On Sun, 20 Sep 2009 22:41:29 -0400, Walter Maxwell
wrote:

My post below is not exactly on target for the thread, but I believe
useful. It's Sec 11.3 from Chapter 11 of Reflections, the whole of
which is available on my web page at www. w2du.com.
The title of the Sec is "A Reader Self-test and Minimum-SWR
Resistance."

Sec 11.3 A Reader Self-Test and Minimum-SWR Resistance

" Everyone knows that when a 50-ohm transmission line is terminated
with a pure resistance of 50 ohms, the magnitude of the reflection
coefficient,, rho , is 0, and the SWR is 1:1. Right? Of course! With
that in mind, here is a little exercise to test your intuitive skill.
If we insert a reactance of 50 ohm in series with the 50-ohm
resistance, the load becomes Z = 50 + j50. The SWR will be 2.618:1.
Now for the question. With this 50-ohm reactance in the load, is the
SWR already at its minimum value with the 50-ohm resistance, or will
some other value of resistance in the load reduce the SWR below
2.618:1? You say the SWR is already the lowest with the 50-ohm
resistance, because, after all, the line impedance, ZC, is 50 ohms?
Sorry, wrong. With reactance in the load, the minimum SWR always
occurs when the resistance component of the load is greater than ZC.
In fact, the more the reactance, the higher the resistance required
for to obtain minimum SWR. For any specific value of reactance in the
load there is one specific value of resistance that produces the
lowest SWR. I call this resistance the "minimum-SWR resistance."
Finding the value of this resistance is easy. First you normalize the
reactance, X, by dividing it by the line impedance, ZC. The normalized
value of X is represented by the lower case x. Thus x = XC / ZC. Then
we solve for the normalized value of resistance r, from Eq 5-1, which
is repeated here.

r = sqrt (x^2 - 1) Eq 5-1

Let's try it on the example above. The normalized value of 50 ohms
of reactance X, is x = 1. Substituting in Eq 5-1, r = sqrt 2 = 1.414.
So the true value of the minimum-SWR resistance is 1.414 x 50 =
70.7ohms. While the 50-ohm resistance yields a 2.618:1 SWR, the
70.7-ohm resistance in series with the 50-ohm reactance yields an SWR
of 2.414:1. Not a great deal smaller, but still smaller than with the
50-ohm resistance.

So let's try a more dramatic example, this time with a 100-ohm
reactance, which has a normalized value x = 2.0. With a 50-ohm
resistance, the SWR is now 5.828:1. However, with the normalized
minimum-SWR resistance, r = sqrt 5 = 2.236. Multiplying by 50, we get
R = 111.8 ohms. With this larger resistance in series with the 100-ohm
reactance, the SWR is reduced from 5.828:1 to 4.236:1. The results of
this exercise didn't turn out quite the way you expected, did it?"

For further proof of this concept I suggest reviewing the remainder of
this Sec using the Smith Chart, available from my web page.

Walt, W2DU

Sorry, I goofed on Eq. 5-1. The corrected eq is r = sqrt (x^2 + 1).

Walt, W2DU