Coaxial Cable Transmission Equations

The characteristic impedance at high frequencies is almost pure resistance and equals²⁴

where R, G, L, and C are the resistance, shunt conductance, inductance, and shunt capacitance in ohms, mhos, henrys, and farads, respectively, for any convenient length, and at the frequencies under consideration. The factor A is the corresponding wavelength.

The difficulty in the use of these equations is that the constants vary, at least to some extent, with frequency because of skin effect. This is particularly true for the losses represented by R and G. If certain reasonable assumptions are made, however, simple relations for the coaxial cable at high frequencies can be written.

If the insulation losses given by the central term of equation 27 are neglected, the attenuation constant, giving the loss in an air-dielectric coaxial cable in nepers, is

where Z₀ is as given by equation 28. The length unit of measure of α will be the same as for R. The value of the phase constant β in radians per meter becomes

where c equals the velocity of light in free space in meters per second (page 195). The value of the resistance R in ohms per centimeter is²⁴

where b is the inner radius of the outer conductor in centimeters, a is the outer radius of the inner conductor in centimeters, and / is the frequency in cycles per second. The value of the inductance L is

where L is in microhenrys per meter and b and a are as shown in Fig. 23 and are measured in the same units. The value of the capacitance C is

where C is in microfarads per meter and 6 and a are as previously considered. When the last two equations are substituted in equation 28, the characteristic impedance in ohms becomes

and substantially equals pure resistance.