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Z0 of a line with unequal wires

  1. Aug 9, 2013 #1
    In case of a transmission line consisting of two wires in parallel (aka Balanced line) the characteristic impedance Z0 is given by this equation:


    where l is half the distance between the wire centres, R is the wire radius and \mu, \epsilon are respectively the permeability and permittivity of the surrounding medium.

    I have two questions I cannot answer myself:

    • what is the characteristic impedance in case of wires of different diameter ?
    • In case of a tube rather than a solid wire: does the characteristic impedance depend soley on the outer diameter or does the inner diameter play a role as well?

    Thanks in advance for any hint.
  2. jcsd
  3. Aug 9, 2013 #2


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    Hi modmix
    welcome to PF

    the formula is very similar but it also takes into account the ratios of the diameters of the inner and outer conductors along with the dielectric constant.
    I dont know how to do latex to do the formula here so will aim you at one of the best sources on the net ....


    have a read down that page ... it should answer your questions

    hope that helps you :)

  4. Aug 9, 2013 #3
    Thanks Dave for both, the welcome and your answer.
    Nice to be back at what I did more than 30 years ago at university ,-)

    I guess I wasn't precise enough - the page you pointed at talks about coax.

    I'm looking for a formular to describe a Twin Line where the diameter '2a' of both wires is different ( eg. 0.8 mm and 2 mm).
    from: Transmission Line Parameter Calculator

  5. Aug 9, 2013 #4


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    ahhh ok

    In all my years of working with radio or other transmission lines, a twin cable or open wire lines are always the same diameter. Am not sure of why you would want or how you would end up with them being different diameters ?
    Coax is the only instance where Im familiar with differences in conductor sizes

    Just curious, is there a specific situation where this is applicable for you ?

  6. Aug 10, 2013 #5
    Sure, in almost 100% the case wires do have the same diameter.

    A while ago I made an SPDIF cable with two wires in parallel. Lucky me, the cable plays music better than all commercial cables I've tried so far. Don't ask me why (but, of course, I would like to know how SPDIF transmission properties do affect sound quality - might be a matter of jitter...).

    As SPDIF signal ground is normally connected to the case of the connected components and, as ground loops due to mains is quite common in such a situation, the idea is to reduce the gound loop impact on the signal by reducing the ground wire resistance (similar to the concept of parallel earth connector (PEC).

    To be able to that, knowing the required geometrie to achieve 75 Ω would be great.

    Just to thicker wire was the first idea. It turned out that the distance between thicker wires for a 75 Ω line are very very small and impedance changes far to much in case of small derivations from the needed geometrie.

    The hope is, to get a less sensitive solution by using just one thick wire.

  7. Aug 10, 2013 #6


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    usually SPDIF uses either coax or optical fibre and maximum cable length is recommended to be ~ 10 metres

    Because its a digital signal, I dont see that there would be any affect on the audio quality unless there is significant degradation to the digital signal. That is if the bit stream leaving the transmitter can be correctly identified and processed at the receiver then the audio wont be affected

  8. Aug 10, 2013 #7
    Apart from the shielding, I guess, this is mainly for the simple reason that controlling the impedance for 75 Ω is a bit tricky with wires in parallel.

    At least in theory.
    In the real world, different digital cables do produce different sound quality - I guess mainly do to different jitter they invoke.
    I did experimental physics - theory comes second ,-)

    That's why I asked about impedance of parallel wires of different diameter - I simply do not know how to get the formular derived from Maxwell's equations ,-)

  9. Aug 10, 2013 #8


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    It is the outside diameter that matters with RF conductors. They are modelled as thin walled conductive cylinders. The current flows only in the surface. With a coaxial line the signal propagates through the dielectric between the outside of the inner and the inside of the outer conductors.

    Audio quality is not degraded when transmitted over a digital link unless there are bit errors. That is better fixed by using a higher quality coaxial line rather than an exposed parallel line. The exposed parallel line will pick up external signals that will on occasion cause more significant bit errors than the original coaxial line. Parallel lines have less dielectric and so are characterised by significantly lower losses than coaxial cables, maybe your line is too long and you need a radio link.

    A parallel transmission line in an audio environment will need to be twisted or it will generate EMI. When it is twisted it will need some dielectric insulation for support. That will again increase the dielectric losses. The distribution of the dielectric will defeat any simple Zo formula and so suggests the use of FEA.

    The characteristic impedance of a transmission line, Zo = Sqrt(L/C); where L and C are the inductance and capacitance per unit length. So to calculate the impedance you must calculate the self and mutual inductance of the two parallel conductors along with their capacitance in the presence of a distribution of dielectric that distorts the electric field. Those are both very involved computations.

