Coaxial cable is a particular kind of
transmission line, so the circuit models developed for general transmission lines are appropriate. See
Telegrapher's equation.
Physical parameters In the following section, these symbols are used: • Length \ell of the cable. • Outside diameter d of
inner conductor. • Inside diameter D of the shield. •
Dielectric constant \epsilon of the insulator. The dielectric constant is often quoted as the relative dielectric constant \epsilon_\text{r} referred to the dielectric constant \epsilon_0 of free space: \epsilon = \epsilon_\text{r} \epsilon_0. When the insulator is a mixture of different dielectric materials (e.g., polyethylene foam is a mixture of polyethylene and air), then the term effective dielectric constant \epsilon_\text{eff} is often used. •
Magnetic permeability \mu of the insulator. Permeability is often quoted as the
relative permeability \mu_\text{r} referred to the permeability \mu_0 of free space: \mu = \mu_\text{r} \mu_0. The relative permeability will almost always be .
Fundamental electrical parameters • Shunt
capacitance per unit length, in
farads per metre: C = \frac{2 \pi \epsilon}{\ln \frac{D}{d}} = \frac{2 \pi \epsilon_0 \epsilon_\text{r}}{\ln \frac{D}{d}}. • Series
inductance per unit length, in
henries per metre, considering the central conductor to be a thin hollow cylinder (due to
skin effect): Z_0 = \sqrt{\frac{R + sL}{G + sC}}, where is the resistance per unit length, is the inductance per unit length, is the conductance per unit length of the dielectric, is the capacitance per unit length, and is the frequency. The "per unit length" dimensions cancel out in the impedance formula. At
DC the two reactive terms are zero, so the impedance is real-valued, and is extremely high. It looks like Z_\text{DC} = \sqrt{\frac{R}{G}}. With increasing frequency, the reactive components take effect, and the impedance of the line is complex-valued. At very low frequencies (audio range, of interest to telephone systems) is typically much smaller than , so the impedance at low frequencies is Z_\text{LF} \approx \sqrt{\frac{R}{sC}}, which has a phase value of −45°. At higher frequencies, the reactive terms usually dominate and , and the cable impedance again becomes real-valued. That value is , the
characteristic impedance of the cable: Z_0 = \sqrt{\frac{sL}{sC}} = \sqrt{\frac{L}{C}}. Assuming that the dielectric properties of the material inside the cable do not vary appreciably over the operating range of the cable, the characteristic impedance is frequency independent above about five times the
shield cutoff frequency. For typical coaxial cables, the shield cutoff frequency is 600 Hz (for RG-6A) to 2,000 Hz (for RG-58C). The parameters and are determined from the ratio of the inner () and outer () diameters and the
dielectric constant (). The characteristic impedance is given by of this frequency. • Peak voltage. The peak voltage is set by the breakdown voltage of the insulator: V_\text{p} = E_\text{d} \frac{d}{2} \ln \frac{D}{d}, where is the peak voltage, is the insulator breakdown voltage in volts per metre, is the inner diameter in metres, is the outer diameter in metres. The calculated peak voltage is often reduced by a safety factor.
Choice of impedance The best coaxial cable impedances were experimentally determined at
Bell Laboratories in 1929 to be 77 Ω for low-attenuation, 60 Ω for high-voltage, and 30 Ω for high-power. For a coaxial cable with air dielectric and a shield of a given inner diameter, the attenuation is minimized by choosing the diameter of the inner conductor to give a characteristic impedance of 76.7 Ω. When more common dielectrics are considered, the lowest
insertion loss impedance drops down to a value between 52 and 64 Ω. Maximum power handling is achieved at 30 Ω. The approximate impedance required to match a centre-fed
dipole antenna in free space (i.e., a dipole without ground reflections) is 73 Ω, so 75 Ω coax was commonly used for connecting shortwave antennas to receivers. These typically involve such low levels of RF power that power-handling and high-voltage breakdown characteristics are unimportant when compared to attenuation. Likewise with
CATV, although many broadcast TV installations and CATV headends use 300 Ω folded
dipole antennas to receive off-the-air signals, 75 Ω coax makes a convenient 4:1
balun transformer for these as well as possessing low attenuation. The
arithmetic mean between 30 Ω and 77 Ω is 53.5 Ω; the
geometric mean is 48 Ω. The selection of 50 Ω as a compromise between power-handling capability and attenuation is in general cited as the reason for the number. 50 Ω also works out tolerably well because it corresponds approximately to the feedpoint impedance of a half-wave dipole, mounted approximately a half-wave above "normal" ground (ideally 73 Ω, but reduced for low-hanging horizontal wires). RG-62 is a 93 Ω coaxial cable originally used in mainframe computer networks in the 1970s and early 1980s (it was the cable used to connect
IBM 3270 terminals to IBM 3274/3174 terminal cluster controllers). Later, some manufacturers of LAN equipment, such as Datapoint for
ARCNET, adopted RG-62 as their coaxial cable standard. The cable has the lowest capacitance per unit-length when compared to other coaxial cables of similar size. All of the components of a coaxial system should have the same impedance to avoid internal reflections at connections between components (see
Impedance matching). Such reflections may cause signal attenuation. They introduce standing waves, which increase losses and can even result in cable dielectric breakdown with high-power transmission. In analog video or TV systems, reflections cause
ghosting in the image; multiple reflections may cause the original signal to be followed by more than one echo. If a coaxial cable is open (not connected at the end), the termination has nearly infinite resistance, which causes reflections. If the coaxial cable is short-circuited, the termination resistance is nearly zero, which causes reflections with the opposite polarity. Reflections will be nearly eliminated if the coaxial cable is terminated in a pure resistance equal to its impedance. == Issues ==