Skin Depth Confusion: Investigating Different Equations for Paul

[paul_harris77]

Dear all

I am slightly confused over the equations for skin depth. My university notes give me the equations:

$$\delta =$$ tan-1 (tan$$\delta$$) = $$\frac{\sigma}{\omega \epsilon}$$ (loss tangent)

where $$\delta$$ is skin depth and $$\sigma$$ is conductivity.

I am also given the equation:

$$\delta =$$ $$\frac{1}{\sqrt{\pi f \mu \sigma}}$$

However, for the situation below, they both yield different skin depths.

f = 1MHz

w = $$2\pi f$$

$$\sigma = 5.8 \times 10^{7}$$ Sm-1

Using the first equation:

$$\delta =$$ tan-1( $$\frac{5.8\times 10^{7}}{2\pi \times 1 \times 10^{6} \times 8.85 \times 10^{-12}} = 1.57m$$)

Using the second equation:

$$\delta =$$ $$\frac{1}{\sqrt{\pi \times 1 \times 10^{6} \times 4\pi \times 10^{-7} \times 5.8 \times 10^{7}}} = 66.09\mu m$$

It seems like the first equation gives 1.57 for all large values of the loss tangent, whereas the second equation gives the correct result.

Is the first equation valid for a certain range of loss tangents only?

Any help would be greatly appreciated.

Many thanks

Regards

Paul

 

[Bob S]

You are using two different definitions for δ.

The angle δ = tan^-1(loss tangent) is in radians

The skin depth δ is in length (e.g., mm).

Bob S