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#101




Thanks, it's becoming clear why we're getting different results.
When analyzing a transmission line, the forward and reflected voltages have a meaning that comes from the solutions to the wave equations. One of the fundamental properties of these waves is that their sum is the total voltage in the transmission line. (Likewise for current.) We assume that these are the only voltage waves on the line, so their sum is necessarily the total voltage. We sum these waves, for example, and look at the resulting maxima and minima to get the standing wave ratio. So it was a basic tenet of my analysis that the sum of the forward and reverse voltage waves at any point on the line (including the end) equals the voltage at that point. (Again, likewise for current, with attention paid to the defined direction of positive flow.) This was explicitly given as equation 5 (for voltage) and 6 (for current) of my analysis. Another property of the waves is that the ratio of V to I of either of the waves equals the Z0 of the cable (equations 1 and 2). Again making sure I see your model circuit correctly, it's a voltage source of voltage a or V+, connected to R in series with Z. If that's correct, then i = a / (Z + R), so your calculated value of b in your analysis below, (Z  R)i, then equals a * (Z  R) / (Z + R). The sum of a and b is a * (1 + (Z  R) / (Z + R)) = 2 * Z * a / (Z + R). While in or at the ends of a transmission line the voltage always equals the sum of the forward and reflected voltages, the sum of a and b, which you call forward and reflected voltages, don't sum to the voltage at the load. The voltage at the load is, from inspection of the circuit, a * Z / (Z + R); the sum of a and b is twice that value. Of course, if we define "forward voltage" and "reflected voltage" to be something other than the waves we're familiar with on a transmission line, while maintaining a definition of voltage reflection coefficient as being the ratio of forward to reverse voltage, we can come up with any number of formulas for reflection coefficient. And this seems to be done. I have only two texts which deal with S parameters in any depth. One, _Microwave Transistor Amplifiers: Analysis and Design_ By Guillermo Gonzalez, consistently uses forward and reverse voltage to mean exactly what they do in transmission line analysis. Consequently, he consistently ends up with the same equation for voltage reflection coefficient I've been using, and states several places that the reflection is zero when the line or port is terminated in its characteristic or source impedance (not conjugate). And this all without an assumption that Z0 or source Z is purely real. The other book, however, _Microwave Circuit Design Using Linear and Nonlinear Techniques_ by Vendelin, Pavio, and Rohde, uses a different definition of V+, V than either of us does, and different a and b than you do. To them, a = V+ * sqrt(Re(Zg)) / Zg* where Zg is the source impedance, and b = V * sqrt(Re(Zg))/Zg. They end up with three different reflection coefficients, Gammav, Gammai, and one they just give as Gamma. Gammav is V/V+, Gammai is I/I+, and plain old Gamma, which they say is equal to b/a, turns out to be equal to Gammai. Incidentally, their equation for Gammav, the voltage reflection coefficient, is: Gammav = [Zg(Z  Zg*)]/[Zg*(Z + Zg)] Which is different from yours, Slick's, and mine. The whole trick seems to be in defining what forward and reflected or reverse voltage mean. In transmission line analysis, the meaning is, I hope, pretty universal and agreed upon. If not, a whole bunch of equations, statements, assumptions, and definitions have to go out the window, along with the idea that those are the only voltages on the line. If you use those widely accepted meanings, you unavoidably end up with the common equation I've been giving for voltage reflection coefficient. (At least nobody reputable so far seems to want to define voltage reflection coefficient as anything but the ratio of reflected to forward voltage, thankfully.) On the other hand, you and the authors of at least one book put a different meaning of forward and reflected voltage (and you two use different meanings), and therefore come up with correspondingly different formulas for reflection coefficient. At this point I have to concede that in some S parameter analysis at least, various different meanings are given to "forward" or "incident", and "reverse" or "reflected" voltage than are used in transmission line analysis. And the two books I have disagree with each other, and both disagree with you, as to what they do mean. With that kind of nonstandardization, it's useless to argue which is "more right" than another. It just points out the importance of carefully defining what you mean by forward and reverse voltages before you begin your analysis  and being very careful about what conclusions you draw from the "reflections" or lack of them. When the definitions are different from those used with transmission lines, the meaning and consequences of reflections and impedance match are correspondingly different. Your analysis is self consistent, given your definitions of forward and reverse voltage. So are the analyses in the two books I have dealing with S parameters. So all three of you reach different but valid conclusions. Unless