Editing Coaxial cable (section)

== Important parameters ==

Coaxial cable is a particular kind of [[transmission line]], so the circuit models developed for general transmission lines are appropriate. See [[Telegrapher's equation]].

[[File:Transmission line element.svg|thumb|Schematic representation of the elementary components of a transmission line]]
[[File:Transmission line schematic.svg|thumb|Schematic representation of a coaxial transmission line, showing the characteristic impedance <math>Z_0</math>]]

=== Physical parameters ===

In the following section, these symbols are used:

* Length of the cable, <math>\ \ell ~.</math>
* Outside diameter of ''inner'' conductor, <math>\ d ~.</math>
* Inside diameter of the shield, <math>\ D ~.</math>
* [[Dielectric constant]] of the insulator, <math>\ \epsilon ~.</math> The dielectric constant is often quoted as the relative dielectric constant <math>\ \epsilon_\mathsf{r}\ </math> referred to the dielectric constant of free space <math>\ \epsilon_\mathsf{o}\ :</math> <math>\ \epsilon = \epsilon_\mathsf{r}\ \epsilon_\mathsf{o} ~.</math> 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 <math>\ \epsilon_\mathsf{eff}\ </math> is often used.
* [[Magnetic permeability]] of the insulator, <math>\ \mu ~.</math> Permeability is often quoted as the ''relative'' permeability <math>\ \mu_\mathsf{r}\ </math> referred to the permeability of free space <math>\ \mu_\mathsf{o}\ :</math> <math>\ \mu = \mu_\mathsf{r}\ \mu_\mathsf{o} ~.</math> The relative permeability will almost always be {{math|1}}.

=== Fundamental electrical parameters ===

* Shunt [[capacitance]] per unit length, in [[farad]]s per metre.<ref name=pozarueng>{{cite book |last=Pozar |first=David M. |year=1993 |title=Microwave Engineering |publisher=Addison-Wesley Publishing Company |isbn=0-201-50418-9}}</ref>
:: <math> C = \frac{2 \pi \epsilon}{\ \ln\left(\frac{\ D\ }{d} \right)\ } = \frac{2 \pi \epsilon_\mathsf{o} \epsilon_\mathsf{r} }{\ \ln\left(\frac{\ D\ }{d} \right)\ } </math>
* Series [[inductance]] per unit length, in [[henry (unit)|henries]] per metre, considering the central conductor to be a thin hollow cylinder (due to [[skin effect]]).
:: <math> L = \frac{\mu}{\ 2 \pi\ }\ \ln\left(\frac{\ D\ }{d} \right)= \frac{\ \mu_\mathsf{o} \mu_\mathsf{r}\ }{ 2 \pi}\ \ln\left( \frac{\ D\ }{d} \right) </math>
* Series [[electrical resistance|resistance]] per unit length, in ohms per metre. The resistance per unit length is just the resistance of inner conductor and the shield at low frequencies. At higher frequencies, [[skin effect]] increases the effective resistance by confining the conduction to a thin layer of each conductor.
* Shunt [[Electrical conductance|conductance]] per unit length, in [[siemens (unit)|siemens]] per metre. The shunt conductance is usually very small because insulators with good dielectric properties are used (a very low [[loss tangent]]). At high frequencies, a dielectric can have a significant resistive loss.

=== Derived electrical parameters ===

<!-- To match above, we could use R/ℓ, etc. -->
*[[Characteristic impedance]] in ohms (Ω). The complex impedance {{mvar|Z}}{{sub|o}} of an infinite length of transmission line is:

