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Then, there is a mistake in 'Practical Rf Circuit Design for Modern Wireless Systems - Vol.I Passive Circuits and Systems', pape 61.
What is the relationship between reflectionless matching and conjugate matching?
For source, TL, and load : Zg, Z0, Zl
The reflectionless matching is Zg=>Z0 & Zl=>Z0
The conjugate matching Zl=>Zg* or Zg=>Zl*, in which reflection does exist.
But in LNA design, we use M2 to make ΓL=Γout*, so it's a conjugate matching. And if M2 is lossless, Γ=0 seeing into M2 from load, so it's also a reflectionless matching?
yes, in LNA design input matching is determined form gain/noise trade-off using available power gain circles. These are valid if output is conjugately matched.
Hi, Vale: Again, I would suggest people not to use complex Zc because it really can lead to unexpected results no matter what definition of reflection coefficient you use. The problem is that waveguide theory is no longer precise for a system with the supposed complex Zc. When "complex Zc" is there, it in fact means that you can not separate the wave into incident and reflected waves. I have posted a detail explanation in the "900 MHz..." a few weeks ago. If you need it, please e-mail to me (jian@zeland.com). Regards.
Actually, S-parameter definition for complex Zo is not standardized. This is the first time I have seen Vale's definition. I would consider the definition by IanP to be correct. The defition similar to Vale's with both Zo's conjugated appears to be used in some microwave design software. That definition can be useful when you want to get a conjugate match, just optimize for minimum reflection coefficient.
Note that selecting a defintion for complex S-parameters DOES NOT change the phyiscial situation. All the currents, etc., are all still the same. When we change the normalizing Zo (and whether or not we choose to conjugate the normalizing Zo), the only thing we are changing is how we look at the exact same physical situation. Yes, it is indeed easy to get numbers that seem wrong if all you are used to seeing is 50 Ohm normalized S-parameters. One should proceed carefully and thoughtfully if working in this area.
All lossy transmission lines (i.e., all transmission lines that can possibly be built) have a complex Zo and a complex propagation coefficient. The only uncertainty is in how we humans define S-parameters normalized to a complex Zo. In most cases, one should normalize to a real Zo, usually 50 Ohms. Then there is no uncertainty as to what definition one is using to normalize. I strongly recommend that so-called "Generalized S-parameters" never be used. These are usually provided by EM guys who do not know what the Zo is, so they give S-parameters that are normalized to whatever the unknown Zo of the line is.
Not only do all actual transmission lines have a complex Zo, but the concept of a complex Zo has been established and in use for over a century. I think it was introduced by Heaviside (undoubtedly in conjunction with his work on the trans-Atlantic cable), but Maxwell very possibly did it too.
Note that Zo = sqrt( (R+jwL)/(G+jwC) ). At low frequency (wL small comared to R or wC small compared to G), the Zo of all physical transmission lines becomes very strongly complex, and waveguide theory still holds just fine.
For silicon RFIC's, Zo is strongly complex at all frequencies (both R and G are large), but Si RFIC's work just fine too.
One thing that is interesting about complex Zo is that if you have the magnitude, that determines the phase. If you have the phase, that determines the magnitude. The magnitude and phase are related by a Hilbert transform. This is under the assumption of causality. Sonnet outputs the Zo as a side result of the de-embedding. When we check to make sure that the Zo phase and Zo magnitude are Hilbert transforms of each other, they match up perfectly. One example is given in a paper I just published on SOC and double delay de-embedding (listed below). I will be glad to email a copy to anyone requesting it.
The paper on the relation between complex Zo magnitude and phase is in the following (if lossy, complex Zo, waveguide theory is invalid, someone had better tell Dylan Williams!):
D. F. Williams, B. K. Alpert, U. Arz, D. K. Walker, and H. Grabinski, “Causal Characteristic Impedance of Planar Transmission Lines,” IEEE Trans. on Avanced Packaging, Vol. 26, No. 2, pp. 165-171, May 2003.
