A coaxial transmission line can operate with different field patterns in it. The most simple is when the electric field is radial from the middle conductor to the outside conductor. This is called the fundamental mode and is a TEM mode and works down to DC and up to any frequency. The wave will travel at the speed of light that is expected in the dielectric that fills the gap.

There are other patterns that are possible, such as with the electric field being upwards at the top, and also upwards at the bottom, and zero to the right and left sides of the centre conductor. This mode will not carry power along the line until the mean circumference of the gap is close to one wavelength in the dielectric filling. The frequency corresponding to this wavelength is called the cutoff frequency for that mode. The cutoff frequency is usually thought of as a lower frequency below which that mode doesn't propagate. There are other modes, with more complicated, patterns, for which the cutoff frequencies are even higher. Actually there are an infinite number of higher modes!

The cutoff mode for the first mode above the TEM mode is the upper cutoff frequency for safe use of the TEM mode, because above this frequency bends and steps and joins can generate other modes which travel at different (slower) speeds and can result in signals getting scrambled.

Below cutoff, a mode doesn't carry power, but the field from it, if you try to send it into the coax, or accidentally create it at a step or bend, looks like the field of an inductor or capacitor, and falls off very quickly along the waveguide. At frequencies only just below cutoff, these may actually go quite a long way along the guide, and seem to appear at the other end with no time lag, just as the inductive near field of a coil seems to increase immediately a small distance away from the coil, each half cycle. This immediate response can only be seen in simulations, where seeing the fields doesn't affect them. As soon as you use something to detect the real fields, it absorbs some power and changes the fields and a time delay is measured.

See https://uspas.fnal.gov/materials/18ODU/11L%20Waveguides.pdf for some more material.

The above post by Aparna Sathya Murthy is mostly incorrect and should not be relied on.

I repeat my warning more strongly. Do not believe what Aparna Sathya Murthy has written. Study transmission line theory indeed, and find out for yourself that he is wrong. I have over 40 years of experience with the theory and practical use of a great many transmission lines.

If you read the wikipedia page on coaxial transmission lines you will find, among other things, that the characteristic impedance Z0 of a transmission line is not the ratio of the inner radius to the outer radius (which always has to be less than 1).

https://en.wikipedia.org/wiki/Coaxial_cable#:~:text=Most%20coaxial%20cables%20have%20a,%2C%2075%2C%20or%2093%20%CE%A9.

This book by Samuel Liao seems good, but does not give the formula for the characteristic impedance of a coaxial transmission line, as far as I can see.

http://referenceglobe.com/kpsslp/support/upload_videos/microwave-devices-and-circuits-samuel-liao_1588230228.pdf

Typical ratios of inner to outer conductor for microwave 50 ohm cables is close to 1/3. It depends on the dielectric between them. 1/3 is not the same as 50 ohms.

Hello Ixs Igreg ,

Let me try and answer your question. A waveguide (hollow tube) has a lower cutoff frequency and basically no real upper cutoff frequency, [1]. According to [1], the low freqency wavelength is λLFCutoff=2a, where a is the width of the waveguide. In other words, if the width of the waveguide is less than or equal to the half-wavelength of the corresponding frequency, then the signal will begin to be attenuated; the smaller the hole, the greater the attenuation. Let us take as an example the lowly microwave oven.

The front door of the microwave oven is a metal plate with an array of round holes of the same diameter. The diameter (width) of these holes is chosen so they are very much less than the half-wavelength of the frequency produced by the magnetron. This arrangement prevents any microwave radiation from leaving the oven and, for instance, cooking your eyeballs. But what about the upper cutoff frequency? The other reason for the perforated metal plate is to allow you to see inside the oven and view the progress of the cooking via the light provided by the oven lightbulb. While the frequency of the magnetron is 2.45 GHz [Giga = 109], the frequency of visible light is in the hundreds of terahertz (~600 THz) [Tera = 1012]. The holes serve as waveguides.

Note, many a PhD candidate has been tripped up by a similar question: "What is the cutoff frequency of this waveguide?" If the candidate gives some high microwave frequency, the questioner will ask the candidate to pick up the waveguide and look through one end of it at the ceiling light.

[1] Anonymous; Basic Electronics, NAVPER 10087; Bureau of Naval Personnel; 1955; pp. 480-490, see in particualr p. 487.

