The active vs passive for diodes seems to depend on the mode of operation. Is it the negative resistance properties of a tunnel diode that makes it a active device or does it have a active mode? Negative resistance can be seen in other processes such as gas discharges, so is an NE-2 lamp a active electronic device?

http://www.oberlin.edu/physics/catalog/demonstrations/em/neonosc.html

http://en.wikipedia.org/wiki/Pearson–Anson_effect

 

It always remains a transmission line, it will become so lossy it is not usable as a transmission line anymore.

If you have a long enough transmission line and an analogue ohmmeter you will see the needle kick down toward the low resistance end of the scale. it will take a very short time returning to infinite resistance. This is not only due to the capacitance between the conductors but it is also due to the distributed inductance in the transmission line.

[HIJACK]
Apologies to the OP. I tried sending a PM to Sue_AF6LJ but it can't go through.

@Sue: I've noticed that you seem pretty knowledgable regarding RF stuff. When I first joined the forum I asked a question regarding normalized impedance and, while there was some good discussion, never resulted in a definitive answer.

I would be most interested in your thoughts on it.

Here's the thread:

http://forum.allaboutcircuits.com/showthread.php?t=68261

By all means, feel free to revive it.
[/HIJACK]

 

[HIJACK]
Apologies to the OP. I tried sending a PM to Sue_AF6LJ but it can't go through.

@Sue: I've noticed that you seem pretty knowledgable regarding RF stuff. When I first joined the forum I asked a question regarding normalized impedance and, while there was some good discussion, never resulted in a definitive answer.

I would be most interested in your thoughts on it.

Here's the thread:

http://forum.allaboutcircuits.com/showthread.php?t=68261

By all means, feel free to revive it.
[/HIJACK]

I'll check it out.....
By the way there are plenty of RF types over on http://forums.qrz.com/forum.php

That's an old thread........
You said.....

I have run across the claim, in numerous texts and online resources, that the real part of the normalized impedance (the ratio of load impedance to the characteristic impedance of the driving transmission line) can never be negative, except in the case of some active devices. The justification I have seen for this in several places is that, for passive circuits, negative resistance has no physical meaning.

This statement is true and in the real world you never see a circumstance where passive parts are used that negative resistance / impedance comes up. While the "J" operator can be ether positive or negative this doesn't make said impedance negative or positive.

And yes when you normalize your impedance you end up with ratios that you are working with. Believe it or not this simplifies the problem, especially since you can now use a Smith chart to work out your reactance values and convert them to tangible parts.
now on to your example.....

More to the point, consider the following example:

Load: ZL = 50+j1000 ohms
Transmission line: Z0 = 50-j3 ohms

It would seem to me that the normalized impedance in this case would be:

Normalized impedance: Zn = ZL/Z0 = -0.2+j20

In fact, it would seem that as long as the transmission line has any reactive component at all, that a passive load could easily be constructed that would result in the real part of the normalized impedance being negative.

So what am I missing?

What would happen if I proceeded to build such a circuit? Is there something that makes such a circuit unrealizable?

In practice the example above would be easily enough solved by providing enough series capacitance to tune out all that inductance at the load.

All normalizing allows you to do is to simplify the problem.


As far as the math goes I'm rusty on the math I had in school when I took electronics technology (thirty seven years ago)

 

Some of the simple transmission line results do not apply when the characteristic impedance is complex.

Starting with a transmission line with characteristic impedance Z0, this is simply the ratio of the voltage to the current in the propagating modes, denoting \(V_+\) as the voltage amplitude of the forward wave, and \(V_-\) as the voltage amplitude of the reflected wave, then the voltage and current at any point on the line are:

\(V = V_+ + V_-\)

\(I = \frac{V_+ - V_-}{Z_0} = (V_+ - V_-)Y_0\)

Now if the line is terminated in ZL, then at ZL

\(\frac{V_-}{V_+} = \frac{Z_L - Z_0}{Z_L + Z_0} = \rho\)

where \(\rho\) is the reflection coefficient.

