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Derivation of Reflection Coefficient vs SWR
On Feb 5, 5:59*am, Cecil Moore wrote:
Keith Dysart wrote: This simple example has a voltage source in series with a 50 ohm resistor, driving 45 degrees of 50 ohm line connected to a 150 ohm load. * * * * * *Rs * * * Vg * * * * * * * * * * Vl * * *+----/\/\/-----+----------------------+ * * *| * *50 ohm * * * * * * * * * * * * * | * * *| * * * * * * * * * * * * * * * * * * / * * Vs * * * * * * * * 45 degrees * * * * *\ * Rl *100 cos(wt) * * * * * 50 ohm line * * * * / 150 ohm * * *| * * * * * * * * * * * * * * * * * * \ *load * * *| * * * * * * * * * * * * * * * * * * | * * *+--------------+----------------------+ * * gnd The voltage source in series with the 50 ohm resistor forms a generator. Vs(t) = 100 cos(wt) So, contrary to Cecil's assertion, an analysis based on 'no reflections at the source', has not resulted in any violation of conservation of energy. This is good. Keith, this special case is covered in my Worldradio energy analysis article at: http://www.w5dxp.com/energy.htm I agree with your special case analysis. The reason that it is a special case is that at the generator terminals: (Vf)^2 + (Vr)^2 = (Vf+Vr)^2 i.e. you have chosen the one special case where the superposition of powers yields a valid result. Quoting my article with some special characters replaced by ASCII characters: "If A = 90 deg, then cos(A) = 0, and there is no destructive/constructive interference between V1 and V2. There is also no destructive/constructive interference between V3 and V4. Any potential destructive/constructive interference between any two voltages is eliminated because A = 90 deg, i.e. the voltages are superposed orthogonal to each other (almost as if they were not coherent)." At the generator terminals, you have chosen the one phase angle between Vf and Vr that will result in zero interference. Whether by accident or on purpose is unclear. The phase angle between Vf and Vr at the generator terminals is 90 degrees resulting in zero interference. Your analysis is completely correct for this special case. Now try it for any length of feedline between 0 deg and 180 deg besides 45 deg and 135 deg. 90 degrees would be very easy to calculate. I choose 45 degrees because it was not 90, the standard value used which leaves all sorts of misleading artifacts when doing analysis. I noticed that a few artifacts also exist at 45 but I had hoped they would not mislead. I see I was wrong. Below, I have redone the analysis with a 35 degree line. The analysis is based on no reflection occurring at the generator, and the energy flows still balance as expected. The analysis for a 35 degree line follow. -------- This simple example has a voltage source in series with a 50 ohm resistor, driving 35 degrees of 50 ohm line connected to a 150 ohm load. Rs Vg Vl +----/\/\/-----+----------------------+ | 50 ohm | | / Vs 35 degrees \ Rl 100 cos(wt) 50 ohm line / 150 ohm | \ load | | +--------------+----------------------+ gnd The voltage source in series with the 50 ohm resistor forms a generator. Vs(t) = 100 cos(wt) or, in phasor notation Vs = 100 /_ 0d where d is degrees or 2*pi/360 when converting to radians is appropriate. When the source is turned on, a wave travels down the line. At the generator terminal (Vg), the forward voltage is: Vf.g(t) = 50 cos(wt) Vf.g = 50 /_ 0d since the voltage divided evenly between the source 50 ohm resistor and 50 ohm impedance of the line. At the load, the voltage reflection coefficient is RCvl = (150-50)/(150+50) = 0.5 The forward voltage at the load is Vf.l(t) = 50 cos(wt - 35d) Vf.l = 50 /_ 35d The reflected voltage at the load is Vr.l(t) = 0.5 * 50 cos(wt - 35d) = 25 cos(wt - 35d) Vr.l = 25 /_ -35d The voltage at the load is Vl = Vf.l + Vr.l = 50 /_ -35d + 25 /_ -35d = 75 /_ -35d Vl(t) = 75 cos(wt - 35d) At the generator, the voltage reflection coefficient is RCvg = (50-50)/(50+50) = 0 so there is no reflection and the system has settled