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#31
August 23rd 03, 09:01 PM
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#32
August 23rd 03, 09:03 PM
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#33
August 23rd 03, 09:22 PM
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Roy Lewallen wrote in message ...
A. The one just posted by Peter, (Zl - Z0conj) / (Zl + Z0conj) I believe this was a typo on Peter's part, as was my typo in the original post. B. Slick's, (Zl - Z0conj) / (Zl + Z0) This is the correct formula. C. The one in all my texts and used by practicing engineers, (Zl - Z0) / (Zl + Z0) This is correct too, but Zo must be purely real. What about the seemingly sound logic that the accepted formula doesn't work for complex Z0 because it implies that a conjugate match results in a reflection? The formula certainly does imply that. And it's a fact -- a conjugate match guarantees a maximum transfer of power for a given source impedance. But it doesn't guarantee that there will be no reflection. We're used to seeing the two conditions coincide, but that's just because we're used to dealing with a resistive Z0, or at least one that's close enough to resistive that it's a good approximation. The fact that the conditions for zero reflection and for maximum power transfer are different is well known to people accustomed to dealing with transmission lines with complex Z0. Wrong. But at least you admit that the "accepted" formula (which is fine for purely real Zo) implies a reflection. The absence of reflection is what makes the maximum power transfer. But doesn't having a reflection mean that some power is reflected and doesn't reach the load, reducing the load power from its maximum possible value? As you might know from my postings, I'm very hesitant to deal with power "waves". But what's commonly called forward power doesn't stay constant as the load impedance is changed, nor does the forward voltage. So it turns out that if you adjust the load for a conjugate match, there is indeed reflected voltage, and "reflected power". But the forward voltage and power are greater when the load is Z0conj than when Zl = Z0 and no reflection takes place -- enough greater that maximum power transfer occurs for the conjugate match, with a reflection present. ?? From a theory point of view, when you cancel series reactances (canceling inductive with capacitive) the series inductor and capacitor are resonant, and will thoeretically have zero impedance, allowing the 50 ohms to feed 50 ohms for max power transfer, WITH THEORETICALLY NO REFLECTIONS. I'd welcome any corrections to any statements I've made above, any of the equations, or the calculations. The calculations are particularly subject to possible error, so should undergo particular scrutiny. I'll be glad to correct any errors. Anyone who disagrees with the conclusion is invited and encouraged to present a similar development, showing the derivation of the alternate formula and giving numerical results from an example. That's how science, and good engineering, are done. And what it takes to convince me. Roy Lewallen, W7EL It's hard to convince anyone who could never admit that they were wrong. Slick |
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#34
August 23rd 03, 09:27 PM
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#35
August 23rd 03, 10:19 PM
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You ARE talking about ME, aren't you Tom, when you say "fine engineer, with a good education and career"? Nope. Most engineers don't write like an attitudinous high-school kid. If you study hard, go to the right schools, get the right job, etc. you might actually become an engineer when you grow up, Slick. Arguing with people who know more than you do on the net won't help you, though. 73, Tom Donaly, KA6RUH |
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#36
August 24th 03, 01:57 AM
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Slick:
[snip] And not the complex conjugate of Z0. : : This is ABSOLUTELY WRONG! The reflection coefficient is zero only when the Zload is the conjugate of the Zo. Go look it up in any BASIC RF book! Slick [snip] Easy now boy! You'r almost as bad as me! It is entirely possible, in fact I know this to be true, that there can be more than one *definition* of "the reflection coefficient". And so... one cannot say definitively that one particular defintion is WRONG. If the definition of the reflection coefficient is given as rho = (Z - R)/(Z + R) then that's what it is. This particular definition corresponds to the situation which results in rho being null when the unknown Z is equal to the reference impedance R, i.e. an "image match". If the definition is given as rho = (Z - conj(R))/(Z + conj(R)) then rho will be null when the unknown Z is equal to the conjugate of the reference impedance conj(R), i.e. a "conjugate match". Nothing is WRONG if the definition is first set up to correspond to what the definer is trying to accomplish. And so one has got to take care when making statements about RIGHT ways and WRONG ways to define things. Everyone is entitled to go to hell their own way if they are the onesmaking the definitions. Just as long as no incorrect conclusions are drawn from the definitions. That may occur when folks don't accept or agree on a definition. OTOH.... Definitions and semantics aside, what we should really be interested in is what is the physical meaning of any particular definition and what are its' practical uses. Clearly if R is a real constant resistance and contains no reactance for all frequencies [R = r + j0] then the two definitions are equivalent i.e. rho = (Z - R)/(Z + R) = (Z - conj(R))/(Z + conj(R)) since R = conj(R). This is the situation for most common amateur radio transmsission line