My paper that includes results for an Si RFIC transmission line (Zo is strongly complex):
Unification of double-delay and SOC electromagnetic deembedding
Rautio, J.C.; Okhmatovski, V.I.;
Microwave Theory and Techniques, IEEE Transactions on
Volume 53, Issue 9, Sept. 2005 Page(s):2892 - 2898
Upon your request, I have e-mailed you the Appendix D of IE3D 11 User's Manual explaining why introduction of complex Zc into waveguide and TLN theory may cause problem. Everything is based upon mathematics. You can also find it from my posting on "Simulate a 900 MHz RFID Tag" a few weeks ago.
Basically, it is not the problem of a circuit or a TLN in transverse direction. A practical circuit certainly has it. The prolem is that, when we introduce incident wave, reflected wave and reflection coefficient based upon complex Zc, we are introducing error or possibly problems. The fundamental is that a wave can't be decomposed into incident wave and reflected wave without approximation in a system with loss in the transverse direction. When we fit it into a system with real Zc, we know there is some approximation there. However, it will not yield ridiculous results. When we use complex Zc, we will introduce approximation. In some cases, it will introduce error. Anyway, please check the e-mail I sent you. Any comment, please let me know. Thanks!
While there is some user flexibility in how to define S-parameters normalized to complex Zo, as described in the beginning of this thread, there is absolutely no approximation in waveguide theory for lossy waveguide or complex Zo.
For example, one could pick a complex Zo and write the S-parameters for a lossless 50 Ohm transmission line normalized to the complex Zo. There is no approximation. The numbers you get will look really strange. If you give the strange results a little thought, you can learn some really interesting things. But if you can't handle the strange numbers, don't throw away waveguide theory, just renormalize back to ordinary, everyday 50 Ohms.
You can do the same for a lossless lumped element. You can still get what Jian describes as "ridiculous" results. The reason the results appear to be ridiculous is because of a lack of understanding of what the results mean.
For example, mag(S11) > 1 might mean you have a tuned circuit (as pointed out in another thread). This would happen if you normalize to a pure imaginary Zo that is inductive and you look at the S11 of a capacitor. Notice that in this case, there is absolutely no loss involved at all, but at and near resonance you get mag(S11) much larger than unity. In fact (with no loss) S11 becomes infinite at resonance! This seems crazy at first, but it makes sense when we realize it is just the fly-wheel effect of an LC circuit. So, should we also throw out imaginary Zo (no loss at all) because we get strange results for this lossless case? No way.
(In fact, this flywheel effect is very useful in RFID when you are trying to generate enough voltage to operate the RFID tag, that is why you need as high a Q as possible in the RFID tag inductor, to get as high a voltage as possible, easy to calculate when you can normalize S-parameters to complex Zo.)
I want to emphasize very strongly that there is no approximation involved when working with complex Zo, we do it all the time, and often with very strongly complex Zo as in the two cases I mentioned in my previous post. Waveguide theory is fully, completely, and exactly valid for lossy and lossless. I am very disappointed to see anyone describing it as approximate because they get results that look strange.
As for incident and reflected wave definitions, I had real difficulty reading Jian's posting on that matter due to notation. Fortunately, that matter is of no importance as far as Sonnet is concerned. We never deal with incident and reflected waves inside Sonnet. We only deal with currents and voltages (and voltages are always across an infinitesimal gap, so voltage definition is not a problem).
Currents and gap voltages are unique, so we can get exact Y parameters, and then convert to 50 Ohm S-parameters when we are all done. I have always had a problem working directly with the incident and reflected waves because of non-uniqueness in definition of line voltage. Approximations are definitely introduced as a direct result. It is because of these approximations that we do not calculate S-parameters based on incident and reflected waves. Even though this approximation is present for both lossy and lossless, it might still be related to Jian's problem with incident and reflected waves, but I do not know.
Internally, Sonnet always uses Y, Z, or ABCD parameters, which are based on current and voltage. Incident and reflected waves are never considered. Conversion to S-parameters is done only as a convenience to the user after all calculations are done and data is being output for use. So, incident and reflected waves do not matter in the least for our work.