So at ZL, the voltage and current are:

\(V = V_+(1 + \rho)\)

\(I = V_+(1 - \rho)Y_0\)

and the power flowing into the load is (assuming RMS amplitudes)

\(P = Re(VI^*) = Re(V_+(1+\rho)V_{+}^{*}(1-\rho^*)Y_{0}^{*})\)

\(P = Re(|V_+|^2 Y_{0}^{*}(1 - |\rho|^2 + 2jIm(\rho)))\)

This expression for the power flowing into the load is the key. If the characteristic impedance is real, then Y0 is real and

\(P = |V_+|^2 Y_{0}(1 - |\rho|^2)\)

and the condition for \(P \geq 0\) is \(|\rho| \leq 1\)
which is equivalent to \(Re(Z_L) \geq 0\).

As Z0 is real, this is the same as the real part of the normalised impedance being >= 0.

So what happens when the characteristic impedance is complex? The expression for power flowing to the load is still valid

\(P = Re(|V_+|^2 Y_{0}^{*}(1 - |\rho|^2 + 2jIm(\rho)))\)

so \(P \geq 0\) means

\(Re((1+\rho)(1-\rho)^*Y_{0}^{*}) \geq 0\)

substitute in

\(\rho = \frac{Z_L - Z_0}{Z_L + Z_0}\)

\(1+\rho = \frac{2Z_L}{Z_L + Z_0}\) and \(1-\rho = \frac{2Z_0}{Z_L + Z_0}\)

then

\(Re((1+\rho)(1-\rho)^*Y_{0}^{*}) \geq 0\)

means

\(Re(\frac{2Z_L2Z_{0}^{*}}{(Z_L+Z_0)(Z_{L}^{*} + Z_{0}^{*})}Y_{0}^{*}) \geq 0\)

\(Re(\frac{4Z_L}{|Z_L+Z_0|^2}) \geq 0\)

i.e. \(Re(Z_L) \geq 0\) as expected.


You could go back to

\(Re((1+\rho)(1-\rho)^*Y_{0}^{*}) \geq 0\)

and work out what the appropriate requirement on \(\rho\) is for complex characteristic impedance (which would not be \(|\rho| \leq 1\) or equivalently in terms of normalised impedance \(Re(z_n) \geq 0\)), but it looks like a lot of work for little benefit.

 

Some of the simple transmission line results do not apply when the characteristic impedance is complex.

Is the normalized impedance one such result?

If so, why not and why don't any references make that point? (I realize that is asking for supposition, but please suppose away!)

Let's be sure that we are trying to answer the question that was asked. To recap:

1) Normalized impedance is defined as: Zn = ZL/Z0

2) It is claimed in many places that Re(Zn) cannot be negative (at least when passive components are used).

Q1) Is this true or is this not true?

Q2) If it is true, why?

The only justification I have found is that a negative resistance has no physical meaning for a passive component.

Fine, let's not debate active/passive or whether a negative resistance is possible. Let's accept that statement at face value and abide by a constraint in which the real part of both the load impedance and the characteristic impedance is nonnegative.

I have shown, with the following simple example

Load: ZL = 50+j1000 ohms
Transmission line: Z0 = 50-j3 ohms

Zn = -0.2+j20

that you do not need to have a negative resistance in either the load impedance nor the characteristic impedance in order to have a normalized impedance that has a negative real part.

Please understand that I am not asking whether practical transmission lines have complex impedances. I am not asking whether, or how, to tune out any complex part of a transmission line's characteristic impedance.

I am asking a very specific question: Given a constraint that the real part of the load and characteristic impedances must be nonnegative, is it true that the real part of the normalized impedance cannot be negative and, if so, why not?

More to the point, if this is true:

Q3) Why is the specific example system shown above not valid?

Q4) Why can't the specific example system shown above be physically constructed?

 

1) Normalized impedance is defined as: Zn = ZL/Z0

2) It is claimed in many places that Re(Zn) cannot be negative (at least when passive components are used).

Q1) Is this true or is this not true?

Q2) If it is true, why?

(2) is clearly false as you have shown. It comes from the power relationship I derived above, and I have shown why it doesn't apply for complex characteristic impedance. If the characteristic impedance is complex, it is clearly easy to come up with passive impedances which normalise to negative real parts.

Given a constraint that the real part of the load and characteristic impedances must be nonnegative, is it true that the real part of the normalized impedance cannot be negative and, if so, why not.

More to the point, if this is true:

Q3) Why is the specific example system shown above not valid?

Q4) Why can't the specific example system shown above be physically constructed?

