Coax cutoff frequency

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New for October 2020. This page was split off from our coax page.

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 criterion 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 π 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:

Update October 2022: Wait, this seems counter-intuitive! Thanks to Xin for bring this up. Everywhere else it seems that wavelength (lambda) is divided by SQRT(ER), not multiplied by it. we assure you this equation is correct, and here is an attempt at an explanation: the cut-off occurs when its wavelength is equal to (D+d)/2. Adding dielectric effectively makes (D+d)/2 longer to an EM wave, so the cutoff wavelength increases (and cutoff frequency decreases).

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., TE₁₁ 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.

But wait, there's many more coax options, at the opposite end of the microwave spectrum! Thanks to Gray. 

For the slow among us that work nearer the opposite end of the frequency and power spectra - there is a 600+-foot run of 9-3/16”, 50 ohm coax next door to my house running approximately 140 KW!

The reason rigid coax (at least in the US) is measured in inches is that invariably the outer conductors are made out of standard Type M copper tubing sizes.  The inners are all custom diameters.

Back in the days when radio engineers were men and UHF was anything above 30 MC, there were some other interesting rigid coax characteristic impedances.  RCA was a great proponent of 51.5 ohm line, produced with your choice of teflon or steatite spacers (and with different OD's of the inners because of the differing permittivities).   Not quite sure where this came from, though I note that nominal 1-1/2” Type K (1.481" ID) and nominal 1/2” (.625" OD) produces 51.7 ohms.

I have a chunk of coax in my garage taken out of an AM directional array built in the late 1940’s that uses nominal 2” Type L copper tubing for the outer (1.505” ID) and nominal 3/8” (.5” OD) for the inner, producing an impedance of 66 ohms.

These are all air dielectric rigid copper transmission line, 50 ohms except as noted.  It's funny how low frequency coax is specified in inches while the true microwave people use millimeters (UE)

Inner diameters and other information can be found here, for example.

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!