    Now the problem is that you cannot make transmission lines with impedances below 200 ohms without significant capacitance which requires a dielectric. That results in losses which in turn increases the bit errors.
  10. Aug 10, 2013 #9
    That is true for lossless lines at high frequencies, I guess. Think about skin effect, for example.
    Here graph from a german manufactuar:

    And bits are bits. Sorry, that's a too simple model. It doesn't fit to my experience.

    What is FEA, pls.

    I'm sorry to tell you that I did both, 75 Ω and 110 Ω using wires instead of coax. 75 Ω isn't twisted. Nevertheless, music is best using this cable.

  11. Aug 10, 2013 #10


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    sorry, perceived personal experience isnt really a scientific answer.
    you would need to provide measurable and repeatable data to support your claims :)

    you seem to be falling into that catagory of those that believe that OFC cables produce better music quality
    their problem is mainly ... that after spending $100's on the cables , they are not likely to admit that it makes no difference, so the myth is perpetuated

    both baluncore and I have told you that unless the bit stream incurs errors there is going to be NO change in the information that is transmitted and received. that is easily measured and can be shown under repeatable tests

    I would love to see how you are getting your low Zo's. show us your test setups and results proving such

  12. Aug 10, 2013 #11


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    The Zo of a transmission line is only important when the length of the line is greater than about one 10th of a wavelength. For a 30m long line that equates to a 300m wavelength which is equivalent to 1MHz. So by the time your frequency rises high enough for the cable to be treated as a transmission line the skin effect depth in copper will be only 66.um = 0.066 mm = 2.6 thousandths of an inch. The digital data rate on your line is above 1MHz so on the German manufacturers marketing graph the horizontal black line Zo = Sqrt(L/C) is the one that counts. If you were not transmitting digital data but analogue audio then the line would be treated differently, as a lump rather than a line. That graph is for analogue signals not for digital data streams. It confuses matching the line with driving and terminating the lump.

    You must differentiate between the transmission of the bits and the initial “analogue to digital” and final “digital to analogue” conversion processes. The transmission will not degrade the audio if there are no bit errors over the link. That is a simple fact. There is jitter in every data stream but the D to A converter at the far end is being clocked with data at a much lower rate than the jitter. 32 bit serial data with 10% jitter is only 1/320 of the sample rate which is 640 times the highest frequency in the audio. If there is loss of audio quality it is because the converters are poorly designed.

    FEA is finite element analysis. It solves for the capacitance and the inductance of a discrete model from which you must compute Zo = Sqrt(L/C). The general transmission line impedance equations use exactly the same Zo = Sqrt(L/C).

    How do you know you “did” those characteristic impedances? Do you really have a time domain reflectometer? I expect you used insulation which has losses so limits the range, just like a coaxial line. Because your signal works once does not mean it will always work since interference is spasmodic, people walk around with mobile electronics. Coaxial lines are used to avoid interference.

    You will believe what you want to believe. I can only give you a guide to understanding the physics of transmission lines. Belief systems are outside the remit of the Physics Forum's peer reviewed science.
    Last edited: Aug 11, 2013
  13. Aug 11, 2013 #12
    Hi Dave,

    I agree completely
    With one remark: personal experience might be a reason for further investitagtion.

    I'm working on that, too.
    It's not so easy to find affordable measurment tools with high enough resolution.

    In a german forum Gert reported some measurements on how jitter depends on an SPDIF cable (the vco voltage at the receiver is recorded).
    Apogee Big Ben als Jittermonitor
    Welches Koaxkabel für S/PDIF? p.5 gives some spectrograms like this

    The like to the recorded files doesn't work anymore.
    Give me your email address in case you would like to get the files.​

    If you agree, I'ld like to come back to the thread question (formular for Z0 of a lecher wire with wires with unequal diameter).

  14. Aug 11, 2013 #13

    Yes. Sort of: an ATtiny13 on a LT Demo Circuit board gives pulses watched with a 400 MHz scope:

    100% agreeded.
    That's why I asked in my previous post to come back to my question, which is pure physics, I'ld say.

    Thanks for your comments.
  15. Aug 11, 2013 #14

    see Bild 4a and equation (7) in mgk1.pdf.

  16. Aug 13, 2013 #15


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    Unless you were getting a significant / measurable difference in error rate then how would Jitter, Noise or frequency response have any effect? Any subtle difference you could be getting, using your connection cable, could very likely be down to some other difference between your system and the comparison system. Did you actually do a Blind A/B comparison between the two systems? If not, then your observation is questionable - along with a large number of other opinions that we read from HiFi enthusiasts.
  17. Aug 13, 2013 #16
    That is exactly the question I'ld like to get an answer for ,-)
    When CD was introduced, I didn't like what I heard as some other did. All CD fans told me: "That's digital.That's just 0 and 1. That must be right! You are wrong by not accepting the truth."

