are you confusing what effect the radials have on transmissionline mode impedance in a sleeve monopole and what lengths we end up with?,
from my first post in this thread,
"sleeve to monopole spacing effects impedance as does element diameter ratio, the closer spacing of the sleeve in the sigma design lowers antenna impedance"
freecells post
"conversely, as the radial elements are raised the input impedance is lowered.
as the radials are raised towards the radiating element something else interesting occurs. the
radiating element becomes less sensitive to the influence of surrounding objects and terrain
in the near field, allowing the antenna to be mounted in locations and at heights above ground
that would be extremely deleterious not only to the feedpoint match but also to the radiated
pattern emanating from the radiating element.
as the radials are swept upwards towards the radiating element capacitance between the radials
and the radiating element increases, effectively raising the resonant frequency of the system
or causing the electrical length of both the radials and the radiating element to be shortened"
further up this page
."secondly as the radials are swept closer to the central monopole impedance drops and resonant frequency of both radial sleeve and monopole goes up,
conventional maths for electrical length go out of the window as you sweep the radials up towards the central monopole,
for instance a gamma fed monopole with 90 degree radials that is electrically 3/4wave will be less than 3/4wave when you sweep the radials up"
from the arrl
If the sleeve elements were brought closer to
the central monopole such that the ratio of the spacing to element diameter was less than 10:1, then the characteristic impedance of the 3-wire transmission
line would drop to less than 250 W. At 28
MHz, ZA remains essentially unchanged, while ZT
begins to edge closer to 52 W as the spacing is reduced.
At some particular spacing the characteristic impedance, as determined by the D/d ratio, is just right to transform the end impedance to exactly 52 W at some frequency. Also, as the spacing is decreased, the frequency where the impedance is purely resistive gradually increases.
again i am not saying this IS how its working , its all i can find that seems even close at the moment, im sure theres more to it,
if you read my first post i used the open sleeve antenna as an example that just because there are currents flowing in opposite directions in the lower 1/4wave it does not mean that radiation from the radial sleeve cannot combine constructively with radiation from the upper 1/2wave to increase gain.
commercial 3/4wave sleeve antennas i have seen have the same claimed gain as the vector with radiation angles @ above or below the horizon, gain is higher and radiation angle lower than a 5/8wave groundplane, the tradeoff seems to be a small reduction in vertical beamwidth.
