**Skin Effect ^{1}**

The “Skin effect” is a phenomenon that occurs for the conductors carrying an AC current where the current density has the maximum value at the outer surface of the conductor, and its magnitude decreases exponentially as we move radially towards the center of the conductor, where the current density has its minimum value at the conductor’s central line (conductor axis).

The frequency and the intensity of this phenomenon are related directly, where increasing the frequency of an AC current increases the skin effect and vice versa. As a result, in the direct current (DC current, zero frequency), practically no skin effect happens, and the conductor’s current density is equal at every point of the conductors.

The following figures show the conductor’s current density magnitude carrying an AC current in the different frequencies.

**The reason of skin effect occurrence**

The theory behind the skin effect is “Lenz’s law^{2}“. We all know that flowing an alternating current in a conductor creates time-varying magnetic fields with concentric lines where their axis is located in the inner and outer of the conductor. According to Lenz’s law, when the magnitude of an alternating current in a conductor changes, new closed electric currents are created, which are orthogonal to the inner magnetic field on both sides of the conductor axis, and the magnetic fields of these currents oppose the changes of the conductor current.

It is also obvious that increasing an alternative current frequency increases the skin effect.

**Proximity Effect ^{3}**

The “proximity effect” is a phenomenon with the origin similar to the “skin effect” which describes the electric current density of conductors near to each other.

The “proximity effect” states that for the two neighboring conductors which are affected by their outer magnetic fields, the electric current density of their opposite side is maximum if their current direction is the same and the electric current density of their counter-opposite side is maximum if their current direction is reversed.

Similar to the rules and principles governing the phenomenon of skin effect, in the effect of the proximity of neighboring conductors, the “proximity effect” is felt more noticeably with the increase in the frequency of the electric current passing through them.

**Relationships and quantities related to skin effect and proximity**

To formulate the skin effect and the proximity effect, some mathematical quantities and expressions are defined, and we will introduce some of them here.

First, we investigate the current distribution through the conductor cross-section. Regarding the previous discussions, as a result of the skin effect, the current density of the outer sections of the conductor is at maximum, and it is decreased gradually by approaching the center of the conductor. This phenomenon is aggravated by increasing the frequency of the current passing through the conductor. Based on the experimental equations, the conductor cross-section current density distribution is exponential and is dependent on the passing current frequency. The following relation calculates the conductor cross-section current density in terms of the distance from its outer surface:

In the above equation, J_{0} is the conductor’s surface current density (the maximum value of current density), and δ is a quantity that is referred to as “skin current penetration depth”, which means by moving away from the conductor’s surface equal to the skin current penetration depth, the conductor current density is decreased to 37% of its conductor surface magnitude. The following relations show that the conductor cross-section current in the skin current penetration depth distance is 63% of the total current.