Hi all,
Thanks for your repies, though some of them I can't catch on. My original question comes from an bbs discuss on conjugate match and Z0 match. I know the correct definition is the one without conjugate, as defined in all RF & MW books. The other definition is presented in Besser's book, in which he say when conjuate match (ZL=ZS*), from the equation Gamma is 0, so there is no reflect wave and source gets the max power. It sound reasonable at first glance, and the very definition had cause a large debate in news group years ago (search google group with 'conjugate match reflection').
Nowadays, to my understanding, when Z0 is real, Z0 match and conjugate match are the same thing. But a problem still puzzles me, that in a real Z0 conjuagte match, in the different point of circuit, different Gamma is calculated (zero or non-zero). How to comprehend it?
Hi, Vale: The conjugate introduced into the formula in Gamma is to avoid Gamma becomes > 1 for passive circuits. However, I don't see it is based upon rigorous derivation from waveguide theory. Again, please try to follow the document I sent you. You will realize that you can not separate the wave precisely into incident wave and reflected wave in a waveguide with loss in the transverse direction. When you can not separate the wave into incident wave and reflected wave, you can not even define the refleciton coefficient. If you define it based upon complex Zc, it is an approximation. It may work when complex Zc has a small imaginary part. It will fail when the imaginary part becomes big. Please let me know if you have any question. Thanks!
Hi jian,
I disagree with your comment on reflection coefficient of complex Zc. For the TL with Z=√(R+jωL)/(G+jωC), which is generally complex, the line voltage is
V(x)=A*exp(-γx)+B*exp(γx), γ=α+jβ
in which the 1st item is incident wave and the 2nd the reflected wave. refection coefficient is defined as
Γ(x)=[B*exp(γx)]/A*exp(-γx)=Γ0*exp(2γx),
whatever β is or is not equal to 0.
The complex conjugate of Zo does not keep mag(S11) from being > 1. For example, with an imaginary Zo, you can always pick a Z such that the denominator goes to zero. Pretty simple to see, it seems to me anyway.
The reason for the complex conjugate is if you are trying to get a conjugate match. Take a practical example, you want to match to the input of a power FET. The gate represents a complex load. You want to have a circuit that matches from 50 Ohms to that complex load. For your matching circuit, normalize port 1 to 50 Ohms and port 2 to your complex load. If your complex Zo S-parameter definition uses the complex conjugate, optimize for mag(S11) and mag(S22) to zero. If your normalization does not use the complex conjugate, take the complex conjugate load and normalize to that. Then, once again, optimize to zero mag(S11) and mag(S22).
Exactly equivalent would be to optimize the output impedance of your matching network to the complex conjugate of the FET input impedance.
Now, for Vale's question: Why complex conjugate? I will illustrate this with my ham radio setup. My antenna, at certain frequencies I use, has a complex input impedance. To maximize power radiated, I have a "match-box" here beside my radio. It has a tunable Pi network (C-L-C). The setup is Radio-Match Box-Long Coax-Antenna. I tune the match-box so that my radio sees a perfect 50 Ohms. What my match-box does is transforms the 50 Ohms from my radio into the complex conjugate of the input impedance of my antenna. The imaginary parts (of the antenna impedance and the 50 Ohms of my radio transformed by the match-box) cancel out completely, because they are of opposite sign. That is why my radio sees a pure real 50 Ohms.
So, what is happening physically? On that long coax, the wave goes from my match box to the antenna. At the antenna, some of that wave gets radiated, some gets reflected. The reflected wave goes back to my match box. The match box is set up just right (tuned to the complex conjugate of the antenna imedance) that all of that wave gets reflected right back towards the antenna where it will have another chance to get radiated.
Returning to the complex conjugate definition of reflection coefficient. When it is zero, it means you have a conjugate match. It does NOT mean you have no reflected wave on the tranmission line. If you take the time to understand (or as you have done, Vale, ask around and be willing to think a little bit), the complex Zo normalization can be quite useful.
As for differentiating incident and reflected waves, seems like it would be quite easy to do physically. Just get a very long line. Then launch a wave. That is the incident wave. Turn the incident wave off a while before the reflection comes bouncing back. Then when the reflection comes bouncing back, there is your reflected wave. Now, you have a reflection coefficient. Seems really simple to me. Perhaps I missed something?
