations. 6. MATLAB Tools & Functions PDE Toolbox: For 2D/3D FEA. FEMM Integration: Link to Finite Element Method Magnetics (via MATLAB scripting). Optimization Toolbox: For design refinement. 7. Limitations & Alternatives MATLAB’s PDE Toolbox is limited for complex 3D geometries. Consider: ANSYS Maxwell: Industry-standard for electrical machines. COMSOL: Multi-physics FEA. Open-Source Tools: FEMM (2D) or Elmer FEM. Sample Results Flux Density Plot: Flux Density Torque vs. Rotor Position: Torque Resources MATLAB PDE Toolbox Documentation FEMM-MATLAB Integration This workflow allows rapid prototyping of electrical machine designs in MATLAB, though complex industrial applications may require specialized FEA software.

Magnetic field analysis in electrical machines (e.g., motors, generators, transformers) is critical for optimizing performance, efficiency, and thermal management. Below is a structured approach to perform magnetic field analysis in MATLAB, leveraging its built-in PDE Toolbox and Finite Element Analysis (FEA) capabilities.

1. Problem Setup

Key Equations

For magnetostatic problems, Maxwell’s equations reduce to:

$\nabla \cdot B = 0 and \nabla \times H = J$

where:

$B = μ H$ (magnetic flux density),
$μ = μ_{0} μ_{r}$ (permeability),
$J$ is the current density.

Typical Applications

Motor/generator stator-rotor flux distribution.
Inductance calculation.
Core losses (hysteresis/eddy currents).
Force/torque computation.

2. MATLAB Workflow

Step 1: Define Geometry

Use MATLAB’s PDE Toolbox to model the machine cross-section (2D analysis is common for simplicity).

% Example: Create a simple motor geometry (stator + rotor + airgap)
model = createpde('electromagnetic','magnetostatic');
% Define circles for stator, rotor, and airgap
stator = [3,4,0,0,0.1,0.1]'; % Outer radius = 0.1 m
rotor = [3,4,0,0,0.05,0.05]'; % Inner radius = 0.05 m
airgap = [3,4,0,0,0.055,0.055]'; % Airgap radius = 0.055 m
gd = [stator, rotor, airgap];
ns = char('stator','rotor','airgap');
g = decsg(gd, 'stator - rotor - airgap', ns');
geometryFromEdges(model, g);

Step 2: Assign Material Properties

Define $μ_{r}$ (relative permeability) for each region:

% Stator/Rotor (Silicon Steel, μr = 2000)
electromagneticProperties(model, 'RelativePermeability', 2000, 'Region', 'stator');
electromagneticProperties(model, 'RelativePermeability', 2000, 'Region', 'rotor');

% Airgap (μr = 1)
electromagneticProperties(model, 'RelativePermeability', 1, 'Region', 'airgap');

Step 3: Define Current Sources

Assign current density $J$ to windings:

% Define a coil with current density J = 1e6 A/m²
electromagneticSource(model, 'CurrentDensity', 1e6, 'Face', [coil_faces]);

Step 4: Boundary Conditions

Dirichlet BC: Set magnetic potential $A_{z} = 0$ at outer boundaries.
Periodic BC: For symmetry in rotating machines.

% Apply Dirichlet BC on outer stator boundary
electromagneticBC(model, 'Edge', [outer_edges], 'MagneticPotential', 0);

Step 5: Generate Mesh

generateMesh(model, 'Hmax', 0.005); % Hmax controls mesh fineness
pdeplot(model)

Step 6: Solve Magnetostatic Problem

result = solve(model);

Step 7: Post-Processing

Visualize magnetic flux density ( $B$ ):

Bx = result.MagneticFluxDensity.x;
By = result.MagneticFluxDensity.y;
B = sqrt(Bx.^2 + By.^2);
pdeplot(model, 'XYData', B, 'FlowData', [Bx, By], 'ColorMap', 'jet');
title('Magnetic Flux Density (B)');

3. Torque Calculation

For motors, compute torque using the Maxwell Stress Tensor:

% Integrate B_tangential * B_normal over the rotor surface
[theta, Br, Btheta] = calculateFluxTangential(result, rotor_boundary);
mu0 = 4*pi*1e-7;
torque = (1/mu0) * trapz(theta, Br .* Btheta) * rotor_radius^2;

4. Example: Permanent Magnet Motor

% Define PM material (μr = 1.05, coercivity Hc = -1e6 A/m)
electromagneticProperties(model, 'RelativePermeability', 1.05, ...
    'Magnetization', [0, -1e6], 'Face', pm_faces);

5. Advanced Features

Transient Analysis: Use solvepde with time-stepping for eddy currents.
Parameter Sweep: Optimize geometry/materials using parfor.
Export to Simulink: Couple FEA results with system-level simulations.

6. MATLAB Tools & Functions

PDE Toolbox: For 2D/3D FEA.
FEMM Integration: Link to Finite Element Method Magnetics (via MATLAB scripting).
Optimization Toolbox: For design refinement.

7. Limitations & Alternatives

MATLAB’s PDE Toolbox is limited for complex 3D geometries. Consider:
- ANSYS Maxwell: Industry-standard for electrical machines.
- COMSOL: Multi-physics FEA.
- Open-Source Tools: FEMM (2D) or Elmer FEM.

Sample Results

Flux Density Plot:
Torque vs. Rotor Position:

Resources

This workflow allows rapid prototyping of electrical machine designs in MATLAB, though complex industrial applications may require specialized FEA software.