Solenoid Design
Theory and Physics
Solenoid Magnetic Field
Professor, what is the formula for the magnetic field of a solenoid?
Internal magnetic field of an infinitely long solenoid:
$n$: Number of turns per unit length [turns/m], $I$: Current [A]. The field is uniform inside and zero outside (in the ideal case).
For a finite-length solenoid, the magnetic field weakens at the ends, right?
At the ends, it's about half the central value. For MRI magnets or Helmholtz coils requiring a uniform field, end-correction design is crucial.
Solenoid Actuator
Electromagnetic valves used in automobiles and industrial machinery are plunger-type solenoids. When current flows through the coil, the iron core plunger is attracted. The attractive force:
The force increases as the gap closes (nonlinear). Calculate the force-stroke characteristic using FEM.
Summary
- $B = \mu_0 n I$ — Ideal solenoid
- Attractive force $\propto B^2$ — Larger for smaller gaps
- Force-stroke curve via FEM — Fundamental for actuator design
Solenoid Physics—Magnetic Energy Conversion Where a Coil "Generates Force"
An electromagnetic solenoid is a device where the magnetic field created by an energized coil attracts a plunger (movable iron core), responsible for converting electrical energy → magnetic energy → mechanical energy. The attractive force F is obtained from the displacement derivative of magnetic energy (F=dW/dx), and the force increases as the air gap length decreases because the magnetic reluctance drops. The theoretical attractive force for a flat-gap type solenoid is given by F=B²A/(2μ₀) (A is the gap area), generating about 4 kN of force per 1 cm² area at 1 T magnetic field. In CAE, the attractive force at each gap position is calculated from FEM magnetic field analysis to design the "stroke-force characteristic curve".
Physical Meaning of Each Term
- Electric field term $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$: Faraday's law of electromagnetic induction. A time-varying magnetic flux density generates an electromotive force. [Everyday example] A bicycle dynamo (generator) generates voltage in a nearby coil by rotating a magnet—a direct application of this law that a changing magnetic field induces an electric field. Induction cooking (IH) heaters also use the same principle, where high-frequency magnetic field changes induce eddy currents in the pot bottom, heating it via Joule heat.
- Magnetic field term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère-Maxwell's law. Current and displacement current generate a magnetic field. [Everyday example] When current flows through a wire, a magnetic field is created around it—this is Ampère's law. Electromagnets operate on this principle, creating a strong magnetic field by passing current through a coil. Smartphone speakers also apply this law: current → magnetic field → force on the diaphragm. At high frequencies (e.g., GHz-band antennas), the displacement current $\partial D/\partial t$ cannot be ignored, describing electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: Indicates that electric charge is the divergence source of electric flux. [Everyday example] Rubbing hair with a plastic sheet creates static electricity, making hair stand up—electric field lines radiate from the charged sheet (charge), exerting force on the light hair. Capacitor design uses this law to calculate the electric field distribution between electrodes. ESD (electrostatic discharge) countermeasures are also based on electric field analysis following Gauss's law.
- Magnetic flux conservation $\nabla \cdot \mathbf{B} = 0$: Expresses that magnetic monopoles do not exist. [Everyday example] Cutting a bar magnet in half does not create a magnet with only an N pole or only an S pole—N and S poles always exist as a pair. This means magnetic field lines form "closed loops with no start or end points." In numerical analysis, the formulation using vector potential $\mathbf{B} = \nabla \times \mathbf{A}$ is used to satisfy this condition, automatically guaranteeing magnetic flux conservation.
Assumptions and Applicability Limits
- Linear material assumption: Permeability and permittivity are independent of magnetic/electric field strength (nonlinear B-H curve needed in saturation region)
- Quasi-static approximation (low frequency): Displacement current term can be ignored ($\omega \varepsilon \ll \sigma$). Common in eddy current analysis
- 2D assumption (cross-section analysis): Effective when current direction is uniform and end effects can be ignored
- Isotropy assumption: For anisotropic materials (e.g., silicon steel rolling direction), direction-specific property definitions are needed
- Non-applicable cases: Additional constitutive relations are needed for plasma (ionized gas), superconductors, nonlinear optical materials
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Magnetic Flux Density $B$ | T (Tesla) | 1T = 1 Wb/m². Permanent magnets: 0.2〜1.4T |
| Magnetic Field Strength $H$ | A/m | Horizontal axis of B-H curve. Conversion from CGS Oe (Oersted): 1 Oe = 79.577 A/m |
| Current Density $J$ | A/m² | Calculated from conductor cross-sectional area and total current. Note non-uniform distribution due to skin effect |
| Permeability $\mu$ | H/m | $\mu = \mu_0 \mu_r$. In vacuum $\mu_0 = 4\pi \times 10^{-7}$ H/m |
| Electrical Conductivity $\sigma$ | S/m | Copper: approx. 5.96×10⁷ S/m. Decreases with temperature rise |
Numerical Methods and Implementation
Solenoid FEM
A 2D axisymmetric model is the most efficient. Vary the plunger position parametrically to automatically calculate the force-stroke curve.
Parametric Analysis
1. Vary the gap $g$ from 0.1mm to 10mm
2. For each $g$, perform magnetic field FEM → calculate electromagnetic force
3. Create a force-stroke curve
4. The intersection point with the spring force is the operating point
JMAG and Maxwell automate this parametric sweep.
Summary
- 2D axisymmetric is standard — Low computational cost
- Parametric analysis — Automatic force calculation for gap variation
Numerical Optimization in Solenoid Design—Parametric Analysis of Coil Turns and Magnetic Core Shape
To maximize solenoid attractive force while minimizing power consumption, simultaneous optimization of coil turns, wire diameter, and magnetic core shape is necessary. Because there are trade-offs between design variables (more turns → increased force & increased heat generation), multi-objective optimization (Pareto optimal) is effective. Execute a parametric sweep with FEM, calculate magnetic flux density, attractive force, and coil resistance for each variable combination, then visualize the optimal solution space by drawing the Pareto front. ANSYS Optimetrics and JMAG-Optimizer have features to perform this multi-objective parametric optimization via GUI.
Edge Elements (Nedelec Elements)
Elements specialized for electromagnetic field analysis. Automatically guarantee continuity of tangential components and eliminate spurious modes. Standard for 3D high-frequency analysis.
Nodal Elements
Used for scalar potential formulations. Effective for scalar potential methods in magnetostatics and electrostatic field analysis.
FEM vs BEM (Boundary Element Method)
FEM: Handles nonlinear materials and inhomogeneous media. BEM: Naturally handles infinite domains (open region problems). Hybrid FEM-BEM is also effective.
Nonlinear Convergence (Magnetic Saturation)
Nonlinearity of B-H curve handled by Newton-Raphson method. Residual criterion: $||R||/||R_0|| < 10^{-4}$ is common.
Frequency Domain Analysis
Reduced to a steady-state problem by assuming time-harmonic conditions. Requires complex number operations, but wideband characteristics are obtained via time-domain analysis.
Time Domain Time Step
Time step less than 1/20 of the highest frequency component is required. Implicit time integration allows larger steps but accuracy must be considered.
Choosing Between Frequency Domain and Time Domain
Frequency domain analysis is like "tuning a radio to a specific frequency"—it can efficiently calculate the response at a single frequency. Time domain analysis is like "recording all channels simultaneously"—it can reproduce transient phenomena containing all frequency components but has high computational cost.
Practical Guide
Practical Applications
Design of solenoid valves, relays, locking mechanisms, injectors.
Checklist
Related Topics
なった
詳しく
報告