The system you have shown is probably valid (I'm no expert on what complex characteristic impedances can be realized), so it can probably be physically constructed. The issue is whether any power will emanate from the passive load because Re(zn) < 0. As I have showed above, this relationship for passivity is only valid for real characteristic impedances.

As I said, when the charactersitic impedance is complex, the requirement on \(\rho\) for a passive load is

\(Re((1+\rho)(1-\rho)^*Y_{0}^{*}) \geq 0\)

and this is NOT the same as \(|\rho| \leq 1\) , so when the characteristic impedance is complex, a passive load is NOT one where \(Re(z_n) \geq 0\).

You could work out what the real requirement is from

\(Re((1+\rho)(1-\rho)^*Y_{0}^{*}) \geq 0\)


but that seems like a lot of work.

If you have a reference that states that when the charactersitic impedance is complex then a passive load is one where \(Re(z_n) \geq 0\) I'd be interested in seeing it. Otherwise it looks as if the misunderstanding comes from applying formulae that are only appropriate for the real characteristic impedance case to the complex characteristic impedance case.

 

In the very first post of this thread you said:

Leaving all of that aside, my confusion stems from the fact that normalized impedance is not an impedance at all, it is the ratio of two impedances. Thus the real part of the ratio is not a "resistance" and the imaginary part is not a "reactance"; they are both dimensionless quantities.

Having said this, and even though you are a "units nazi" (your words)

you yourself continue to use the phrase "normalized impedance". If this thing is not an impedance, what shall we call it? Just "normalized ______"?

What about the fact, often disparaged by Ratch, that the phrase "flow of current" suffers from the same technical defect--bad units.

Current is already a flow--a flow of charge, so when one says "current flow", one is saying "charge flow flow". Yet, most practitioners of the art seem to know what is meant.

We speak of a river flowing to the sea, yet it would seem that the river pretty much stays put--it's the water that flows to the sea.

These things are tropes, customary ways of speaking. Customary locutions are sometimes not technically accurate.

I believe the answers to your questions 1 and 2 are rooted in custom as I explained in post #30. When calculating normalized impedance (there it is again), the divisor is customarily taken to be the nominal characteristic impedance, a pure real number. If the divisor is pure real, then the normalized impedance will never have a negative real part if only passive circuit elements (such as a load) are present.

Even though, as you shown in your example, it is possible to get a negative real part in the "normalized impedance" if you have a characteristic impedance with a suitable reactive part, this doesn't imply generation of energy, since as you point out, "normalized impedance" is not really an impedance.

 

In the very first post of this thread you said:



Having said this, and even though you are a "units nazi" (your words)

you yourself continue to use the phrase "normalized impedance". If this thing is not an impedance, what shall we call it? Just "normalized ______"?

I continue to use the phrase because that is what it is called.

Also, while part of me would love to call it something else, I can't claim that it isn't proper to call it "normalized impedance" -- the key is the word "normalized", which in most measurement contexts means the dimensless ratio of two things. So a quantity that is the "normalized whatnot" is the ratio of some "whatnot" to some reference "whatnot"; Thus as soon as you use the adjective "normalized" you are probably (not necessarily always, but very probably) talking about a unitless quantity. The physics E&M folks, in particular, seem enthralled with using normalized parameters for everything. I can see the attraction, but I hate giving up the units that help keep the meaning of the various quantities straight.

What about the fact, often disparaged by Ratch, that the phrase "flow of current" suffers from the same technical defect--bad units.

Current is already a flow--a flow of charge, so when one says "current flow", one is saying "charge flow flow". Yet, most practitioners of the art seem to know what is meant.

We speak of a river flowing to the sea, yet it would seem that the river pretty much stays put--it's the water that flows to the sea.

These things are tropes, customary ways of speaking. Customary locutions are sometimes not technically accurate.

I'm not talking about the NAME that is used! Call it "Fred's number" if you want. I am talking about the DEFINITION of the quantity, which is the ratio of two impedances and, hence, whatever you want to call it, is dimensionless and, therefore, the real part IS NOT a resistance and the imaginary part IS NOT a reactance and the whole thing IS NOT an impedance, no matter what you choose to call it.

I believe the answers to your questions 1 and 2 are rooted in custom as I explained in post #30. When calculating normalized impedance (there it is again), the divisor is customarily taken to be the nominal characteristic impedance, a pure real number.

