Microwave TL coax impedance

[nix85]

I been researching how to impedance match microwave antenna and i am not sure about (characteristic) impedance of coax TL. Namely, we all know CI is not length dependent as all units determining it are per unit length and thus cancel out, but what about frequency?

Yesterday i stumbled upon this article from IetLabs and they clearly state:

Although it can be represented in terms of inductors, capacitors and resistors, characteristic impedance is a complex number that is highly dependent on the frequency of the applied signal. Zo is not a function of the cable length. At high frequencies (> 100kHz), the characteristic impedance is almost purely resistive. At mid-range frequencies (1kHz), Zo is affected by capacitance (ωC) and at low frequencies (DC – 100Hz), Zo is influenced by conductance (G). Refer to Figure 2.

https://www.ietlabs.com/pdf/application_notes/5-Characteristic%20Cable%20Impedance-Digibridge.pdf

What does this mean, that coax at 100KHz+ behaves purely like a resistor? If so is this because reactance of the coax is mainly due to capacitive reactance which drops with frequency?

In practical example, if we use coax TL to transmit let's say 1.2GHz signal to the antenna, what impedance will the signal source see at the coax input?

Few general reactance/resonance formulas.

XL= 2πfL
XC= 1/2πfC
Z = sqrt(R² + (Xc - Xl)²)
F = 1/6.28(LC)
F = 1/2π√LC

[T3sl4co1l]

Zo is the terminal impedance of an infinite transmission line, or of an infinite equivalent, i.e., terminated into the conjugate impedance.  It would indeed look like a resistor in this case, or at least, even if it doesn't turn your applied power into heat, it's power you're never getting back out (of the infinite TL).

For other cases, Zo is the impedance used to transform the load and source impedances according to the relevant relations.  You can use a 2-port transmission (ABCD) matrix and trig identities (for the AC steady-state characteristics), or the equivalent simplified formula,
https://en.wikipedia.org/wiki/Transmission_line#Input_impedance_of_transmission_line
or if you require time domain, you can use a ladder diagram to analyze the interference of wavefronts, or...

Tim

[nix85]

Ok thnx, but can you or someone say from experience, what would be the APROXIMATE impedance of coax at 1.2GHz.

[T3sl4co1l]

"Of coax"?

You'll have to fill in all the missing variables, mentioned above.

Impedance
Length
Anything else particular (losses, dispersion, leakage)
Source impedance
Load impedance

Tim

[nix85]

Impedance - let's say standard 75Ohm

Length - let's say 1.5m (multiple of 25 cm wavelength)

Anything else particular (losses, dispersion, leakage) - unknown

Source impedance - let's say standard 50Ohm

Load impedance - helical antenna so let's say standard 14OHm

I am sorry but i have no more detail, i dont expect an exact answer anyway, i just want to hear from someone with experience of matching impedance at particular frequency to share their experience.

[vk6zgo]

I been researching how to impedance match microwave antenna and i am not sure about (characteristic) impedance of coax TL. Namely, we all know CI is not length dependent as all units determining it are per unit length and thus cancel out, but what about frequency?

Yesterday i stumbled upon this article from IetLabs and they clearly state:

Although it can be represented in terms of inductors, capacitors and resistors, characteristic impedance is a complex number that is highly dependent on the frequency of the applied signal. Zo is not a function of the cable length. At high frequencies (> 100kHz), the characteristic impedance is almost purely resistive. At mid-range frequencies (1kHz), Zo is affected by capacitance (ωC) and at low frequencies (DC – 100Hz), Zo is influenced by conductance (G). Refer to Figure 2.

https://www.ietlabs.com/pdf/application_notes/5-Characteristic%20Cable%20Impedance-Digibridge.pdf

What does this mean, that coax at 100KHz+ behaves purely like a resistor? If so is this because reactance of the coax is mainly due to capacitive reactance which drops with frequency?

In practical example, if we use coax TL to transmit let's say 1.2GHz signal to the antenna, what impedance will the signal source see at the coax input?

Few general reactance/resonance formulas.

XL= 2πfL
XC= 1/2πfC
Z = sqrt(R² + (Xc - Xl)²)
F = 1/6.28(LC)
F = 1/2π√LC

Coaxial, or any other transmission line, may be analysed as a network of series inductance & parallel capacitance.

wikipedia has a fairly good explanation:-
https://en.m.wikipedia.org/wiki/Characteristic_impedance

In the real world, however, characteric impedance is calculated using  physical dimensions of the cable such as conductor radius, spacing, etc.


