I agree that it's an obvious geometrical thing...and it could be that
I'm simply not understanding what you're trying to communicate.
But...do a simulation of the "dipole1.ez" that ships with EZNec, after
first changing the number of segments to 31 so it doesn't complain.
Find the frequency between 300MHz and 600MHz which maximized the
reactance. I believe you will find it at about 494MHz, and the
reactance is +j866.0 ohms. At that frequency, the resistive part is
1085 ohms. Put that on a Smith chart whose reference resistance is 50
ohms. Draw a constant-SWR circle (35.542; magnitude of rho = .9453)
through that point. Note that there are two constant-reactance arcs
tangent to that SWR circle, and note that they are NOT tangent at the
antenna's impedance point. In fact, it's reasonably easy to calculate
them as +/-j887.8. Now change the reference impedance to 2000 ohms,
and note that the constant SWR circle which passes through the antenna
max-reactance impedance point is nowhere near the point that SWR
circle is tangent to a reactance arc. The SWR for 1085+j866 referred
to 2000 ohms is 2.296, magnitude of rho = .3932. The constant SWR
circle in this case is tangent to reactance arcs of +/-j1860.28.

I certainly agree that the point where a reactance arc is tangent with
an SWR circle is the maximum reactance on that SWR circle, but that's
not necessarily (and generally is NOT) also a point on the curve
representing antenna feedpoint impedance versus frequency. -- Did I
miss something fundamental here? Are we not discussing how the
antenna feedpoint impedance changes with frequency, and specifically
the frequency between half and full wave resonances at which the
antenna feedpoint reactance is maximum?

And still, all this does NOTHING to tell us WHY the antenna's
reactance reaches maximum at that particular frequency. Again, maybe
I missed it, but I didn't see anything in your swr circles and
reactance arcs explanation that even mentioned frequency.



Example: Cecil Moore wrote in message ...
Tom Bruhns wrote:
I'm sorry, Cecil, but you lost me there. For any given SWR circle,
there are only two (complex conjugate) points at which reactance arcs
are tangent. Why would we think that the point of max reactance on
the antenna impedance curve will necessarily be at the point of
tangency?

Note: I am talking about the frequencies between the 1/2WL resonant
point and the one-wavelength (anti)resonant point for a fixed dipole.
There will exist a maximum reactance point between those two
frequencies. By definition of the bi-linear transformation rules
involving the Smith Chart, the maximum reactance point will be
located at the point where the SWR circle is tangent to the
reactance arc. It simply cannot be located anywhere else.

It's an obvious geometrical thing, Tom. The SWR circle is centered at
the center of the Smith Chart. The reactance arc (circle) is centered
somewhere else outside of the Smith Chart. Where these two circles are
tangent, the reactance is at a maximum, by definition. If the two circles
are not tangent and not touching, then that cannot possibly be the maximum
reactance point. If the two circles are not tangent and intersect at two
points, then that cannot possibly be the maximum reactance point. In the
latter case, the maximum reactance point lies between those two intersection
points.

The antenna impedance arc of the simple dipole I modelled
indeed does not lie tangent to the max reactance arc at the same point
as the SWR circle that's tangent that reactance arc.

Sorry, you did something wrong or don't understand what I am
saying. It is impossible for the maximum reactance point not to be
tangent to the reactance arc at the maximum reactance point. On the
inductive top part of the Smith Chart, between 1/2WL and 1WL, if the
circles intersect at more than one point, you are not at the maximum
reactance point. If the circles intersect at one and only one point,
they are tangent, by definition, and you are at the maximum reactance
point. If they don't intersect at all, you are not at the maximum
reactance point.

In any event, I don't see that this tells us anything about _why_ the
dipole shows max reactance at that particular frequency.

Because it's an obvious geometrical thing, Tom. It simply cannot be any
other frequency and can be proved with relatively simple geometry.

EXAMPLE: 1/2WL resonant feedpoint impedance is 50+j0 ohms.
One-wavelength (anti)resonant feedpoint impedance is 5000 ohms.
Maximum reactance point has a feedpoint impedance of 2500+j2500 ohms.

The SWR circle (at the frequency of maximum reactance) will pass through
the 2500+j2500 ohm point. Do you disagree?

The reactance arc (at the frequency of maximum reactance) will pass
through the 2500+j2500 ohm point. Do you disagree?

Certainly the +j2500 ohm reactance arc will pass through 2500+j2500.
But that reactance arc is not tangent to the SWR circle that passes
through that point, necessarily. Draw the chart with Zref set to
10,000 ohms, and you'll instantly see this. Draw the chart with
Zref=50 ohms, and it will be difficult to see, but in fact if you go
through the calcs, I believe you'll find a slight discrepancy even
there! The point of tangency is not exactly where R=X, though it's
close for high SWR. But try it for a low SWR circle, and it will be
visually obvious on the Smith chart. For example, on a 50 ohm chart,
SWR=1.63 will be tangent to +/-j25 ohm reactance arcs, but at roughly
55 ohms resistive, no where near 25 ohms resistive.


ERGO: The SWR circle will be tangent to the reactance arc at the
2500+j2500 ohm point no matter what Z0 is being used. Do you
disagree?

Sure do. See numerical examples above.