If the divisor is pure real, then the normalized impedance will never have a negative real part if only passive circuit elements (such as a load) are present.

Then please provide some kind of a reference to something that says (1) that a pure real number should be taken to be the nominal characteristic impedance of a transmission line, even if that transmission line has a significant reactive component, and (2) what purely real number should be used for a transmission line that has a significant reactive component in its characteristic impedance. For instance, should it be the real part or should it be the magnitude of the impedance?

Even though, as you shown in your example, it is possible to get a negative real part in the "normalized impedance" if you have a characteristic impedance with a suitable reactive part, this doesn't imply generation of energy, since as you point out, "normalized impedance" is not really an impedance.

I'm not talking about generation of energy. I've never made any claim about any relationship between normalized impedance and what it may or may not imply about energy.

Nor have I seen anywhere that hints that the nominal characterstic impedance is a pure real number. Quite the opposite, I have asked several people that have designed or had to deal with transmission lines on PCBs and other non-ideal places who have told me that the charactersitic impedance for such lines quite often has a significant reactive component that has to be taken into account. So no matter how "customary" it might be to assume that the reactive component of the transmission line's characteristic impedance has been made small enough to neglect, I have a hard time just accepting that it is always reasonable to do so.

But, even more specifically, I am trying to get at the explicit assertion that I have seen in several places, including textbooks, that the real part of the normalized impedance, for passive systems, cannot be negative. Most of these don't even attempt to offer a concrete reason for the assertion, but at least one text that I have specifically states that this is because the real part is a resistance and a passive resistance can't be negative. This statement is made in the part of the text that precedes any mention about the characterisic impedance generally being real for most designed transmission lines. It is THAT author that is interpretting the real part of a normalized quantity as if it were a physical resistance and not a dimensionless normalized quantity. Now, I'm open to the claim that this particular author blew it on the reason, but the fact remains that numerous other references make the same claim about the real part of the normalized impedance.

 

But, even more specifically, I am trying to get at the explicit assertion that I have seen in several places, including textbooks, that the real part of the normalized impedance, for passive systems, cannot be negative.

Unless the normalizing impedance is real and positive, then this is clearly wrong, as your simple example shows. Any reference that states this is true for a complex characteristic impedance is obviously wrong.

I'm not sure what more you want, I demonstrated that the passivity of \(Z_L\) requires \(Re(Z_L/Z_0) \geq 0\) only for real \(Z_0\). I showed that for complex \(Z_0\) the relationship is much more complicated, and your example shows that a passive \(Z_L\) can have \(Re(Z_L/Z_0) \lt 0\) if \(Z_0\) is complex.

So clearly any reference that asserts that the passivity of \(Z_L\) requires \(Re(Z_L/Z_0) \geq 0\) for complex \(Z_0\) is wrong!

 

Then please provide some kind of a reference to something that says (1) that a pure real number should be taken to be the nominal characteristic impedance of a transmission line, even if that transmission line has a significant reactive component, and (2) what purely real number should be used for a transmission line that has a significant reactive component in its characteristic impedance. For instance, should it be the real part or should it be the magnitude of the impedance?

I can't provide a reference that says that "a pure real number should be taken to be the nominal characteristic impedance of a transmission line, even if that transmission line has a significant reactive component"; I can only provide a reference that says that is "usual practice". Whether it should be the practice is not for me to say; I only report the custom.

I quote from my earlier post #30:

I have never seen a numerical example where the actual Zo of a real cable was used, such a Zo typically having a small non-zero complex part. I've always seen the nominal impedance used.

And, the nominal impedance of a cable is usually taken to be a pure resistance. See, for example: http://en.wikipedia.org/wiki/Nominal_impedance

At the very beginning of that article is said "It is usual practice to speak of nominal impedance as if it were a constant resistance,[1] that is, it is invariant with frequency and has a zero reactive component, despite this often being far from the case.", where this is taken from a text:

Nicholas M. Maslin, HF communications: a systems approach, CRC Press, 1987 ISBN 0-273-02675-5.

Nor have I seen anywhere that hints that the nominal characterstic impedance is a pure real number.

It could be seen in post #30, and now, just above.

