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#1




Derivation of Reflection Coefficient vs SWR
It is not too hard to use the concept of traveling waves and reflections
to derive the familiar reflection coefficient to SWR relationship. SWR is a measurable and useful relationship that most hams are familiar with. A clear path between SWR and traveling waves should make the concepts more understandable and believable. Power placed on a transmission line is placed over time. No matter how small the time span interval we might want to examine, the span will always be wide enough to include some quantity of power or energy. If we desire, we can eliminate the time consideration and just consider energy, but there is no need to do that. In this derivation, the distinction between power and energy will be ignored. We will assume that neither power nor energy can be stored at the discontinuity in amounts greater than the natural storage capacity of the lines. This assumption fixes the impedance of any waves to the impedance of the transmission lines. Begin the derivation by assuming that power is applied to a transmission line with impedance Zo. A traveling wave moves down the transmission line to a discontinuity which is composed of a second transmission line or resistor with impedance Zl. The junction between the two lines is like a window or thin plane, with Zo on one side and Zl on the other. Upon encountering the discontinuity, the lead edge of the wave (and all following energy levels) follow a "conservation of energy" rule that requires energy to be preserved at all times. In other words, the energy that has been conveyed to the junction by some interval of applied power is not lost to heat, radiation, or storage, but will leave the junction as fast as it arrives, and can be located, maintaining time shape. The following equation will be valid, Pf = Pl + Pr where Pf = power forward, Pl = power to load, and Pr = power reflected. Use the voltage equivalent, (Vf^2)/Zo = (Vl^2)/Zl + (Vr^2)/Zo where Vf = forward voltage, Vl = load voltage, and Vr = reflected voltage. The reflected wave will travel back down the main line with impedance Zo. Simplify the equation by rearranging and substitute SWR = Zl/Zo (Vf^2)/Zo  (Vr^2)/Zo = (Vl^2)/Zl SWR(Vf^2  Vr^2) = (Vl^2) Change the Vl into terms of Vf and Vr. Vl = Vf + Vr. We can do this because at a reflection, traveling waves double back over one another, adding voltage. Substitute Vl = Vf + Vr SWR(Vf^2  Vr^2) = (Vf + Vr)^2 Factor the polynomial on the left above SWR(Vf  Vr)(Vf + Vr) = (Vf + Vr)^2 Divide both sides by (Vf + Vr) SWR(Vf  Vr) = Vf + Vr Divide both sides by Vf SWR(1  Vr/Vf) = 1 + Vr/vf Vr/Vf = Reflection coefficient Ro, substitute SWR(1  Ro) = 1 + Ro Rearrange to put Ro on one side Ro + Ro*SWR = SWR  1 Factor out Ro and rearrange Ro = (SWR  1)/(SWR + 1) We have found the familiar relationship for the Reflection Coefficient (Ro) and SWR using traveling wave logic. Using identical logic but using current instead of voltage, the same relationship can be found from Zo*If^2 = Zl*Il^2 + Zr*Ir^2 By examining this derivation, the reader can see that power and energy is reflected when a wave encounters a discontinuity. The reader can also see that more power is present on the transmission line than is delivered to the load. Here is a link to additional information about transmission lines: http://www.astrosurf.com/luxorion/qs...sionline2.htm 73, Roger, W7WKB 
#2




Derivation of Reflection Coefficient vs SWR
Roger Sparks wrote in
: ... The reader can also see that more power is present on the transmission line than is delivered to the load. The notion that "power is present" is a different one. Owen 
#3




Derivation of Reflection Coefficient vs SWR
Owen Duffy wrote:
Roger Sparks wrote in : ... The reader can also see that more power is present on the transmission line than is delivered to the load. The notion that "power is present" is a different one. Owen It's reasonable, though. Looking at demo 4 with the TLVis1 program, you can see that there's power all along the line except at specific nodal points (where I or V is always zero), yet there's no power at all being delivered to the load. Roy Lewallen, W7EL 
#4




Derivation of Reflection Coefficient vs SWR
Roy Lewallen wrote in news:13pirk5h1cpt4f5
@corp.supernews.com: Owen Duffy wrote: Roger Sparks wrote in : ... The reader can also see that more power is present on the transmission line than is delivered to the load. The notion that "power is present" is a different one. Owen It's reasonable, though. Looking at demo 4 with the TLVis1 program, you can see that there's power all along the line except at specific nodal points (where I or V is always zero), yet there's no power at all being delivered to the load. Roy, my though was that on anything but a lossless line with VSWR=1, instantaneous power (being the rate of flow of energy) varies with time and location, so to make the statement that "power is present" and to quantitatively compare it with the power at a point (being the end of the line where the load is attached) seems to not be so reasonable. If the statement is about average power in both cases, then it is reasonable, obvious even, that power decreases with distance from the source. Perhaps "power is present" is an avoidance of the somewhat tautological form "power flows to the load". Owen 
#5




