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 = (ZloadZo*)/(ZloadZo) 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 162, 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 BytheSea, FL. 

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 = (ZaZo*)/(Za+Zo) [Eq 6, page 247] 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 = (ZLZo)/(ZL+Zo) then rho = 0, since the numerator evaluates to 0+j0. However, if the formula is rho = (ZLZo*)/(ZL+Zo) then rho evaluates to something other than 0, since the numerator evaluates to 0j2Xo. To use an example with real numbers: RG174, 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=50j2.396. If ZL = Zo = 50j2.396, 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) / (1rho), 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 RG174 as stated above (Zo=50j2.396), 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 nonnegative 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 41.40j31.29. 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 310), 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 (ZLZo)/(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 RG174 from above (Zo=50j2.396), with a load of ZL=Zo*=50+j2.396. Then (ZLZo)/(ZL+Zo) = 0.0479 = small voltage traveling wave reflection coefficient while (ZLZo*)/(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' = (ZZo*)/(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 RG174 Zo from above, Zo=50j2.396. 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 247] back to the classic "nonZo*" form, since rho is used in this context as the voltage reflection coefficient. 2. Equation 11 on page 249 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 RG174, 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 2420 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 

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(1ja/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 
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