problems and so in these simple cases it clearly doesn't matter which definition one takes. But the question of definitions for rho is even broader than that. We amateurs usually only examine a very small class of problems, and there are many more and usually much more interesting and challenging problems that require the use of reflection [scattering] parameters. Now for broadband problems where the reference impedance R is in general a complex function of frequency, e.g. R(w) = r(w) + jx(w), one is faced with the problem of creating a definition for rho which will be practical and useful and easy to measure... For an example of a practical consideration, with R a constant resistance it is easy to manufacture wide ranging reflectometers, like the Bird Model 43 since all it needs inside is a replica of the R, simply a the equivalent of a garden variety 50 Ohm resistor. But if the reference impedance R needs to be a complex function of frequency it is not so easy to design an instrument to measure the reflection coefficient over a broad band. In fact if the reference impedance R(w) = r(w) + jx(w) corresponds to the driving point impedance of a real physical system such as 18,000 feet of telephone line operated over DC to 50MHz, then it can be proved using network synthesis theory that one cannot exactly physically create the conjugate R(w) = r(w) - jx(w) for such a line. How then to make a satisfactory reflectometer for this application? To synthesize the reference impedance for such a "broad band Bird" one would have to be approximated over some narrow band, etc... Not an easy problem... Actually, under the general so-called Scattering Formalism, the reference impedance can be chosen arbitrarily, and often is, to make the particular physical problem being addressed easy to solve. Within the general Scattering Formalism the so-called port "wave variables" a and b [a is the incident wave and b is the reflected wave] are nothing more than linear combinations of the port "electrical variables" v and i [v is the port voltage and i is the port current]. Thus each port on a network has an electrical vector [v, i]' and a wave vector [a, b]' and these two vectors are related to each other by a simple linear transformation matrix made up of the sometimes arbitrarily chosen reference impedance(s). For example for the "normal" case we are all used to where r is a fixed constant then... [a, b]' = M [v, i]' where M is the matrix of the transformation. Specifically... b = v - ri a = v + ri and the 2x2 matrix M relating the "waves" to the "electricals" has the first row [1, -r] and second row [1, r], i.e. M is equal to: |1 -r | |1 +r| It is easy to show with simple algebra that this definition of the relation M betweent the waves and the electricals yeilds the common defiintion of the reflection coefficient rho = b/a = (Z - r)/(Z + r). The way linear algebraists view this is that the vector of waves a and b is just the vector of electricals rotated and stretched a bit! In other words the waves are just another way of looking at the electricals. Or... the waves and the electricals are just different manifestations of the same things, their specific numerical values depend only upon your viewpoint, i.e. what kind of measuring instruements you are using, i.e. voltmeters and ammeters or reflectometers with a particularly chosen reference impedance. All that said, it should be clear that one can arbitrarily chose the matrix transformation [reference impedance] which relates the waves to the electricals to give you the kinds of wave variables that makes your particular physical problem easy to solve. i.e. it dictates the kind of reflectometer you must use to make the measurements. The Bird Model 43 is only one such instrument and it is useful only for one particular and common kind of narrow band set of problems. For broad band problems one needs an entirely different set of definitions, etc... And so... In transmission line problems it us usual to choose the characterisitic impedance Zo of the transmission media to be the reference impedance for the system under examination, but that is certainly not necessary, only convenient. And... if you want a null rho to correspond to a "conjugate match" you must choose the reference impdance in your reflectometer to be the conjugate of the reference impedance of the system under examination, and if you want a null rho to correspond to an "image match" then you must choose the reference impedance in your reflectometer to be identical to the reference impedance of the system under examination. Every one is entitled to go to hell their own way when defining the wave to electrical variable transformations required to make their measurements and solve their problems and this will result in a variety of definitions for the scattering [reflection] parameters. Nothing more nothing less. Others may not agree with your tools, methodologies and definitions, but just be careful to follow through and be consistent with your definitions, measurements, algebra, and arithmetic and you will always get the right answers. Thoughts, comments? -- Peter K1PO Indialantic By-the-Sea, FL. |
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#37
August 24th 03, 03:44 AM
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Yes indeed. I hope no one has interpreted all this as meaning that I
