Unless the load impedance equals the line impedance there will be a reflected component.

Surely the negative sign implies that the 'reflected' wave is greater than the incident?

And is the reflected wave not travelling the other way?

 

The negative normalized resistance does, indeed, imply that the magnitude of the reflection coefficient is greater than one. But the reflected wave is travelling the other way regardless of whether the normalized resistance is positive or negative. And, remember, we are talking about the sign of the "normalized resistance", which can be negative even if both the transmission line characteristic impedance and the load impedance have positive resistive components.

 

Unless the load impedance equals the line impedance there will be a reflected component.

Surely the negative sign implies that the 'reflected' wave is greater than the incident?

And is the reflected wave not travelling the other way?

You are correct that there will be a reflected wave if the load is not matched, and the reflected wave is traveling the other way by definition. And correct that negative real part of normalized resistance means reflection coefficient with magnitude greater than one.

A negative sign in the reflation coefficient is just a relation of the phase of the reflected wave compared to the incident wave. We see this in some of the nice animations you referred us too. A positive wave pulse gets reflected and returns as a negative wave pulse.

The magnitude of the reflection coefficient is what determines the size of the reflected wave amplitude. Typically, the value is less than one. A reflection coefficient magnitude greater than one would normally indicate the reflected wave is greater, and normally we don't expect this - at least if Zo is real, and the medium has no gain.

The OP is worried that his example shows the magnitude is greater than one, which seems strange. The first impulse is to say that this violates energy conservation, but we must remember that the field is not the same as the power. Typically, field squared (or voltage squared) indicates power, but the case of complex Zo is a little different because the current and the voltage are not in phase any more. Hence, I think we need to be more careful in defining the wave power.

There are two things to question here. First, is the formula for reflection coefficient correct for complex Zo, and second, if it is correct, how is wave power calculated in this case.

I think the answer simply requires carefully stepping through the usual derivations. I've run across field theory derivations that did not obey energy conservation before. In particular, it was a mode coupling theory applied to coupled electromagnetic waves. The violation was the result of approximations made in the derivation. Hence, the OP is asking a very good and appropriate question.

 

The OP is worried that his example shows the magnitude is greater than one, which seems strange. The first impulse is to say that this violates energy conservation, but we must remember that the field is not the same as the power. Typically, field squared (or voltage squared) indicates power, but the case of complex Zo is a little different because the current and the voltage are not in phase any more. Hence, I think we need to be more careful in defining the wave power.

I think it more like your were saying before with a resonant circuit, the unloaded gain can appear to be > unity but as soon as real power is drawn from it we see normal circuit behavior. So if we have a complex Zo (input impedance contains a negative real part) we can see this strangeness but it's not really strange if we understand the whole picture with something like a tunnel diode circuit where at some frequencies it's input immittance might have it's real part negative.

 

I've tried going to cable manufacturer's websites and getting numbers for {L,C,G,R} per foot and haven't had much luck. Of course, I have nomincal L and C numbers from the geometry and can probably estimate G and R values from knowing the materials and dimensions.

I'm still thinking about this question. I've made some progress, but am holding off on giving an answer until I'm sure. However, here are some values I ran across in "The Electromagnetic Field" by Albert Shadowitz, chap 16.

"Typical values for an open wire transmission line are R=4 mOhm/m, L=2 μH/m, G=0.2 nMho/m, C=6pF/m, alpha=3 μNp/m, beta=20 μrad/m."

Note that these values indicate a wave speed of 0.962 times the speed of light in vacuum and a characteristic impedance of 577 Ohms. Obviously there are transmission lines that are significantly different than this, but this gives one example to consider.

 

Here is my present conclusion on this question.

When I do a verification of energy conservation for the case of complex characteristic impedance, I find the system fails to obey energy conservation.

If we call Ro the real part, and Xo the imaginary part of the complex impedance Zo, then Xo must be zero for energy to be conserved and for the average reflected power plus the average transmitted power to equal the incident power at the interface.

An interesting side result in the mathematics is that there is a special case for nonzero Xo when XL*Ro=Xo*RL. In this case the energy conservation is obeyed. A special case of this is when the line is shorted and XL=RL=0, which allows all the energy to be reflected and none to be transmitted.

