75-ohm Coaxial Cable

A few comments.

Just to eliminate confusion, the OP asked about 75 Ohm cable, and the Cat answered about 50 Ohm cable. The Cat should have said 75 Ohms, to eliminate confusion. 75 Ohm is standard in the cable TV industry, whereas 50 Ohms is standard in instrumentation and test equipment.

On to the background of what this is all about.

The cable impedance must match the source and load impedances in order to get power to flow with maximum efficiency from source to load, when the cable electrical length exceeds about a tenth wavelength, or thereabouts. If there is mismatch anywhere, some power is reflected from the mismatch, which reduces the power delivered to the load, and (if we are looking at high power transmission), can actually damage the source.

Electrical length means the ratio of the cable length to a wavelength at the frequency of interest. The wavelength, in air, in meters, is given by:

λ = 300/f_MHz,

where f_MHz is the frequency in MHz.

However, most coax has a dielectric material between the center conductor and shield that is different than air, and has a relative dielectric constant that is different than unity. The wavelength of the signal traveling down the coax is reduced by the square root of the relative dielectric constant ε_r:

λ_m = λ_air / √ε_r,

where λ_m is the wavelength in the medium.

To give you a feel for why you use coax for cable TV, consider the following. Channel 2, the lowest channel distributed by the cable industry, covers 54-60 MHz. That translates into a wavelength in air between 5.55 and 5 meters. And that's in air. A wavelength in the coax will be shorter by the square root of the relative dielectric constant. And you are looking at a tenth of that for your decision of when to use coax, so that if you need to efficiently transit energy on a wire at channel 2, if the cable length needs to be greater than about one-half meter, you need to use coax.

The characteristic impedance of the cable is equal to the square root of the per unit length cable inductance L in units of Henries per meter divided by the per unit length capacitance in units of Farads per meter.

Z_{characteristic} = √(L/C)

That is why the characteristic impedance is independent of cable length, as the Cat pointed out - the per unit length dependence divides out.

To give you a feel for the numbers, per unit length inductance for coax is in the tens of nanohenries per meter, while per unit length capacitance is on the order of tens of picofarads per meter.

If your inductance per meter happens to be 50 nH, and the capacitance per meter happens to be 50 pF, then the characteristic impedance is

Z_C = √(50e-9 / 50e-12) = 31.62 Ω.

Which is pretty close to 50 Ohms, so you can see the actual values of L & C are in that ballpark.

Finally, the equation for the characteristic impedance of coax in terms of the coax physical parameters is:

Z_C = (138 / √ε_r) log₁₀ (D/d),

where D is the diameter of the coax shield, and d is the diameter of the coax center conductor. That equation is just a result of dividing the expressions of per unit length inductance and capacitance, and taking the square root.

To round out this discussion, there are also equations for how much power is reflected if you have a mismatch, based on the mismatched impedances. And coax cable has losses which are separate from the whole matching issue. The losses are frequency dependent (they always increase exponentially with increasing frequency) and are mainly based on losses in the dielectric medium. Hence at microwave frequencies (well above the spectrum of cable TV signals), you will see the use of very low loss dielectrics, at very high cost compared to the RG-59 you get at Radio Shack.

And if you go high enough, you have to use waveguide instead of coax in order to get efficient power transmission of microwave signals. Waveguide is very expensive. That is why you will see a satellite TV receiver system using waveguide right out of the dish, but then an immediate down-conversion to cable TV frequencies before transmission from the dish to your house.