Coaxial cable looks like a simple wire wrapped in a shield, but it's actually a precisely tuned transmission line. At frequencies above a few MHz, wires stop behaving like wires — signals become traveling waves, and the geometry of the conductor pair determines whether that wave reaches the far end intact or bounces back as a reflection that destroys your signal.

What characteristic impedance actually is: Z₀ is the ratio of voltage to current for a wave traveling down the line. It's not resistance — an infinitely long lossless cable draws 50Ω worth of current from a 50Ω source, but no power is dissipated in the cable itself. The energy is stored in the distributed inductance (series) and capacitance (shunt) of the conductor geometry.

For coax with inner radius a, outer radius b, and dielectric constant ε_r:

Z₀ = (138 / √ε_r) × log₁₀(b/a)

With polyethylene dielectric (ε_r ≈ 2.3), a b/a ratio of about 3.5 gives 50Ω; a ratio of about 6.6 gives 75Ω.

Why two standards exist:

• 50Ω minimizes the geometric mean of two competing optima: lowest attenuation occurs near 77Ω (air dielectric), and maximum power-handling occurs near 30Ω. 50Ω is the compromise — RF transmitters, test equipment, WiFi, cellular, and radar all standardized here.
• 75Ω is optimized for lowest attenuation with solid dielectric, which matters for long cable TV runs and broadcast video where signal loss over hundreds of feet dominates over power delivery.

What happens if impedance is wrong: Connect a 50Ω transmitter to a 75Ω antenna and part of the wave reflects back at the mismatch. The reflection coefficient Γ = (Z_L − Z₀)/(Z_L + Z₀) = 0.2, meaning 4% of your power bounces back. In a transmitter, that reflected power hits the finals and can destroy them — which is why VSWR meters exist and why ham radios shut down above VSWR 3:1.

Real-world example: An oscilloscope probe with a 1m coax cable and a BNC connector uses 50Ω throughout. If you probe a fast digital signal without terminating the far end into 50Ω, you get ringing — the edge travels down the cable, hits the high-impedance scope input, reflects back at nearly full amplitude, and creates the characteristic overshoot-and-decay pattern engineers mistake for circuit ringing.

Rule of thumb: Cable length matters when it exceeds λ/10. At 100 MHz in coax (velocity factor ~0.66), that's about 20cm. Below that, treat it as a wire; above, treat it as a transmission line and match the impedance at both ends.