Among the many answers to this question, one available is, Ahaneku and Duru [DURU]
Following, with MATLAB, the part explaining how Z0(b/a) is optimised, that explains common values of coaxial Z0:
the alpha of the conductor, the attenuation caused by metal is
After plugging Cu conductivity
sigma_Cu=5.183e7 % S/m
the expression with the ratio to optimise is
Ed is |E| to be reached in order to disrupt dielectric.
For case of air
Ed_air=3e6 % V/m
here ignored but also depending upon air humidity and air pressure.
Note [DURU]'s b and a are ∅ while [POZAR] Microwave Engineering 4th edition chapter 3 coaxial b and a are radii.
text(z(nzmin),20+f2(nzmin),['optim b/a = exp(zmin) =' num2str(b_over_a_optim)])
mur=1;er=1 % no filling material
mu0=4*pi*10^-7; % H/m
e0=8.853*10^-12; % F/m
no fill-up or fill-up with materials that have er close to 1, that tend to be cheaper
than those with er>>1 when fillip up transmission lines
when coaxial fill-up is PTFE (Teflon)
Now let's have a look at real 75Ω coaxial specifications:
since these specs read 'foam PE' what is the relative permittivity of the filling material?
a=1.016;b=4.512; % both mm
matching the relative permittivity of Foam Polyethylene shown in table [PRAD]
Yet one has to be aware of the wide frequency span that supplier claims the coaxial is operative because the attenuation varies widely depending upon what frequencies used.
sigma_Cu=5.183e7 % S/m
f1=5e6;f2=1.8e9;f0=(f1+f2)/2; % chosen centre band in specs
a=1.016;b=4.512; % both mm, Commscope F677TSW EXPRESS
mur=1;er=1.2 % cheap coax
and directly from the previous manufacturer's datasheet:
Lcoax75=[1.9 5.25 6.4 6.46 9.35 9.84 10 10.82 11.64 12.63 13.61 14.43 15.29 16.08 16.73 18.54 20.01 21.49 23.66 24.71 25.71 26.68 27.63 28.54 29.39 29.44];
We can start feeling the need to test, if not all, at least all the critical components of any expensive system. Theoretical models? datasheets?
Test the one you buy.
when systems are really expensive one of the strategies is to send someone to factory to test what you buy right at the factory just before shipping, and then test again just on arrival at your premises before any further assembly, with agreed heavy penalties on contract.
MATLAB scripts cannot be uploaded as attachment to forum posts yet, so I am sending copy of the .m script to anyone asking by email, check my profile for contact details.