If the transmission line was infinitely long it would always appear as purely resistive at its characteristic impedance.
For instance, an infinitely long 50 Ohm CI coax cable could be replaced by a perfect resistor of that value.

Unfortunately, we can't make infinitely long cables, nor would we have any use for them, so we must use real lengths of cables.

if we now place our perfect 50 Ohm resistor scross the far end of our practical cable, it "looks like" an infinitely long cable at the input end.

We can't make perfect 50 Ohm resistors, be we can come pretty damn close!
If the 50 Ohm cable is now terminated in its Characteristic Impedance, the input will look like 50 Ohms resistive to any source, regardless of frequency, up to the design limits of the cable & the termination.

Terminate the cable in any other impdance, or a short or open circuit, the input impedance to that cable will no longer be independent of the applied frequency

[nix85]

vk6zgo, thank you but none of that is new to me and does not answer my question regarding FREQUENCY. I'm already subbed to Stan's channel and own his book and have read the wiki article on CI.

I am asking how does CI vary relative to frequency, to quote IetLabs again, they don't mention an imaginary infinite line, it seems they are talking about real finite length TL.

"Although it can be represented in terms of inductors, capacitors and resistors, characteristic impedance is a complex number that is highly dependent on the frequency of the applied signal. Zo is not a function of the cable length. At high frequencies (> 100kHz), the characteristic impedance is almost purely resistive. At mid-range frequencies (1kHz), Zo is affected by capacitance (ωC) and at low frequencies (DC – 100Hz), Zo is influenced by conductance (G). Refer to Figure 2."

[rfeecs]

The IET labs article is maybe badly worded at best.  Figure 2 seems like it might be complete bullshit:


I believe the article is explaining how they recommend to extract the line impedance by making measurements at very low frequencies with an LCR meter.  They are saying that the impedance goes way off when you go to low frequencies.  I don't think this is really the case.  It is just based on the crude lumped model they are using for this measurement.  I would disregard this article for your question of impedance matching at high frequencies.

[Kalvin]

If you have a 50 ohm coax which is specified for 1.2GHz operation and which is terminated by 50 ohm resistor, the signal generator (transmitter) will see 50 ohm resistive impedance. If you want to double check, just measure the impedance using a VNA or spectrum analyzer+tracking generator+rf bridge.

[nix85]

That was my first reaction when seeing those formulas, but don't rush to conclusions. This is a professional company specialized in such matters, i would NOT call BS on them. https://www.ietlabs.com/

[nix85]

If you have a 50 ohm coax which is specified for 1.2GHz operation and which is terminated by 50 ohm resistor, the signal generator (transmitter) will see 50 ohm resistive impedance. If you want to double check, just measure the impedance using a VNA or spectrum analyzer+tracking generator+rf bridge.

Yea, and to be safe i'll make cable multiple of half wavelength long (taking into account velocity factor) so CI DOES NOT REALLY MATTER, since load impedance repeats at half wavelengths down the TL.

[nix85]

RoGeorge Double check your assumptions because i know what CI is, how it's calculated, why 50 Ohm compromise between 36 and 73 ohm, why impedance match is important, reflections, standing wave ratios, what happens when you got mismatch, both cases, i know what reactance is, what's complex impedance, real + imaginary (-capacitive, +inductive)...characterstic AKA surge impedance is closely related to complex impedance, in fact they are the same if there are no reflections. For long i understood this as a general principle comparable to water flowing through a tube, it is of course desirable that diameter of the tube remains constant cause otherwise ratio of pressure (voltage) and volume (amps) changes, and when they fall out of phase we get reflections that is energy losses...etc etc.

As for IET labs article, don't dimiss their article just because you are not sure what it's about. I am not 100% sure, but looking at the Complex Equation for CI, it seems frequency does not cancel out.

[T3sl4co1l]

The formulas (corrected for formatting, it would seem?) are approximations that only hold in certain regimes, and not on the transitions between them.

For example, for very long lines (many km, as a telephone operator or power distributor might be concerned with), resistance and capacitance tend to dominate, and the inductance isn't so much of a problem (the electrical length may be less than a wavelength).  It turns out, for normal twisted pair at telephone frequencies (500-3000Hz), 600 ohms is a closer match; preamps and frequency and phase compensation networks are needed.

This is not relevant at 1.2GHz where 1.5m of cable has many wavelengths of electrical length, and normal high frequency losses apply.

