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Lots of questions, maybe a little bit of useful answers from me.
First, I doubt there is any BOG that will cover 8 octaves with a useful 
pattern.  Unobtanium, IMO. 
Second, your question did cause me to investigate improving the RDF of a 
240' BOG which is on my to try list.  So using EZNEC/Pro4 I found a BOG 
with 9x 20uH loads can improve the RDF to a little over 11db.  That is a 
nice improvement in the RDF value and it also lowered the peak gain to 
25 deg (from 40 deg) and has much lower Z gain.  I think worth trying. 
BOG at +2.4", two 10' orthogonal radials at each end buried 2.4", 
termination 1k ohm, two full cycles of current on the 16ga insulated 
wire. Gain -26dbi.  Having not traversed this modeling territory before, 
the termination value does cause concern and it is a sensitive value +/- 
10%. 
Third, playing with the 240' model for 500KHz, 100uH  9x does yield 
directivity and RDF around 8db with a 40 deg peak gain of -42dbi.  So 
there is no free lunch and it's possibly not a useful antenna at that 
gain level.  As I said at first, this antenna does not perform at higher 
frequencies, e.g. 160m. 
What real inductors will do re resistive losses is TBD.  I have to 
wonder why you chose the double E core with a gap geometry? 
Grant KZ1W
On 8/23/2019 11:40 AM, Mikek wrote:
 
 ??I want a BOG that will have a proper pattern over 500kHz to 4Mhz.
 ??As I understand it,
"loaded Beverages show increased directivity because velocity of 
propagation along the antenna is /*decreased*/." 
I assume that is for antennas that are shorter than what is required for 
that frequency. 
Info from here, https://www.w8ji.com/slinky_and_loaded_beverages.htm
 ???? Is there any reason I can't load a BOG to make it directive at 500kHz?
My BOG length limit is 250ft.
Assuming the answer is yes.
If I do add the proper inductances along the length of the antenna
 ??then at 4MHz the pattern suffers. (previously I used nine inductors)
The inductance must be reduced as frequency goes up.
To that end I'm searching for a specific ferrite core geometry, like this,
https://www.dropbox.com/s/2bwsxrc58g51yu8/E%20Core%20Ferrire%20DB.jpg?dl=0
??Note, I dropped this and chipped the important part in the center, but 
before that, it did what I wanted. 
 ??This material was not good at the frequencies I'm interested in.
So I need a material good to at least 4MHs and with an AsubL of 2000 to 
3000, or more, with a similar 
geometry. Or any geometry that necks down to a smaller cross section.
Where can I buy such a core?
 ?????????????????????????????? Thanks, Mikek
??PS, if anyone here wants to model a 250ft bogit over that frequency 
range, you would have my undying gratitude. 
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