How does coaxial cable maintain a constant impedance over a range of frequencies

[sonnytiger]

Is it a function of the inductance and capacitance? Ie the capacitive impedance increases but inductive impedance decreases resulting in an overall impedance of 50 or 75 or whatever ohms?

[WarSim]

Is it a function of the inductance and capacitance? Ie the capacitive impedance increases but inductive impedance decreases resulting in an overall impedance of 50 or 75 or whatever ohms?

The short answer it doesn't.  The cable is designed to express a relatively constant AC impedance but it varies with frequency.  Depending on the quality ergo cost of the cable will determine how close to the rated impedance it will be over it's rated frequency range.  The losses due to changes in impedance are expressed as insertion losses.  Not the same as cable losses due to cable length. 

[c4757p]

Even in an ideal cable, keep in mind that the characteristic impedance is not the same as the impedance. Even an ideal 50 ohm cable will not measure 50 ohms if you stick it on an ohmmeter.

There is a decent explanation here.

In short, keep in mind that a transmission line has a propagation delay. If you short out one end of the line (or load it "properly") and apply voltage to it, once you wait for the current to flow, after that delay, the current will be determined by the DC resistance of the cable.

But imagine a transmission line infinitely long. If you apply voltage to one end, current will definitely begin to flow. Even though you can't see the other end, there's still capacitance in the cable local to you, and charge will flow into that, slowly beginning to propagate down the line. This impedance seen by a voltage step, before the changes propagate down to the other end and back, is the characteristic impedance. It is determined by the series resistance, leakage conductance, series inductance and parallel capacitance, or in the case of a lossless line, just the series inductance and parallel capacitance. (If you have a good enough meter, you can even measure them: )

Loading the cable at the other end causes the current flowing to continue flowing at the same rate even when it reaches the end, thereby removing the difference between step response (high frequency) and steady-state behavior (low frequency).

Here is a plot showing the impedance looking into an improperly loaded cable, 50 ohms, 50 ns propagation delay, 400 ohm load, with a 0-1V step. As you can see, until double the propagation delay, the signal source sees a 100 ohm impedance (source termination plus Z₀), no matter what the load is. As soon as the signal reflects back, it spikes to near the sum of the total impedance for a time period representing the signal rise time (this is due to the reflection from the bad load), then settles to the DC steady state, which is the actual load impedance of 400 ohms plus the source termination.

Because characteristic impedance deals in step response rather than frequency-dependent behavior, it is ideally frequency independent. But WarSim is dead on, real cables do not have the same impedance at all frequencies. Among other things, that capacitance will be significantly dependent on frequency, as are all real capacitors.