Tim

[vk6zgo]

The IET labs article is maybe badly worded at best.  Figure 2 seems like it might be complete bullshit:


I believe the article is explaining how they recommend to extract the line impedance by making measurements at very low frequencies with an LCR meter.  They are saying that the impedance goes way off when you go to low frequencies.  I don't think this is really the case.  It is just based on the crude lumped model they are using for this measurement.  I would disregard this article for your question of impedance matching at high frequencies.

I would agree.
At the frequencies they used, the short will look like zero Ohms, & the O/C like infinity.
Both capacitance & inductance are going to be so small at low frequencies for practical lengths of cable that they can be ignored.

It seems that IET make  LCR bridges---period!

Such devices are not the instrument of choice for measuring the Zo of transmission lines.
More effective methods are a simple RF signal generator used with an Oscilloscope & a variable resistance  termination, a Time domain reflectometer, used with the same termination, or to be a bit more upmarket, a VNA.

Perhaps it may be a good idea to look for application notes from HP, Rohde & Schwarz, or even venerable old General Radio.

In an ideal world, Zo does not vary with frequency, but real cables will differ from ideal as they approach their maximum useable frequency, as losses will accumulate, & affect the measurable Zo.
( those series resistances & parallel conductances we ignored at lower frequencies come back into the picture).

[vk6zgo]

If you have a 50 ohm coax which is specified for 1.2GHz operation and which is terminated by 50 ohm resistor, the signal generator (transmitter) will see 50 ohm resistive impedance. If you want to double check, just measure the impedance using a VNA or spectrum analyzer+tracking generator+rf bridge.

Yea, and to be safe i'll make cable multiple of half wavelength long (taking into account velocity factor) so CI DOES NOT REALLY MATTER, since load impedance repeats at half wavelengths down the TL.

Sorry, but that's not quite so!
All is rosy at the input end of the cable, except that, using your figures, you still have a matching problem between 50 Ohms & 14 Ohms.

At every other point of the cable, apart from the halfwave points, the impedance looking towards the load is different than 14 Ohms.
For instance, at 1/4 wavelength back from the termination, the impedance is 401 Ohms, again using your figures.
You are still transferring the same power, so at that point, the line current will be low, and the voltage across the line, high.

This can, at high power levels cause dielectric breakdown, damaging the cable, as well as being a source of losses.
That is for a point where the impedance is purely resistive.

At other points along the cable the impedance may be highly reactive, causing large variations in both voltage & current.
High currents introduce greater I^2R losses, & as above, high voltages can cause losses or perhaps, break down.

For these reasons, the simple use of half wavelength transmission line is not widely recommended, especially using normal flexible coaxial cable.

Hams often "get away with it" by using Open Wire Parallel line, as its air dielectric, wide spacing, & low series resistance, allow the user to ignore to a large extent, the problems referred to above.

[nix85]

At every other point of the cable, apart from the halfwave points, the impedance looking towards the load is different than 14 Ohms.

You mean 140 Ohm. And so what? I mean that's the point to terminate it exactly at halfwave point to get that load impedance repeated. Are you saying other factors will make those points not exactly where one would expect them to be so this kind of matching is not practical or?

[nix85]

https://postimg.cc/PC5ByKLT

Is this even correct, i never seen CI calculated this way. How do you even measure impedance of open circuit?

[rfeecs]

This seems to come out of Figure 2:
Start with the "Complex Equation":


This comes right out of the expression for Z0 from the telegrapher's equations:
https://en.wikipedia.org/wiki/Characteristic_impedance#Transmission_line_model

They are saying this is equivalent to the "Resistive Measurement" from Fig 2:


So for a short length of transmission line (no distributed effects, much shorter than a wavelength)
Z_SC ≈ R+jωL
Z_OC ≈ 1 / (G+jωC)

It does make some sense that this BS looking equation would be approximately true for low frequencies.

[vk6zgo]

At every other point of the cable, apart from the halfwave points, the impedance looking towards the load is different than 14 Ohms.

You mean 140 Ohm. And so what? I mean that's the point to terminate it exactly at halfwave point to get that load impedance repeated. Are you saying other factors will make those points not exactly where one would expect them to be so this kind of matching is not practical or?

You, in fact, wrote 14OHm, which it would be reasonable to read as 14

The half wavelength points will be as you assumed, & the load impedance will appear at the cable input.

If you read the rest of my posting, you would have realised that, at various points, along the transmission line, the relationships between current through, & voltage across the cable will not be the same as at its input or output.

Your antenna & Transmitter may be happy, although you still need to match 50 to 140 , but remember, the transmission line exists at all these other lengths in between.

At some point, the current will be high, causing I^2R losses, & at others, the voltage will be high, with dielectric losses & the risk of breakdown.,

Losses in a transmission line are specified when correctly matched.
In other situations, all bets are off!

