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Generally, as long as the OD of the center conductor, the ID of the
shield, the concentricity between the center conductor and the shield
and the dielectric constant of the insulation between the center
conductor and the shield don't change, the impedance of the cable will
remain constant and will suffer only a frequency dependent attenuation
which increases as length and frequency increase.
I was referring to the twisted pair's characteristic impedance. However,
since you described a coax, your comment that the characteristic impedance
is constant is not true, unless you apply 'engineering' tolerances to
it...The characteristic impedance of a coax changes with frequency.
For a 'word argument': Characteristic impedance is approximately the
square root of the ratio of inductance per length over capacitance per
length. At low frequencies, the whole center conductor's cross section is
involved in carrying current and has a value of inductance per length. At
high frequencies, skin effect rears its ugly head and you see the
increased loss that you mentioned, but also the carriers are traveling
along the outside of the center conductor, which changes the center
conductor's inductance per length, thus the characteristic impedance of
the coax changes with frequency.