Reg Edwards wrote:

For those of you who may be unfamiliar with the mathematics of
transmission lines, download program TWOLINES and enter the following
practical values -

Line-1 represents the loading coil.
Line-2 represents the whip to be pruned for 1/4-wave resonance.

Enter Zo1 = 6000 ohms, a practical value.
Enter electrical length L1 = 0.15 wavelengths.
Enter attenuation = 0.1 dB for wire loss in coil.

Ignoring losses and starting at 0-j0, plotting this value
on a Smith Chart normalized to 6000 ohms, the normalized
value is -j1.38

-j1.38 * 6000 = -j8280 ohms

Enter Zo2 = 400 ohms, a practical value.
Enter electrical length L2 = 0.0077 wavelengths.
Enter attenuation = 0.001 dB

Starting at 0-j*infinity and plotting backwards, this value
on a Smith Chart normalized to 400 ohms is -j20.7

-j20.7 * 400 = -j8280 ohms

But L1 + L2 = 0.1577 wavelengths = only 57 degrees.

The abrupt jump from -j1.38 to -j20.7 provides a 66 degree
phase shift in the angle of the reflection coefficient as
read off the Smith Chart.

The missing 90 - 57 = 33 degrees are just not needed for resonance.
So why is everybody making such a fuss about it?

The missing 33 degrees are there, provided by the phase shift
in the reflection coefficient. The angle of the reflection
coefficient in the stinger at the impedance discontinuity is
~6 degrees. The angle of the reflection coefficient in the
coil at the impedance discontinuity is ~72 degrees.

72 - 6 = a 66 degree jump in the angle of the reflection
coefficient.

That equates to the missing 33 degrees of antenna. The wave
that is only 3 degrees back from the tip reflection point
is suddenly only 54 degrees from the current maximum point.

It appears to me that the interference of the forward and
reflected waves at the impedance discontinuity causes an
actual 33 degree phase shift. How to analyze what happens
at such an impedance discontinuity is covered in my energy
analysis article at:

http://www.qsl.net/w5dxp/energy.htm

I will try to do that analysis sometime today.
--
73, Cecil http://www.qsl.net/w5dxp