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A Subtle Detail of Reflection Coefficients (but important to know)
Actually, my first posting was right all along, if Zo is always real.
From Les Besser's Applied RF Techniques: "For passive circuits, 0=[rho]=1, And strictly speaking: Reflection Coefficient = (Zload-Zo*)/(Zload-Zo) Where * indicates conjugate. But most of the literature assumes that Zo is real, therefore Zo*=Zo." And then i looked at the trusty ARRL handbook, 1993, page 16-2, and lo and behold, the reflection coefficient equation doesn't have a term for line reactance, so both this book and Pozar have indeed assumed that the Zo will be purely real. That doesn't mean Zload cannot have reactance (be complex). Try your calculation again, and you will see that you can never have a [rho] (magnitude of R.C.)greater than 1 for a passive network. How could you get more power reflected than what you put in (do you believe in conservation of energy, or do you think you can make energy out of nothing)? If you guys can tell us, we could fix our power problems in CA! But thanks for checking my work, and this is a subtle detail that is good to know. Slick |
"Dr. Slick" wrote
Actually, my first posting was right all along, if Zo is always real. =============================== What a pity - it never is. |
Dr. Slick:
[snip] What a pity - it never is. Makes for a good enough approximation for most simulations, though. : : Slick [snip] Heh, heh... No it's not! Unless perhaps the only use you ever make of transmission line dynamics Engineering is for simple narrow band ham radio problems! Try solving some real [i.e. broadband] problems where Zo is not real, not even close, and you'll see how important it is to use the whole danged complex expression. rho = (Z - R)/(Z + R) is a complex function as are both the driving point impedances the load termination Z and the reference impedance of the transmission line R. rho is complex! Get over it. -- Peter K1PO Indialantic By-the-Sea, FL. |
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It is my opinion that the confusion over whether to use Zo or Zo*
(conjugate of Zo) in the computation for reflection coefficient arises because there are two different meanings for the reflection coefficient itself: one applies to voltage or current waves and the other applies to "power waves." I do not have the Besser text mentioned above but a 1965 IEEE paper by Kurokawa also uses the Zo* term to calculate the reflection coefficient. However, Kurokawa makes it clear that he is referring to "power waves" and not voltage or current waves. The Kurokawa paper was given as the justification for what I believe is an erroneous equation in the 19th edition of the ARRL Antenna Book. In all previous editions (at least the ones that I have) the formula for reflection coefficient uses the normal Zo term. In the 19th edition the formula was changed to use the Zo* (Zo conjugate) term. I did some research on this and exchanged emails with some smart folks, including Tom Bruhns and Bill Sabin. Then I wrote a note to Dean Straw, editor of the Antenna Book, explaining why I thought the new formula in the 19th edition was wrong. Here's a copy of that note: =========================================== Email to Dean Straw, 10/5/01 Dear Dean, A week or so ago I wrote you concerning the formula for rho in the 19th Antenna Book: rho = (Za-Zo*)/(Za+Zo) [Eq 6, page 24-7] where Za is the impedance of the load, Zo is the line characteristic impedance, and Zo* is the complex conjugate of Zo. You replied that the justification for using Zo* in the numerator is explained in the 1965 IEEE paper by Kurokawa, and that it didn't really make much difference whether the "classic" formula (Zo in numerator) or the "conjugate" formula (Zo* in numerator) was used at SWR levels under 100 or so. I obtained and studied the Kurokawa paper, did some research on the Internet, exchanged some emails with some folks who know more about this stuff than I do, and read through all the other technical literature I have concerning rho. I'm afraid I disagree with both of your statements (1. Justified by Kurokawa; 2. Doesn't matter for normal SWR levels). Here's why: An -infinitely- long line will have zero reflections (|rho|=0). If a line of -finite- length is terminated with a load ZL which is exactly equal to the Zo of the line, the situation will not change, there should still be zero reflections. So if the formula for rho is rho = (ZL-Zo)/(ZL+Zo) then |rho| = 0, since the numerator evaluates to 0+j0. However, if the formula is rho = (ZL-Zo*)/(ZL+Zo) then |rho| evaluates to something other than 0, since the numerator evaluates to 0-j2Xo. To use an example with real numbers: RG-174, Ro=50, VF=.66, Freq=3.75 MHz, matched line loss = 1.511 dB/100 ft (at 3.75 MHz), which yields a calculated Zo (at this Freq) of Zo=50-j2.396. If ZL = Zo = 50-j2.396, then: |rho(Zo)| = 0 [classic formula, Zo in the numerator] and |rho(Zo*)| = 0.0479 ["conjugate" formula, Zo* in the numerator] Using SWR=(1+|rho|) / (1-|rho|), these two rho magnitude values evaluate to SWR 1:1 and SWR 1.10:1, respectively. So if a line is terminated with a load equal to Zo, which is equivalent to an infinite line, the "conjugate" formula results in a rho magnitude greater than 0 and an SWR greater than 1. This doesn't seem to make intuitive sense. This same anomaly may be extended to loads of other than Zo and to points other than just the load end of the line. Using the Zo for RG-174 as stated above (Zo=50-j2.396), and an arbitrary but totally realistic load of ZL=100+j0 ohms (nominal 2:1 SWR), I used the full hyperbolic transmission line equation to calculate what the Zin would be at points along the line working back from the load from 0° to 360° (one complete wavelength) in 15° steps. I then calculated the magnitude of both rho(Zo) [classic formula] and rho(Zo*) [conjugate formula] using the Zin values, and plotted the results. Here's the plot: http://www.qsl.net/ac6la/adhoc/Rho_C..._Conjugate.gif (The scale for rho is on the right. The left scale is normally used for R, X, and |Z|, but those plot lines have been intentionally hidden in this case just to reduce the chart clutter.) Note that the plot line for rho(Zo) [classic formula] progresses downward in a smooth fashion as the line length increases, as expected. The rho(Zo*) [conjugate formula] swings around, and even goes above the value at the load point until a line length of about 75° is reached. Again, this doesn't seem to make intuitive sense, and I can think of no physical explanation which would result in the voltage reflection coefficient magnitude "swinging around" as the line length is increased. Of course, the same data may be used to calculate and then plot SWR. Here's the plot: http://www.qsl.net/ac6la/adhoc/SWR_C..._Conjugate.gif Again, this doesn't seem intuitive, and this is for a load SWR much less than 100. Now if the intent of the "conjugate" formula was to always force rho to be = 1 and therefore to avoid the "negative SWR" problem, it appears