Litz Wire
Theory and Physics
Principle of Litz Wire
Professor, why can Litz wire reduce AC losses?
By twisting together multiple thin insulated strands, the diameter of each strand is made smaller than the skin depth $\delta$, thereby suppressing the skin effect and proximity effect.
If the strand diameter $d_{strand}$ is sufficiently smaller than $\delta$:
Is the twisting method also important?
The purpose of twisting is to ensure each strand passes through all positions equally within the conductor cross-section (transposition). Incomplete twisting leads to uneven current distribution among strands (circulating currents), increasing losses. The hierarchical structure of bundle twisting (Type 1, 2, 3) is important.
Summary
- $d_{strand} < \delta$ — Condition for skin effect suppression
- Twisting (Transposition) — Equalization among strands
- Applications — High-frequency transformers, wireless power transfer coils
The Invention of Litz Wire—Bundles of Thin Wires that Supported Early 20th Century Wireless Telegraphy
Litz wire originates from the German word "Litzendraht" (stranded wire) and was first used in the early 1900s for the transmitting coils of high-frequency wireless communication (spark-gap transmitters). The idea of insulating and twisting thin wires to suppress increased high-frequency resistance due to the skin effect was born from practical necessity during the Marconi wireless telegraph era. Modern applications in wireless power transfer (WPT) and high-frequency inductors are examples of a 100-year-old technology returning to the cutting edge in a new form.
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) produces voltage in a nearby coil by rotating a magnet—a direct application of this law stating that a changing magnetic field induces an electric field. Induction heating (IH) cooktops also use this principle, where high-frequency magnetic field changes induce eddy currents in the pot bottom, heating it via Joule heating.
- Magnetic field term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère-Maxwell's law. Electric 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, passing current through a coil to create a strong magnetic field. Smartphone speakers also apply this law: current → magnetic field → force on the diaphragm. At high frequencies (e.g., GHz-band antennas), the displacement current $\partial D/\partial t$ becomes significant and describes 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 a plastic sheet on hair makes hair stand up due to static electricity—the charged sheet (electric charge) radiates electric field lines outward, 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 the non-existence of magnetic monopoles. 【Everyday Example】Cutting a bar magnet in half does not create a magnet with only a N or S pole—they always exist in pairs. 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.
- Isotropic assumption: For anisotropic materials (e.g., silicon steel rolling direction), direction-specific property definitions are needed.
- Non-applicable cases: Additional constitutive laws are required 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 Oersted (Oe): 1 Oe = 79.577 A/m |
| Current Density $J$ | A/m² | Calculated from conductor cross-section 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
Litz Wire Modeling in FEM
Modeling hundreds of strands individually must be tough, right?
Individual strand models are accurate but computationally expensive. Practically, a homogenized model is used.
- JMAG: Automatically calculates equivalent properties of Litz wire using the FEM Coil function.
- COMSOL: Sets equivalent conductivity using the Homogenized Multi-Turn Coil feature.
- Extended Dowell's Formula: Analytically calculates the AC resistance factor $F_r$ from strand diameter, number of strands, and twist pitch.
How do you handle circulating current losses?
Circulating current losses due to imperfect twisting are difficult to capture in homogenized models. JMAG has a method using a "partial model" that individually models a representative few strands to evaluate circulating currents.
Summary
- Homogenized Model — Practical. ~90% accuracy.
- Individual Strand Model — High accuracy but high cost.
- Circulating Currents — Additional losses when twisting is imperfect.
FEM for Litz Wire—How to Model Tens of Thousands of Thin Wires
Litz wire has a structure bundling tens to tens of thousands of thin wires. Faithfully modeling all strands leads to an enormous mesh and computational failure. A practical approach is the "homogenization method," which replaces the structure with an equivalent anisotropic conductivity tensor considering the volume fill factor and orientation of the strands, allowing analysis with macroscopic FEM. The accuracy of this homogenized model depends on the number of strands, twist count, and frequency; errors increase at high frequencies, requiring careful judgment of its applicable range.
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 non-homogeneous media. BEM: Naturally handles infinite domains (open boundary 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 typical.
Frequency Domain Analysis
Reduced to a steady-state problem by assuming time-harmonic conditions. Requires complex number arithmetic, but broadband characteristics are obtained via time-domain analysis.
Time Domain Time Step
Time step smaller than 1/20 of the highest frequency component is required. Implicit time integration allows larger steps but requires attention to accuracy.
Choosing Between Frequency Domain and Time Domain
Frequency domain analysis is like "tuning a radio to a specific frequency"—it can efficiently compute 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 at a higher computational cost.
Related Topics
なった
詳しく
報告