Quite the opposite, I have asked several people that have designed or had to deal with transmission lines on PCBs and other non-ideal places who have told me that the charactersitic impedance for such lines quite often has a significant reactive component that has to be taken into account. So no matter how "customary" it might be to assume that the reactive component of the transmission line's characteristic impedance has been made small enough to neglect, I have a hard time just accepting that it is always reasonable to do so.

Of course in a real design any non-zero reactive part of the characteristic impedance will be taken into account. But, this doesn't mean that the nominal impedance can't be taken as pure real.

You may not think it reasonable, but it is customary to do so.

Transmission lines on PCBs are designed with an ideal in mind, often 50 ohms. Even if the lines have a non-zero reactive part, their nominal characteristic impedance is taken to be pure real. As the Maslin quote above says "It is usual practice to speak of nominal impedance as if it were a constant resistance,[1] that is, it is invariant with frequency and has a zero reactive component, despite this often being far from the case."

 

First, I want to point out that when I said, "Then please provide some kind of reference ..." I wasn't trying to be argumentative or challenging. I was truly asking for a reference if you happened to know of one. My phrasing may not have relayed that connotation well.

The problem I have, in the context of what I am trying to gain an understanding of, is that the reference you mention seems to be talking about cables, as in cables that are specifically designed to be used as transmission lines and, hence, have been designed to have sufficiently low reactance in the characteristic impedance that it is reasonable to ignore it. Within that context, I don't find it surprising at all that it would be customary to ignore it.

The same might be the case for PCB traces and the like that are DESIGNED to be transmission lines. But what about the traces that weren't designed to be transmission lines and only after the fact, perhaps when other changes in the system raise the frequencies of the signals on those lines, have to be analyzed and treated as transmission lines perhaps that can't be changed because they aren't physically accessible?


I don't understand what you mean by "Of course in a real design any non-zero reactive part of the characteristic impedance will be taken into account. But, this doesn't mean that the nominal impedance can't be taken as pure real." If you are using a nominal impedance as your characteristic impedance that doesn't take it into account, then how can you be said to be taking it into account?

Again, this is not a flippant or combative question. I haven't done these kinds of things enough to resolve that seeming contradiction. If I measure or calculate a characteristic impedance that has a significant reactive component and I then proceed to normalize my load impedance to a nominal characteristic impedance in which the reactive part has been neglected, at what point in the subsequent design process do I take the reactive component of the charactersitic impedance into account?

Moving on, the quote that ends with "despite this often being far from the case" is a lot more telling, although even then it would seem that it would be more proper to say that a normalized impedance having a negative real part is seldom encountered not because it doesn't happen, but rather as an artifact of the customary practice of ingoring any reactive part of the characteristic impedance that the load impedance is being normalized to.

 

The problem I have, in the context of what I am trying to gain an understanding of, is that the reference you mention seems to be talking about cables, as in cables that are specifically designed to be used as transmission lines and, hence, have been designed to have sufficiently low reactance in the characteristic impedance that it is reasonable to ignore it. Within that context, I don't find it surprising at all that it would be customary to ignore it.

The same might be the case for PCB traces and the like that are DESIGNED to be transmission lines. But what about the traces that weren't designed to be transmission lines and only after the fact, perhaps when other changes in the system raise the frequencies of the signals on those lines, have to be analyzed and treated as transmission lines perhaps that can't be changed because they aren't physically accessible?

To be honest this frames your question very poorly.

Let's start from the beginning, if you are talking about a characteristic impedance you are talking about solutions to the telegrapher's equation http://en.wikipedia.org/wiki/Telegrapher's_equations.

There is nothing magic about lines "designed as transmission lines", simply that there are a class of arrangements of conductors which can be usefully analyzed as transmission lines.

The Telegraphers Equation has solutions in the form of travelling waves, one forward and one reverse. Any waveform on the line can be represented as a sum of these travelling waves, including when the line is terminated. You should work through the maths, (I did most of it for you) and you will see that your questions are obvious.

This is not a semantic problem, simply one of a transmission line with complex characteristic impedance terminated in a load.

 

It's instructive to measure some coax with a VNA and display the impedance on a Smith chart. The VNA has been calibrated with a precise 50+j0 ohm reference so that the center of the Smith chart represents a normalized 50+j0 ohms.