that this has the effect of "throwing the baby out with the bathwater." That is, it may make it possible to calculate a rho value less than 1 and hence a non-negative SWR value in an "extreme load" situation like ZL=1+j1000 ohms (even though SWR is pretty meaningless in that case). However, it also changes the rho and SWR values for completely reasonable loads, such as the example above. At a line length of 45°, the impedance at the input end of the line is 41.40-j31.29. Using the "classic" rho formula results in calculated rho and SWR values of rho=0.3095 SWR=1.90 while the "conjugate" formula gives rho=0.3569 SWR=2.11 Note that these results are for a perfectly reasonable load on a perfectly reasonable line at a perfectly reasonable frequency, but the results differ by an unreasonable amount. Another point. In the William Sabin article, "Computer Modeling of Coax Cable Circuits" (QEX, August 1996, pp 3-10), Sabin includes the Kurokawa paper as a reference. Even with that reference, Sabin gives the "classic" formula for rho (called gamma in his paper) as Eq 31. When I asked him recently about this, he stated that the article is correct and he stands by the given formula for rho. Given these various intuitive arguments as to why computing rho with the Zo conjugate formula doesn't make sense, where did it come from? Well, so far I have two candidates: 1) A QST Technical Correspondence article by Charlie Michaels (Nov 1997, pg 70). Michaels gave a formula for computing the portion of the loss on a line that is due to standing waves. That loss formula involves calculating rho by using the "conjugate" formula. The SWR dB loss result, when added to the normal matched line loss number, gives exactly the same figure for total power loss as do other formulas that use completely different techniques (such as in papers by Sabin and Witt). However, the Michaels QST article never said that the rho "conjugate" formula should be used to calculate rho in the general case, only that it should be used as part of an intermediate step to calculate a dB number. 2) The 1965 IEEE paper by Kurokawa, "Power Waves and the Scattering Matrix." Kurokawa does indeed show a formula for -a- reflection coefficient that uses Z conjugate in the numerator. However, in Section I of his paper he explains that he is talking about "power waves" and takes pains to explain that these waves are not the same as the more familiar voltage and current traveling waves. He then goes on to give a mathematical description of these power waves. In Section III he defines the power wave reflection coefficient as s = (ZL - Zi*) / (ZL + Zi) where ZL is the load impedance and Zi is the internal impedance of the source. In a footnote he makes it clear that "s" is equal to the voltage reflection coefficient only when Zi is real (no jX component). Finally, in Section IX ("Comparison with Traveling Waves") he explains that when the line Zo is complex the calculations that apply to voltage and current waves are not the same calculations used to determine the power delivered. He ends this section with this statement: "Further, since the traveling wave reflection coefficient is given by (ZL-Zo)/(ZL+Zo) [note no conjugate] and the maximum power transfer takes place when ZL=Zo*, where ZL is the load impedance, it is only when there is a certain reflection in terms of traveling waves that the maximum power is transferred from the line to the load." To put some actual numbers with this statement, consider the RG-174 from above (Zo=50-j2.396), with a load of ZL=Zo*=50+j2.396. Then |(ZL-Zo)/(ZL+Zo)| = 0.0479 = small voltage traveling wave reflection coefficient while |(ZL-Zo*)/(ZL+Zo)| = 0 = zero power wave reflection, meaning maximum power transfer. Now it seems to me that this clears up the confusion. It looks like it is necessary to consider -two different meanings- for rho. One is for the voltage (or current) traveling wave reflection coefficient. The classic formula to compute that still holds (as Kurokawa states), and that is the rho that should be used when talking about voltage (or current) standing waves on a line. Specifically, that is the rho that should be used in the formula for SWR, SWR = 1 + |rho| / 1 - |rho| possibly with an explanation that this formula is only applicable when |rho| 1. The -other- meaning for rho is used when dealing with "power waves" or with power and loss calculations as in the Michaels QST formula, and -that- rho (call it rho prime, or maybe some other letter ala Kurokawa) may be defined as rho' = (Z-Zo*)/(Z+Zo) However, rho' does -not- have to do with voltage (or current) traveling waves, and may -not- be used to compute SWR. If it is understood that there are two different "reflection coefficients" then everything starts to fall in place, including the last part of the Kurokawa quote above saying that there is a situation when the "voltage" reflection coefficient is slightly greater than 0 while the "power wave" reflection coefficient is exactly 0. Note that no matter what combination of values for Zo and ZL are used, |rho'| will never be greater than 1 (although it can be equal to 1 for purely reactive loads), thus satisfying the intuitive understanding that there can never be more power reflected -from- a (passive) load than is delivered -to- a (passive) load. But at the opposite extreme, as shown above, "no reflected power" but does -not- necessarily mean "no standing waves." A further example of the importance of making a distinction between the voltage reflection coefficient and the power reflection coefficient would be the following: Assume a load of ZL=1+j1000 with the RG-174 Zo from above, Zo=50-j2.396. Then |rho| = 1.0047 (voltage wave reflection coefficient) and |rho'| = 0.9999 (power wave reflection coefficient) This shows that the reflected voltage is slightly greater than the incident voltage, at least at the point of reflection, before the line loss has caused the calculated (or measured) rho to decay. It further shows that the reflected power is still less than the incident power, thus not violating the principal of conservation of energy. Robert Chipman, "Theory and Problems of Transmission Lines," presents a mathematical proof of this, including this quote from page 138: "... a transmission line can be terminated with a [voltage] reflection coefficient whose magnitude is as great as 2.41 without there being any implication that the power level of the reflected wave is greater than that of the incident wave." In summary, I think a distinction must be made between the "voltage" reflection coefficient and the "power" reflection coefficient, and therefore I think the following changes should be made to the 19th edition of the Antenna Book: 1. Revise the initial equation for rho [Eq 6, page 24-7] back to the classic "non-Zo*" form, since rho is used in this context as the voltage reflection coefficient. 