A few antenna cable formulas

The radio frequency characteristics of coaxial antenna cables are determined by properties of the materials used to create the cable, along with its dimensions. If you know some of these properties it may be possible to determine other properties of a cable, which might be useful:

  • Inner diameter of shield (outer conductor)
  • Outer diameter of center (inner conductor)
  • Characteristic impedance (typically 50 or 75 Ohms)
  • Velocity factor
  • Dielectric constant/relative permittivity

For symbol names, please see the end of this post.

 

The characteristic impedance is determined by the diameter of the cable’s conductors, and can be found using this formula:

Z_{0} = \frac{1}{2\pi}\sqrt{\frac{\mu}{\epsilon}} \times ln(\frac{D}{d})\approx\frac{60\Omega}{\sqrt{\epsilon_{r}}} \times ln(\frac{D}{d}) \quad [\Omega]

Note that precisely measuring the needed thicknesses may be difficult, but you should easily be able to tell if it’s a 50 or 75 Ohm cable you’re dealing with.

 

The velocity factor is determined by the dielectric constant (relative permittivity) of the dielectric material making up the insulation inside the cable:

V_{f} = \frac{1}{\sqrt{\epsilon_{r}}}

 

The velocity factor can be found if you know the characteristic impedance of the cable and the diameter of its conductors:

V_{f} = \frac{1}{\frac{60\Omega}{Z_{0}} \times ln(\frac{D}{d})}

Note that this assumes an insulating material with relative permeability = 1, which is a reasonable assumption in most cases.

 

The dielectric constant of the insulation inside the cable can be found if you know the cable’s velocity factor:

\epsilon_{r} = \frac{1}{V_{f}^2}

 

If you’re rolling your own, you can find the ratio of conductor diameters for a cable with Z_{0} \Omega impedance using dielectric insulation of \epsilon_{r} relative permittivity:

\frac{D}{d} = 10^{(\frac{Z_{0}}{60} \sqrt{\epsilon_{r}})}

If you’re making an air-insulated hardline/cable (\epsilon_{r} = 1), the ratios are 3.490 for 75 Ohm cable and 2.301 for 50 Ohm cable.

 

The ratio of conductor diameters for a cable with Z_{0} \Omega impedance and a velocity factor of V_{f}:

\frac{D}{d} = 10^{(\frac{Z_{0}}{60} \frac{1}{V_{f}})}

 

The resistive loss at a given frequency per length of cable is determined by the permeability and the conductivity/resistivity of the cable’s conductors, plus their diameters, and can be found using this formula:

\alpha_{R} = (\frac{f \mu_{0} \epsilon_{r}}{\pi})^\frac{1}{2} \times (\frac{(\mu_{rD} \times \rho_{D})^\frac{1}{2}}{D} + \frac{(\mu_{rd} \times \rho_{d})^\frac{1}{2}}{d}) \times \frac{1}{6 ln(10) ln(\frac{D}{d})} \quad [dB/m]

This is usually the main contributor to loss of signal strength in a normal antenna cable.

 

The dielectric loss at a given frequency per length of cable is determined by the dielectric constant and the loss tangent, and can be found using this formula:

\alpha_{D} = 92.0216 \sqrt{\epsilon_{r}} \times tan(\delta) \times f_{GHz} \quad [dB/m]

This is usually small enough that you do not need to worry about it. A cheap Polyethylene-insulated cable with a loss tangent of 0.00031 at 3 GHz, works out to a loss of less than 0.01 dB per 100 meter at a frequency of 1 GHz.

 

Z_{0} = characteristic impedance
\mu = permeability
\mu_{0} = vacuum permeability (1.2566370614…×10−6 H/m)
\mu_{r} = relative permeability
\epsilon = permittivity
\epsilon_{0} = vacuum permittivity (8.854 187 817… x 10−12 F/m)
\epsilon_{r} = relative permittivity (dielectric constant)
V_{f} = velocity factor
ln = natural logarithm
D = inner diameter of shield (outer conductor)
d = outer diameter of center (inner conductor)
f = frequency (Hz)
f_{GHz} = frequency (gigahertz)
\rho = resistivity
\alpha_{R} = resistive loss
\alpha_{D} = dielectric loss
tan(\delta) = loss tangent
dB = decibel