after one round trip. The voltage at the generator is, therefore, Vg = Vf.g + Vr.g = 50 /_ 0d + 25 /_ (-35d -35d) = 50 /_ 0d + 25 /_ -70d = 63.087640 /_ -21.862219d Vg(t) = 63.087640 cos(wt - 21.86221d) Currents can be similar computed... If.g(t) = 1 cos(wt) If.g = 1 /_ 0d RCil = -(150-50)/(150+50) = -0.5 If.l(t) = 1 cos(wt - 35d) If.l = 1 /_ -35d Ir.l(t) = -0.5 * 1 cos(wt - 35d) = -0.5 cos(wt - 35d) Ir.l = -0.5 /_ -35d Il = If.l + Ir.l = 1 /_ -35d - 0.5 /_ -35d = 0.5 /_ -35d Il(t) = 0.5 cos(wt - 35d) And just to check, from V = I*R R = V / I = Vl(t) / Il(t) = (75 cos(wt-35d)) / (0.5 cos(wt-35d)) = 150 as expected. RCig = -(50-50)/(50+50) = 0 Ig = If.g + Ir.g = 1 /_ 0d - 0.5 /_ (-35d -35d) = 1 /_ 0d - 0.5 /_ -70d = 0.952880 /_ 29.543247d Ig(t) = 0.952880 cos(wt + 29.54324d) Now let us see if all the energy flows balance properly. The power applied to the load resistor is Pl(t) = Vl(t) * Il(t) = 75 cos(wt - 35d) * 0.5 cos(wt - 35d) Recalling the relation cos(a) * cos(b) = 0.5(cos(a+b)+cos(a-b)) Pl(t) = 75 * 0.5 * 0.5 (cos(2wt-70d)+cos(0)) = 18.75 cos(2wt-70d) + 18.75 cos(0) = 18.75 cos(2wt-70d) + 18.75 Since the average of cos is 0, the average power is Pl.avg = 18.75 To confirm, let us compute using Pavg = Vrms * Irms * cos(theta) Pl.avg = (75 * 0.707) * (0.5 * 0.707) * cos(-35d-(-35d)) = 18.75 * 1 = 18.75 Agreement. Now at the generator end Pg(t) = Vg(t) * Ig(t) = 63.087640 cos(wt - 21.862212d) * 0.952880 cos(wt + 29.543247d) = 63.087640 * 0.952880 * 0.5 (cos(2wt+7.681035)+cos(-51.405466)) = 30.057475 cos(2wt) + 30.057475 * 0.623805 = 30.057475 cos(2wt) + 18.750004 Pg.avg = 18.75 Good news, the average power into the line is equal to the average power delivered to the load, as required to achieve conservation of energy. Now let us check Pf and Pr Pf.avg = Vfrms^2 / Zo = (50*.707)^2 / 50 = 25 Pr.avg = Vrrms^2 / Zo = (25*.707)^2 / 50 = 6.25 Pnet.avg = Pf.avg - Pr.avg = 25 - 6.25 = 18.75 Again, as expected, the average energy flow in the line is equal to the average power into the line and the average power out of the line. So, contrary to Cecil's assertion, an analysis based on 'no reflections at the source', has not resulted in any violation of conservation of energy. This is good. It is instructive to continue the cross-checking. Let us validate the input impedance of the line. Using a Smith chart (http://education.tm.agilent.com/index.cgi?CONTENT_ID=5), and going towards the source from a 150 ohm load on a 50 ohm line yields Zin.smith = 41.28 - j51.73 = 66.18 /_ -51.41 Dividing the voltage by the current at the input to the line... Zin = Vg / Ig = 63.087640 /_ -21.862219d / 0.952880 /_ 29.543247d = 66.207329 /_ -51.405466 = 41.300466 -j51.746324 yields the same result as the Smith chart. Lastly let's compute the power provided by the source... Ps(t) = Vs(t) * Is(t) = 100 cos(wt) * 0.952880 cos(wt+29.543247d) = 95.288000 * 0.5 (cos(2wt+29.543247d) + cos(-29.543247d)) = 47.644000 cos(2wt + 29.543247) + 47.644000 * 0.869984 = 47.644000 cos(2wt + 29.543247) + 41.449507 Ps.avg = 41.449507 and that dissipated in the source resistor Prs(t) = Vrs(t) * Irs(t) = (Vs(t)-Vg(t)) * Ig(t) = (100 cos(wt) - 63.087640 cos(wt - 21.862219d)) * 0.95288 cos(wt + 29.543247d) = 47.643989 cos(wt + 29.543247d) * 0.952880 cos(wt + 29.543247d) = 47.643989 * 0.952880 * 0.5 (cos(2wt + 59.086480d) + cos(0)) = 22.699502 cos(2wt + 59.086480d) + 22.699502 Prs.avg = 22.699502 The power provided by the source is equal to the power dissipated in the source resistor and the power dissipated in the load resistor. 41.44950 = 22.699502 + 18.75 So all the energy is accounted for, as expected. When the source impedance is the same as the line impedance, there are no reflections at the source and all the energy is properly accounted. There is no violation of the conservation of energy principle. ...Keith PS. It is worthwhile to note that there are a large number (infinite) of source voltage and source resistor combinations that will produce exactly the same final conditions on the line. But if the source resistor is not 50 ohms, then there will be reflections at the source and it will take infinite time for the line to settle to its final state. |