Derivation of Reflection Coefficient vs SWR
Owen Duffy wrote:
Perhaps "power is present" is an avoidance of the somewhat tautological form "power flows to the load". Want to muddy the waters even more? Ramo & Whinnery say: "Another very important case is that of a perfect conductor, which by definition must have a zero tangential component of electric field at its surface. Then ^P^ [Poynting vector] can have no component normal to the conductor and there can be no power flow through the perfect conductor."  73, Cecil http://www.w5dxp.com 
#6




Derivation of Reflection Coefficient vs SWR
On Fri, 25 Jan 2008 05:32:33 GMT
Owen Duffy wrote: Roy Lewallen wrote in news:13pirk5h1cpt4f5 @corp.supernews.com: Owen Duffy wrote: Roger Sparks wrote in : ... The reader can also see that more power is present on the transmission line than is delivered to the load. The notion that "power is present" is a different one. Owen It's reasonable, though. Looking at demo 4 with the TLVis1 program, you can see that there's power all along the line except at specific nodal points (where I or V is always zero), yet there's no power at all being delivered to the load. Roy, my though was that on anything but a lossless line with VSWR=1, instantaneous power (being the rate of flow of energy) varies with time and location, so to make the statement that "power is present" and to quantitatively compare it with the power at a point (being the end of the line where the load is attached) seems to not be so reasonable. If the statement is about average power in both cases, then it is reasonable, obvious even, that power decreases with distance from the source. Perhaps "power is present" is an avoidance of the somewhat tautological form "power flows to the load". Owen Nothing mysterious was hinted with the words "power is present". As I finished writing the post, I wanted to call attention to the assumption that the reflected power is true power and adds to the amount of energy "stored" on the transmission line. But "stored" is a word that implies static conditions, and static conditions are not found on a transmission line. So I substituted "present" for "stored. 73, Roger, W7WKB 
#7




Derivation of Reflection Coefficient vs SWR
Roger Sparks wrote:
As I finished writing the post, I wanted to call attention to the assumption that the reflected power is true power and adds to the amount of energy "stored" on the transmission line. But "stored" is a word that implies static conditions, and static conditions are not found on a transmission line. So I substituted "present" for "stored. The amount of energy existing in a transmission line is exactly the amount required to support the measured forward power and reflected power. If the steadystate forward power is 200 watts, the reflected power is 100 watts, and the lossless transmission line is one microsecond long, it contains 300 microjoules of energy. I don't think that is a sheer coincidence. :)  73, Cecil http://www.w5dxp.com 
#8




Derivation of Reflection Coefficient vs SWR
Roger Sparks wrote:
Nothing mysterious was hinted with the words "power is present". As I finished writing the post, I wanted to call attention to the assumption that the reflected power is true power and adds to the amount of energy "stored" on the transmission line. But "stored" is a word that implies static conditions, and static conditions are not found on a transmission line. So I substituted "present" for "stored. A better reason to avoid "stored" is that power isn't stored at all, anywhere. Anyone who believes so should be able to tell us how many watts of power are stored in a 50 Ah, 12 volt battery. Roy Lewallen, W7EL 
#9




Derivation of Reflection Coefficient vs SWR
On Fri, 25 Jan 2008 16:15:33 GMT
Cecil Moore wrote: Roger Sparks wrote: As I finished writing the post, I wanted to call attention to the assumption that the reflected power is true power and adds to the amount of energy "stored" on the transmission line. But "stored" is a word that implies static conditions, and static conditions are not found on a transmission line. So I substituted "present" for "stored. The amount of energy existing in a transmission line is exactly the amount required to support the measured forward power and reflected power. If the steadystate forward power is 200 watts, the reflected power is 100 watts, and the lossless transmission line is one microsecond long, it contains 300 microjoules of energy. I don't think that is a sheer coincidence. :)  73, Cecil http://www.w5dxp.com Yep, and if we quickly replaced the source with a termination having the impedance of the transmission line, 100 watts of power would continue to be delivered to the load for one microsecond, delivering 100 microjoules of energy. 100 watts of power would be delivered to the reflected wave termination for two microseconds, delivering 200 microjoules of energy. The transmission line was a dynamic power storage device for two microseconds after the power source was disconnected. 73, Roger, W7WKB 
#10




Derivation of Reflection Coefficient vs SWR
Roger Sparks wrote in
: On Fri, 25 Jan 2008 05:32:33 GMT Owen Duffy wrote: Roy Lewallen wrote in news:13pirk5h1cpt4f5 @corp.supernews.com: Owen Duffy wrote: Roger Sparks wrote in : ... The reader can also see that more power is present on the transmission line than is delivered to the load. The notion that "power is present" is a different one. Owen .... Nothing mysterious was hinted with the words "power is present". As I finished writing the post, I wanted to call attention to the assumption that the reflected power is true power and adds to the amount of energy "stored" on the transmission line. But "stored" is a word that implies static conditions, and static conditions are not found on a transmission line. So I substituted "present" for "stored. Roger, If you were wanting to mean "stored", perhaps it is energy that is stored (over a nonzero length of line) rather than power. In that sense, energy is "present" on the line, and the load may store energy (only if it has reactive elements, and irrespective of whether it looks resistive at its terminals). Owen 
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