Editing Skin effect (section)

===== Coaxial cable =====
Let the dimensions {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} be the inner conductor radius, the shield (outer conductor) inside radius and the shield outer radius respectively, as seen in the crossection of figure&nbsp;'''{{sc|A}}''' below.

[[File:Coax and Skin Depth.svg|center|thumb|800px|Four stages of skin effect in a coax showing the effect on inductance. Diagrams show a cross-section of the coaxial cable. Color code:
{{legend|black|black - overall insulating sheath}}
{{legend|tan|tan - conductor}}
{{legend|white|white - dielectric}}
{{legend|lime|green - current into the diagram}}
{{legend|teal|teal - current coming out of the diagram}}

Dashed black lines with arrowheads are [[magnetic flux]] ('''{{math|B}}''').

The width of the dashed black lines is intended to show relative strength of the magnetic field integrated over the circumference at that radius. The four stages are:

{{bulleted list
| '''{{sc|A}}''' non-energized
| '''{{sc|B}}''' low frequency
| '''{{sc|C}}''' middle frequency
| '''{{sc|D}}''' high frequency}}

There are three regions that may contain induced magnetic fields: the center conductor, the dielectric and the outer conductor. In stage  '''{{sc|B}}''', current covers the conductors uniformly and there is a significant magnetic field in all three regions. As the frequency is increased and the skin effect takes hold ('''{{sc|C}}''' and  '''{{sc|D}}''') the magnetic field in the dielectric region is unchanged as it is proportional to the total current flowing in the center conductor. In '''{{sc|C}}''', however, there is a reduced magnetic field in the deeper sections of the inner conductor and the outer sections of the shield (outer conductor). Thus there is less energy stored in the magnetic field given the same total current, corresponding to a reduced inductance. At an even higher frequency, '''{{sc|D}}''', the skin depth is tiny: All current is confined to the surface of the conductors. The only magnetic field is in the regions between the conductors; only the ''external inductance'' remains.]]

For a given current, the total energy stored in the magnetic fields must be the same as the calculated electrical energy attributed to that current flowing through the inductance of the coax; that energy is proportional to the cable's measured inductance.

The magnetic field inside a coaxial cable can be divided into three regions, each of which will therefore contribute to the electrical inductance seen by a length of cable.<ref name="Hayt434">{{Harvtxt|Hayt|1981|p=434}}</ref>

*The inductance  <math> L_\text{cen} \, </math> is associated with the magnetic field in the region with radius <math> r < a \, </math>, the region inside the center conductor.
*The inductance  <math> L_\text{ext} \, </math> is associated with the magnetic field in the region <math> a < r < b \, </math>, the region between the two conductors (containing a dielectric, possibly air).
*The inductance  <math> L_\text{shd} \, </math> is associated with the magnetic field in the region <math> b < r < c \, </math>, the region inside the shield conductor.

The net electrical inductance is due to all three contributions:
<math display="block"> L_\text{total} = L_\text{cen} + L_\text{shd} + L_\text{ext}\, </math>

<math> L_\text{ext} \, </math> is not changed by the skin effect and is given by the frequently cited formula for inductance ''L'' per length ''D'' of a coaxial cable:
<math display="block"> L/D = \frac{\mu_0}{2 \pi} \ln  \left( \frac {b}{a}  \right)   \, </math>

At low frequencies, all three inductances are fully present so that <math> L_\text{DC} = L_\text{cen} + L_\text{shd} + L_\text{ext}\, </math>.

At high frequencies, only the dielectric region has magnetic flux, so that  <math> L_\infty = L_\text{ext}\, </math>.

Most discussions of coaxial transmission lines assume they will be used for radio frequencies, so equations are supplied corresponding only to the latter case.

As skin effect increases, the currents are concentrated near the outside the inner conductor (''r''&nbsp;=&nbsp;''a'') and the inside of the shield (''r''&nbsp;=&nbsp;''b''). Since there is essentially no current deeper in the inner conductor, there is no magnetic field beneath the surface of the inner conductor. Since the current in the inner conductor is balanced by the opposite current flowing on the inside of the outer conductor, there is no remaining magnetic field in the outer conductor itself where  <math> b < r < c \, </math>. Only <math> L_\text{ext} </math> contributes to the electrical inductance at these higher frequencies.

Although the geometry is different, a twisted pair used in telephone lines is similarly affected: at higher frequencies, the inductance decreases by more than 20% as can be seen in the following table.