Cu vs. Al Resistance Considering Skin Effect
This short research topic came up through discussion with my coworkers. We all knew that copper had lower resistance than aluminum and that it was due to resistivity, but didn't know the details of that. We realized that none of us could answer the question “Why does copper have a smaller skin depth than Aluminum?” To gain a complete understanding, we naturally turned to first-principles physics.
The DC Case:
Copper has a lower intrinsic resistivity than aluminum, which is a multiplier in the formula for resistance:
R = ρ * (L / A) where:
R is resistance (Ohms)
ρ is resistivity (Ohm-meters)
L is length (meters)
A is cross-sectional area (square meters)
Lower resistivity means the material is a better conductor.
When an electric field is applied across a conductive material, it causes electrons to move in a specific direction relative to the field. Electrons of equal magnitude are the charge carrier in both copper and aluminum.
Consider the equation for approximating conductivity (Drude Model), which is the inverse of resistivity:
σ ≈ (n * q² * τ) / m (m is the effective mass of the electron)
τ is the mean free time, which is longer in copper due to its atomic structure resulting in less scattering of electrons compared to aluminum.
Understandably, electron mobility is the main factor in the resistivity of materials. Copper's electrons have the ability to move more freely, leading to higher conductivity or inversely lower resistivity.
Room Temperature Resistivity values:
Copper: ≈ 1.68 x 10⁻⁸ Ohm-meters
Aluminum: ≈ 2.65 x 10⁻⁸ Ohm-meters
Aluminum is roughly 1.58 times more resistive than copper.
The AC Case (skin effect):
Let's look at the equation for skin depth:
δ = 1 / √(π * f * μ * σ) Where:
δ = skin depth (meters)
f = frequency
μ = magnetic permeability ≈ μ₀ = 4π x 10⁻⁷ H/m (for both copper and aluminum)
σ = electrical conductivity (S/m)
As described in the DC section, copper has a higher conductivity than aluminum. Looking at the formula above we can see that larger conductivity value will result in lower skin depth. As a result, the skin depth of copper at any given frequency is going to be less than aluminum.
The question remains, "what causes skin effect?"
With AC current, the direction and magnitude of the flow of electrons is constantly changing. Due to Ampere's law, the changing current creates a fluctuating magnetic field around the conductor. This changing magnetic field induces a voltage within the conductor (Faraday's Law), which in turn drives eddy currents. Lenz's law dictates that the induced eddy currents always oppose the change in magnetic field. The changing magnetic field is strongest in the center of the conductor. Consequently, the eddy currents in the center flow opposite to the main current, which create an opposing magnetic field that pushes the main current away to the edges of the conductor. This situation can be thought of as the center of the conductor having an inductive impedance. Current wants to flow in the path of least resistance, therefor redistributing to the surface of the conductor.
Summary:
What we really wanted was to understand why skin effect happens. Working through the cause-and-effect using Maxwell's equations helped us get there and provided the understanding we were looking for. It's now clear that the skin effect is the direct reason for power losses in conductors at high frequencies – those "AC losses" that are so important to consider when designing electromagnetic systems.
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