Talk:Characteristic impedance

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Does the definition of characteristic impedance in terms of permittivity and permeability belong here?[edit]

Pozar defines the characteristic impedance of a transmission line in terms of the RLGC parameters of the transmission line, and uses the term intrinsic impedance to refer specifically to the relation between the magnitudes of the electric and magnetic fields of a plane wave traveling in an optical medium.

Should this article use the RLGC parameters consistently throughout, perhaps with a statement that the characteristic impedance of a transmission line is analogous to the intrinsic impedance of an optical medium?

Pozar, David (February 2004). Microwave Engineering (3rd edition ed.). 

IJW 01:59, 14 September 2006 (UTC)

Sounds reasonable. And if you believe this should be here, then be bold and edit the article!
Atlant 13:08, 14 September 2006 (UTC)

I have edited the article to define the characteristic impedance in terms of RLGC parameters and linked to Medium (optics) instead of defining characteristic impedance in terms of permittivity and permeability. I also removed the section on frequency dependence because it is misleading; R and G are not constant and the frequency dependence of Z_0 is not as simple as the previous version of the article stated. At AC and higher frequencies R \sim \sqrt{\omega} and G \sim \omega. Only at very low frequencies (where the thickness of the conductors is comparable to the skin depth) is R relatively constant, but the frequency dependence of G remains. I will revisit this section to include these facts and give a more complete treatment.

IJW 14:41, 14 September 2006 (UTC)

Following is the section on frequency dependence that I deleted:

=== variation with frequency ===

The impedance of a real lossy transmission line is not constant, but varies with frequency. At low frequencies, when

\omega L \ll R and \omega C \ll G,

the characteristic impedance of a transmission line is

Z_0 = \sqrt{R/G}.

At high frequencies where

\omega L \gg R and \omega C \gg G,

then the characterstic impedance is

Z_0 = \sqrt{L/C}.

So there are two distinct characteristic impedances for every line. Usually G is very small so the low-frequency impedance is high, whereas the high-frequency impedance is low. The break points in the impedance frequency graph are at \omega_1 = G/C and \omega_2 = R/L (where \omega =2 \pi f). If R/G \gg L/C, it is obvious that \omega_2 \gg \omega_1. Between these two break frequencies the cable impedance decreases smoothly.

Example[edit]

Take the case of a 50Ω coaxial cable with polyethylene dielectric. R is about 100 mΩ/m and G < 20 pS/m (based on measurements of leakage resistance in a 1 m length). Using L=CZ^2, L can be calculated at about 250 nH/m. So,

ω2 = R/L = 200 krad/s (f2 = 30 kHz)

and

ω1 = G/C = 0.2 rad/s (f1 = 30 millihertz)
At 100 Hz the 50 ohm coaxial cable will have an impedance of about 900 ohms, only reaching 50 ohms at about 30 or 40 kHz. The phase angle of the impedance between the two break frequencies is leading (the cable looks capacitive).

IJW 17:11, 14 September 2006 (UTC)

Four comments[edit]

