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William E. Sabin wrote:
Dan wrote: Now that the various typo mistakes have been corrected, and putting aside for the moment the name calling and ad hominem arguments, could it be that _both_ sides in this discussion are correct? Camp 'A' says that the reflection coefficient is computed the classical way, without using Zo conjugate, and offers various mathematical proofs and discussions of infinitely long lines. Camp 'B' says the reflection coefficient is computed with Zo* (Zo conjugate) in the numerator, and offers explanations dealing with the conservation of energy and maximum transfer of power. Both sides may be correct since they are talking about _two different_ meanings for the term "reflection coefficient." One has to do with voltage (or current) traveling waves and the other has to do with power. If a lossy line is terminated with the complex Z0, there is no reflection from the load, but the maximum possible power is not delivered. If the *load* is made equal to the complex conjugate of Z0 the maximum *forward power* is delivered but there is a reflected power (VSWR is not 1:1). It is difficult to say that the maximum *power* is delivered without knowing the generator impedance, since it is involved in any so-called "conjugate match". For a lossy line, the idea of conjugate match is, at best, very approximate anyway. And generator impedance is a mystery in most, but not all, transmitter PA situations. One possible exception: a large amount of negative feedback helps to determine, to some extent, output impedance, for a signal with time-varying amplitude (e.g. SSB). It seems to me to be clear that the use of Z0* in the reflection coefficient equation has not been corroborated (see Roy's post), but the use of ZL=ZO* has been. The two ideas are not equivalent. After looking at some examples, using the exact complex hyperbolic equations with Mathcad, it is obvious that a line must be very lossy to make a significant difference whether ZL or ZL* Correction: Z0 or Z0* is used to terminate the line. Still, it is important to understand the basic principles involved, so this exercise is not foolishness at all. A word about "credentials". We all respect established and competent authors. But I have noticed on several occasions that blind faith has some exceptions. As an experienced author, I am personally familiar with this problem. G. Gonzalez (highly respected) "Microwave Transistor Amplifiers" second edition, has a good discussion of Power Waves, based on Kurokawa (I also have his article). There are no transmission lines, and the term ZS* (ZS=generator impedance) is used. In particular, a power wave reflection coefficient is defined: Gp = (ZL-ZS*)/(ZL+ZS) which looks quite familiar, with ZS replacing Z0. Also, a voltage reflection coefficient: Gv = [Zs/Zs*] x [(ZL-ZS*)/(ZL+ZS)] and a current reflection coefficient Gv = Gp. Correction: Gi = Gp The author also defines two-port scattering parameters in terms of power waves, in which ZS* and ZL* appear. For the purposes of the present topic, involving transmission lines, it seems best to stay away from power waves, without a lot more studying. Bill W0IYH |