Apparently, when ReZin > or = 0 and ReZc > 0 we will have |ReN| < |ReD| and |ImN| = |ImD|. Therefore, |N| < |D| and G < 1. Only when ReZin =0 and we will get G = 1. The conjugate formula can certainly guarantee |G|<= 1 when ReZin >= 0 or the circuit is passive. From this aspect, I can see the conjugate definition is more reasonable in some sense. Again, it is just some approximated formula for the real situation. Mathematically, we can not decompose a wave inside a waveguide with loss in the transverse direction into incident wave and reflected wave. Here, “loss in the transverse direction” means that there is some power dissipated in the transverse direction beyond the main domain of the waveguide. A typical example is a rectangular waveguide with non-PEC walls. Basically, when a wave hit the wall, most of the power will be reflected back. However, there is a small amount of power go into the wall. Such a situation will cause the wave no being able to be decomposed into incident wave and reflected wave. Basically, for such a situation, precisely we have to use Poynting vector to investigate the power flow and we have to include region beyond the walls because there is power flow. Fortunately, for rectangular waveguide structures, the walls are very close to PEC. Traditional waveguide theory is more than enough for a typical analysis because there is little power goes into the walls. For very lossy situations such as TLNs, RFIC and RFID in semi-conductor process. There is much loss in it. In such a situation, we should try to avoid using s-parameters normalized to complex Zc for it. Using s-parameters normalized to a real number is as good as using Y and Z-parameters.
A precise definition of reflection coefficient should be in terms of incident/reflected waves. It should come along with specifications for these two waves (plane wave/guided mode etc.)
One can use voltage/current/power to define the reflection or other coefficient. But the problem (to me) is how to make the "voltage/current" precise. I have learned that the concepts of voltage and current can be very tricky in EM field analysis. They are meaningful only within some specific context, under some assumptions.
For the above reasons, it is very surprising to read that Sonnet never deal with incident and reflected waves (exact quote: "We never deal with incident and reflected waves inside Sonnet").
It is more troublesome to read that "Currents and gap voltages are unique". I think these refer to the lumped source (delta gap source). and I think one should not expect the voltage/curent at a lumped source to be unique. (e.g., it would depend on the mesh at the lump port. exact how long is an infinitesimal delta gap?)
Since it is difficult to make the "voltage/current" precise when one has complex wave or wave with complex characteristic impedance, the classical microwave network theory (more precisely, network formulism) is with approximation.
Jian: Take a very simple example. Forget using any transmission lines to directly measure incident and reflected waves. Throw away the connecting Zo transmission lines and substitute lumped Zo loads. Exactly equivalent situation. We can still use the equation for Gamma. And to further simplify, we will use nothing but lossless lumped circuits, inductors and capacitors. It is now all just lumped element circuit theory. None of that confusing EM stuff. No transmission line, no loss, so all your problems with certain lossy transmission lines goes completely away!
Now calculate your G for Zin = 0 + jX and Zo = 0 - jX. D is zero and G is infinite. There is NO loss, there is NO transmission line. What we have is the reflection coefficient of an inductor (Zin = 0+jX) in a system normalized to a pure imaginary impedance (Zo=0-jX). The reflection coefficient is infinite. Does not matter if you conjugate the numerator or not.
Now, why is it infinite? Reasonable question. Recall, that we are just using pure lumped circuit theory. If this ridiculous infinite Gamma is an approximation, we have BIG problems. Instead, let's try to understand it.
Remember that S-parameters are measured with each port terminated in the normalizing impedance, in this case Zo = -jX. You put a -jX (=Zo) in parallel with a +jX (= Zin) and you have a perfect open circuit. This is a C in parallel with an L. This is a resonant circuit. You excite this lossless LC with a sinewave at the resonant frequency and the voltage on the LC circuit grows to infinity (force 1.0 A into an open circuit, you calculate the voltage!). This is what the infinite Gamma means, you have a resonant circuit and you are exciting it at the resonant frequency. There is no loss in this case, so discussion of loss is pointless. There is no transmission line either, so discussion of incident and reflected waves is also pointless.
