There's good reason for confusion.

Resonance is defined as the frequency at which the reactance is zero.

Antenna resonance is the frequency at which the feedpoint reactance is zero.

At the input end of a feedline, resonance will occur at the same
frequency as antenna resonance only if the antenna resistance is the
same as the feedline characteristic impedance, or if the feedline is a
multiple of a quarter wavelength (yes, quarter) long. So resonance at
the input end of a feedline is often at a different frequency than
antenna resonance. (You could call that "antenna system resonance".)

The reading on an SWR meter is a function of the both the resistance and
reactance of the load presented to the meter.

As the frequency changes, both the feedpoint resistance and reactance
change. So if the resistance is closer to the SWR meter's Z0 when
there's a slight reactance, the SWR meter reading at the feedpoint can
be better than at resonance, when the reactance is exactly zero. When
measured at the feedline input, there's the additional factor of
impedance transformation that can modify the reactance and therefore
resonance.

In practice, though, the point of minimum SWR is nearly always very
close to resonance, at the antenna feedpoint. This is because for most
antennas, the reactance changes much faster with frequency than the
resistance does. And, when the feedpoint resistance is roughly equal to
the feedline Z0, the transformation by a mismatched feedline isn't
extreme, so again the point of minimum SWR is usually close to
resonance. Where you're likely to see a noticeable difference between
resonance and lowest SWR is with antennas with several coupled elements
(where feedpoint R can change more rapidly with frequency), or severely
mismatched feedlines, like when using open wire line to feed a dipole on
several bands.

In the end, it really doesn't matter. There's nothing at all magical
about resonance, so there's no need to try and achieve it. What you're
usually interested in doing is matching the rig to the input of the
feedline, and the common measure of the quality of that match is the
reading on an SWR meter. So the common, and valid, practice is to prune
the antenna for lowest SWR meter reading. If you do that, there's no
need to worry about where resonance might be.

Roy Lewallen, W7EL

chuck wrote:

Ken, I guess I'm still confused.

As I understand it, one cannot reliably determine the exact resonance of
a dipole by finding the point of minimum SWR. Until this measurement
issue is resolved, there would seem to be little benefit to seeking an
explanation of why the formula appeared not to work.

It is probably too late now, but if you had used an impedance bridge
(MFJ or Autek, for example) you could have found resonance at the point
of zero reactance. All within the limits of the instruments, of course.

Some of the posts suggest other reasons why the formula might not work,
but it is not yet evident to me that it didn't work.

Sorry my earlier post was not more clear. (I'm even sorrier for this
post if you actually used an impedance bridge! Hi.)

73,

Chuck
NT3G