# Talk:Reflection coefficient

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## Reflection Coefficient for Complex Characteristic Impedances

In many RF/MW reference books Characteristic Impedance is usually presumed to lack a reactive component (it's almost always a 50-ohm transmission line for measurements) and hence the famous-but-not-complete Γ =(ZL-Z0)/(ZL+Z0) formula. I thought it's a common knowledge now, that first Z0 is actually a conjugate! It seems there is still some confusion and I've been asked to provide a source for this claim.

The math and analysis could be found in K. Kurokawa IEEE paper, “Power waves and the scattering matrix” IEEE Transactions on Microwave Theory and Technique, vol. MTT-13, no. 3, pp. 194–202, Mar. 1965. I don't have any access to these old documents but there are other papers (fortunately available online) referencing this. “Power Reflection Coefficient Analysis for Complex Impedances in RFID Tag Design” IEEE Trans., VOL. 53, NO. 9, Sep 2005 is a pretty good example with the complete formula in the intro.

I personally lack mathematical skill to demonstrate it with any rigor, but considering these are peer reviewed IEEE papers (Kurokawa paper has been cited many times in other publications) and consensus among simulation gurus (Les Besser, the guy behind Compact Software, briefly explained this confusion in his book "Practical RF Circuit Design for Modern Wireless Systems - VOL. I" page 61.) I'm convinced myself. Anoniphile (talk) 09:57, 21 October 2012 (UTC)

So if I have got this right you have changed the article on the basis of two sources, one of which you cannot read and the other you cannot understand? The derivation of ${\displaystyle \scriptstyle \Gamma }$ does not depend on an assumption of ${\displaystyle \scriptstyle Z_{0}}$ being real, on the contrary, it is quite general, assuming only an incident wave, ${\displaystyle \scriptstyle V^{+}}$, and a reflected wave, ${\displaystyle \scriptstyle V^{-}}$. The voltage and current at the load are given by,
${\displaystyle V=V^{+}+V^{-}}$
${\displaystyle I=I^{+}-I^{-}}$
From the definition of ${\displaystyle \scriptstyle Z_{0}}$,
${\displaystyle Z_{0}={V^{+} \over I^{+}}={V^{-} \over I^{-}}}$
and
${\displaystyle Z_{L}={V \over I}}$
${\displaystyle \Gamma \triangleq {V^{-} \over V^{+}}={I^{-} \over I^{+}}={\frac {Z_{L}-Z_{0}}{Z_{L}+Z_{0}}}}$
A version of this derivation is given in two books on my shelf;
• F.R. Connor, Wave Transmission, Edward Arnold Ltd., 1972 ISBN 0-7131-3278-7. Page 30.
• Matthaei, G.; Young, L.; Jones, E. M. T., Microwave Filters, Impedance-Matching Networks, and Coupling Structures McGraw-Hill 1964. Page 34.
Matthaei et al. is an extremely well regarded source and widely cited. They are careful to distinguish when they are talking about specifically resistive sources so when they use Z in an expression one can be sure they mean a general impedance.
Numerous other sources can be found which support this;
The latter source explicitly declare they are deriving the expression in the case of complex values of ${\displaystyle \scriptstyle Z_{0}}$.
Besser is no good as he does not give a derivation and may simply be following Kurokawa. Kurokawa gets a different result due to the odd way in which he defines incident and reflected waves, explained at Scattering parameters#Definition. This effectively makes his version of reflection coefficient true by definition. I suspect Kurokawa is interested in maximum power transfer, for which the load is the complex conjugate of the source. He talks about power waves and power reflection coefficient whereas other authors talk about voltage reflection coefficient.
I intend to revert this soon (assuming you do not self-revert) if there is not a response or some better explication of the sources. By the way, the convention here is bold, revert, discuss not bold, revert, redo. SpinningSpark 17:39, 22 October 2012 (UTC)
Also note that Kurokawa is discussing ${\displaystyle \scriptstyle S_{11}}$ not ${\displaystyle \scriptstyle \Gamma }$ and hence has differentiated them. SpinningSpark 19:55, 22 October 2012 (UTC)

