This antenna was very quick (one evening!) designed, follows to the good article of G6LVB on his home page.
The difference is only in turns numbers - 21 in my case. All parameters of this antenna was calculated with a
helping of online W0FMS Helix Antenna Calculator.

The matching strip-line was designed regarding to G6LVB article.
No more adjustments. Main dimensions are:
Circumference:                  13.91 cm        
Diameter:                             4.43 cm
Spacing Between Turns:     3.16 cm 
Length of Each Turn:         14.26 cm 
Theoretical gain:                16.41 dbi

At the first evening new antenna was mounted inside the room
and I found AO-40 through the closed window. I was surprised to hear MB
with a 2 S-units over the noise level! 
Then I found several stations from
Europe, VK6XH and UN7PV. They all were loud enough!
Range to the satellite was 62 000 km and squint 12.
My station was in the other room, so I can not call to somebody, but I catch my own CW signal.
All was perfect!

On the next day I have mounted  this helix antenna together with a converter on the mast and was totally confused to hear only terrible
gnash, chirps, whistles, noise and so on.... What it was?  At the distance about 100 meters, on the roof of  the neighbors building  are
installed antennas for commercial radiolinks, for cellular telephones and something else. May be it was the source of troubles.
I can change antenna azimuth and reduce these troubles, but satellite was not in that direction!
Such a troubles I have never heard with a dish antenna. So I understood, this helix antenna has a very wide receiving range and I can
 use that in my living area only with a high quality  band pass filter, or only in a country side!

Any way, I take away antenna from the mast and then install outside of the window, so the building's wall was like a screen
for the troubles from the "enemy" antennas. Yes, it helped me so much, that I can work with a many stations at this day
and later.
This 21 turns helix antenna was build specially for portable and mobile operation via AO-40.
Antenna has only 66 cm in length and about 80 cm together with a converter.

This story will be continued after testing helix antenna from the car, or other portable situation.

   And, of course, when AO-40 will come back to the life again or we will give in a future some other satellites with S-band downlink.