2. Equation 11 on page 24-9 is rho = sqrt(Pr/Pf) where Pr and Pf are the reflected and forward power levels. Intuitively this seems to be the "second" definition for rho, namely the power reflection coefficient, although my math skills are not up to the task of proving that this formula is the equivalent of the Kurokawa formula for the power reflection coefficient 's'. Perhaps this formula should have a footnote indicating that it refers to the "power" and not "voltage" reflection coefficient, and that the two are technically equal only when the Xo component of the line Zo is ignored. (The same point as is made in the Kurokawa footnote referred to above.) Given the precision to which most amateurs can measure power, and the fact that under normal circumstances the line loss and hence the Xo value is much smaller than that of RG-174, of course this point is moot in a practical sense. 3. If Equation 11 is for the "power" reflection coefficient, then Equation 12 is a mixing of apples and oranges. Perhaps the second equal sign could be replaced with an "almost equal" sign. 4. Equation G in Table 2 "Coaxial Cable Equations" on page 24-20 should remain as is, since it obviously is referring to the voltage reflection coefficient. Thanks for looking this over, Dean. I would certainly welcome any comments or feedback you might have. 73, Dan Maguire AC6LA =========================================== I don't know if Dean has changed the formula in later printings of the 19th edition or in the upcoming 20th edition. He responded that he was busy with other matters and would get back to me later. He never did and I let the matter drop. I have the Kurokawa paper in pdf format. If anyone would like a copy, drop me a private email and I'll be glad to send it to you. The two charts mentioned above were produced with a modified version of the XLZIZL Excel application. XLZIZL is available free from www.qsl.net/ac6la. Dan, AC6LA |
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A big deal is being made of the general assumption that Z0 is real.
As anyone who has studied transmission lines in any depth knows, Z0 is, in general, complex. It's given simply as Z0 = Sqrt((R + jwL)/(G + jwC)) where R, L, G, and C are series resistance, inductance, shunt conductance, and capacitance per unit length respectively, and w is the radian frequency, omega = 2*pi*f. This formula can be found in virtually any text on transmission lines, and a glance at the formula shows that Z0 is, in general, complex. It turns out that R is a function of frequency because of changing skin depth, but it increases only as the square root of frequency. jwL, the inductive reactance per unit length, however, increases in direct proportion to frequency. So as frequency gets higher, jwL gets larger more rapidly. For typical transmission lines at HF and above, jwL R, so R + jwL ~ jwL. G represents the loss in the dielectric, and again for typical cables, it's a negligibly small amount up to at least the upper UHF range. Furthermore, G, initially very small, tends to increase in direct proportion to frequency for good dielectrics like the ones used for transmission line insulation. So the ratio of jwC to G stays fairly constant, is remains very large, at just about all frequencies. The approximation that jwC G is therefore valid, so G + jwC ~ jwC. Putting the simplified approximations into the complete formula, we get Z0 ~ Sqrt(jwL/jwC) = Sqrt(L/C) This is a familiar formula for transmission line characteristic impedance, and results in a purely real Z0. But it's very important to realize and not forget that it's an approximation. For ordinary applications at HF and above, it's adequately accurate. Having a purely real Z0 simplifies a lot of the math involving transmission lines. To give just a couple of examples, you'll find that the net power flowing in a transmission line is equal to the "forward power" minus the "reverse power" only if you assume a real Z0. Otherwise, there are Vf*Ir and Vr*If terms that have to be included in the equation. Another is that the same load that gives mininum reflection also absorbs the most power; this is true only if Z0 is assumed purely real. So it's common for authors to derive this approximation early in the book or transmission line section of the book, then use it for further calculations. Many, of course, do not, so in those texts you can find the full consequences of the complex nature of Z0. One very ready reference that gives full equations is _Reference Data for Radio Engineers_, but many good texts do a full analysis. Quite a number of the things we "know" about transmission lines are actually true only if the assumption is made that Z0 is purely real; that is, they're only approximately true, and only at HF and above with decent cable. Among them are the three I've already mentioned, the simplified formula for Z0, the relationship between power components, and the optimum load impedance. Yet another is that the magnitude of the reflection coefficient is always = 1. As people mainly concerned with RF issues, we have the luxury of being able to use the simplifying approximation without usually introducing significant errors. But whenever we deal with formulas or situations that have to apply outside this range, we have to remember that it's just an approximation and apply the full analysis instead. Tom, Ian, Bill, and most of the others posting on this thread of course know all this very well. We have to know it in order to do our jobs effectively, and all of us have studied and understood the derivation and basis for Z0 calculation. But I hope it'll be of value to some of the readers who might be misled by statements that "authorities" claim that Z0 is purely real. Roy Lewallen, W7EL |
One more thing. I've never seen that conjugate formula for voltage
reflection coefficient and can't imagine how it might have been derived. I've got a pretty good collection of texts, and none of them show such a thing. If anyone has a reference that shows that formula and its derivation from fundamental principles, I'd love to see it, and discover how the author managed to get from the same fundamental principles as everyone else but ended up with a different formula. Roy Lewallen, W7EL |
Roy Lewallen wrote:
A big deal is being made of the general assumption that Z0 is real. As anyone who has studied transmission lines in any depth knows, Z0 is, in general, complex. It's given simply as Z0 = Sqrt((R + jwL)/(G + jwC)) where R, L, G, and C are series resistance, inductance, shunt conductance, and capacitance per unit length respectively, and w is the radian frequency, omega = 2*pi*f. This formula can be found in virtually any text on transmission lines, and a glance at the formula shows that Z0 is, in general, complex. A good approximation to Z0 is: Z0 = R0 sqrt(1-ja/b) where Ro = sqrt(L/C) a is matched loss in nepers per meter. b is propagation constant in radians per meter. The complex value of Z0 gives improved accuracy in calculations of input impedance and losses of coax lines. With Mathcad the complex value is easily calculated and applied to the various complex hyperbolic formulas. Reference: QEX, August 1996 Bill W0IYH |
(Dr. Slick) wrote in message . com...