believe it has any direct relevance to typical amateur antenna applications. It doesn't. As Bill and quite a few others have stated, the output Z of the PA isn't important at all for our applications. And for nearly any calculation you care to do at HF, the assumption that Z0 is purely real is entirely adequate. The precise Z0 might possibly be important if very precise measurements are being made, but that's not something done by most amateurs. But there was information posted that's incorrect, even if it's not directly relevant to most of us, and that's what prompted my posting in response. Roy Lewallen, W7EL William E. Sabin wrote: Roger to that. In the special case of conjugate matching generator to load, via a Z0 line, if we know the generator impedance we can do that. But for PAs the generator impedance is "who knows what?" so the best we can do is make the load equal to the complex Z0. Then forward power is all there is and reflected power is zero. My Bird meter then tells me that the calculated VSWR is 1.0:1.0. which is what my PA is designed for. If my coax gets so lossy that I have to worry about stuff like this, I will buy new (better) coax. Bill W0IYH |
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#38
August 24th 03, 03:49 AM
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I apologize if it sounded like my analysis was original. I had assumed,
apparently mistakenly, that readers realized it was simply a statement of very well known principles, and had no intention whatsoever to claim or imply originality. I did mention in a followup posting that a similar analysis can be found in many texts. Please amend my posting from: "I agree entirely, and it follows from my analysis and my conclusion." to "I agree entirely, and it follows from the analysis and conclusion I posted." I do take credit for posting it on this newsgroup, something neither Reg nor anyone else has, to my knowledge, taken the trouble to do. Roy Lewallen, W7EL Reg Edwards wrote: Roy Sez - That's fine. I agree entirely, and it follows from my analysis and my conclusion. A similar analysis can be found in many texts. My offering to provide a large number of references has brought forth no interest from the most vocal participants, and they've also showed a lack of willingness to work through the simple math themselves. So I felt that it might be a good idea to post the derivation before more converts are made to this religion of proof-by-gut-feel-and-flawed-logic. =============================== YOUR analysis ! Oliver Heaviside worked it all out 120 years back. . . . |
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#39
August 24th 03, 03:56 AM
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I'm eagerly awaiting your analysis showing how and why it's wrong. Or
simply which of the statements and equations I wrote are incorrect, and what the correct statement or equation should be and why. Or even a simple numerical example that illustrates the relationship between reflection and power transfer. It appears that there are two groups of readers: those who are convinced by authoritative sounding statements not backed up by any evidence, and those who require a solid basis for believing a statement. The first group I can't help at all. But hopefully I've reached at least some people in the second group. Roy Lewallen, W7EL Dr. Slick wrote: . . . Wrong. But at least you admit that the "accepted" formula (which is fine for purely real Zo) implies a reflection. The absence of reflection is what makes the maximum power transfer. . . . |
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#40
August 24th 03, 04:06 AM
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This is interesting. But how did it lead you to the equation you
determined must be correct? That is, what definition of reflection coefficient did you start with, where did you get it, and how did you get from there to the reflection coefficient equation you presented? I assume that, consistent with the admonition in the last paragraph of your posting, you were "careful to follow through and be consistent with your definitions, measurements, algebra, and arithmetic". It would be very instructive for us to be able to follow the process you did in coming to what you feel is the "right answer". Roy Lewallen, W7EL Peter O. Brackett wrote: Easy now boy! You'r almost as bad as me! It is entirely possible, in fact I know this to be true, that there can be more than one *definition* of "the reflection coefficient". And so... one cannot say definitively that one particular defintion is WRONG. If the definition of the reflection coefficient is given as rho = (Z - R)/(Z + R) then that's what it is. This particular definition corresponds to the situation which results in rho being null when the unknown Z is equal to the reference impedance R, i.e. an "image match". If the definition is given as rho = (Z - conj(R))/(Z + conj(R)) then rho will be null when the unknown Z is equal to the conjugate of the reference impedance conj(R), i.e. a "conjugate match". Nothing is WRONG if the definition is first set up to correspond to what the definer is trying to accomplish. And so one has got to take care when making statements about RIGHT ways and WRONG ways to define things. Everyone is entitled to go to hell their own way if they are the onesmaking the definitions. Just as long as no incorrect conclusions are drawn from the definitions. That may occur when folks don't accept or agree on a definition. OTOH.... Definitions and semantics aside, what we should really be interested in is what is the physical meaning of any particular definition and what are its' practical uses. . . . Every one is entitled to go to hell their own way when defining the wave to electrical variable transformations required to make their measurements and solve their problems and this will result in a variety of definitions for the scattering [reflection] parameters. Nothing more nothing less. Others may not agree with your tools, methodologies and definitions, but just be careful to follow through and be consistent with your definitions, measurements, algebra, and arithmetic and you will always get the right answers. |
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