The OP mentioned that the distributed circuit approximation may be the issue here, and I think this is likely to be the correct explanation. The exact equivalence of the voltage/current analysis with a rigorous field analysis seems to be linked to the case of TEM waves which are the lossless case where Zo is real. Several text books mention this without a formal proof. However, my own calculations that show energy conservation is not obeyed seems like a very significant piece of evidence, and is enough for me to hang my hat on also. I'm at 95 % confidence level and am willing to let this question rest unless anyone has counter arguments.

 

The problem I have is that we know that, in general, the characteristic reactance of a real transmission line does not have to be zero and, it is my understanding from people that work with RF PCB designs, that the characteristic reactance can be significant. I wanted to hang my hat on the claim that as soon as you have a characteristic reactance that you no longer have TEM waves, however I have found several places where the discussion implies that it is still valid to talk about lossy transmission lines under the assumptions of TEM waves and that non-TEM waves start developing only at much higher frequencies (and that the point at which this happens is a limiting parameter for the transmission line).

 

The problem I have is that we know that, in general, the characteristic reactance of a real transmission line does not have to be zero and, it is my understanding from people that work with RF PCB designs, that the characteristic reactance can be significant. I wanted to hang my hat on the claim that as soon as you have a characteristic reactance that you no longer have TEM waves, however I have found several places where the discussion implies that it is still valid to talk about lossy transmission lines under the assumptions of TEM waves and that non-TEM waves start developing only at much higher frequencies (and that the point at which this happens is a limiting parameter for the transmission line).

I think that, strictly, loss means the wave is not TEM, but it is usually a very good approximation to consider them to be TEM, if the lossless solution is TEM. In general there can be nonTEM solutions in transmission lines and waveguides of various sorts, even in the lossless case.

It's not clear what you mean about the high frequency. Are you referring to higher order modes which may naturally be nonTEM, independent of the loss, or are you saying that the loss is much more significant at higher frequency, thus making an otherwise TEM solution significantly nonTEM?

 

It is instructive to look at the particular example you gave in the first post in this thread. Lets use your example with Zo=50-j3 Ohms and compare to the same example with Zo=Ro=50 Ohms. The load impedance was 50+j1000 Ohms.

First, do Zo=Ro=50 Ohms and ZL=50+j1000. The results are ...

Voltage Reflection Coefficient: 0.9901+j0.099 (magnitude = 0.995)
Voltage Transmission Coefficient: 1.9901+J0.099 (magnitude =1.9926)
Power Reflection Coefficient: 0.9901
Power Transmission Coefficient: 0.0099
Power Reflection + Transmission: 1.0000

Note that energy conservation is perfectly valid. However, the transmitted voltage is almost twice the incident voltage and the reflected voltage is almost equal to the incident voltage. Intuition might tell you that energy conservation is not met, but it is an issue of real power versus reactive power that gives the illusion.

Next, do Zo=50-j3 Ohms and ZL=50+j1000. The results are ...

Voltage Reflection Coefficient: 0.996+j0.0999 (magnitude =1.001)
Voltage Transmission Coefficient: 1.996+J0.0999 (magnitude =1.9985)
Power Reflection Coefficient: 1.002
Power Transmission Coefficient: 0.01
Power Reflection + Transmission: 1.012

Notice that energy conservation is not met in this case and there is a 1.2 percent extra amount of power, and more power is reflected than is incident. Still, the error is small and the reflected power is about equal to the incident power in both cases and the transmitted power is about equal 1 percent of the incident power in both cases.

Hence, if you are careful to interpret the numbers carefully, the approximation is still useful for engineering purposes. In most practical cases, the imaginary part of the characteristic impedance is much smaller than your example, so the approximation is usually better than this.

 

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.

I think that when the various texts say to divide by Zo, they don't mean that you should keep the units attached to that denominator. I think of normalization as a scaling process; I take the divisor to be just a number which is the numeric magnitude of Zo.

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.

Edit:

Just for grins I checked our Vector Network Analyzer. I knew that it allow one to specify a system impedance other than 50 ohms for the center of the Smith chart and I wondered if it would allow the entry of an impedance with a non-zero complex part; it doesn't.

 

However, here are some values I ran across in "The Electromagnetic Field" by Albert Shadowitz, chap 16.