You seem to be ready to argue with every point brought up in response to your OP.
If you feel you already "know it all", why post the query in the first place?

[radiolistener]

You can measure your coax cable capacitance (with open second cable end) and it's inductance (with short circuit on the second cable end), then you can use this formula to get the cable impedance [Ohm]:

Z = sqrt( L / C )

also you can calculate wave propagation speed of your cable by using this formula [m/s]:

NOTE: for this formula you're need to divide measured L and C values by cable length in order to get distributed L and distributed C.

v = 1 / sqrt( L * C )


and then calculate velocity factor of your cable by using this formula:

vf = v / 299792458

L should be in [H/m] and C should be in [F/m].

You can transform these formulas for L in [uH/m] and C in [pF/m]:

Z = 1000 * sqrt( L / C )
v = 1000000000 / sqrt( L * C )


But note, you're needs to measure L and C for your cable at the working frequency, because coax cable impedance depends on the frequency. Coax cable has too high impedance deviation at very low frequency (smaller than 500 kHz). So, you're needs to use LC meter with working frequency for about 1 MHz or above that. Usually coax cable has stable impedance from specification at 5-10 MHz and more. At low frequency coax cable has too high impedance deviation which depends on the exact frequency value.

Also it will be hard to measure too short piece of coax cable, because it has too small capacitance (just some pF) and too small inductance (just some nH) and your your probes will affect your measurements. For long cables (several meters and more) this measurement method works very well.

PS: in the same way you can calculate impedance and wave propagation speed of any environment by measuring distributed impedance and distributed capacitance of that environment.

For example you can calculate impedance and wave propagation speed in the free space vacuum by using it's distributed inductance (also known as magnetic constant or vacuum permeability μ0) and distributed capacitance (also known as electric constant or vacuum permittivity ε0):

L = μ0 = 4 * pi * 1e-7 [H/m]
C = ε0 = 8.854187812813e-12 [F/m]

Z = sqrt( 4 * pi * 1e-7 / 8.854187812813e-12 ) = 376.7303136 [Ohm]
v = 1 / ( 4 * pi * 1e-7 * 8.854187812813e-12 ) = 299792458 [m/s]

these values known as free space impedance (377 Ohm) and speed of light (299792458 m/s)

[nix85]

At every other point of the cable, apart from the halfwave points, the impedance looking towards the load is different than 14 Ohms.

You mean 140 Ohm. And so what? I mean that's the point to terminate it exactly at halfwave point to get that load impedance repeated. Are you saying other factors will make those points not exactly where one would expect them to be so this kind of matching is not practical or?

You, in fact, wrote 14OHm, which it would be reasonable to read as 14

The half wavelength points will be as you assumed, & the load impedance will appear at the cable input.

If you read the rest of my posting, you would have realised that, at various points, along the transmission line, the relationships between current through, & voltage across the cable will not be the same as at its input or output.

Your antenna & Transmitter may be happy, although you still need to match 50 to 140 , but remember, the transmission line exists at all these other lengths in between.

At some point, the current will be high, causing I^2R losses, & at others, the voltage will be high, with dielectric losses & the risk of breakdown.,

Losses in a transmission line are specified when correctly matched.
In other situations, all bets are off!

You seem to be ready to argue with every point brought up in response to your OP.
If you feel you already "know it all", why post the query in the first place?

First of all, i have no intention to argue nor i feel "know it all", i came here FOR ANSWER. It's just that people often answer without thinking through or really knowing.

Again, helical antenna radiation resistance at resonance is ~140 Ohm, i don't know where you saw 14.

"If you read the rest of my posting, you would have realised that, at various points, along the transmission line, the relationships between current through, & voltage across the cable will not be the same as at its input or output."

You keep bringing that irrelevant and obvious fact up although it is assumed cable is rated for given current/voltage.

I did not say i will just make cable multiple of halfwavelength, i said i will do it as additional measure to impedance matching.

[nix85]

You can measure your coax cable capacitance (with open second cable end) and it's inductance (with short circuit on the second cable end), then you can use this formula to get the cable impedance [Ohm]:

Z = sqrt( L / C )

also you can calculate wave propagation speed of your cable by using this formula [m/s]:

NOTE: for this formula you're need to divide measured L and C values by cable length in order to get distributed L and distributed C.

v = 1 / sqrt( L * C )


and then calculate velocity factor of your cable by using this formula:

vf = v / 299792458

L should be in [H/m] and C should be in [F/m].

