Derivation of an Approximate Formula

S =n(ro -rj )is the cross-sectional area[m ] is the inner radius of the conductor [m] 

it is well known that currents concentrate nearby the outer surface area of the conductor when the frequency of an applied (source) voltage (or current) to the conductor is high. This phenomenon is called skin effect. The depth d of the cross-sectional area where most of the currents flow is given approximately as the (complex) penetration depth or the so-called skin depth in the following form  

The penetration depth is physically defined as the depth for an electromagnetic wave penetrating into a conductor when the wave hits the conductor surface. The physical concept of the penetration depth is very useful to explain the behavior of a current and a voltage on a conductor and also to derive impedance and admittance formulas of various conductor shapes and geometrical configuration. However, it should be reminded that the concept is based on TEM wave propagation and thus is not applicable to non-TEM propagation. Also, remind that it is just an approximation. 

By adopting the penetration depth, the internal impedance Z in a high-frequency region can be derived in the following manner. 

Following the definition of the conductor resistance in Equation 1.3, the internal impedance is given by the ratio of the resistivity and the cross-sectional area S, which is evaluated as 

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