"Typical values for an open wire transmission line are R=4 mOhm/m, L=2 μH/m, G=0.2 nMho/m, C=6pF/m, alpha=3 μNp/m, beta=20 μrad/m."

Note that these values indicate a wave speed of 0.962 times the speed of light in vacuum and a characteristic impedance of 577 Ohms. Obviously there are transmission lines that are significantly different than this, but this gives one example to consider.

There are some values for a number of cables here:

http://en.wikipedia.org/wiki/Primary_line_constants#Special_cases

Note that the last column in the table titled "Typical values for some common cables" lists values for Zo without any complex part; they're showing nominal values: http://en.wikipedia.org/wiki/Nominal_impedance

 

I think that when the various texts say to divide by Zo, they don't mean that you should keep the units attached to that denominator. I think of normalization as a scaling process; I take the divisor to be just a number which is the numeric magnitude of Zo.

There is no need to just "throw away units" -- doing so is unjustified and, as far as I'm concerned, unforgivable. When you normalize to a reference impedance, such as Zo, then the units will go away automatically because normalizing (usually) involves dividing the non-normalized value by the reference value. When it doesn't, then the surviving units must be tracked. For instance, the cable parameters (R,G,L,C) are normalized on a per-unit-length basis. But in this case the normalization does NOT result in a units cancellation and it is critical that the units be properly tracked. The results will be very different if someone used L in Henries/meter and C in Farads/foot.

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.

Yes, and I imagine that cable manufacturers go to great lengths to make cables for which this is a perfectly reasonable assumption to make. But there are transmission lines that are not carefully designed and engineered cables, such as microstrip lines on a PCB made using tracks. My understanding, again from talking with people that have experience with them, is that these can have significant reactive components to the characteristic impedance.

I've kinda set this problem aside for now, but intend to come back to it when I am in a position to make some meaningful measurements.

 

FYI: I am just getting into electromagnetics, so please keep explanations close to the basics and fundamentals, if possible. Thanks.

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.

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.

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?

It would indeed be difficult to plot a negative real resistance on a Smith Chart, since the left extreme is 0 ohms (whether normalized or not). As you suggest, for passive devices, negative resistance has no meaning. A signal generator, on the other hand, can be interpreted as a negative resistance device. Also certain active devices exhibit negative resistance over some part of their operating range (see references on "tetrode kink").

Eric

 

It would indeed be difficult to plot a negative real resistance on a Smith Chart, since the left extreme is 0 ohms (whether normalized or not). As you suggest, for passive devices, negative resistance has no meaning. A signal generator, on the other hand, can be interpreted as a negative resistance device. Also certain active devices exhibit negative resistance over some part of their operating range (see references on "tetrode kink").

Eric

Certain passive devices (tunnel diodes) also exhibit negative resistance regions.

But the point is that saying that negative resistance has no meaning for a passive device is entirely beside the point because normalized impedance, despite the use of the word impedance, is NOT an impedance and the real part of it is NOT a resistance. If you have a characteristic impedance that has a positive resistive component and if you have a load that has a positive resistive component, it is still possible to have a normalized impedance with a negative real part.

We've pretty well established that the likelihood of encountering this situation in practice is slim (but I'm not even too sure about that when dealing with circuit board transmission lines), but it is still of theoretical/academic relevance.

 

I forgot where my train of thought was on this....but here's another interesting point. An ideal transformer has no real (dissipative) resistance, and yet if you terminate it with a real resistance, the input impedance is REAL.

Another case is that of an infinite length transmission line. Although there is no "real" resistance to burn up any power, the input impedance is real.

Just some more intuitive thoughts.

Eric

 

An ideal transformer has no real (dissipative) resistance, and yet if you terminate it with a real resistance, the input impedance is REAL.

Is it? If I have a huge transformer that is unloaded, it will look like a large inductive load to the source, right? I would not expect that to change much if I barely load it by terminating it with a very large resistor, hence I would expect it to still look like a large mostly inductive load, wouldn't I? I would expect the input impedance to go asymptotically toward a real load only as the load get heavier and heavier (R approached zero) and even then only if there is perfect coupling between the primary and secondary.