I usually refrain from tinkering with a formula in wiki. This is simply beyond me especially when it's the discussion of the fundamentals. This time I got a little bold because of the frustration I endured for this little formula over the past few days. I was trying to work out a mismatch loss for a rudimentary complex source/load situation and my numbers wouldn't add up till I finally see that little star in Besser book.
Looking at your sources, you provided just one author that apparently explicitly stated that the formula covers Sources with reactive impedances. And you assumed other authors would of course mention this although it's customary to presume the characteristic impedance to be real. It seems you have access to Kurokawa paper. I thought he was talking about a single port network. And of course in that paper he proposes a method to map complex gen/load impedances over smith chart (power reflection). He wasn't trying to present a new fundamental concept for reflection coefficient. He just used the conventional complete formula (with conjugate) to get to his point.
Now I'm trying to figure out if you would be more disappointed, knowing you've wasted all that algebra on someone, a hobbyist who already had stated that couldn't get the math, or all those practicing microwave engineers that have worked in the industry in the past couple of decades and trusted Besser's COMPACT (now Ansoft) everyday when they finally realize the author actually never understood, you put it so eloquently, a "simple algebra"! I can't imagine the unease people like Guillermo Gonzalez who supposedly proofread Besser/Gilmore book and missed that egregious error might feel and the wretched reaction at IEEE Transactions on Microwave Theory and Techniques Journal, eventually finding out distinguished members where publishing papers based on misconceived and rather eccentric interpretation of RF fundamentals.
Feel free to revert the edit by the way. I'm trying to improve my math maybe someday we could have a productive discussion. But I would remain wary of the day my grasp of math, outstrips my commonsense. Anoniphile (talk) 00:33, 23 October 2012 (UTC)
I'm presuming you were trying to maximize the power transfer to your load. For that purpose Z* of the source impedance will indeed give you the right answer. But did you observe an actual reflected wave when terminating with Zs instead of Zs* ? SpinningSpark 09:41, 23 October 2012 (UTC)
The difference is clear in Kurokawa's book An introduction to the theory of microwave circuits, available at Srcibd. At page 14 eq. 1.28 he gives the conventional formula for reflection coefficient. At pages 56-57 eq. 1.56 he gives the formula for power reflection coefficient (an s-parameter) using the complex conjugate form. SpinningSpark 10:35, 23 October 2012 (UTC)