someone comes up with a convincing argument that one definition of forward and reverse voltages is better or more meaningful than another in that context, I have to agree that any of these three, or any of an infinite number of other possibilities, is equally valid for S parameter analysis. But not for transmission lines. There, forward and reverse voltages do have real meaning and a rigorous derivation. So anyone redefining them in that context is certainly deviating from very well understood usage. Consequently, I don't see how it would be productive, or add insight to the problem of voltages, currents, and reflections in a transmission line, to bring in S parameter terminology which obviously differs and in an inconsistent way from author to author. I'll be glad to continue discussing transmission lines, which is what this discussion originally involved, with the first step being for someone disagreeing with the formula for voltage reflection coefficient to point out which of the equations or assumptions preceding it are incorrect. Your analysis disagrees with equations 5 and 6 and, I believe, 1 and 2, by virtue of your different definition of forward and reverse voltage. I don't believe this can be justified within and at the end of a transmission line  if the forward and reverse voltages on the line don't add up to the total, then you have to come up with at least one more voltage wave you assume is traveling along the line (which summed with the others does equal the total), and justify its existence. Roy Lewallen, W7EL Peter O. Brackett wrote: Roy: [snip] b = v  Ri = Zi  Ri = (Z  R)i Volts. First of all, you're speaking of a circuit with a source impedance R and load impedance Z, rather than a terminated transmission line. Forward and reflected wave terminology is widely used in S parameter analysis, which also uses this model, so I'll be glad to follow along to see if and how S parameter terminology differs from the transmission line terminology we've been discussing so far. Please correct me where my assumptions diverge from yours. Your "classical definition" of b isn't one familiar to me. v + Ri would of course be the source voltage (which I'll call Vs). So v  Ri is Vs  2*Ri. Where does this come from and what does it mean? [snip] Yes it should be familiar to you because it is the most common definition and one you seem to agree with. I presume that you are not used to seeing the use of the symbols "a" and "b" for those quantities. The use of "a" and "b" is widely used in Scattering Formalism and is less confusing to many than using subscripts. In your terminology above the "Vs" symbol is nothing more than the incident voltage usually given the symbol "a" in the Scattering Formalism, or the symbol V with a "+" sign subscript in many developments. Often you will find authors use a V with a plus sign "+" as a subscript to indicate the "a" voltage and a V with a minus sign "" subscript to idicate the "b" voltage. Personally I find the use of math symbols "" and "+" or other subscripts to variables to be confusing, I much prefer the use of "a" and "b" for forward or incident and reflected voltages. Simply put, if a generator with "open circuit" voltage "a" and "internal impedance" R is driving a load Z [Z could be a transmission line driving point impedance, for instance Z would be the characteristic or surge impedance Zo of a transmission line if the generator was driving a semiinfinite line.] then v is the voltage drop across Z and I is the current through Z, and so... a = v + Ri = Zi  Ri = (Z  R)i is simply the [usual] forward voltage or incident voltage applied by the generator to the to the load Z, which may be a lumped element load or if you prefer to talk about transmission lines, Z can be just the driving point impedance of a transmsision line, whatever you wish. Then its'just a simple application of Ohms Law tosee that b = v  Ri = Zi  Ri = (Z  R)i is the [usual] reflected voltage. b is just the difference between the voltage across Z which is calculated as Zi and the voltage that would be across Z if Z was actually equal to R. i.e. the reflected voltage b is just the voltage that would exist across Z if there was an "image match" between Z and R. [If Z is the Zo of a semiinfinite transmsision line you could call this a Zo match]. Taking the ratio of "b" to "a" just yeilds the [usual] reflection coefficient as b/a = (Z  R)i/(Z + R)i = (Z  R)/(Z + R). A well known result. Simple? [snip] From your equation, and given source voltage Vs, i = Vs/(R+Z). Therefore, your "classical definition" of reflected voltage b is, in terms of Vs, Vs*((ZR)/(Z+R)). and the incident voltage a would be the Thevinins equivalent voltage across the sum of Z and R, i.e. a = (Z + R)i [snip] Yep you got it all right! [snip] Since i = Vs(Z+R), you're saying that a = the source voltage Vs (from your two equations). So what you're calling the "incident voltage" is simply the source voltage Vs. [snip] Yes, mathematically "a" = "Vs", what else would it be? Nothing mysterious about that. The incident voltage is always simply the open circuit voltage of the source. In words a is not the source voltage because the source is a Thevinin equivalent made up of the ideal voltage generator Vs = a behind the "internal" source impedance R. A better way to describe Vs = a in words would be the incident voltage a is