1. Skin Effect 
The skin effect occurs when alternating current (AC) flows primarily near the surface of a conductor at higher frequencies. This reduces the effective cross-sectional area and increases resistance.

Skin Depth Formula: 
δ = sqrt(2 * ρ / (μ * ω))

Where: 

• δ: Skin depth (m)
• ρ: Resistivity of the conductor (Ω·m)
• μ: Magnetic permeability (H/m), μ = μ₀ * μᵣ
• ω: Angular frequency, ω = 2 * π * f (rad/s)
• f: Frequency (Hz)

Skin Effect Resistance Increase: 
R_ac = R_dc * F_skin

For a single round conductor:
F_skin = I₁(u) / I₀(u)
 
Where u = r / δ and I₀, I₁ are modified Bessel functions. 

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2. Proximity Effect 
The proximity effect arises when magnetic fields from nearby conductors redistribute the current in a conductor, leading to increased resistance.

Proximity Effect Resistance Increase: 
R_ac = R_dc * F_proximity
 
Where F_proximity depends on conductor spacing, diameter, and frequency.

For bundles of wires:
F_proximity ∝ (wire spacing / wire diameter)²
 
Smaller spacing between wires increases proximity losses.

Proximity Effect in Litz Wires: 
In Litz wires:

• Divide the total wire into smaller bundles.
• Twist the bundles uniformly to distribute magnetic interactions.

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3. Combined Losses in Litz Wires 
The total AC resistance of a Litz wire is given by:
R_ac = R_dc * (1 + F_skin + F_proximity)

Minimizing Losses: 

• Skin Effect: Ensure strand diameter (d) is smaller than 2 * δ (skin depth).
• Proximity Effect: Optimize twist pitch (lay length) of Litz wire bundles.

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4. Practical Example 

Given Data: 

• Conductor material: Copper
• Frequency: f = 1 MHz
• Resistivity: ρ = 1.68 × 10⁻⁸ Ω·m
• Relative permeability: μᵣ = 1

Step 1: Calculate Skin Depth (δ) 
δ = sqrt(2 * ρ / (μ₀ * μᵣ * 2πf))
 
Substitute the values: 
δ = sqrt(2 * 1.68 × 10⁻⁸ / (4π × 10⁻⁷ * 2π * 10⁶))
δ ≈ 66 μm

Step 2: Select Strand Diameter (d) 
Ensure d < 2 * δ: 
d < 2 * 66 μm = 132 μm

Step 3: Total AC Resistance (R_ac) 
Using the combined formula:
R_ac = R_dc * (1 + F_skin + F_proximity)
 
Where F_skin and F_proximity are calculated based on strand geometry and arrangement.

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5. Key Takeaways 

• The skin effect becomes significant at high frequencies; use fine strands with diameters smaller than the skin depth to reduce losses.
• The proximity effect depends on wire spacing and arrangement; proper bundling and twisting are critical to minimizing losses.
• Litz wires are highly effective for applications like transformers, inductors, and high-frequency motors.

Conclusion 
By understanding and applying the principles of skin and proximity effects, engineers can optimize Litz wire designs for superior performance in high-frequency applications. Proper calculations and design strategies ensure minimal resistance and efficient energy transfer.

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