There is a
frequency region between around 600 MHz and 1500 MHz where the selfresonant frequencies of
discrete components make filter design very difficult, and where the physical dimensions of
transmission lines and cavity filters are too large for practical implementation.
One of the most prominent ISM bands (900 MHz) falls squarely in the middle of the region.
Thanks to the wireless revolution, there are a plethora of SAW and dielectric filters available
for the 840 – 980 MHz band, but that is about it, and they are only rated for relatively low
powers (maybe 20 dBm). Helical filters fill that gap nicely, and are a combination of all
three formats.
Like cavity filters, the Q of the helical resonators is very high if constructed properly.
That is because at the frequencies of operation, the skin thickness is getting very small
and most of the current is flowing on the surfaces. Plating the cavity walls and helix with
a high conductivity material increases the Q even more that bare of tinned copper.
Zverev was the first to publicly publish the equations
for helical filter design (link to book on Amazon). While the book goes into more detail,
the basics of those equations are replicated here, and should serve as a good enough springboard
to get you going on a design. I actually worked with, but did not design, a multicavity helical
filter as part of a transmitter for a remote utility meter reading transponder back in the
early 1990s, produced by Itron. Choosing shield and helix sizes is largely an iterative task
once you know what the resonant frequency for each cavity will be.
Sorry, but I do not have equations for the tuning slug's diameter or how it affects the
resonant frequency as it is advanced into the helix. A wag would be that if the outside diameter
of the tuning slug is very close to the inside diameter of the helix, then the resonant frequency
is modified as if the total height of the helix and shield are shorter than when the tuning
slug is fully retracted.
The graphs illustrate how the equations are affected when the diameter of the helix, d,
is varied. Notice that the wire diameter, g, increases with frequency so that an adjustment
of the helix diameter, d, is needed to keep the wire size and space between the turns reasonable.

When the resonator is entirely filled with a dielectric, these parameters are substituted
as shown here: 

d = Mean helix diameter (cm)
C = Inside diameter of circular shield (cm)
S = Length of square shield side (cm)
H = Height of shield (cm)
f = Resonant frequency (MHz)
Q = Unloaded Q of an airfilled resonator
Q_{d} = Unloaded Q of a dielectricfilled resonator
N = Number of turns on helix
p = Helix pitch (turns/cm)
Z_{0} = Characteristic impedance of airfilled helical transmission line (Ω)
Z_{d} = Characteristic impedance of dielectricfilled helical transmission line
(Ω)
g = Wire diameter and also space between turns (cm)tan(δ) = Loss tangent of dielectric
material
Example of a commercial Helical Filter by Toko Type 5CHW, 450 MHz
Custom 7:1 Helical Filter Assembly "High Performance Helical Resonator Filters" by Ming
Yu and Van Dokas