the "open circuit source voltage". [snip] Let's do a consistency check. The voltage at the load should be a + b = Vs + Vs*((ZR)/(Z+R)) = Vs*2Z/(Z+R). Inspection of the circuit as I understand it shows that the voltage at the load should be half this value. So, we already diverge. Which is true: [snip] No, the voltage at the load is not (a + b) rather it is [the quite obvious by Ohms Law] v = Zi. and the sum of the incident and reflected voltage is simply a + b = (v + Ri) + (v  Ri) = 2v = 2Zi Now if there is an "image match" and the "unknown" Z is actually equal to R, i.e. let Z = R in all of the above, then... a = Vs b = 0 a + b = 2Ri and i = Vs/2R = a/2R. [snip] 1. I've goofed up my algebra (a definite possibility) [snip] Only a little :) [snip] 2. I've misinterpreted your circuit, or [snip] No you have it correct! [snip] 3. The voltage at the load is not equal to the sum of the forward and reflected voltages a and b, as you use the terms "forward voltage" and "reflected voltage". If v isn't equal to a + b, then what is the relationship between v, a, and b, and what are the physical meanings of the forward and reflected voltages? [snip] I showed those relationships above. There is nothing new here... these are the [most] widely accepted definitions of incident and reflected voltages. [snip] I'd like to continue with the remainder of the analysis, but can't proceed until this problem is cleared up. Roy Lewallen, W7EL [snip] OK, let's carry on.  Peter K1PO Indialantic BytheSea, FL. 
#102




Roy:
[snip] Consequently, I don't see how it would be productive, or add insight to the problem of voltages, currents, and reflections in a transmission line, to bring in S parameter terminology which obviously differs and in an inconsistent way from author to author. I'll be glad to continue discussing transmission lines, which is what this discussion originally involved, with the first step being for someone disagreeing with the formula for voltage reflection coefficient to point out which of the equations or assumptions preceding it are incorrect. Your analysis disagrees with equations 5 and 6 and, I believe, 1 and 2, by virtue of your different definition of forward and reverse voltage. I don't believe this can be justified within and at the end of a transmission line  if the forward and reverse voltages on the line don't add up to the total, then you have to come up with at least one more voltage wave you assume is traveling along the line (which summed with the others does equal the total), and justify its existence. Roy Lewallen, W7EL [snip] Thanks for following along with my development. I like to develop stuff from first principles, instead of quoting often questionable sources. Roy apart from a few typos and a small factor of 1/2 it seems that we agree on everything! Sorry if I messed up a little with typos, and... there is that potentially confusing but unimportant factor of 1/2 that appears in my definitions of the incident "waves" a and b. Your claim as to my definitions being quite different from the "mainstream" of transmission line theory was a bit hasty, because in fact with a closer look I believe that you will see that my definitions and yours [Which are hhe mainstream transmission line definitions and equations] only differ by a simple numerical factor of 2. I don't use that factor of 1/2 because it drops out whenever you take a ratio of waves anyway. Perhaps I should have defined my a to be a = 1/2(v + Ri) and b = 1/2(v  Ri) instead of a = v + Ri and b = v  Ri as I have done. Including that factor of 1/2 in my a and b makes them identical to your incident and reflected voltages. This factor of 1/2 is a very minor "scaling" difference in the "waves" and there is absolutely no difference in reflection coefficient, or indeed in Scattering Matrix definitions since the factor of 1/2 in each of "my" wave definitions simply cancels out in rho = b/a, when you divide b by a to get rho or indeed in calculating with wave vectors and Scattering Matrixs of any order. My definition of the scaling factor [1/2] for the wave variables works as well as any other as long as consistency is maintained. Like you I have found that different Scattering Theory books often use a variety of different scaling factors in defining the waves, for instance some use a factor of 1/2*sqrt(R) = 1/2*sqrt(Zo) in the definition of the waves, etc... It simply doesn't matter as long as you are consistent. In any case, the scaling value in the definition of the "waves" was not my point. The point I was trying to make was to address the subtle point initiated by Slick, i.e. the one about the definition of rho which is the ratio of reflected to incident waves, where the scaling factors drop out, and whether the CONJUGATE of the reference impedance should be used in the definition for rho or not. My point was that the conventional definition of rho = (Z  R)/(Z + R) without the CONJUGATE is in fact the "natural" definition for wave motion whether on transmission lines or in impedance matching to a generator. And... That with the conventional definition of rho in the case of a general Zo the reflected voltage will *NOT* be zero at a conjugate match. At a conjugate match, the classical rho and b will only be null in the special case of the reference impedance Zo being a pure real resistance.  Peter K1PO Indialantic BytheSea, FL. 