Here's the result for a decent quality 25 foot length of RG-58 cable, swept from 10 MHz to 200 MHz, measured at one end, open at the other end. An ideal cable with no loss would present an impedance plot consisting of mutiple circular rotations around the periphery of the Smith chart. What we're seeing is the open circuit at the far end transformed in impedance by the cable. For a lossy cable, the curve is a spiral instead of multiple circles, due to the losses. If the cable were very long, the curve would spiral inward ending in a single dot at the characteristic impedance:

Now, here is the same cable, same sweep range, with a precise 50 ohm load at the other end. This load is part of a calibration fixture, rated to be 50+j0 ohms up to 8 GHz, so I trust that it is a pure resistance up to 200 MHz. If the cable were perfect (no losses and a characteristic impedance of exactly 50+j0 ohms) we would see a little yellow dot right in the middle of the Smith chart. For a real cable, if the characteristic impedance were constant with frequency, but not 50 ohms, we would see some small circles (a spiral actually) with the characteristic impedance in the middle of the spiral. We see a blob; the blob is a little to the right of the center of the Smith chart, so the characteristic impedance is a little more than 50 ohms. The characteristic impedance varies with frequency and we could say that for the 10 MHz to 200 MHz range, the characteristic impedance at a particular frequency is somewhere in that blob:

Here's the same situation as the previous one, but zoomed in to the middle of the Smith chart so we can see that the measured impedance is quite variable with frequency, with a reactive part that varies:

 

Here are the same sweeps but with an inferior cable. I bought this 25 foot cable on eBay. It is labeled RG-58, which should be a 50 ohm cable, but as can be seen, it's not very good.

The cable in the previous post had an end-to-end DC resistance of the center conductor of .33 ohms. This inferior cable's center conductor measures 1.85 ohms end-to-end; much lossier than the previous cable.

Here's the sweep of the cable with the far end open circuited. We can see right away that the spiral is not centered on the Smith chart. It is offset to the right and downward a little. This means that the characteristic impedance is not 50+j0 ohms:

Here's the sweep with a variable resistor at the far end, adjusted to make the smallest blob possible. If the cable's characteristic impedance didn't vary with frequency, it would be possible to achieve a small dot rather than a blob, that dot representing the characteristic impedance.

Here's the zoomed in view. I turned on a marker and tried to place it near the center of the blob. You can see near the top left the impedance at the position of the marker, 66.6-j3.36 ohms. This cable would be better called RG-59, a 75 ohm cable.

The upshot of all this is to show that not only can a real cable have a reactive part to its characteristic impedance, but that reactive part varies with frequency.

The impedances of amplifiers, matching networks, antennas, etc., show similar non-idealities.

But, we would like the total environment of a radio frequency system to be at some ideal characteristic impedance, often 50+j0 ohms.

It makes sense to let the normalization impedance be 50+j0, the ideal. Then the real impedance can be displayed on a Smith chart and the deviation from the ideal is readily apparent.

 

Nicely done.
A picture is always worth a Killoword

 

For a lossy cable, the curve is a spiral instead of multiple circles, due to the losses. If the cable were very long, the curve would spiral inward ending in a single dot at the characteristic impedance:

I always appreciate posters that go to the effort to post measurements or give references, and I did like the spirals, but I have to take issue with the statement above. This would be correct if the characteristic impedance was constant with frequency, but for lossy lines this is never the case. If you increased the length of your line the spirals will shrink and eventually look like a series of large circular markers on the probably strange looking plot of characteristic impedance vs frequency. If you actually ran your experiment on an infinite line (don't ask me for the budget) there would be no spirals and you would actually get the plot of characterisitic impedance vs frequency.

As the characteristic impedance at any given frequency is the input impedance of an infinite line, the way you would get a spiral centred on that point would be to plot the reflection coefficient as you kept the frequency constant and increased the line length.

I'm just being pedantic, I've never had to worry about the complex nature of the characteristic impedance in all my RF work, but WBahn's original question did pique my interest. It turns out that strange things can happen in the analysis of lossy lines.

 

You raise a number of complicating effects that i didn't mention because I didn't want to stray too far from the question at hand, namely, why normalize with 50+j0.

I always appreciate posters that go to the effort to post measurements or give references, and I did like the spirals, but I have to take issue with the statement above. This would be correct if the characteristic impedance was constant with frequency, but for lossy lines this is never the case.

I was thinking of a cable ideal in every way other than lossiness.