#2
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Derivation of Reflection Coefficient vs SWR
Keith Dysart wrote:
So, contrary to Cecil's assertion, an analysis based on 'no reflections at the source', has not resulted in any violation of conservation of energy. This is good. Again your analysis violates the conservation of energy principle. The only way to balance your energy equation is for your source to supply 25 joules/sec. IF THE REFLECTED WAVE IS NOT REFLECTED FROM THE SOURCE, IT FLOWS THROUGH THE SOURCE RESISTOR AND IS DISSIPATED IN THE SOURCE RESISTOR. THAT REFLECTED WAVE ENERGY IS THEREFORE NOT AVAILABLE TO THE FORWARD WAVE. You cannot eat your reflected wave and have it too. The standard equation is: Psource = Pfor - Pref = Pload But you have taken away the reflected energy and dissipated it in the source resistor. So your new equation becomes: Psource = Pfor + ???????? 18.75w = 25w + ________ You say the source power is 18.75 joules/sec and the forward power is 25 joules/sec. If none of the reflected energy is available to the forward wave, where did the extra 6.25 joules/sec come from? Is it sheer coincidence that the reflected wave is associated with 6.25 joules/sec that are now missing from the above equation? Here is an example of the reflected wave flowing through a circulator resistor and being dissipated. Source---1---2----45 deg 50 ohm feedline---150 ohm load 25w \ / 18.75w | 50 ohms 6.25w But in this example Psource = Pfor in order to satisfy the conservation of energy principle which your example does not. -- 73, Cecil http://www.w5dxp.com |
#3
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Derivation of Reflection Coefficient vs SWR
On Feb 5, 10:44*am, Cecil Moore wrote:
Keith Dysart wrote: So, contrary to Cecil's assertion, an analysis based on 'no reflections at the source', has not resulted in any violation of conservation of energy. This is good. Again your analysis violates the conservation of energy principle. The only way to balance your energy equation is for your source to supply 25 joules/sec. IF THE REFLECTED WAVE IS NOT REFLECTED FROM THE SOURCE, IT FLOWS THROUGH THE SOURCE RESISTOR AND IS DISSIPATED IN THE SOURCE RESISTOR. THAT REFLECTED WAVE ENERGY IS THEREFORE NOT AVAILABLE TO THE FORWARD WAVE. Not sure that shouting helps. I do not find any unbalances with the energy. See below. You cannot eat your reflected wave and have it too. The standard equation is: Psource = Pfor - Pref = Pload Yes. Indeed. And just changing to use my terminology... Pgenerator = Pfor - Pref = Pload For the example at hand... Pgenerator = 18.75 Pfor - Pref = 18.75 Pload = 18.75 And the source provides 50 W of which 31.25 is dissipated in the source resistor and 18.75 is delivered to the line (Pgenerator, above). All seems well with world and no energy is left unaccounted. But you have taken away the reflected energy and dissipated it in the source resistor. So your new equation becomes: Psource = Pfor + ???????? 18.75w *= 25w *+ ________ As your original equation above Psource = Pfor - Pref 18.75 = 25 - 6.25 I do not see any issues. When the source was first turned on, 25 J/s flowed from the generator into the line. Pfor = 25 W After the reflected wave makes it back to the generator, 25 - 6.25 - 18.75 J/s are flowing in the line. Pfor - Pref = 18.75 W And the generator output has also reduced from 25 W to 18.75 W, so all is still in balance. I am having difficulty determining where you think there is a violation of the conservation of energy principle. ...Keith PS Do not be fooled by the numerology where 25 + 6.25 gives 31.25. For the 35 degree line, the corresponding numbers are Ps = 41.449507 Prs = 22.699502 Pg = 18.75 Pf = 25 Pr = 6.25 Pl = 18.75 Pg = Pf - Pr = Pl Ps = Prs + Pg All as expected. No missing energy. |
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