#103




Roy Lewallen wrote:
I have only two texts which deal with S parameters in any depth. One, _Microwave Transistor Amplifiers: Analysis and Design_ By Guillermo Gonzalez, consistently uses forward and reverse voltage to mean exactly what they do in transmission line analysis. Consequently, he consistently ends up with the same equation for voltage reflection coefficient I've been using, and states several places that the reflection is zero when the line or port is terminated in its characteristic or source impedance (not conjugate). And this all without an assumption that Z0 or source Z is purely real. The other book, however, _Microwave Circuit Design Using Linear and Nonlinear Techniques_ by Vendelin, Pavio, and Rohde, uses a different definition of V+, V than either of us does, and different a and b than you do. To them, a = V+ * sqrt(Re(Zg)) / Zg* where Zg is the source impedance, and b = V * sqrt(Re(Zg))/Zg. They end up with three different reflection coefficients, Gammav, Gammai, and one they just give as Gamma. Gammav is V/V+, Gammai is I/I+, and plain old Gamma, which they say is equal to b/a, turns out to be equal to Gammai. Incidentally, their equation for Gammav, the voltage reflection coefficient, is: Gammav = [Zg(Z  Zg*)]/[Zg*(Z + Zg)] This formula is Gonzalez's definition of voltage reflection coefficient, based on *power wave* theory (not *transmission line* theory) on page 48 of his *second edition*. We need to keep in mind that a power wave is a different kind of wave than the ones that we are used to thinking about in transmission lines (Gonzalez's words). If you don't have his second edition, I suggest get online and buy it. It has a lot of stuff not found in the first edition. The discussion of power waves is excellent and readable, with some mental suffering. The power wave concept is quite valid. We need to come to grips with this and learn to accept it. It is the actual basis for microwave simulation programs. In these programs transmission lines are treated as "circuit elements" with certain properties and calculated scattering parameters. But we must wear a different "hat" when dealing with it. The idea of "power wave" requires some meditation. I discussed some of this in a previous post. Bill W0IYH 
#104




William E. Sabin wrote:
Roy Lewallen wrote: I have only two texts which deal with S parameters in any depth. One, _Microwave Transistor Amplifiers: Analysis and Design_ By Guillermo Gonzalez, consistently uses forward and reverse voltage to mean exactly what they do in transmission line analysis. Consequently, he consistently ends up with the same equation for voltage reflection coefficient I've been using, and states several places that the reflection is zero when the line or port is terminated in its characteristic or source impedance (not conjugate). And this all without an assumption that Z0 or source Z is purely real. The other book, however, _Microwave Circuit Design Using Linear and Nonlinear Techniques_ by Vendelin, Pavio, and Rohde, uses a different definition of V+, V than either of us does, and different a and b than you do. To them, a = V+ * sqrt(Re(Zg)) / Zg* where Zg is the source impedance, and b = V * sqrt(Re(Zg))/Zg. They end up with three different reflection coefficients, Gammav, Gammai, and one they just give as Gamma. Gammav is V/V+, Gammai is I/I+, and plain old Gamma, which they say is equal to b/a, turns out to be equal to Gammai. Incidentally, their equation for Gammav, the voltage reflection coefficient, is: Gammav = [Zg(Z  Zg*)]/[Zg*(Z + Zg)] This formula is Gonzalez's definition of voltage reflection coefficient, based on *power wave* theory (not *transmission line* theory) on page 48 of his *second edition*. We need to keep in mind that a power wave is a different kind of wave than the ones that we are used to thinking about in transmission lines (Gonzalez's words). If you don't have his second edition, I suggest get online and buy it. It has a lot of stuff not found in the first edition. The discussion of power waves is excellent and readable, with some mental suffering. The power wave concept is quite valid. We need to come to grips with this and learn to accept it. It is the actual basis for microwave simulation programs. In these programs transmission lines are treated as "circuit elements" with certain properties and calculated scattering parameters. Mason's Rule is then applied to the collection of circuit elements to get the system response. But we must wear a different "hat" when dealing with it. The idea of "power wave" requires some meditation. I discussed some of this in a previous post. Bill W0IYH Bill W0IYH 
#105




Ian White, G3SEK wrote:
