Because high frequency signals do not penetrate well into good conductors, the resistance associated with a conductor at these high frequencies will be higher than the dc resistance. This effect is know as the skin effect since the high frequency current flows in a thin layer near the surface of the conductor. The formula to determine the effective skin depth for a conductor is shown below.
for pure copper
In this case, d is the skin depth (in m), f is the frequency of interest (in Hz), m is the permeability of the material (m_{o}, or 1.2566E-6 H/m for most materials),s is the conductivity of the material (in Siemens/m or 1/r where r is the resistivity in ohm-m).
If a circular wire is used with radius a, the effective resistance of the wire can be calculated as shown below where l is the length of the wire and the other variables are defined as above.
when a >> d
The above equation applies for those cases where the skin depth is between 0 and the wire radius, a. If the skin depth is larger than the wire radius, then the equivalent ac resistance of the wire is no different than the dc resistance and is merely determined by the standard formula using the entire wire cross-sectional area. As the frequency approaches zero (dc), the skin depth becomes infinite and as the freqency increases, the skin depth becomes smaller and smaller.The following table shows how the skin depth varies with some example conductor materials (pure copper and pure aluminum) for typical pulsed power and power conditioning frequencies in the range of 1 kHz to 1 GHz.
Frequency (Hz) |
Skin Depth in Copper (cm) |
Skin Depth in Aluminum (cm) |
1000 |
2.09E-1 |
2.68E-1 |
10,000 |
6.61E-2 |
8.46E-2 |
100,000 |
2.09E-2 |
2.68E-2 |
1,000,000 |
6.61E-3 |
8.46E-3 |
10,000,000 |
2.09E-3 |
2.68E-3 |
100,000,000 |
6.61E-4 |
8.46E-4 |
1,000,000,000 |
2.09E-4 |
2.68E-4 |
The calculator below can be used to determine the skin depth for a given material alloy and operating frequency. It will also display the resistivity, conductivity, and permeability assumed for the chosen material. It is suggested that users double check this information since material resistivity can vary depending upon the exact composition, tempering, etc. of the selected material alloy. Credit for the initial Javascript code used in the calculator is given to Ray Allen who has a number of similar useful calculators on his website, Pulsed Power Portal.
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