You can transform these formulas for L in [uH/m] and C in [pF/m]:

Z = 1000 * sqrt( L / C )
v = 1000000000 / sqrt( L * C )


But note, you're needs to measure L and C for your cable at the working frequency, because coax cable impedance depends on the frequency. Coax cable has too high impedance deviation at very low frequency (smaller than 500 kHz). So, you're needs to use LC meter with working frequency for about 1 MHz or above that. Usually coax cable has stable impedance from specification at 5-10 MHz and more. At low frequency coax cable has too high impedance deviation which depends on the exact frequency value.

Also it will be hard to measure too short piece of coax cable, because it has too small capacitance (just some pF) and too small inductance (just some nH) and your your probes will affect your measurements. For long cables (several meters and more) this measurement method works very well.

PS: in the same way you can calculate impedance and wave propagation speed of any environment by measuring distributed impedance and distributed capacitance of that environment.

For example you can calculate impedance and wave propagation speed in the free space vacuum by using it's distributed inductance (also known as magnetic constant or vacuum permeability μ0) and distributed capacitance (also known as electric constant or vacuum permittivity ε0):

L = μ0 = 4 * pi * 1e-7 [H/m]
C = ε0 = 8.854187812813e-12 [F/m]

Z = sqrt( 4 * pi * 1e-7 / 8.854187812813e-12 ) = 376.7303136 [Ohm]
v = 1 / ( 4 * pi * 1e-7 * 8.854187812813e-12 ) = 299792458 [m/s]

these values known as free space impedance (377 Ohm) and speed of light (299792458 m/s)

Yea, i was considering that but i also thought these values should be known by the cable manufacturer. I'm planning to buy this LCR meter

https://www.ebay.com/sch/i.html?_from=R40&_nkw=Peak+LCR45&_sacat=0&LH_TitleDesc=0&_sop=15

(for some reason link above starts listing with $156.66 unit altho original link displays cheaper ones starting at 61$ + 17$ shipping)

but i think it can only measure up to 200KHz, far from 5-10MHz. And LCRs that can measure that high are quite expensive, will consider those too tho.

And tnx for additional explanations of free space impedance and speed of light based on LC values of free space.

[rfeecs]

They are saying that the impedance goes way off when you go to low frequencies.  I don't think this is really the case.

Uh, yeah I was wrong about that.  But you do have to typically go down below about 1 MHz before you see this effect:

"Transmission Lines at Audio Frequencies, and a Bit of History"
http://audiosystemsgroup.com/TransLines-LowFreq.pdf

Not a concern for matching an antenna at 1.2GHz.

Not really a concern unless you are dealing with miles long cables at audio frequencies.

Section 5.9 of Keysight Impedance Measurement Handbook shows the exact same method as EIT Labs for measuring cable impedance with an impedance analyzer:
https://literature.cdn.keysight.com/litweb/pdf/5950-3000.pdf

[radiolistener]

I'm planning to buy this LCR meter

but i think it can only measure up to 200KHz, far from 5-10MHz. And LCRs that can measure that high are quite expensive, will consider those too tho.

it's too expensive and has too small working frequency. And doesn't have way to connect picofarad capacitors and nanohenry inductors with no wires (to avoid influence).

I recommend you to buy vector antenna anlyzer:
https://www.aliexpress.com/item/Lusya-4-3-inch-LCD-Mini600-HF-VHF-UHF-Antenna-Analyzer-0-1-600MHz-SWR-Meter/32912877741.html

it will allow you to measure L, C, SWR, complex impedance (R,X), S11 and smith chart at 0.5 - 450 MHz range.
Also it will help you to check or tune your antenna.
Also you can use it as TDR reflectometer to check RF cables.
And you can use it as a simple RF generator for 0.5 - 450 MHz.

With custom firmware you can also use it as spectrum analyzer, frequency meter, crystal Q-factor tester and other


If you want just a cheap and simple LC meter, you can buy this one:
https://www.aliexpress.com/item/Digital-LCD-Capacitance-meter-inductance-table-TESTER-LC-Meter-Frequency-1pF-100mF-1uH-100H-LC100-A/32829227933.html

it works at about 700 kHz

[radiolistener]

They are saying that the impedance goes way off when you go to low frequencies.  I don't think this is really the case.

it really goes way off at low frequency. You can connect a piece of coax cable to a vector analyzer and make sure that this is a truth.

here is example how 10 meters of Chinese RG316 cable behave at low frequency.
Unfortunately I cannot show you frequencies below 500 kHz, but I know that it goes to even higher impedance deviation.