I think we're getting somewhere. And I'm glad you've agreed that stigmatized equation in fact would give correct answers for power transfer. And nevertheless it's all about power transfer.
If you deliver all the power in your incident wave, there will be no reflection to have a coefficient. You've asked if I really observed an actual reflected wave when using Zs instead of Zs* in the circuit. In other words, you're saying the Zs* indeed gives correct answer of zero-reflection (maximum power transfer) and still, you expect to see no observable reflection with 180° phase shift in source impedance? Well no, there will be reflections since you agreed we won't have full power transfer (you've accepted conjugated expression is good at that kind of prediction) how can we have no reflection when power is not all transferred? Our network consists of lossless reactive components, all those waves, not radiated or dissipated, have to go somewhere. They bounce back of course. Conventional expression with non-conjugate Zs simply can't see this because it's not meant to work with non-real sources.
Kurokawa's case, I think is a long walk to the same conclusion. He uses conventional expression when he's describing the way Smith chart maps impedances. Smith chart is a good example of prevailing norm in the field: It's conventionally designed to work with purely Real reference (Z0) impedances. Kurokawa, in later pages is actually trying to overcome this inherent limitation and for that he proposes the concept of power reflection just to enable us to map complex references (sources) into the chart. Again another case showing conventional expression presumes real sources only.
I took a closer look at the Inder Bahl book that you said explicitly derived the non-conjugated expression with complex source impedances in mind. In my view though, like many others in the literature, no complex source impedance where considered as he was giving the conventional expression.
Bahl description is actually quite nice. That single page summarizes our confusion. He first explains how we need to get rid of reactive component for max power transfer then he demonstrates (average power transfer) that ZL should equals Zs* for full power deliver. And at the end he provides the conventional expression for S11 (no explicit mention of complex reference impedance). Now, you're good with arithmetic, just put two equal complex numbers in the numerator of that expression and we know they should be complex conjugate of each other for full power transfer. You see we can never have zero-coefficient after subtraction if Zs is not (re) conjugated! It's that simple!
And Wai Kai Chen Handbook is actually a treat. In that he precisely calls the conventional expression, standard transmission line equation. Well, I've been saying that for the past few days here. It deals with real reference impedance.
All those S parameter measurements (in simulation or in lab) are done through transmission lines. It's not a crime to assume reference impedance to be real. It's customary and practical. It does however cause confusion among students and endless online discussions like this to say the least. Look at my longwinded struggle here; I think we're overanalyzing it to the point of incomprehension.
What is striking to me about this discussion is that you're trying to understand this thoroughly before submitting any changes to the page. I personally appreciate the effort, I'm really doing the same here but this is all very unwikipedian! You're a very active editor here and I'm sure you're aware that contributors are discouraged to get technical and get into nitty-gritty parts to prove a claim, when there's a disagreement it'd all boil down to source credibility.
To repeat myself, you've yet to provide any source that clearly (no assumptions, no between-the-line readings and no mathematical deductions from our side) states conventional expression covers complex reference impedances. On the other hand Besser/Gilmore book irrefutably states, in simple English, that Zs is a conjugate and explains why normally we don't see this in literature. Those guys are big in the field. They're putting more than three decades of their profession in the business and academia on the line for this. These are fundamentals. No book is following any IEEE paper. Although we see fundamentals simply are being used when applicable; as in Kurokawa's paper for example.
To salvage the wiki page, and to be fair, I think we should keep the expression in the conventional from BUT we need to add a couple of lines explaining that conventional expression assumes reference impedance is Real. Anoniphile (talk) 16:06, 24 October 2012 (UTC)
I don't really have an answer to your points, but I can't get past Kurokawa using both expressions without saying that one is a special case of the other. he actually introduces the reflection coefficient (Γ) after he has introduced the a(z) and b(z) parameters, but without using them in the definition of reflection coefficient. He then defines what he calls the power reflection coefficient (s1) where he now uses a1 and b1 and arrives at the conjugate form. He then explicitly compares this with the expression for Γ (page 37) but still fails to say that really they are the same thing. It is quite clear to me that Kurokawa at least, considers Γ and s1 to be two entirely different parameters and that Γ is as defined in our article. At the moment we have no other sources which give the derivation of these quantities (Besser just states it without explanation so we don't know which of the two he means) and without sources we really can't say anything. SpinningSpark 19:43, 24 October 2012 (UTC)
I'm completely confused now. I'll get someone else to comment here. SpinningSpark 21:00, 24 October 2012 (UTC)
This book has a detailed explanation. Note especially figure 6.30b showing a transmitter connected to an antenna via a transmission line. The transmitter is connected to the line with a conjugate impedance matching network for maximum power transfer into the line. There is no question of a reflection here since this is the point the wave is generated. The load (antenna), however, is connected to the line with a reflectionless matching network. Note also that in the limiting case where the transmission line length is reduced to zero, we are left with a generator connected to the load with conjugate matching. SpinningSpark 11:04, 30 July 2014 (UTC)
Further evidence that sources are not simply ignoring the possibility of complex Z0: this source derives the reflection coefficient then explicitly points out its variance to conjugate matching. This source, makes clear that a conjugate matched antenna is still scattering some electric field. Many antenna books define a conjugate matched reflection coefficient. This is a useful concept, but it does not actually represent the real reflection on a line. The article could give this definition to solve the confusion, but it needs to be made clear that this is just a convenient definition, not an actual reflection. SpinningSpark 18:11, 26 October 2014 (UTC)

The conjugate complex version is correct. The reference Handbook of Smart Antennas refers to structural reflections inside the antenna. This has nothing to do with reflection due to mismatch. The Engineering Electro magnetics derives the reflection coefficient for elements connected to a transmission line. Since transmission lines always have a pure real impedance you can omit/miss the conjugate complex part. So - I hope I can finish the discussion with an example. We assume ${\displaystyle Z_{1}=50+j100}$ and ${\displaystyle Z_{2}=50-j100}$. They are conjugate and we know (and I can demonstrate this also in practice), that conjugate matching leads to perfect matching with ${\displaystyle \Gamma =0}$