Actually, my first posting was right all along, if Zo is always real. From Les Besser's Applied RF Techniques: "For passive circuits, 0=[rho]=1, And strictly speaking: Reflection Coefficient = (Zload-Zo*)/(Zload-Zo) Where * indicates conjugate. But most of the literature assumes that Zo is real, therefore Zo*=Zo." Fascinating... Please have a look at the following reply I got from Besser... I still wish people would go through the simple math themselves, and make up their own minds what's correct and what isn't. I gather that Slick has made up his own mind, though see no evidence that it's on the basis of the simple calcs from what I believe he already agrees with. Oh, well, not MY problem. (This is twice now, recently, that I've followed up on other people's references and found them to be at best questionable in some way.) Cheers, Tom =-=-=-=-=-=-=-=-= Beginning of quoted material =-=-=-=-=-=-=-=-=-=-=- Hello Tom- Thank you for your message. I do not know which specific course was referenced by the person you mentioned in your message, but I did check the notes for our more popular course which covers linear RF circuits. In the manuals for that course, the formula is given as you described, except that the Zo term in the numerator is _not_ the complex conjugate. Thus the formula in the manual reads: Gamma = Vr/Vf = (Zl-Zo)/(Zl+Zo) This is in agreement with Guillermo Gonzalez's text, "Microwave Transistor Amplifiers," which is one of the references used in writing the course. Please let me know if this information addresses your concern. Have a good day. Regards, Rex From: ] Sent: Thursday, August 21, 2003 11:11 AM To: Subject: Other concern/question Below is the result of your feedback form. It was submitted by ) on Thursday, August 21, 2003 at 14:10:53 ------------------------------------------------------------------------ --- name: Tom Bruhns body: I have recently seen someone attribute to Besser Associates training a formula for reflection coefficient at a load Zl as Vr/Vf = (Zl-Zo*)/(Zl+Zo), where Zo* is the complex conjugate of the line characteristic impedance Zo. I'm curious if this is actually what you teach, as it is counter to what is commonly in texts, and is also counter to the commonly accepted boundary conditions on a TEM line at such a load. Yours in the interest of accurate models, Tom Bruhns -------- .... Rex Frobenius Engineering Director Besser Associates 650-949-3300 650-949-4400 FAX www.besserassociates.com =-=-=-=-=-=-=-=-= End of quoted material =-=-=-=-=-=-=-=-=-=-=- |
Roy:
[snip] "Roy Lewallen" wrote in message ... One more thing. I've never seen that conjugate formula for voltage reflection coefficient and can't imagine how it might have been derived. I've got a pretty good collection of texts, and none of them show such a thing. [snip] You are absolutely correct Roy, that formula given by "Slick" is just plain WRONG! rho = (Z - R)/(Z + R) Always has been, always will be. Where does that "Slick" guy get his information? And where does "Slick" get off with all of his "potifications"??? I dunno... *He* thinks "Besser", "Pozar" and ARRL are authoritative sources for transmission line technology!!! Me? I have made a living as a professional Engineer designing transmission equipment over the past four decades, currently more than $4BB gross shipped to world wide markets, where the Zo I used is neither real, nor a constant! And what is more... I have never consulted any of those three authorities referenced by "Slick". I certainly don't think of them as being authoritative, "cream skimmers" perhaps, but not certainly not authoritative. I believe that "Slick" has gotta stop "pontificating" and start reading in better circles... much better! -- Peter K1PO Indialantic By-the-Sea, FL. |
William E. Sabin wrote:
Roy Lewallen wrote: A big deal is being made of the general assumption that Z0 is real. As anyone who has studied transmission lines in any depth knows, Z0 is, in general, complex. It's given simply as Z0 = Sqrt((R + jwL)/(G + jwC)) where R, L, G, and C are series resistance, inductance, shunt conductance, and capacitance per unit length respectively, and w is the radian frequency, omega = 2*pi*f. This formula can be found in virtually any text on transmission lines, and a glance at the formula shows that Z0 is, in general, complex. A good approximation to Z0 is: Z0 = R0 sqrt(1-ja/b) where Ro = sqrt(L/C) a is matched loss in nepers per meter. b is propagation constant in radians per meter. The complex value of Z0 gives improved accuracy in calculations of input impedance and losses of coax lines. With Mathcad the complex value is easily calculated and applied to the various complex hyperbolic formulas. Reference: QEX, August 1996 Bill W0IYH The usage of complex conjugate Z0* becomes significant when calculating very large values of VSWR, according to some authors. But for these very large values of standing waves, the concept of VSWR is a useless numbers game anyway. For values of VSWR less that 10:1 the complex Z0 is plenty good enough for good quality coax. W.C. Johnson points out on page 150 that the concept: Pload = Pforward - Preflected is strictly correct only when Z0 is pure resistance. But the calculations of real power into the coax and real power into the load are valid and the difference between the two is the real power loss in the coax. For these calculations the complex value Z0 for moderately lossy coax is useful and adequate. The preoccupation with VSWR values is unfortunate and excruciatingly exact answers involve more nitpicking than is sensible. Bill W0IYH |
Roy Lewallen wrote in message ...