  • A definition should give the method to measure the item defined. If you define the characteristic impedance as the ratio of voltage to current in the line, you simply cannot measure it. The only place where the measure can be done is at the end of a semi-infinite line. Then, instead of talk about waves, why not define the characteristic impedance as the impedance measured at the end of a semi-infinite line?
    Of course, in the two definitions, there is always the problem of reflections if the line is finite. You can replace the "last infinite length" of the line with an impedance equal to the characteristic impedance. The proposed definition allows this.
    You can also use a finite length of line and make the measure before the arrival of reflections (a pulse generator and an oscilloscope are enough).
  • In a transmission line, you cannot have a voltage wave without current or a current wave without voltage. Any wave is voltage plus current. You can write two equations, one for voltage and one for current but they form the same wave. There is not "a pair of waves".
  • You can hardly talk of "transmission line" or "characteristic impedance" when the length of the line is negligible compared to the wavelength in the line. You have just a conductor with negligible inductance and stray capacitance. This is the case of IJW example of the coaxial cable at 30 kHz. It just begins to be a transmission line at a length of a few hundreds of meters and then it behaves more as a resistance than as a transmission line. The characteristic impedance depends on frequency. However, when this is the case, in low frequency, the transmission line is no more interesting. This is not the most interesting aspect of transmission lines. LPFR 12:17, 15 September 2006 (UTC)
  • The vacuum characteristic impedance should be mentioned here,related to the rf domain where the carcateristical impedance is very important. —The preceding unsigned comment was added by 194.138.39.55 (talkcontribs) .
Impedance of a transmission line or impedance of an electrical circuit is the ratio of a voltage divided by a current, both of them measurable quantities. Impedance of vacuum or impedance o a substance is something (\scriptstyle{\sqrt{\mu\over \varepsilon}}) related to the properties of the substance in an electromagnetic field. It just happened that the units of this value are ohms and people could not avoid calling it "impedance". Impedance of a line and impedance of vacuum are very different things. The difference is still greater if you think of transmission lines built with discrete inductors and capacitors used (in the past) as delay lines. LPFR 12:05, 1 October 2006 (UTC)

Electrical and electromagnetic impedances[edit]

As Mebden himself wrote in the page "intrinsic impedance", electrical impedance and electromagnetic impedance should not be confused. The impedance of a transmission line is an electrical impedance and the impedance of a medium is an electromagnetic impedance. LPFR 08:51, 23 October 2006 (UTC)
Impedance of a transmission line or impedance of an electrical circuit is the ratio of a voltage divided by a current, both of them measurable quantities. Impedance of vacuum or impedance o a substance is something (\scriptstyle{\sqrt{\mu\over \varepsilon}}) related to the properties of the substance in an electromagnetic field. It just happened that the units of this value are ohms and people could not avoid calling it "impedance". Impedance of a line and impedance of vacuum are very different things. The difference is still greater if you think of transmission lines built with discrete inductors and capacitors used (in the past) as delay lines. LPFR 12:05, 1 October 2006 (UTC)

Meaning of high or low characteristic impedance[edit]

Would it be accurate to add this (e.g. to the introduction): "A high-quality (high conductance) transmission line tends to have a low characteristic impedance, and vice versa." (Or is it the other way around?) --Coppertwig 13:18, 11 January 2007 (UTC)

Electrical conductance is pretty orthogonal to impedance; you can design a transmission line in a wide variety of impedances (to suite the need) although some impedances are a lot "easier" (natural for the materials employed?) than others. So I guess I disagree with your proposed addition.
Atlant 13:41, 11 January 2007 (UTC)
My goal here is to have this page and related pages improved to the point that a person similar to myself can quickly and correctly understand the concepts being presented. So if something seems unclear or contradictory, that means it needs to be edited.
For now, I'm thinking in terms of transmission lines with zero conductance and zero inductance. The equation given for characteristic impedance in that situation seems to me to reduce to being equal to the resistance of a unit length of the transmission line.
I really like the mention of the infinitely long transmission line in the opening paragraph: it appeals to the intuition in a simple, relatively easily understandable way. However, by itself it isn't enough; I'd like at least one more sentence with similar simplicity and appeal but providing complementary information.
Problem: The first paragraph seems to be claiming that the characteristic impedance is equal to the impedance (resistance, in the case I'm considering) of an infinitely long piece of transmission line, while the equation seems to reduce (in the case I mentioned) to the resistance of a unit length of transmission line. Those can't both be true, can they? It seems to me that there's something wrong. (If they can both be true, this needs to be explained in the article.)
Also, if you're talking about a transmission line with two conductors, (such as often plug into electrical appliances), then with the infinite one you're attaching to both of the conductors, whereas when measuring the impedance of a unit length, it would seem to make sense to measure only one of the conductors at a time. This clouds the issue.
Question I'd like to see answered in the first or second paragraph of the article: which has a larger characteristic impedance as a transmission line: a pair of thick copper wires, or a pair of thin copper wires (straight, separated by an insulator)? Again, I'm thinking in terms of the resistive part of the impedance. (After I understand that, I might tackle the imaginary part.) Doubling the cross-section of the copper wire cuts its resistance in half, I believe. Or do they both have the same characteristic impedance? (I don't think they do.) The answer to this question is basically the same thing as the sentence I proposed at the beginning of this discussion.
Could somebody just give a few examples? Would a coaxial cable tend to act as a capacitor at high frequencies, for example? What happens to the characteristic impedance when you double the thickness of the conductor of a coaxial cable?
I mean: if there's anybody out there who understands what characteristic impedance is, could you please explain it more fully and illustrate it with examples? Thanks. --Coppertwig 04:44, 12 January 2007 (UTC)

