Click here to go to our main page on microwave transmission lines

Click here to go to our main page on microwave connectors

Click here to go to our page on RF cables

Click here to go to our page on waveguide-to-coax transitions

Click here to learn about the undesirable TE11 mode in coax

**New for April 2014:** thanks to Daniel, we have images of TE01 and TE11 modes using CST, check it out!

Coax symbol

The schematic symbol image for coax shown here is provided in our free schematic symbols download which you can find here.

Coax is unique in that it is the easiest transmission line to analyze, most coax properties can be described with exact closed-form equations. Not like stripline, microstrip or coplanar waveguide...

Go to our download area and get the new Coax.xls spreadsheet. It calculates and plots (over frequency) three types of loss (metal, loss tangent, and dielectric conductivity), and uses a "more exact" metal loss calculation, so it is more accurate (and way more convenient and infinitely cheaper) than EDA software such as Agilent ADS. It just might be our coolest spreadsheet yet! There is also a new multi-dielectric coax spreadsheet in the download area for your enjoyment, thanks to Alex!!

Here are the coax topics covered on this page and other related pages:

Capacitance and inductance per unit length

Characteristic impedance of coax

Cutoff wavelength and frequency

Some industry-standard coax cables

Off-center coax impedance (separate page)

Rectangular (and square) coax (separate page)

Coax power handling (separate page)

Coax loss calculations (separate page)

A more exact calculation of coax loss due to metal (separate page)

The derivation of coax loss due to dielectric loss tangent (separate page)

Loss due to dielectric conduction (separate page)

Why fifty ohms? (separate page)

Multi-dielectric coax (separate page)

### Coax overview

Coaxial cable is the solution to many problems, from wide bandwidth, to low loss and high isolation. Ask your cable company how many miles of it they string just so you can enjoy the next Tyson-Jameson Encounter for 15 seconds. Thanks for the Tyson update, James! Wow, that is a dated reference...

Coax provides the very desirable transverse-electromagnetic (TEM) mode of transmission. The filling factor for coax is unity, and "Keff" is equal to ER. Coax has no lower cutoff frequency (like waveguide does).

A coax transmission line (we prefer the nickname here at Microwaves101 instead of the more academic term "coaxial") consists of two round conductors in which one completely surrounds the other, with the two separated by a continuous solid dielectric (or sometimes by periodic dielectric spacers), as shown below:

### Capacitance and inductance per unit length

These formulas are the exact calculations for capacitance and inductance per length for coax cable. We'll try to stick with the "prime" nomenclature whenever we are talking about quantities that are normalized per unit length.

Note that the units of D and d don't matter, both calculations only use the ratio D/d. We can simplify them, noting that _{R} is usually =1 for any dielectric we might be interested in. Then the equations can be expressed in SI units (per meter):

or in English units (per foot)

### Characteristic impedance

Characteristic impedance is always the square-root of the ratio of inductance per length to capacitance per length:

This can be simplified to the familiar equation that is shown in almost all coax cable and feedthrough catalogs:

Or, perhaps more elegantly,

Where b and a and the radii of the conductors. Feel free to replace them with D and d!

### Coax cutoff wavelength and frequency

What is meant by the cutoff frequency f_{c}? The desirable TEM mode is allowed to propagate at all frequencies, but at frequencies above f_{c} the first higher-order mode called TE11 is also allowed to propagate. Higher-order mode that will screw up your loss and VSWR and as they have different propagation velocity than the TEM mode and will interfere with it. Higher modes will be excited at small imperfections, bends, etc., but below cutoff they rapidly disappear along straight sections of coax. To be sure that only one mode propagates, thus keeping the signal clean, you will need to stay below f_{c}. To obtain good performance at higher frequencies, smaller diameter cables are required to stay* below* the cutoff frequency (thanks for the correction, Gary!) This is the reason that precision air-dielectric connector families have progressed from 3.5mm, to 2.9mm, to 2.4mm, to 1.85mm and now to 1mm as microwave applications have moved from X-band to W-band frequencies. For more info on connector species, check out our section on microwave connectors!