Quite a number of the things we "know" about transmission lines are actually true only if the assumption is made that Z0 is purely real; that is, they're only approximately true, and only at HF and above with decent cable. Among them are the three I've already mentioned, the simplified formula for Z0, the relationship between power components, and the optimum load impedance. Yet another is that the magnitude of the reflection coefficient is always = 1. That would be only into a passive network. As people mainly concerned with RF issues, we have the luxury of being able to use the simplifying approximation without usually introducing significant errors. But whenever we deal with formulas or situations that have to apply outside this range, we have to remember that it's just an approximation and apply the full analysis instead. Tom, Ian, Bill, and most of the others posting on this thread of course know all this very well. We have to know it in order to do our jobs effectively, and all of us have studied and understood the derivation and basis for Z0 calculation. But I hope it'll be of value to some of the readers who might be misled by statements that "authorities" claim that Z0 is purely real. Roy Lewallen, W7EL No one ever said that Zo is always purely real. But many texts do approximate it this way. Even the ARRL "bible". Slick |
So another way for the lurkers to check all this: assume a line Zo =
50-j5, and a load Zload = 1+j100. Assume some convenient Vf at the load. Calculate rho = Vr/Vf from the equation quoted below. Now find Vr, and from the line impedance and Vf and Vr, find If and Ir. Add the V terms and I terms to get the net line voltage and current at the load. Does that correspond to the expected load current for the given Zload? If so, fine; if not, where does the difference in current come from? If you assume the line current is correct from your If and Ir calcs, and the load current is correct as the net line voltage = net load voltage, and use Zload to get Iload, does the line power dissipation plus the load power dissipation equal the power fed in from a generator? Try all those calcs after revising the Vr/Vf formula to match what Besser is now teaching, and see if things line up a bit better. The truth is all there to be seen with just a bit of work. Cheers, Tom (yeah, I've done it, as you might guess. And so have a lot of others.) (Dr. Slick) wrote in message . com... Actually, my first posting was right all along, if Zo is always real. From Les Besser's Applied RF Techniques: "For passive circuits, 0=[rho]=1, And strictly speaking: Reflection Coefficient = (Zload-Zo*)/(Zload-Zo) Where * indicates conjugate. But most of the literature assumes that Zo is real, therefore Zo*=Zo." And then i looked at the trusty ARRL handbook, 1993, page 16-2, and lo and behold, the reflection coefficient equation doesn't have a term for line reactance, so both this book and Pozar have indeed assumed that the Zo will be purely real. That doesn't mean Zload cannot have reactance (be complex). Try your calculation again, and you will see that you can never have a [rho] (magnitude of R.C.)greater than 1 for a passive network. How could you get more power reflected than what you put in (do you believe in conservation of energy, or do you think you can make energy out of nothing)? If you guys can tell us, we could fix our power problems in CA! But thanks for checking my work, and this is a subtle detail that is good to know. Slick |
(Dr. Slick) wrote in message . com...
(Tom Bruhns) wrote in message om... Fascinating... Please have a look at the following reply I got from Besser... I still wish people would go through the simple math themselves, and make up their own minds what's correct and what isn't. I gather that Slick has made up his own mind, though see no evidence that it's on the basis of the simple calcs from what I believe he already agrees with. Oh, well, not MY problem. (This is twice now, recently, that I've followed up on other people's references and found them to be at best questionable in some way.) I have no problem admitting i am wrong, when i am wrong. But you haven't given me any reason to think so. Well, you may not think I have, but... What is your definition of a conjugate match? When do you think max. power transfer occurs? I'd be happy to answer this more directly after you show us the steps, as I suggested, to get from the basic TEM transmission line relations and the load boundary conditions to Vr/Vf. But for now, I'll let you consider, if you wish, the case where you have a long transmission line with reactive Zo, terminated so you have no reflected wave. Rho = 0. SWR = 1:1. I trust you'll agree you "see" an impedance equal to Zo looking into the source-end of the line. Now imagine that you have cut this line at some point; you also see Zo looking into that cut, right? (The side with the load attached, that is.) So, can you simply connect those two pieces back up and still see no reflection on the piece on the source side? I _do_ believe that the line can't tell whether the impedance it's connected to is a load right there, or the impedance presented by another length of line, so it should be obvious from that what I believe the line must be connected to, to get rho=0. Cheers, Tom |
"Peter O. Brackett" wrote in message nk.net...