Characteristic Impedance? What's infinity got to do with it? ____________________________________________________________ —Preceding unsigned comment added by 92.40.34.110 (talk) 11:25, 6 September 2009 (UTC)


I wish people would avoid talking about "infinite" lines when discussing Z0. Has anybody ever seen one?

Its true, that Zin of a line is equal to "Z0" multiplying a quotient, containing the hyperbolic tangent of the product of length of the line and the propagation co-efficient (easily derived from the transfer matrix of a line). If you let the length of the line tend to infinity, then the quotient tends to unity and one is left with Zin = Z0. Which is all very well mathematically, but it has never been shown practically, because of the problem of obtaining, for example, an infinite length of 50 ohm coaxial cable!

There is a much better definition of Z0, but which requires knowledge of iterative impedance and image impedance. As follows.

One can always find an impedance which when connected to the output terminals of any two port network (including a transmission line), that will give the same impedance, measured at the input terminals. This is called the "iterative impedance" of the network Zit1. Similarly one can always find a suitable generator, whose source impedance, when placed at the input terminals of a two port network will give the same impedance, measured at the output terminals of the network. This is also an iterative impedance, Zit2.

If the network is symmetrical, i.e the determinant of the transfer matrix is unity, then Zit1 = Zit2 = Zit.

Similarly the "image impedance" of a two port network, is that input impedance (and is the complex conjugate)of the generator source impedance, due to a load at the output terminals, and causes maximum power to be transferred from the generator to the network, Zim1. Similarly if the output impedance of the network is equal to (and is the complex conjugate of) the load impedance, then maximum power will be transferred from the network to the load, Zim2. For a symmetrical network, Zim1 = Zim2 = Zim.

And now for the definition. If (and only if) for a symmetrical network, the case that the "iterative impedance" is equal to the "image impedance", this is known as the "characteristic impedance" of the network, and is given the symbol Z0. Z0 = Zim = Zit.

Note, it doesn't matter if the network is a piece of coax cable a mile long, or three resistors connected in a "T" configuration, the definition is still the same. This is charactersitic impedance, and doesn't require the mention of the word infinity.

Phil Robinson —Preceding unsigned comment added by 92.40.34.110 (talk) 11:09, 6 September 2009 (UTC)

However much you might not like it, the infinite line is a commonly accepted way of defining characteristic impedance and it is not Wikipedia's place to change the world. Do any reliable sources use your definition? I have some problems with it - if Zit is always equal to Zim then it is overcomplicated, on the other hand if you are claiming there are cases when Zit != Zim then that implies there are cases when characteristic impedance is undefined, which is nonsensical. An alternative definition which is widely found in the sources, and does not require infinite lines, is that characteristic impedance is the impedance encountered by a wave travelling in a single direction. This is the definition that the article opens with so I don't really see the problem. SpinningSpark 08:14, 1 June 2010 (UTC)

Another way of looking at/understanding Characteristic impedance[edit]

This impedance, remains the same, no matter how long the line is, because the ratio of voltage applied to the current, remains the same, but their actual values reduce, along the length, of a lossy line.

Is the above statement false?, please tell me how to correct it, is the one below better?