:: <math> Z_\mathsf{o} = \sqrt{\frac{R + sL\ }{G + sC\ }\ } </math>
: Where {{mvar|R}} is the resistance per unit length, {{mvar|L}} is the inductance per unit length, {{mvar|G}} is the conductance per unit length of the dielectric, {{mvar|C}} is the capacitance per unit length, and {{math|''s'' {{=}} ''jω'' {{=}} ''j''2''πf''}} is the frequency. The "per unit length" dimensions cancel out in the impedance formula.
: At [[direct current|DC]] the two reactive terms are zero, so the impedance is real-valued, and is extremely high. It looks like
:: <math> Z_\mathsf{DC} = \sqrt{\frac{\ R\ }{G}\ } ~.</math>
: 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) {{mvar|G}} is typically much smaller than {{mvar|sC}}, so the impedance at low frequencies is
:: <math> Z_\mathsf{Low Freq} \approx \sqrt{\frac{R}{\ sC\ }\ }\ ,</math>
: which has a phase value of -45&nbsp;degrees.
: At higher frequencies, the reactive terms usually dominate {{mvar|R}} and {{mvar|G}}, and the cable impedance again becomes real-valued. That value is {{math|''Z''<sub>o</sub>}}, the ''characteristic impedance'' of the cable:
:: <math> Z_\mathsf{o} = \sqrt{\frac{sL}{\ sC\ }\ }= \sqrt{\frac{L}{\ C\ }\ } ~.</math>
: Assuming 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 [[Cutoff frequency#Waveguides|shield cutoff frequency]]. For typical coaxial cables, the shield cutoff frequency is 600&nbsp;Hz (for RG-6A) to 2,000&nbsp;Hz (for RG-58C).<ref>{{cite book |last=Ott |first=Henry W. |year=1976 |title=Noise Reduction Techniques in Electronic Systems |publisher=Wiley |isbn=0-471-65726-3}}</ref>
: The parameters {{mvar|L}} and {{mvar|C}} are determined from the ratio of the inner ({{mvar|d}}) and outer ({{mvar|D}}) diameters and the [[dielectric constant]] ({{mvar|ε}}). The characteristic impedance is given by<ref>{{cite book |last1=Elmore |first1=William C. |last2=Heald |first2=Mark A. |year=1985 |title=Physics of Waves |publisher=Courier Corporation |isbn=0-486-64926-1 |url=https://archive.org/details/physicsofwaves0000elmo |url-access=registration}}</ref>
:: <math> Z_\mathsf{o} = \frac{1}{\ 2\pi\ }\sqrt{\frac{\mu}{\ \epsilon\ }\ }\ \ln\left( \frac{D}{\ d\ } \right)  \approx \frac{\ 59.96\ \mathsf{\Omega}\ }{\sqrt{\epsilon_\mathsf{r}\ } } \ln\left( \frac{D}{\ d\ } \right)  \approx \frac{\ 138\ \mathsf{\Omega}\ }{\sqrt{\epsilon_\mathsf{r}\ } } \log_{10}\left( \frac{D}{\ d\ } \right) ~.</math>
* Attenuation (loss) per unit length, in [[decibel]]s per meter. This is dependent on the loss in the dielectric material filling the cable, and resistive losses in the center conductor and outer shield. These losses are frequency dependent, the losses becoming higher as the frequency increases. Skin effect losses in the conductors can be reduced by increasing the diameter of the cable. A cable with twice the diameter will have half the skin effect resistance.  Ignoring dielectric and other losses, the larger cable would halve the dB/meter loss. In designing a system, engineers consider not only the loss in the cable but also the loss in the connectors.
* [[Velocity of propagation]], in meters per second. The velocity of propagation depends on the dielectric constant and permeability (which is usually {{math|1}}).
:: <math> v = \frac{1}{\ \sqrt{\epsilon \mu\ }\ } = \frac{c}{\ \sqrt{\epsilon_\mathsf{r} \mu_\mathsf{r}\ } \ } </math>