If you calculate the reflection coefficient for the same inductor in a 50 Ohm system, you take out the capacitor and substitute a 50 Ohm resistor (still no transmission line). Now you get a complex Gamma. This is what we are all used to. In a pure real system (all ports same Zo) mag(Gamma) <= 1 for passive structures. This condition no longer holds when normalizing to complex Zo, as shown above. It also no longer holds when normalizing different ports to different Zo's.
All the above is pure lumped circuit theory. No transmission lines at all. No loss at all. But we still get your "ridiculous" Gamma. Me thinks what is needed is understanding the meaning of the result, it is nothing to do with loss and nothing to with transmission lines.
Jian, you continue to state that you can't distinguish between the incident and reflected waves on certain lossy transmission lines. You did not address the simple counter example I suggested (long line, turn off the excitation before the reflected wave arrives). If your ideas are correct, it has to work for that situation too.
Loucy: I don't know how familar you are with MoM. If I go over your head, let me know and I will simplify. Basically, all the planar MoM software out there that I am familar with solves for current in the metal. It does not solve for incident and reflected waves. It solves for the current under the condition that the voltage on the metal must be zero (or proportional to the current if there is loss).
OK, we got the current in the metal (true for all MoM software), now what do we do? In Sonnet's case, we have perfectly conducting sidewalls. We put an infinitely small gap between the sidewalls and the circuit and impress a voltage across that gap. Because the gap is infinitely small, our path of integration is infinitely short and we get one unique value for the voltage. The sidewalls are a perfect short circuit. Thus, the relation between the port voltages and port currents are the Y-parameters of the circuit. For a single given EM solution, there is only one voltage for each port and only one current. This is what I mean by unique. To within numerical precision and to within the accuracy of the underlying EM analysis, we get the exact answer.
For unshielded tools, one way to proceed is to extend a length of open circuit stub on the ground side of the port. Now, the port voltage is across a gap between the stub (on one side of the port) and the circuit (on the other side of the port). You still have to excite it with a voltage across the gap. But now, you can look at the standing wave (by viewing the current) that results and extract an incident and reflected wave. Now you can get S-parameters directly. I have a feeling that this is where Jian encounters his difficulty in certain lossy cases.
I have never explored that because we don't use this technique. This technique does indeed introduce approximations (due to voltage definition ambiguities, which require an integration of some kind from line to "ground"). I wonder if these might be the approximations that Jian is talking about, however, these approximations are for all cases except lossless, homogenous (same dielectric everywhere). Even though these approximations are usually fairly small, we do not tolerate them, so we do not proceed in this manner.
Basically, Loucy, a precise defintion of reflection coefficient can not come from incident and reflected waves (except for lossless, homogenous) because of ambiguity in determining the "line voltage". If you want precision, you have to use perfect short circuit terminations and calculate Y-parameters, and then covert to whatever form is desired by the user (usually S-parameters).
After we are all done in Sonnet, we convert the Y parameters, as desired, to S parameters by pure circuit theory. Couldn't care less (for the analysis output) as to what are incident and reflected waves. It is really quite a beautiful approach. I think gradually even those working in unshielded analysis will come to realize its significance.
Don't give something out of the topic. I was talking about the case with ReZc > 0. There is never a case with ReZc = 0 for any practical applications. If you like to make everything unreasonable, you can assume Zc = 0 too. Regards.
Let me start by saying that I am still a learner of MoM. I am constantly confused by many questions, and I appreciate your comment.
To my understanding, the "infinitely small gap" you mentioned has the size of one rooftop or at least a half rooftop (a unit cell in Sonnet's Manhatan mesh). When you change the cell size, the size of the source is changed, the resulting current/Zin will change accordingly. I can understand if you said the "current" is unique "To within numerical precision and to within the accuracy of the underlying EM analysis". However, simply saying the current is unique is very troublesome to me.
Furthermore, in addition to "unique to within an unknown accuracy", your earlier statement of "Currents and gap voltages are unique" seems to suggest that the current at the "gap" is well defined, regardless of whether there are well defined incident/reflected waves. (otherwise if there are two definitions A & B, then one can reasonly expect two different values.)