By the way, the idea that the transmission line (two separate conductors: coax, twin lead, etc.) or waveguide (hollow pipe) looks like a LPF (Low Pass Filter) with no lower cutoff frequency but with an upper cutoff frequency because it is basically N RLCG sections in tandem, is completely wrong. When you go to the limit as each section becomes infinitely small, the behavior of the resulting sysem changes qualitatively (in character) and not just quantitatively (in amount). All of this is explained in [2].

[2] Everard M. Wlliams, James B. Woodford, Jr.; Transmission Circuits; The Macmillan Company; 1957; p. 127

The formula for the characteristic impedance (actually, resistance) of a coaxial cable with R = 0Ω [resistance of the inner and outer conductors] and G = 0mho [conductance = 1/(leakage) resistance of the dielectric] is given in [3]:

Z0 = √(Ll/Cl) = {[√(μ/ε)]/2π}loge(b/a)

where Ll and Cl are the inductance and capacitance per unit length, respectively, and μ and ε are the permeability and permittivity of the dielectric. Note, b/a is the ratio of the radius of the inside of the outer conductor, b, and the radius of the outside of the inner conductor, a: b > a.

[3] Walter C. Michels; Electrical Measurements and Their Applications; D. Van Nostrand Company, Inc.; 1957; p. 188.

Thanks for the qreat question,

Tom Cuff

Hello Ixs Igreg ,

Initially, when I wrote the previous answer I erroneously set G → ∞mho (dead short leakage), instead of G = 0mho (no leakage). I apologize for the mistake.

Regards,

Tom Cuff

Hello Ixs Igreg ,

After some checking with other books [1-2], I have determined that the expression for the characteristic impedance in terms of the distributed capacitance and distributed inductance was misprinted in the book by Michels, it should read,

Z0 = √(Ll/Cl)

The expression for Z0 in terms of radii a and b is correct.

[1] Anonymous; Basic Electronics, NAVPER 10087; Bureau of Naval Personnel; 1955; p. 462.

[2] Robert A. Chipman; Transmission Lines; Schaum's Outline Series, McGraw-Hill Book Company; 1968; p. 96.

Regards,

Tom Cuff

Aparna Sathya Murthy

The formula in 2.2.6 in the reference you give is not Z0=D/d.

formula 2.6.6 says Z0 = 60 * ln (b/a) sqrt(mr/er)

= 60 * ln (R/r) sqrt(mr/er)

=60 * ln (D/d) sqrt(mr/er)

All other textbooks will say the same.

This is not Z0=R/r

Aparna Sathya Murthy wrote "Considering the lossy media the Z0 is the ratio of inner radius to the outer radius"

Perhaps you should read this ( unless you talking about different things e.g. high freq cut-off caused by increase in losses with freq or cut-off arising in simulation based on lumped lement model) https://www.microwaves101.com/encyclopedias/coax-cutoff-frequency

The cutofffr equency for coax lines is definded as the frequency where higher order modes appear..in principle there is not cutoff frequency for the TEM mode but in practice one should not use TEM lines beyond this frequency when higher order modes come in.

I think a coaxial cable has three main electrical elements i.e. inductance-capacitance-resistance .when this cable is powered by a current with a range of frequency into it , in the simplest case , it is modeled by a series inductance and a parallel capacitance. and it will have a cut off frequency.

thank you

Reza Ghorbanpour Taheri

Because the coaxial cable does not have lumped capacitance or inductance, but all the elements are continuously distributed, there is no cutoff frequency as you describe, or, equivalently, the cutoff frequency is infinite.

Thank you Mr. Engineer Malcolm White

I think for short line and low frequency , the response of line is as like as a line with lumped parameters, and will have a cut off frequency.

for long line with distributed elements ,in the frequency response, we obtain multiple cut off frequency with frequency sweeping, or for each pole in response.

it depends on the length of the line.

thank you

Reza Ghorbanpour Taheri

You are mistaken.

A straight coaxial line, excited in the TEM mode, has no cutoff frequencies and transmits all frequencies without distortion between a matched transmitter and matched load at the two ends. There is no effect due to the length of the line except attenuation. The only effect of frequency is an increase in attenuation with increasing frequency due to skin depth, and changing attenuation with frequency due to the dielectric between the conductors. This continues to very high frequencies until the conductors start to be transparent to the electromagnetic radiation.

hello Malcolm White

Ithink if in a cable "c"and"l" cancell each other,the line will be a ohmic line and will not have cut off frequency,but when the other condition like length-frequency-invironment and...change, and the line is not in a matched condition,

it will have cut off frequency

this is why one should match the line to pass all frequency,otherwise it will cancell some frequency because of cut off.

thank you