    Later it turned out that jitter is important and that first CD players didn't do it very well.
    I'm used to being told being wrong when I do believe in what I (and other people) do listen ,-)

    What if the difference can be heard in my system just by changing one cable?
    What if the same happens in other systems doing the same change?

    I only have one eye ,-)

    To be serious:

    Once a person building quite good analog stuff got a digital cable - just to get his opion. He didn't like digital audio much, was just starting to get a setup which did well enough for his eras. He put the cable into his system. Later, his wife went in to call him for lunch. She asked quite irritated: "What did you do? All the treble is gone!" - blind enough?
    Not the treble but the artefacts due to jitter were almost gone - something one must in many cases learn to appreciate ,-)
    Once a friend played some warm and frozen ripped CDs. CDs I never had heard before. After an hour or so, I had a kind of signature related with either warm or frozen. Many hours later we where listing to the reproduction by a prototype amplifier - the focus was clearly on the amp. Within the middle of that session, I asked for confirmation that the played track is from a frozen rip. Was confirmed. - blind enough?

    We did a check with a β-version of the Mutec MC-3+ smart clock used as a re-clocker. I had two different verisions at that time. I simply wanted to get a second opinion on that gear. In a first run, the result was reverse to mine. Then we flipped the digital cables (the said better cable now between re-clocker and dac). Now the result was 'correct' ,-)
    The friend left the room. I stayed for further listening.
    As it wasn't the best heard configuration, I stood up and changed to the best configuration. Well, it wasn't as good, as I've heard this before - may be I'm simply tiered, I thought.
    The friend came back and immediately started complaining loudly about the bad sound. After a while, I asked him: "Do you know that you are telling me, that the so far best heard configuration doesn't play well?" "No! that cannot be the case!". He went to see what configuration we were listenig to. Yes, seemed to be the best one. He went back for further listening. As he was still sure that that wasn't right, he went again to see what's going on. "What did you do?" he suddenly asked. "Nothing except for going back to the best configuration..." - "Hee, hee, you made a mistake! the digital cable is not connected in the right direction!"
    After having changed that, sound was indeed better.
    My friend didn't know that I had changed the setup.
    I didn't know about the direction of the digital cable.
    - blind enough?

    I do stop here.

    And I do admit that the listening capability does vary among persons.

  18. Aug 13, 2013 #17


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    When you digitally encode an analogue signal, you build in inherent distortion by the sampling process and, of course there will be imperfections introduced in the implementation of the ADC. They use so many bits and sample at the frequency they do in order to reduce the impairments to a level that is imperceptible / acceptable (based on groups of 'experts' with golden ears).The distortions and noise which are associated with this process can be made measurably less than any analogue tape recording or with vinyl pressing - factors that are always ignored by the anti-digital brigade. If you do nothing with the resulting digital signal except reproduce it with no errors then the information will not, by definition, be changed in any way. The channel is indistinguishable from a short piece of copper wire.
    If you play out that digital signal with a high enough quality DAC, then there will be no additional impairment. The digital part of the channel is totally transparent unless errors have been introduced by some fault.
    It is a fact that the distortions due to tape and the characteristics of vinyl recording are very audible and identifiable and people have got used to them enough to like them. This is along the same lines as the fondness some people have for valve amplifiers and their 'sound'. It's subjective and perfectly ok. But the quality of such systems is due to the original program signal having been modified. It's not the same as when it started out.
    Once you take a 'CD quality' digital signal and try to compress it into a channel with a lower data rate, there is a risk (certainty in most cases) that the quality will be impaired. 'Digital' is often sold as 'perfect' even when it's an mp3 signal at 48kbps. Look at all those dodgy adverts.

    @modmix Do you have any way of measuring the error rates in your digital system? If the error rates are not the same then you might expect the programme to sound different. (Fair testing and all that.) There are good and bad systems for masking and correcting digital errors.

    Probably best just to stick to the Physics or Engineering on PF; things that you can actually measure. There are plenty of other forums where the subjective aspects of media are discussed without wet blankets like me.
  19. Aug 13, 2013 #18
    As far as I remember, I simply asked for a certain formular - quite a matter of "Physics or Engineering", right?
    May be it was not a good idea to answer off topic questions... I just wanted to be polite.
    I offline here.
  20. Aug 13, 2013 #19


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    You made a statement about performance, too, with an implied claim. That was what I was responding to. (Post 5)
  21. Aug 13, 2013 #20
    #4 asked an off topic question:
    That was answered by me in #5 - sorry for trying to be polite,

    #1 has my original question.
    #3 some clarification

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