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?

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?
--
73, Cecil http://www.qsl.net/w5dxp



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