* Single-mode band. In coaxial cable, the dominant mode (the mode with the lowest [[cutoff frequency]]) is the [[TEM mode]],<ref name=jackson/> which has a cutoff frequency of zero; it propagates all the way down to DC. The mode with the next lowest cutoff is the TE{{sub|11}} mode. This mode has one 'wave' (two reversals of polarity) in going around the circumference of the cable. To a good approximation, the condition for the TE{{sub|11}} mode to propagate is that the wavelength in the dielectric is no longer than the average circumference of the insulator; that is that the frequency is at least
:: <math>f_\mathsf{c} \approx \frac{1}{\ \pi \left( \frac{D + d}{2} \right) \sqrt{\mu \epsilon\ } } = \frac{ c }{\ \pi \left(\frac{D + d}{2}\right) \sqrt{\mu_\mathsf{r} \epsilon_\mathsf{r}\ } } ~.</math>
: Hence, the cable is single-mode from DC up to this frequency, and might in practice be used up to 90%<ref>{{cite book |last=Kizer |first=George Maurice |year=1990 |title=Microwave communication |publisher=Iowa State University Press |isbn=978-0-8138-0026-4 |url=https://books.google.com/books?id=T2fI766k2R0C&pg=PA312 |page=312}}</ref> of this frequency.
* Peak Voltage. The peak voltage is set by the breakdown voltage of the insulator.:<ref>{{cite web |title=Coax power handling |url=http://www.microwaves101.com/encyclopedia/coax_power.cfm |archive-url=https://web.archive.org/web/20140714170552/http://www.microwaves101.com/ENCYCLOPEDIA/coax_power.cfm |archive-date=2014-07-14 |df=dmy-all}}</ref>
:: <math> V_\mathsf{p} =\ E_\mathsf{d} \ \frac{d}{2}\ \ln \left( \frac{\ D\ }{d} \right) </math>
:: where
::: {{mvar|V}}{{sub|p}} is the peak voltage
::: {{mvar|E}}{{sub|d}} is the insulator's breakdown voltage in volts per meter
::: {{mvar|d}} is the inner diameter in meters
::: {{mvar|D}} is the outer diameter in meters
: 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&nbsp;Ω for low-attenuation, 60&nbsp;Ω for high-voltage, and 30&nbsp;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&nbsp;Ω.<ref name="Why 50 Ohms">{{cite web |title=Why 50 Ohms? |publisher=Microwaves 101 |date=2009-01-13 |df=dmy-all |url=http://www.microwaves101.com/encyclopedia/why50ohms.cfm |archive-url=https://web.archive.org/web/20140714170552/http://www.microwaves101.com/ENCYCLOPEDIA/why50ohms.cfm |archive-date=2014-07-14}}</ref> When more common dielectrics are considered, the lowest [[insertion loss]] impedance drops down to a value between 52 and 64&nbsp;Ω. Maximum power handling is achieved at 30&nbsp;Ω.<ref>{{cite web |title=Coax power handling |publisher=Microwaves 101 |date=2008-09-14 |url=http://www.microwaves101.com/encyclopedia/coax_power.cfm |archive-url=https://web.archive.org/web/20120128192901/http://www.microwaves101.com/encyclopedia/coax_power.cfm |archive-date=2012-01-28 |df=dmy-all}}</ref>

The approximate impedance required to match a centre-fed [[dipole antenna]] in free space (i.e., a dipole without ground reflections) is 73&nbsp;Ω, so 75&nbsp;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 [[Cable television|CATV]], although many broadcast TV installations and CATV headends use 300&nbsp;Ω folded [[dipole antenna]]s to receive off-the-air signals, 75&nbsp;Ω coax makes a convenient 4:1 [[balun]] transformer for these as well as possessing low attenuation.

The [[arithmetic mean]] between 30&nbsp;Ω and 77&nbsp;Ω is 53.5&nbsp;Ω; the [[geometric mean]] is 48&nbsp;Ω. The selection of 50&nbsp;Ω as a compromise between power-handling capability and attenuation is in general cited as the reason for the number.<ref name="Why 50 Ohms"/> 50&nbsp;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&nbsp;Ω, but reduced for low-hanging horizontal wires).

RG-62 is a 93&nbsp;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 (television)|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.