I did say in post #54, "If the cable's characteristic impedance didn't vary with frequency, it would be possible to achieve a small dot rather than a blob, that dot representing the characteristic impedance."

Let me be a little more pedantic. You said "This would be correct if the characteristic impedance was constant with frequency, but for lossy lines this is never the case.", but I don't think this is true.

Just being lossy is not necessarily enough to cause the characteristic impedance to vary with frequency.

For example, if the characteristic impedance of a transmission line is given by:

\(Zo=\sqrt{\frac{R+j\omega L}{G+j\omega C}}\)

and if R/L = G/C with R, L, G, and C invariant with frequency, then the characteristic impedance won't vary with frequency even if the line is lossy. This is the classic distortionless line.

If you increased the length of your line the spirals will shrink and eventually look like a series of large circular markers on the probably strange looking plot of characteristic impedance vs frequency.

For a distortionless line the spirals would shrink to a dot. Good quality coax is close enough to a distortionless line that for a reasonably long line, the spirals do shrink to quite small circular arcs. See this image of the sweep on a 100 foot length of RG-174 (the limited number of points in the FFT are apparent):

If you actually ran your experiment on an infinite line (don't ask me for the budget) there would be no spirals and you would actually get the plot of characteristic impedance vs frequency.

As the characteristic impedance at any given frequency is the input impedance of an infinite line, the way you would get a spiral centered on that point would be to plot the reflection coefficient as you kept the frequency constant and increased the line length.

All true, just inconvenient!

I only swept up to 200 MHz and Zo doesn't vary all that much over that range. Terminating the line with a resistance adjusted to give the smallest "blob" gives a good approximation to Zo.

I'm just being pedantic, I've never had to worry about the complex nature of the characteristic impedance in all my RF work, but WBahn's original question did pique my interest. It turns out that strange things can happen in the analysis of lossy lines.

Here is an example of the very different behavior of a line that is far from the distortionless case. I had a scope probe fail and I salvaged the coax. Scope probe coax has a center conductor that is very small (looks like about .001 inch in diameter) and made of resistance wire. The end-to-end DC resistance of the center conductor of a 1 meter length of this cable is 230 ohms. A sweep of the input impedance with the other end open is very different from the sweep of a low loss cable. The curve shown is not the characteristic impedance, but the characteristic impedance does vary wildly with frequency. We don't see the same large spirals seen with low loss cable. Here the "spirals" are "drifting" around the chart. One could guess that the actual Zo vs frequency is a curved line connecting the centers of the drifting "spirals". Zo could also be determined by the Zo=Sqrt(Zsc*Zoc) method.

 

Let me be a little more pedantic. You said "This would be correct if the characteristic impedance was constant with frequency, but for lossy lines this is never the case.", but I don't think this is true.

Just being lossy is not necessarily enough to cause the characteristic impedance to vary with frequency.

For example, if the characteristic impedance of a transmission line is given by:

\(Zo=\sqrt{\frac{R+j\omega L}{G+j\omega C}}\)

and if R/L = G/C with R, L, G, and C invariant with frequency, then the characteristic impedance won't vary with frequency even if the line is lossy. This is the classic distortionless line.

OK, ideally this exists, but to be pedantic again , you can't make one. R in particular will vary according to the skin depth at high frequencies.

Making R/L roughly equal to G/C in order to reduce distortion is the basis for loading coils in telephone lines.

Here is an example of the very different behavior of a line that is far from the distortionless case. I had a scope probe fail and I salvaged the coax.

BRILLIANT!

 

OK, ideally this exists, but to be pedantic again , you can't make one. R in particular will vary according to the skin depth at high frequencies.

That's not being pedantic. That's being practical.

To carry it further, how about an ultra precision air line, but operating in a vacuum, made of superconducting material, at frequencies not so high nor temperature so low that the anomalous skin effect kicks in (http://www.nature.com/nature/journal/v165/n4189/abs/165239b0.html). That could get pretty close to the ideal. Still not practical.

The cable in post #54 puzzles me. I can't determine for sure why it's so bad. I bought several of them and I cut the BNC connector off one end of one of them to have a look. The center conductor is solid and appears to be copper plated. A non-copper core could explain the high DC resistance, but why so lossy at RF? Apparently the copper plating on the center conductor is so thin as to be useless. The outer braid is skimpy, and there must be significant loss by leakage. I can see the impedance sweep on the VNA vary significantly when I grasp the outside of the cable!! As is so often the case, you get what you pay for.