The way I have always explained it, intuitively is that, unloaded, it looks like a purely reactive load. But if the output is loaded with a real resistance, then the resistance extracts energy from the secondary winding, which then has to be replaced with energy from the primary winding, which then has to be replaced with energy from the source. Hence the overal load seen by the source picks up a real component which means the input impedance of the transformer now has a real component.

Another case is that of an infinite length transmission line. Although there is no "real" resistance to burn up any power, the input impedance is real.

Is this true even if the characteristic impedance of the transmission line has a reactive component?

 

I forgot where my train of thought was on this....but here's another interesting point. An ideal transformer has no real (dissipative) resistance, and yet if you terminate it with a real resistance, the input impedance is REAL.

Another case is that of an infinite length transmission line. Although there is no "real" resistance to burn up any power, the input impedance is real.

Just some more intuitive thoughts.

Eric

Basically, any time there is real power transfer away from the electrical circuit, whether it be energy dissipation, energy propagation or energy conversion (i.e. sound, light, electromagnetic wave etc.), it must appear to be a real impedance to the electrical source.

"Reactive power" is all about power that flows to and fro, while "real power" is all about power that goes away and never comes back to the circuit.

Certain passive devices (tunnel diodes) also exhibit negative resistance regions.

By the way, I always thought of a tunnel diode as an active device. I guess the definition is debatable.

 

Basically, any time there is real power transfer away from the electrical circuit, whether it be energy dissipation, energy propagation or energy conversion (i.e. sound, light, electromagnetic wave etc.), it must appear to be a real impedance to the electrical source.

I'm really struggling with this notion that both you and KL7AJ seem to be putting forth. I fully agree and understand that the impedance must have a real component commensurate with the real power that is extracted, but I do not understand why the impedance is now real; Why can't it still have a reactive component, as well?

By the way, I always thought of a tunnel diode as an active device. I guess the definition is debatable.

That's the sense I'm getting. The definition I was taught (back in the very first course that introduced transistors) was that "active" meant that it had the ability to increase the power in the signal, which in turn meant that there had to be some means of taking power from an external source and adding that to the signal. But as I was just looking on the internet, it does seem like there are different definitions and some of them say that it is also active if it has the ability to amplify a signal in any way or to electrically control the flow of current (i.e., has 'on' and 'off' modes that are electrically determined). Yet on those same sites, some claim that regular diodes are passive while others say that all semiconductor devices, including LEDs, are active. Based on the snippets in the Google results page, it appears that some definitions base it on whether the small-signal model of the device includes any supplies (independent or dependent).

Basically, I've just learned that "active" and "passive" need to be added to the list of words that I should automatically clarify whether both parties are speaking the same language.

 

I'm really struggling with this notion that both you and KL7AJ seem to be putting forth. I fully agree and understand that the impedance must have a real component commensurate with the real power that is extracted, but I do not understand why the impedance is now real; Why can't it still have a reactive component, as well?

I'm just latching on to the idea from KL7AJ, that this is an "interesting point".

Consider that the case of a lossless transmission line has inductance and capacitance (yet no resistance) both conspiring to generate a real impedance, as seen by the generator. As he points out, if the line is infinite (or equivalently, if it is terminated perfectly), there is only a real component of impedance to describe the fact that the wave is carrying energy away and never allowing it to return.

Also, consider the case of a lossy line with complex impedance. The addition of resistance (loss) into the line, gives it a reactive component. Yet, the loss associated with the resistance is a dissipative process that does not return energy. There is a real dichotomy between the circuit view and the field veiw here.

In the case of an infinite lossy transmission line, the effect of reactance of the line is a bit harder to understand. The line itself does not return any energy. It transmits the wave out into space, and eventually all power is dissipated through resistance, until the wave is gone. If the line was terminated, then the reactance could be understood as a circuit effect. However, an infinite line does not strictly provide a circuit. (of course, we try to model that way)

I just view this as nourishing food for thought. I think it may all tie back to a point we made earlier in the thread. Namely, that the voltage/current/circuit viewpoint is and approximate one (probably) derived from the lossless assumption. There may be radiation effects, or other effects not accounted for when lossy lines are considered. It is not a trivial field problem to solve exactly.

 

I'm just latching on to the idea from KL7AJ, that this is an "interesting point"....There is a real dichotomy between the circuit view and the field veiw here.

I will certainly agree with that!