There in one sentence is the whole problem with the "power" approach. For a complete solution including phase conditions, you have to assume a Z0match, and even Cecil acknowledges that is only true for "most ham systems". But Ian, what we have been arguing about for two years is what happens to the energy around a *Z0match point*. Now you admit that, for a Z0 matched system (which most ham systems are) the power approach yields a "complete solution including phase conditions". We are making progress. As Reg says, this is because the power approach throws away the phase information at the start, and if you want it back again, you have to make assumptions. For an energy analysis, you don't need the phase information. Energy, like SWR, is the same for 50+j50 and 50j50. So the problem is not that the "power" approach cannot give a complete solution, but that it cannot do it for all cases. In other words, it isn't completely general  and that flaw is fatal. You don't understand the purpose of a "power" approach. It is not to solve for the phases. It is to analyze the energy flow. For that purpose, like SWR, it is not necessary to know the sign of the reactance. I have specifically said that the power approach does not attempt to replace conventional approaches. It augments conventional approaches to determine the details of the energy flow, something the conventional approach sadly lacks.  73, Cecil http://www.qsl.net/w5dxp = Posted via Newsfeeds.Com, Uncensored Usenet News = http://www.newsfeeds.com  The #1 Newsgroup Service in the World! == Over 100,000 Newsgroups  19 Different Servers! = 
#106




Roy Lewallen wrote:
Consequently, I don't see how it would be productive, or add insight to the problem of voltages, currents, and reflections in a transmission line, to bring in S parameter terminology which obviously differs and in an inconsistent way from author to author. A good reference is HP's application note, AN 951, available for download from Agilent: http://contact.tm.agilent.com/Agilen...51/index.html Another good reference is: _Fields_and_Waves_... by Ramo, Whinnery, and Van Duzer, section 11.09. The basic equations are pretty simple: b1 = s11*a1 + s12*a2 b2 = s21*a1 + s22*a2  73, Cecil http://www.qsl.net/w5dxp = Posted via Newsfeeds.Com, Uncensored Usenet News = http://www.newsfeeds.com  The #1 Newsgroup Service in the World! == Over 100,000 Newsgroups  19 Different Servers! = 
#107




William E. Sabin wrote:
The power wave concept is quite valid. In fact, power waves are used almost exclusively in the field of optics, the "other" EM waves. Since the phase of light is extremely hard to measure, it is deduced from the irradiance (power) patterns. The same deductive logic can be used with RF transmission lines (if anyone actually cares about logical thought).  73, Cecil http://www.qsl.net/w5dxp = Posted via Newsfeeds.Com, Uncensored Usenet News = http://www.newsfeeds.com  The #1 Newsgroup Service in the World! == Over 100,000 Newsgroups  19 Different Servers! = 
#108




In transmission line analysis, we're not free to rescale the forward and
reverse voltage waves, unless we also scale all the voltages, currents and powers accordingly. The forward and reverse waves have to add to the total voltage in the line and at its ends, and the ratio of each component to the corresponding current component has to equal the Z0 of the line. It's quite apparent that in S parameter analysis you're quite free to scale them as you wish, as you have. Vendelin et al didn't just scale them, but chose a set of V+ and V which aren't even related to a and b by the same constant. By defining V+ and V as we wish, we can make the reflection coefficient V/V+ to be zero when there's a Z0 match, when there's a conjugate match, or when any other impedance of our choice is used as a termination. And when we relieve the requirement that the sum of V+ and V add to the total voltage, we can have any value of V+ we choose, when V is zero. The various analyses I've seen have made different choices, and arrived at different V+, V, and voltage reflection coefficient values. Again, though, when dealing with a transmission line we don't have the luxury of choosing any definitions of V+ and V we want. Consequently, in a transmission line, the ratio of V/V+, universally defined at the voltage reflection coefficient, can be calculated with the familiar nonconjugate formula. The formula can be derived as I did it, from basic principles. And from it or other methods, we can conclude that when a transmission line is terminated in its characteristic impedance, there is no reflection of the voltage (or current) wave. When it's terminated in the complex conjugate of its characteristic impedance, or any other impedance except its characteristic impedance, there is a reflection. Roy Lewallen, W7EL Peter O. Brackett