With the wrong formular we get:

${\displaystyle \Gamma ={{Z_{1}-Z_{2}} \over {Z_{1}+Z_{2}}}={{((50+j100)-(50-j100)} \over {(50+j100)+(50-j100)}}={j200 \over 100}=2j}$

This is obviously wrong since it would be generating power out of a passive device. (You can build a circuit out of a resistor and capacitor/inductor with such reflection coefficients)

However - with the conjugate complex version:

${\displaystyle \Gamma ={{Z_{1}-Z_{2}^{*}} \over {Z_{1}+Z_{2}}}={{((50+j100)-(50+j100)} \over {(50+j100)+(50-j100)}}={0 \over 100}=0}$

This is correct and what you also see in experiments. The characteristic impedance of a (TEM) transmission line itself is pure real by physics. The key in the discussion is, that every time you assume a transmission line (infinite long or matched at the end) in your calculation (on the "left side") you imply a pure real input impedance for ${\displaystyle Z_{2}}$ for this transmission line. Since most of the books/paper refer to this type of reflection they can omit the conjugate complex part also in the derivation. — Preceding unsigned comment added by 2001:4dd0:ff00:c66::2 (talkcontribs)

I've reverted that. Your first source does not really say anything relevant (at least not on the page linked) and the second source flat out contradicts your claim. It directly supports the original formula in the article. Your claim that "transmission lines always have a pure real impedance" is certainly wrong. See here for instance. SpinningSpark 19:41, 7 June 2015 (UTC)

## Editorializing in article

Scrap view of scaled and inverted image

I reverted this edit buy user:Sandalsandsocks which has the edit summary "Added comment by first figure which appears to be incorrect. The figure shows the transmitted wave being an amplitude-scaled version of the incident wave, which is correct, but the reflected wave clearly has been distorted." Problems in the article should be discussed on this talk page, not in the article itself. SpinningSpark 16:22, 21 April 2014 (UTC)

Please advise on the best way to proceed. The article contains a figure which is clearly in error. Should I have just deleted it? Sandalsandsocks (talk) 18:13, 21 April 2014 (UTC)

First explain what you think is wrong with it. You say that it is distorted, but if I invert and rescale the amplitude of the reflected wave it fits perfectly over the incident wave, see the scrap view. SpinningSpark 19:38, 21 April 2014 (UTC)

Quite right. My mistake--the error appears to actually be on the transmittance side. If you scale that and overlay on the incident wave you will see that it doesn't match. I have the graphic, but I'm not sure how to post it.Sandalsandsocks (talk) 21:07, 21 April 2014 (UTC)

Never mind. It's the difference in propagation speed on the other side of the discontinuity which compresses the pulse. Sorry for the confusion.Sandalsandsocks (talk) 21:12, 21 April 2014 (UTC)

## Recent change to introduction

I feel the recent changes to the first paragraph, which introduced the word electrodynamic for a wave on a transmission line, will be misleading for general readers. Electrodynamic redirects to Classical electromagnetism, which is uninformative about transmission line propagation. I understand the motivation, which was to differentiate a space electromagnetic wave discussed in the previous sentence from a wave on a transmission line. I'd be open to other wording. But the current state of the sentence, which pipes the phrase electromagnetic wave to the term electrodynamic, is an WP:EASTER EGG which needs to be corrected. --ChetvornoTALK 02:31, 3 June 2015 (UTC)

I agree that it is an Easter egg, but that is not the fundamental problem. The real problem is that the redirect should be to an article that discusses electromagnetic propagation on copper wires, but the target article does not do that however you try to pipe it. There should be an article at TEM mode covering this, but that is an equally hopeless redirect as well. No link at all is a better option than a useless one in my opinion. If you want to insert a link specifically about reflections on copper lines though, as opposed to just waves, there is Reflections of signals on conducting lines which I created some time ago. SpinningSpark 15:06, 3 June 2015 (UTC)
Looks good; I'll change the link. It's still an Easter egg, but at least it's a relevant Easter egg. --ChetvornoTALK 03:27, 5 June 2015 (UTC)