## Skin, Symbols, and General Consideration Effect

On: 07 Jul, 2017

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Symbols

A = correction coefficient
D = diameter of conductor in inches
f = frequency in hertz
Rac = resistance at frequency f
RdC = direct-current resistance
Rsq = resistance per square
T = thickness of tubular conductor in inches T1 = depth of penetration of current
= skin depth
A = free-space wavelength in meters Pr = relative permeability of conductor material

(ILr = I for copper and other nonmagnetic materials)
p = resistivity of conductor material at any temperature
= resistivity of copper at 20 °C = 1.724 microhms-centimeter

Skin Depth
The skin depth is that distance below the surface of a conductor where the current density has diminished to lie of its value at the surface. The thickness of the conductor is assumed to be several (perhaps at least three) times the skin depth. Imagine the conductor replaced by a cylindrical shell of the same surface shape but of thickness equal to the skin depth, with uniform current density equal to that which exists at the surface of the actual conductor. Then the total current in the shell and its resistance are equal to the corresponding values in the actual conductor.

The skin depth and the resistance per square (of any 6 = (1.50 X 104A”2)k1
size), in meter-kilogram-second (rationalized) units, = (2.60/f 2)k1 inch
are 6m = (2.60/fmc112)ki mu
Rsq = (4.52 x I03/A”2)k2
8 (A/ircric)U2 = (2.61 X 10f112)k2 ohm
= [(l/iLr)P/Pj1’
Rsq = 1/8o k2 = (I4P/P)
k1, k2
= unity for copper
where,
Example: What is the resistance/foot of a cylindrical
6 = skin depth in meters, copper conductor of diameter D inches?
R = resistance per square in ohms,
c = velocity of light in vacuo R = (l2/3TD)Rsq
= 2.998 X iO meter/second,
= X iO !Lr hen/meter, = (l2/D) X 2.61 X iO_?(fU2)
I/o- = 1.724 X 10—8 p/p ohm-meter.
= 0.996 X 106(f”2)/D ohm/foot
For numerical computations
6 = (3.82 X iO A”2)k1 If D = 1.00 inch andf= 100 X 106 hertz, then R =
= (6.61/f’2)k1 centimeter 0.996 x 10-6 x iO 1 x 10-2 ohm/foot.

General Considerations
Fig. 7 shows the relationship of Rac/RdC versus D(f”2) for copper, or versus D(f02)(pp/p)”2 for any conductor material, for an isolated straight solid Cond uctor of circular cross section. Negligible error in the equations for Rac results when the conductor is spaced at least lOD from adjacent conductors. When the spacing between axes of parallel conductors carrying the same current is 4D, the resistance Rac is increased about 3 percent, when the depth of penetration is small. The equations are accurate for concentric lines due to their circular symmetry.

For values of D(f112)(I.LrPc/p)”2 greater than 40
Rac/Rdc = 0.0960 D(fU2)(,.L,p/p)2 + 0.26 (Eq. 1)

The high-frequency resistance in ohms/foot of an isolated straight conductor (either solid or tubular for T <D/8 or T1 <D/8) is given in Eq. (2). If the current is along the inside surface of a tubular conductor, D is the inside diameter.

Rac = A[(f112)/D1[I.Lr(P/Pc)1 1/2 x 106

(Eq. 2)

The values of the correction coefficient A for solid conductors and for tubular conductors are given in Table 4.
The value of T(f02)(/Irpc/P)02 that just makes A = I indicates the penetration of the currents below the surface of the conductor. Thus, approximately

T1 = [3.5/(f2)l(P/prPc)U2

(Eq. 3)

where T1 is in inches.
When T1 <D/8, the value Of Rac as given by Eq. (2) (but not the value of Rac/Rdc in Table 4, “Tubular Conductors”) is correct for any value T  T1.
Under the limitation that the radius of curvature of all parts of the cross section is appreciably greater than T1,

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