This impedance, remains the same, no matter how far along a uniform line, you measure it, because the ratio of voltage applied to the current, remains the same, but their actual values reduce, along the length of a lossy line.

or

This impedance, remains the same, no matter how long the line is, even if it is infinite, because the ratio of voltage applied to the current, remains the same, but their actual values reduce, along the length, of a lossy line.

Bookbuddi (talk) 17:05, 15 April 2012 (UTC)

At the place you tried to add this the article defines characteristic impedance as the input impedance of an infinitely long line (also uniform line is meant but not stated). It makes no sense at all to talk about measuring the characteristic impedance of other lenghts of line if we are taking that as the definition - by definition it is constant since it is defined in terms of an infinite line for any length.
If you were to measure the impedance some way along an infinite line you would not measure Z0; if the line extended to infinity in both directions you would measure Z0/2, otherwise some other impedance. If you were to cut the line some distance from the source and then measure the impedance looking to the right you would indeed measure Z0, but that is still the input impedance looking into an infinite line. The impedance looking to the left (the finite portion of line) would not be Z0 unless the line was terminated in Z0 - that is, you would need to know the quantity you were trying to measure prior to measuring it.
I think what you are trying to say is something like what is already in the lede "Z0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections." The ratio remains constant because Z0 is constant, not the other way around. The limitation to waves travelling in one direction only is essential. Applied voltages and currents will generally not be in the ratio Z0 because of reflections. SpinningSpark 19:01, 15 April 2012 (UTC)


I am trying to give a practical example of what happens with real non infinite lines (to make the concept easier to understand, as we don't have any infinite lines to measure). so can I say:

On a real, non-infinite terminated line, we can see that the impedance remains the same, no matter how far along the line you measure it, because the ratio of voltage applied to the current remains the same, but their actual values reduce, along the length of a lossy line. Bookbuddi (talk) 19:32, 15 April 2012 (UTC)

Does it depend on the length of the cable?[edit]

Does the Characteristic Impedance depend (like resistance) on the length of the transmission line, or is it (like resistivity) independent of the length? Thank you.CountMacula (talk) 11:06, 22 September 2012 (UTC)

It is independent of length. But unlike resistivity, will change if the cross-section of the line is changed (or any other physical characteristic of the line is changed). In future, please go to WP:RDS to ask questions unrelated to improving the article. SpinningSpark 12:17, 22 September 2012 (UTC)
Thank you for that explanation. In future, please think rather than assume that a question is unrelated to improving the article. And in future, when a worthy improvement is pointed out with sublety, please do the actual improvement by editing the article rather than by responding only in the talk page. The article should blurt out right away that characteristic impedance is a property of the type of cable and is independent of length. Thanks again.CountMacula (talk) 00:54, 23 September 2012 (UTC)
I have made some improvements. -—Kvng 14:34, 25 September 2012 (UTC)
You wrote: "Characteristic impedance is determined by the geometry and materials of the transmission line and is not dependent on its length." Well said---perfectly clear right off the bat. Thank you.CountMacula (talk) 00:03, 27 September 2012 (UTC)

Edits by Zolot[edit]

Zolot, the characteristic impedance of a lossless transmission line is real, i.e., resistive. For example:

The impedance is determined by the speed of the signal and the capacitance per length of the pair of conductors, both intrinsic properties of the line. This intrinsic impedance is termed the characteristic impedance of the line (Z0).
If a measurement is made at one end of the line in a short time compared to the round trip time delay, the line behaves like a resistor with a resistance equal to the characteristic impedance of the line. — Preceding unsigned comment added by Alfred Centauri (talkcontribs)
I just reverted this edit, which appears to be aimed at clarifying Zolot's misunderstanding. My edit summary was truncated by Twinkle because it was too long. For the record, the full summary was "It is not essential to introduce complex numbers to summarize this phenomenon and doing so requires more prior knowledge from the reader. Thinking that Z0 is connected to line resistance is mistaken and "resistive" helps the reader understand this mistake." SpinningSpark 18:05, 23 November 2013 (UTC)