Editing Skin effect (section)

===Impedance===

==== Resistance ====
The most important effect of skin effect on the impedance of a single wire is the increase of the wire's resistance, and consequent [[Copper loss|losses]]. The effective resistance due to a current confined near the surface of a large conductor (much thicker than {{mvar|δ}}) can be solved as if the current flowed uniformly through a layer of thickness {{mvar|δ}} based on the DC resistivity of that material. The effective cross-sectional area is approximately equal to {{mvar|δ}} times the conductor's circumference.
Thus a long cylindrical conductor such as a wire, having a diameter {{mvar|D}} large compared to {{mvar|δ}}, has a resistance ''approximately'' that of a hollow tube with wall thickness {{mvar|δ}} carrying direct current. The AC resistance of a wire of length {{mvar|ℓ}} and resistivity <math>\rho</math> is:
<math display="block">R\approx
{{\ell \rho} \over {\pi (D-\delta) \delta}}
\approx
{{\ell \rho} \over {\pi D \delta}}
 </math>

The final approximation above assumes <math>D \gg \delta</math>.

A convenient formula (attributed to [[Frederick Terman|F.E. Terman]]) for the diameter {{mvar|D}}{{sub|W}} of a wire of circular cross-section whose resistance will increase by 10% at frequency {{mvar|f}} is:<ref>{{harvnb|Terman|1943|p=??}}</ref>
<math display="block">D_\mathrm{W} = {\frac{200~\mathrm{mm}}{\sqrt{f/\mathrm{Hz}}}}</math>

This formula for the increase in AC resistance is accurate only for an isolated wire. For nearby wires, e.g. in a [[Electrical cable|cable]] or a coil, the AC resistance is also affected by [[proximity effect (electromagnetism)|proximity effect]], which can cause an additional increase in the AC resistance.
The ''internal'' [[Electrical impedance|impedance]] per unit length of a segment of round wire is given by:<ref name="Walter_Weeks"/>{{rp|p=40}}
<math display="block"> \mathbf{Z}_\text{int} =   \frac { k \rho } { 2 \pi R }      \frac {  J_0(k R) } { J_1(k R) }.</math>

This impedance is a [[complex number|complex]] quantity corresponding to a resistance (real) in series with the [[Electrical reactance|reactance]] (imaginary) due to the wire's internal self-[[inductance]], per unit length.

==== Inductance ====

The portion of a wire's inductance that can be attributed to the magnetic field ''inside'' the wire itself is called the ''internal inductance'', which accounts for the inductive reactance (imaginary part of the impedance) given by the above formula. In most cases this is a small portion of a wire's inductance which includes the effect of [[Electromagnetic induction|induction]] from magnetic fields ''outside'' of the wire produced by the current in the wire. Unlike that ''external'' inductance, the internal inductance is reduced by skin effect, that is, at frequencies where skin depth is no longer large compared to the conductor's size.<ref name="Hayt303">{{Harvtxt|Hayt|1981|pp=303}}</ref> This small component of inductance approaches a value of <math> \frac \mu { 8 \pi } </math> (50 nH/m for non-magnetic wire) at low frequencies, regardless of the wire's radius. Its reduction with increasing frequency, as the ratio of skin depth to the wire's radius falls below about 1, is plotted in the accompanying graph, and accounts for the reduction in the telephone cable inductance with increasing frequency in the [[#Characteristics of telephone cable as a function of frequency|table below]].
[[File:Wire Internal Inductance.png|thumb|300px|left|The internal component of a round wire's inductance vs. the ratio of skin depth to radius. That component of the self inductance is reduced below ''μ''/8{{pi}} as skin depth becomes small (as frequency increases).]]
[[File:Wire AC Resistance vs skin depth.png|thumb|300px|The ratio AC resistance to DC resistance of a round wire versus the ratio of the wire's radius to the skin depth.   As skin depth becomes small relative to the radius, the ratio of AC to DC resistance approaches one half of the ratio of the radius to the skin depth.]] {{clear}}

Refer to the diagram below showing the inner and outer conductors of a coaxial cable. Since skin effect causes a current at high frequencies to flow mainly at the surface of a conductor, it can be seen that this will reduce the magnetic field ''inside'' the wire, that is, beneath the depth at which the bulk of the current flows. It can be shown that this will have a minor effect on the self-inductance of the wire itself; see Skilling<ref name="Skilling157_159">{{Harvtxt|Skilling|1951|pp=157–159}}</ref> or Hayt<ref name="Hayt434_439">{{Harvtxt|Hayt|1981|pp=434–439}}</ref> for a mathematical treatment of this phenomenon.

The inductance considered in this context refers to a bare conductor, not the inductance of a coil used as a circuit element. The inductance of a coil is dominated by the mutual inductance between the turns of the coil which increases its inductance according to the square of the number of turns. However, when only a single wire is involved, then in addition to the ''external inductance'' involving magnetic fields outside the wire (due to the total current in the wire) as seen in the white region of the figure below, there is also a much smaller component of ''internal inductance'' due to the portion of the magnetic field inside the wire itself, the green region in figure B. That small component of the inductance is reduced when the current is concentrated toward the skin of the conductor, that is, when skin depth is not much larger than the wire's radius, as will become the case at higher frequencies.

For a single wire, this reduction becomes of diminishing significance as the wire becomes longer in comparison to its diameter, and is usually neglected. However, the presence of a second conductor in the case of a transmission line reduces the extent of the external magnetic field (and of the total self-inductance) regardless of the wire's length, so that the inductance decrease due to skin effect can still be important. For instance, in the case of a telephone twisted pair, below, the inductance of the conductors substantially decreases at higher frequencies where skin effect becomes important. On the other hand, when the external component of the inductance is magnified due to the geometry of a coil (due to the mutual inductance between the turns), the significance of the internal inductance component is even further dwarfed and is ignored.

===== 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.

===== Telephone cable =====

Representative parameter data for 24 gauge PIC telephone cable at {{convert|21|C|F}}.
:{| class="wikitable" style="text-align:right;"
|-
! Frequency<br/>(Hz) !! R<br/>(Ω/km) !! L<br/>(mH/km) !! G<br/>(μS/km) !! C<br/>(nF/km)
|-
| 1&nbsp;&nbsp;Hz   || 172.24 || 0.6129 || 0.000 || 51.57
|-
|   1&nbsp;kHz   || 172.28 || 0.6125 || 0.072 || 51.57
|-
|  10&nbsp;kHz   || 172.70 || 0.6099 || 0.531 || 51.57
|-
| 100&nbsp;kHz   || 191.63 || 0.5807 || 3.327 || 51.57
|-
|   1&nbsp;MHz   || 463.59 || 0.5062 || 29.111 || 51.57
|-
|   2&nbsp;MHz   || 643.14 || 0.4862 || 53.205 || 51.57
|-
|   5&nbsp;MHz   || 999.41 || 0.4675  || 118.074 || 51.57
|}
More extensive tables and tables for other gauges, temperatures and types are available in Reeve.<ref name="Reeve558">{{Harvtxt|Reeve|1995|p=558}}</ref>
Chen<ref name="Chen26">{{Harvtxt|Chen|2004|p=26}}</ref> gives the same data in a parameterized form that he states is usable up to 50&nbsp;MHz.

Chen<ref name="Chen26" /> gives an equation of this form for telephone twisted pair:
<math display="block"> L(f) = \frac {\ell_0 + \ell_\infty \left(\frac{f}{f_m}\right)^b }{1 + \left(\frac{f}{f_m}\right)^b} \, </math>