wrote: Roy: [snip] Consequently, I don't see how it would be productive, or add insight to the problem of voltages, currents, and reflections in a transmission line, to bring in S parameter terminology which obviously differs and in an inconsistent way from author to author. I'll be glad to continue discussing transmission lines, which is what this discussion originally involved, with the first step being for someone disagreeing with the formula for voltage reflection coefficient to point out which of the equations or assumptions preceding it are incorrect. Your analysis disagrees with equations 5 and 6 and, I believe, 1 and 2, by virtue of your different definition of forward and reverse voltage. I don't believe this can be justified within and at the end of a transmission line  if the forward and reverse voltages on the line don't add up to the total, then you have to come up with at least one more voltage wave you assume is traveling along the line (which summed with the others does equal the total), and justify its existence. Roy Lewallen, W7EL [snip] Thanks for following along with my development. I like to develop stuff from first principles, instead of quoting often questionable sources. Roy apart from a few typos and a small factor of 1/2 it seems that we agree on everything! Sorry if I messed up a little with typos, and... there is that potentially confusing but unimportant factor of 1/2 that appears in my definitions of the incident "waves" a and b. Your claim as to my definitions being quite different from the "mainstream" of transmission line theory was a bit hasty, because in fact with a closer look I believe that you will see that my definitions and yours [Which are hhe mainstream transmission line definitions and equations] only differ by a simple numerical factor of 2. I don't use that factor of 1/2 because it drops out whenever you take a ratio of waves anyway. Perhaps I should have defined my a to be a = 1/2(v + Ri) and b = 1/2(v  Ri) instead of a = v + Ri and b = v  Ri as I have done. Including that factor of 1/2 in my a and b makes them identical to your incident and reflected voltages. This factor of 1/2 is a very minor "scaling" difference in the "waves" and there is absolutely no difference in reflection coefficient, or indeed in Scattering Matrix definitions since the factor of 1/2 in each of "my" wave definitions simply cancels out in rho = b/a, when you divide b by a to get rho or indeed in calculating with wave vectors and Scattering Matrixs of any order. My definition of the scaling factor [1/2] for the wave variables works as well as any other as long as consistency is maintained. Like you I have found that different Scattering Theory books often use a variety of different scaling factors in defining the waves, for instance some use a factor of 1/2*sqrt(R) = 1/2*sqrt(Zo) in the definition of the waves, etc... It simply doesn't matter as long as you are consistent. In any case, the scaling value in the definition of the "waves" was not my point. The point I was trying to make was to address the subtle point initiated by Slick, i.e. the one about the definition of rho which is the ratio of reflected to incident waves, where the scaling factors drop out, and whether the CONJUGATE of the reference impedance should be used in the definition for rho or not. My point was that the conventional definition of rho = (Z  R)/(Z + R) without the CONJUGATE is in fact the "natural" definition for wave motion whether on transmission lines or in impedance matching to a generator. And... That with the conventional definition of rho in the case of a general Zo the reflected voltage will *NOT* be zero at a conjugate match. At a conjugate match, the classical rho and b will only be null in the special case of the reference impedance Zo being a pure real resistance.  Peter K1PO Indialantic BytheSea, FL. 
#109




W5DXP wrote: I don't think there's anything to argue about. From _Optics_, by Hecht: "We define the reflectance R to be the ratio of the reflected power to the incident power." He certainly wasn't including Max Born and Emil Wolf when he said "We". They define reflectance in terms of indices of refraction i.e. (n1n2)/(n1+n2). 73, ac6xg 
#110




Roy Lewallen wrote:
It's quite apparent that in S parameter analysis you're quite free to scale them as you wish, as you have. The scaling is defined by the sparameter specification. Nobody is free "to scale them as you wish". For instance, a1 is defined as: (V1+I1*Z0)/2*Sqrt(Z0)  73, Cecil http://www.qsl.net/w5dxp = Posted via Newsfeeds.Com, Uncensored Usenet News = http://www.newsfeeds.com  The #1 Newsgroup Service in the World! == Over 100,000 Newsgroups  19 Different Servers! = 
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