How can PCB trace have 50 ohm impedance regardless of length and signal frequency?

There is a simple hand-waving explanation why the effective impedance of a (ideal) transmission line is a constant. Other explanations leave some confusion on how do we “select” Li and Ci in the model of transmission line. What are these Li and Ci exactly?

First, once we say “transmission line”, we are talking about long wires. How long? Longer than the length of an electromagnetic wave that is transmitted along the line. Therefore, we are talking either about very long lines (miles and miles), or a very high frequencies. But the concept of wavelength relative to trace length is fundamentally important.

Now, as people mentioned, a trace has certain inductance per unit of length, and, correspondingly, certain capacitance, again proportional to length. These L and C are inductance and capacitance per unit length. So, the actual inductance of a wire segment would be L = L * length; same for C.

Now consider a sine wave coming into the trace. Waves propagate at the speed of light (in particular dielectric/air media it is about 150ps/inch). At every and each moment the particular charge deviation (waveform) interacts with a section of wire equal to the corresponding length of this wave. Slower frequencies have longer wave lengths, while faster frequency components have proportionally shorter lengths. So, what do we have? Longer waves “see” a longer trace and therefore a bigger L, and bigger capacitance C. Shorter (higher frequency) waves “see” the shorter effective line length, and therefore smaller L and C. So, both effective L and C are proportional to wavelength. Since the impedance of the line is Z0=SQRT(L/C), the dependence of L and C on length cancels, and that’s why waves with different frequencies “see” the same effective impedance Z0.