In order to minimize losses due to skin depth, you want to use the BIGGEST coax cable you can that won't support TE11 mode . The criteria for cutoff is that the circumference at the midpoint inside the dielectric must be less than a wavelength. Note: this is an APPROXIMATION of a transcendental equation which must be solved numerically. If you are interested in reading about the true solution, we suggest you pick up a copy of Pozar's book.

If you have half a brain, like us, you can easily prove to yourself that the average circumference is just times the average of the inner and outer diameters. Therefore the cutoff wavelength for the TE11 mode is:

Here the units must be consistent, so use meters for d and D to get cutoff wavelength in meters. In the above equation, we didn't take into account the reduction in wavelength when a dielectric (or magnetic) material is used as the coax insulator. So the cutoff wavelength for arbitrary dielectric is:

Now let's convert that to cutoff frequency instead of cutoff wavelength:

Last, we offer simplifications for cutoff frequency in both SI and English units, with the assumption that μ_{R} is usually =1 for any dielectric we might be interested in. We've also tweaked the equations so units for frequency are GHz (instead of Hertz).

For November 2015 we have a frequency plot illustrating cut-off frequency, thanks to Purushothaman. What does the frequency response look like for the first two modes? For a coaxial line with D = 5 mm and d = 1mm and air as dielectric (ER=1), the cut-off frequency equation solves to 31.81 GHz.In the plot below, ‘1’ in parenthesis represents the first mode (i.e., TEM mode) and ‘2’ represents the second mode or the first non-TEM mode (i.e., TE

_{11}mode). Just as predicted, TE11 starts propagating at 32 GHz; it is highly attenuated depending on how far below cutoff you operate. This plot was produced using CST's electro-magnetic solver. Note that TEM continues to propagate in the region above 32 GHz, but the two modes will interfere with each other and you may get unexpected results. Thus, it is a good idea to stay below the TE11 cutoff, unless you have a good reason and know what you are doing.Why would you risk operation anywhere near TE11 cutoff? Attenuation due to metal is reduced by making coax larger and larger. A 7mm air-line will have 1/2 the loss of a 3.5mm air-line, but above 19 GHz, 7mm coax propagates two modes and should not be used. You can evaluate cut-off frequency using our coax calculator.

Microwaves101 Rule of thumb #117.

The 90% rule: coax is never specified to operate beyond 90% of its TE11 cutoff frequency. Below is a list of commercial connector species' cutoff frequencies we calculated using our downloadable coax spreadsheet, and the ratio of recommended to actual cutoff frequencies.

Connector species | Calculated TE11 cutoff frequency (GHz) | Recommended maximum frequency (GHz) | % cutoff frequency at max recommended frequency (GHz) |
---|---|---|---|

7mm | 19.02 | 18 | 95% |

3.5mm | 38.0 | 26.5 | 70% |

2.92mm | 45.6 | 40 | 88% |

2.4mm | 55.4 | 50 | 90% |

1.85mm | 71.9 | 60 | 83% |

1mm | 133 | 110 | 83% |

**New for April 2014: **we tried to compare the coax TE11 mode to TEM in Powerpoint. Not a very successful effort, there are much better sketches dating back to WWII RadLab. What did that third grade teacher tell us on the playground 50 years ago? The important thing is *we tried.* Nah, that's BS, we *failed....*

TEM mode, cable sliced at maximum E-field strength

TE11 mode, cable also sliced at maximum E-field strength.

What we failed to capture is that there are field lines on the east and west side that start and stop on the outer jacket and by looping out and back. Note that in TE11 mode, the magnetic fields are no longer purely transverse, they point in radial and longitudinal directions, and as always, forming closed loops.

A reader came to the rescue. The images below arrived from Daniel in March 2014. First, here's two images of the desirable TEM mode:

And here's the undesired TE11 mode:

Thanks!

### Some industry-standard coax cables

This material was moved here.

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