R in particular will vary according to the skin depth at high frequencies.

Yes, at sub-GHz frequencies I think R is the parameter that varies most with frequency; losses of modern dielectrics are pretty good in that range.

Anyway, we've strayed off-topic a bit. It might be interesting to start a new thread on the topic of transmission lines and their non-idealities.

 

I thought I'd summarise what I've learnt in this thread:

What caught my eye for the OP's question was that for any transmission line terminated in an impedance ZL, the ratio of the reflected wave amplitude to the forward wave amplitude is given by

\(\rho = \frac{\frac{Z_L}{Z_0} - 1}{\frac{Z_L}{Z_0} + 1}\)

or

\(\rho = \frac{z_n - 1}{z_n + 1}\)

ignoring for the moment what we call \(z_n\) , it is easy to show that

\(|\rho| \leq 1\) is the same as \(Re(z_n) \geq 0\)

which, for real Z0 means that passive loads have \(|\rho| \leq 1\) which makes sense - more power flowing into the load than flows out. This much I knew.

When Z0 is complex though, these equations clearly implied that you can have passive loads which give |ρ|>1 ! This bothered me. The analysis I did in post 44 shows that the power flow in a line with complex characteristic impedance not only depends on the amplitudes of the forward and reflected waves, but also their relative phases. Indeed |ρ| can be greater than 1 for passive loads. I could see how the waves could interact, the reflected wave could affect the voltages across or currents through the line losses, thus affecting power flow. I went looking to see if there were alternative or more elegant explanations.

I found a few references:

Ref 1:
Vernon, R.J.; Seshadri, S.R., "Reflection coefficient and reflected power on a lossy transmission line," Proceedings of the IEEE , vol.57, no.1, pp.101,102, Jan. 1969

Abstract: It is shown that the magnitude of voltage reflection coefficient |ρ| occurring in transmission line theory can exceed unity even for a passive load, ...


This confirms that indeed |ρ| can be > 1. Also note that in regard to the issue of normalising, in this paper they don't bother normalising at all.



Ref 2:
Graham, P.J.; Distler, R.J., "Use of the Smith Chart with Complex Characteristic Impedance," Education, IEEE Transactions on , vol.11, no.2, pp.144,146, June 1968

"In most transmission-line applications of the chart the characteristic impedance is assumed to be real and is used as the normalizing factor. With passive impedances, then, the normalized impedance z will be in the closed right half-plane."
....
"There is fundamentally no requirement that the normalizing factor be real, except that a complex factor can lead to normalized passive impedances appearing in the left half-plane, and consequently off the Smith chart."


They give an example where this may be of use:
A 100 mile long transmission line whose parameters at 60 Hz are
Zo = 711<-14.20 ohms
α = 0.068 dB/mile
β = 1.645°/mile
is terminated by an impedance ZL =j5OO ohms. Find the input to output voltage ratio.

In this paper they normalize using the complex impedance and get |p|>1 for a passive load. They then modify their process of using the smith chart so that every time |ρ|>1 they invert ρ -> 1/ρ. It's complicated. Note that this was back before computers, so they were stretching their graphical computational aids.


Ref 3:
Gago-Ribas, E.; Gonzalez-Morales, M.J.; Garcia-Vazquez, C.; Pecharroman-Hernandez, A.; Perez-Baraja, S.; Fernandez-Perez, J.C.; Gonzalez-Rodriguez, M.C., "Software tool for transmission line analysis," Frontiers in Education Conference, 2000. FIE 2000. 30th Annual , vol.2, no., pp.S1E/10,S1E/16 vol.2, 2000

They normalise to |Z0|, but then they have to modify their smith chart and they get something really weird (and they are selling software to generate - and do manipulations on - a smith chart which is warped depending on the phase angle of Z0):

So the bottom line seems to be:
1. there is no 'standard' way of normalising when the characteristic impedance is complex
2. some results for lossless lines do not apply to lossy lines, particularly that |ρ| can be greater than 1 for passive loads, i.e. \(Re(\frac{Z_L}{Z_0})\) can be negative for passive loads.
3. Whilst the results are interesting (to me at least), nothing I have ever done in RF and microwaves has needed them.