Roy: [snip] "Roy Lewallen" wrote in message ... One more thing. I've never seen that conjugate formula for voltage reflection coefficient and can't imagine how it might have been derived. I've got a pretty good collection of texts, and none of them show such a thing. [snip] You are absolutely correct Roy, that formula given by "Slick" is just plain WRONG! rho = (Z - R)/(Z + R) Always has been, always will be. As long as they are all purely real. Roy disagrees even when he is wrong, because too many people read this NG, and it might make him look bad (i.e., Not the All-Knowing Guru he pretends to be). Where does that "Slick" guy get his information? And where does "Slick" get off with all of his "potifications"??? I dunno... *He* thinks "Besser", "Pozar" and ARRL are authoritative sources for transmission line technology!!! Bwa! HAah! Much, much, MUCH more than you will ever be! Me? I have made a living as a professional Engineer designing transmission equipment over the past four decades, currently more than $4BB gross shipped to world wide markets, where the Zo I used is neither real, nor a constant! I feel sorry for your customers... And what is more... I have never consulted any of those three authorities referenced by "Slick". I certainly don't think of them as being authoritative, "cream skimmers" perhaps, but not certainly not authoritative. Dr. Besser kicks your ass backwards when it comes to RF knowledge. And the ARRL is extremely well known. Pozar not so much, but the guy is out there on the PhD level. I don't give a Sh** who you think is an authority. Look them up, they have way more credentials than either you or I. I believe that "Slick" has gotta stop "pontificating" and start reading in better circles... much better! Much better than the likes of you, then yes, you would certainly be correct! The conjugate formula is correct. If you believe in cancellation of reactance. Why else would the magnitude rho (numerator of Reflection Coefficient) be zero when Zload=Zo*??? Slick |
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Seems to be, indeed, though you never know. The lurkers may well have learned a thing or two. The one reference you did post now disavows the form you posted. You've been invited to do some simple math that would show you the truth and you apparently refuse. You are posting any number of ideas contrary to what's easy to show from fundamentals and what's in a large number of published papers and texts and what has been posted here by many contributors recently and over the years. It's been done with both symbolic math and specific examples. Several inconsistencies demonstrate clearly that Vr/Vf does NOT equal (Zload-Zo*)/(Zload+Zo), and of course most certainly a (Zload-Zo) denominator is going to get you quickly into trouble. The inconsistencies have been pointed out here by me and by others, but apparently you've missed them. I'm sure it's apparent to most lurking where the communications is breaking down. But the formulas you posted above have given me a good laugh tonight, at least! Thanks! Cheers, Tom When a fine engineer, with a good education and a distinguished career, stoops to argue with an anonymous fellow who doesn't have a firm grasp of even the most basic ideas of wave mechanics, the result is bound to be a certain amount of frustration. You might want to ask yourself, Tom, whether Slick is arguing in good faith, or whether he has other motives. 73, Tom Donaly, KA6RUH |
Slick:
[snip] [snip] You are absolutely correct Roy, that formula given by "Slick" is just plain WRONG! rho = (Z - R)/(Z + R) Always has been, always will be. [snip] After consideration, I must agree with Slick. Slick is RIGHT and I was WRONG! Slick please accept my apologies!!! I was wrong, and I admit it! Indeed, the correct formula for the voltage reflection coefficient "rho" when computed using a "reference impedance" R, which is say the, perhaps complex, internal impedance R = r + jx of a generator/source which is loaded by a perhaps complex load impedance Z = ro + j xo must indeed be: rho = (Z - conj(R))/(Z + conj(R)) = (Z - r + jx)/(Z + r - jx) For indeed as Slick pointed out elsewhere in this thread, how else will the reflected voltage equal zero when the load is a conjugate match to the generator. Slick thanks for directing the attention of this "subtlety" to the newsgroup, and again... Slick, please accept my apologies, I was too quick to criticize! Good work, and lots of patience... :-) Regards, -- Peter K1PO Indialantic By-the-Sea, FL. |
I have made a living as a professional Engineer designing transmission
equipment over the past four decades, currently more than $4BB gross shipped to world wide =============================== There's an old adage "With radio any bloody thing will work". |
Let's run a quick calculation as Tom has suggested several times.
Consider a transmission line with Z0 = 50 - j10 ohms connected to a 50 + j0 ohm load. As I hope will be evident, it doesn't matter what's connected to the source end of the line or how long it is. We'll look at the voltage V and current I at a point within the cable, but very, very close to the load end. Conditions are steady state. When numerical V or I is required, assume it's RMS. I hope we can agree of the following. If not, it's a waste of time to read the rest of the analysis. 1. Vf / If = Z0, where Vf and If are the forward voltage and current respectively. 2. Vr / -Ir = Z0, where Vr and Ir are the reverse voltage and current respectively. The minus sign is due to my using the common definition of positive Ir being toward the load. (If this is too troublesome to anyone, let me know, and I'll rewrite the equations with positive Ir toward the source.) 3. Gv = Vr / Vf, where Gv is the voltage reflection coefficient. This is the usual definition. Likewise, 4. Gi = Ir / If 5. V = Vf + Vr 6. I = If + Ir Ok so far? Combine 1 and 2 to get 7. Vf / If = -(Vr / Ir) = Vr / Vf = -(Ir / If) From 3, 4, and 7, 8. Gv = -Gi From 3 and 5, 9. V = Vf(1 + Gv) and similarly from 4 and 6, 10. I = If(1 + Gi) From 8 and 10, 11. I = If(1 - Gv) Dividing 9 by 10, 12. V / I = (Vf / If) * [(1 + Gv) / (1 - Gv)] Combining 12 with 1, 13. V / I = Z0 * [(1 + Gv) / (1 - Gv)] And finally we have to observe that, by inspection, the voltage V just inside the line has to equal the voltage Vl just outside the line and, by Kirchoff's current law, the current I just inside the line has to equal the current Il just outside the line: 14. Vl = V and Il = I, where Vl and Il are the voltage and current at the load. From 13 and 14, and noting that Zl = Vl / Il, then, 15. Zl = Z0 * [(1 + Gv) / (1 - Gv)] Now let's test the formulas for Gv that have been presented. We have three to choose from: A. The one just posted by Peter, (Zl - Z0conj) / (Zl + Z0conj) B. Slick's, (Zl - Z0conj) / (Zl + Z0) C. The one in all my texts and used by practicing engineers, (Zl - Z0) / (Zl + Z0) We'll plug the numbers into Peter's (A) first, giving us: Gv = (50 - 50 - j10) / (50 + 50 + j10) = -j10 / (100 + j10) = -0.009901 - j0.09901 Plugging this into the right side of 15, we get: Zl = 46.15 -j19.23 This obviously isn't correct -- Zl is known to be 50 + j0. Let's try Slick's (B): Gv = (50 - 50 - j10) / (50 + 50 - j10) = -j10 / (100 - j10) = 0.009901 - j0.09901 from which we calculate, from 15, Zl = 48.00 - j20.00 Again, obviously not right. Finally, using the universally accepted formula (C): Gv = (50 - 50 + j10) / (50 + 50 - j10) = j10 / (100 - j10) = -0.009901 + j0.09901 from which, from 15, we get Zl = 50.00 + j0 This is the value we know to be Zl. Getting the correct result for one numerical example does *not* prove that an equation is correct. However, getting an incorrect result for even one numerical example *does* prove that an equation isn't correct. So we can conclude that variations A and B aren't correct. ------------------ Actually, the accepted formula (C) can be derived directly from equation 15, so if all the steps to that point are valid, so is the accepted formula. Why aren't Peter's or Slick's formulas correct? The real reason is that they aren't derived from known principles by an orderly progression of steps like the ones above. There's simply no way to get from known voltage and current relationships to the conjugate equations. They're plucked from thin air. That's simply not adequate or acceptable for scientific or engineering use. (I challenge anyone convinced that either of those equations is correct to present a similar development showing why.) 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. 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. 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 Peter O. Brackett wrote: Slick: [snip] [snip] You are absolutely correct Roy, that formula given by "Slick" is just plain WRONG! rho = (Z - R)/(Z + R) Always has been, always will be. [snip] After consideration, I must agree with Slick. Slick is RIGHT and I was WRONG! Slick please accept my apologies!!! I was wrong, and I admit it! Indeed, the correct formula for the voltage reflection coefficient "rho" when computed using a "reference impedance" R, which is say the, perhaps complex, internal impedance R = r + jx of a generator/source which is loaded by a perhaps complex load impedance Z = ro + j xo must indeed be: rho = (Z - conj(R))/(Z + conj(R)) = (Z - r + jx)/(Z + r - jx) For indeed as Slick pointed out elsewhere in this thread, how else will the reflected voltage equal zero when the load is a conjugate match to the generator. Slick thanks for directing the attention of this "subtlety" to the newsgroup, and again... Slick, please accept my apologies, I was too quick to criticize! Good work, and lots of patience... :-) Regards, -- Peter K1PO Indialantic By-the-Sea, FL. |
Methinks someone is confused between the conditions for maximum power
transfer and no reflections. ....Keith "Peter O. Brackett" wrote: Slick: [snip] [snip] You are absolutely correct Roy, that formula given by "Slick" is just plain WRONG! rho = (Z - R)/(Z + R) Always has been, always will be. [snip] After consideration, I must agree with Slick. Slick is RIGHT and I was WRONG! Slick please accept my apologies!!! I was wrong, and I admit it! Indeed, the correct formula for the voltage reflection coefficient "rho" when computed using a "reference impedance" R, which is say the, perhaps complex, internal impedance R = r + jx of a generator/source which is loaded by a perhaps complex load impedance Z = ro + j xo must indeed be: rho = (Z - conj(R))/(Z + conj(R)) = (Z - r + jx)/(Z + r - jx) For indeed as Slick pointed out elsewhere in this thread, how else will the reflected voltage equal zero when the load is a conjugate match to the generator. Slick thanks for directing the attention of this "subtlety" to the newsgroup, and again... Slick, please accept my apologies, I was too quick to criticize! Good work, and lots of patience... :-) Regards, -- Peter K1PO Indialantic By-the-Sea, FL. |
Roy Lewallen wrote:
Let's run a quick calculation as Tom has suggested several times. Consider a transmission line with Z0 = 50 - j10 ohms connected to a 50 + j0 ohm load. As I hope will be evident, it doesn't matter what's connected to the source end of the line or how long it is. We'll look at the voltage V and current I at a point within the cable, but very, very close to the load end. Conditions are steady state. When numerical V or I is required, assume it's RMS. I hope we can agree of the following. If not, it's a waste of time to read the rest of the analysis. 1. Vf / If = Z0, where Vf and If are the forward voltage and current respectively. 2. Vr / -Ir = Z0, where Vr and Ir are the reverse voltage and current respectively. The minus sign is due to my using the common definition of positive Ir being toward the load. (If this is too troublesome to anyone, let me know, and I'll rewrite the equations with positive Ir toward the source.) 3. Gv = Vr / Vf, where Gv is the voltage reflection coefficient. This is the usual definition. Likewise, 4. Gi = Ir / If 5. V = Vf + Vr 6. I = If + Ir Ok so far? I am looking at W.C. Johnson, pages 15 and 16, Equations 1.22 through 1.25, where he shows, very concisely and elegantly, that the reflection coefficient is zero only when the complex terminating impedance is identically equal to the the complex value of Z0. And not the complex conjugate of Z0. In other words (read carefully), the complex terminating impedance whose value is Z0 is equivalent to an *infinite extension* of the coax whose complex value is Z0. This is the "clincher". Observe also the the reflection coefficient for current is the negative of the reflection coefficient for voltage. Bill W0IYH |
Roy Lewallen wrote:
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. The "primary" waves are voltage and current, and the power value involved is "derived" from the primary waves. Pick a point on the line and calculate the instantaneous power at that point. Over time, that value of power "travels" along the line in concert with the voltage and current waves. There is nothing wrong with this. All of the hulabaloo over this subject isn't worth a hill of beans. Everyone understands what power "flow" and "power outage" mean. Bill W0IYH |
William E. Sabin wrote:
I am looking at W.C. Johnson, pages 15 and 16, Equations 1.22 through 1.25, where he shows, very concisely and elegantly, that the reflection coefficient is zero only when the complex terminating impedance is identically equal to the the complex value of Z0. And not the complex conjugate of Z0. In other words (read carefully), the complex terminating impedance whose value is Z0 is equivalent to an *infinite extension* of the coax whose complex value is Z0. This is the "clincher". 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. Observe also the the reflection coefficient for current is the negative of the reflection coefficient for voltage. Likewise. Bill W0IYH One other thing. If the reflection coefficient is zero, then all of the real power that is dumped into the coax is being delivered to the load, minus the losses in the coax. The system is as good as it can get. Bill W0IYH I agree with this only in the sense that the "system is as good as it can get" means only that the loss in the coax is minimized for a given delivered power. It doesn't guarantee that the maximum possible power will be delivered to the load. After some thought, I see I was in error in stating that terminating the line in its complex conjugate necessarily results in the maximum power transfer to the load. If the 50 - j0 had been a source impedance instead of a transmission line impedance, that would be true. However, what results in the maximum power from the source in a system like this is that the *source* be conjugately matched to the impedance seen at the input of the line. Although terminating the line in its characteristic impedance minimizes the line loss, it doesn't guarantee maximum load power. Of course, the conjugate matching theorem says that if the line is lossless, a load which is the conjugate of the impedance looking back toward the source from the load will result in a conjugate match at the source and everywhere else along the line, so that will effect maximum power transfer. But the impedance looking back toward the source from the load isn't by any means necessarily equal to the Z0 of the line, so the conjugate of Z0 isn't necessarily the optimum impedance for power transfer, as I erroneously stated. And I don't believe that the conjugate match theorem applies to a lossy line. I certainly don't want to start up an argument about this topic, though, and will simply state for certain that the maximum net power will be delivered to the line when the impedance seen looking into the line is equal to the complex conjugate of the source impedance. It should be easy to set up a couple of simple numerical examples to illustrate this. Unfortunately, I'm pressed for time at the moment and have to run. I apologize for the error regarding terminating impedance and maximum power transfer. Thanks for spurring me to re-think the conditions for maximum power transfer. I apologize for the error. Roy Lewallen, W7EL |
"William E. Sabin" sabinw@mwci-news wrote in message ...
.... In other words (read carefully), the complex terminating impedance whose value is Z0 is equivalent to an *infinite extension* of the coax whose complex value is Z0. This is the "clincher". Exactly, Bill! And with that, you don't need any equations. You only need to realize that if you cut a line which had no reflection on it, you can weld it back together without introducing reflections (assuming you do a perfect job restoring it to its original state). That should be obvious and easy to grasp even for a math-o-phobe. And thanks to Roy for taking the trouble to actually go through the algebra and post it. I honestly hoped that S. would do that, and still cling to the belief that there may have been a lurker or two who also did it. Cheers, Tom |
"William E. Sabin" sabinw@mwci-news wrote in message ...
.... The usage of complex conjugate Z0* becomes significant when calculating very large values of VSWR, according to some authors. But for these very large values of standing waves, the concept of VSWR is a useless numbers game anyway. For values of VSWR less that 10:1 the complex Z0 is plenty good enough for good quality coax. My working definition for SWR is (1+|rho|)/|(1-|rho|)|. (Note the overall absolute value in the denominator, so it never goes negative.) Rho, of course, is Vr/Vf = (Zload-Zo)/(Zload+Zo), no conjugates. In that way, when |rho|=1, that is, when |Vr|=|Vf|, SWR becomes infinite. When |rho|1, SWR comes back down, and corresponds _exactly_ to the Vmax/Vmin you would observe _IF_ you could propagate that Vr and Vf without loss to actually establish standing waves you could measure. I agree with Bill that all this is academic, but my working definition seems to me to be consistent with the _concept_ of SWR, and does not require me to be changing horses in mid-stream and remembering when to use a conjugate and when not to. I _NEVER_ use Zo* in finding rho. To me it's not so much a matter of nitpicking an area of no real practical importance as coming up with a firm definition I don't have to second-guess. Except on r.r.a.a., I don't seem to ever get into discussions about such things, but my definitions are easy to state up front so anyone I'm talking with can understand where I'm coming from on them. W.C. Johnson points out on page 150 that the concept: Pload = Pforward - Preflected is strictly correct only when Z0 is pure resistance. But the calculations of real power into the coax and real power into the load are valid and the difference between the two is the real power loss in the coax. For these calculations the complex value Z0 for moderately lossy coax is useful and adequate. The preoccupation with VSWR values is unfortunate and excruciatingly exact answers involve more nitpicking than is sensible. Agreed! And thanks for the reference re powers. Cheers, Tom |
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. For those antagonists who are not willing, or are unable, or are too frightened, or are just too plain lazy to work through the simple ARITHMETIC themselves, download in a few seconds program COAXPAIR from website below and run immediately. It is designed to do exactly the calculations everybody is at war about. (And many more). Use any ordinary coax with an overal length of 30 dB or more at 1000 Hz (to ensure a nice negative angle of Zo) and terminate the line with its own calculated input impedance = Zo = Ro+jXo. Then terminate the line with its conjugate. Observe what happens to the reflection coefficient, SWR and actual overall loss. Observe what happens as line length is reduced. Program COAXPAIR (with other transmission line progs) has been sitting there for 2 years or more. --- ======================= Regards from Reg, G4FGQ For Free Radio Design Software go to http://www.g4fgq.com ======================= |
"William E. Sabin" sabinw@mwci-news wrote in message ...
I am looking at W.C. Johnson, pages 15 and 16, Equations 1.22 through 1.25, where he shows, very concisely and elegantly, that the reflection coefficient is zero only when the complex terminating impedance is identically equal to the the complex value of Z0. 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 |
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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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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 |
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. |
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 |
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. . . . |
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. . . . |
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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