I've learned that the delta voltage gap source will excite a whole spectrum of radiation modes and discrete modes, each of them can propagate in two directions, requiring at least two complex "magnitudes" to represent its contribution. In terms of microwave network, the delta voltage gap source would generate an array of "voltages" and "currents"--the degree of freedom is infinite. So the "voltage and current" at the lumped port is not uniquely defined.
I think one reason people discuss/ask about the definition of reflection coefficient (or other terms) is that they want to compare results--they want to verify the "measurements" of the same thing. The word "reflection" means that there is something coming in/going out. If there is no transmission line, only "a C in parallel with an L", it is poitless to talk about a "reflection" coefficient--what is being reflected?
Although I haven't completely understood Jian's argument, I can imagine that a complex Zc is involved with microstrip structure with lossy substrate or a leaky wave supported by "lossless" waveguide. In these cases, a big question is whether it is possible and how to decompose the field into a finite number of discrete modes. If one can not separate out an individual forward/backward wave, there is no "voltage/current" associated with it, and therefore it is impossible to characterize the physical process by one "reflection coefficients".
Finally, there is no question that complex Zc is useful in many situations.
Ahhh... OK. Fair enough. Let's make real(Zc) = 1.0e-9 and X = 1.0. Same example otherwise. Instead of infinite Gamma, infinite voltage, and infinite Q, you now have a really big Q and really big Gamma and a really big voltage. Now you have to either call these things approximation errors and offer an explanation that does not involve transmission lines (remember, everything is lumped) or understand what the numbers really mean. Your choice. (Do I understand correctly from your comment that you now have no position on infinite values of Gamma when loss is exactly zero?)
As for practical...you could really build this with lumped elements from your local parts supply store and excite it with a signal generator. In this case, Re(Zo) would be maybe 1.0e-4, but a few watts at resonance and you will still get beautiful arcs and sparks!
You like transmission lines. You could build this using transmission lines by using rectangular or circular waveguide (with the capacitive Zo realized by operating the waveguide well below cutoff so Zo is almost pure imaginary. then terminate it with an above cut-off shorted waveguide for the inductor), with really big Q. (Ever see a high Q cylindrical waveguide cavity arc over when putting just a few watts into it? I have. On space flight hardware. This is why.)
And you still have to explain the infinite Gamma at zero loss. (My explanation is that the value is correct and indicates resonance.) The fact we can't quite build it is unimportant. We can come really really close, esp. with superconductors.
I think part of the problem is that you seem to have this fascination for actually seeing incident and reflected waves. For that you do need a transmission line. As I show in the above simple counter example, there is no need for transmission lines to figure out what the S-parameters are. Just take out the transmission lines and substitute a lumped load. If you know the impedance (or, in the case of Sonnet, the admittance) of the one port (or impedance/admittance matrix for a multi-port), then you can calculate S-parameters by simple arithmetic (matrix arithmetic for multi-ports). No transmission lines and their associated line voltage uncertainties needed. As I said, it is really quite a beautiful idea, side-stepping very neatly around the traveling wave voltage ambiguity. I think there are a lot of people out there who do not yet understand this.
Jian...I don't want to get you all angry at me (again), but I really think you need to carefully reconsider your postion in light of the counter examples I have suggested.
We go through discussions like this all the time here at Sonnet. In fact we just spent a few hours in one this morning, throwing out ideas and then trying to find counter examples to see if we can break the ideas. It can be a very enlightening exercise when everyone knows how to play the game. I know I changed my position on the topic of discussion several times this morning in light of counter examples presented by my employees that easily broke ideas I once thought were true. And then we came up with a really spectacular understanding of a long standing problem as a direct result. Boy was that a good feeling. But first I had to let go of the ideas that just couldn't cut it.
Don't make me wrong. I am not angry. I just repeat to you the scope of my discussions have been . I have been discussing the case when we have a complex Zc in a waveguide and TLN and we are using the conjugate defintion. In a real waveguide or TLN, the ReZc must be larger than 0. In such a condition, the gamma in the conjugate definition of a passive component will be guaranteed not to exceed 1 and you said it would be possible to exceed 1. I just proved it would under the condition I have.
I think you should try not to judge people by your own baseless thought. Thank you.
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