2. Why Waveguides at Microwave?

2.1 The death of coax above 30 GHz

Coax is the universal RF interconnect from DC up to about 18 GHz. Above that, two effects kill it.

Conductor loss rises with frequency. Skin depth shrinks as $1/\sqrt{f}$, so AC resistance per unit length grows like $\sqrt{f}$. A coax dropping 0.3 dB/m at 1 GHz drops 1 dB/m at 10 GHz and 3+ dB/m at 100 GHz. Dielectric loss in the polyethylene or PTFE gets worse with frequency, scaling like $f \cdot \tan\delta$. The combined attenuation at 30 GHz makes coax a non-starter for runs longer than a few centimeters.

A subtler problem: as the coax cross-section becomes a non-trivial fraction of a wavelength, higher-order modes start propagating alongside the desired TEM mode. The cable behaves erratically, with mode conversion at every connector. Smaller-diameter coax pushes this onset higher but increases per-meter loss. You cannot win.

2.2 The waveguide alternative

A waveguide is a hollow metal pipe, usually rectangular or circular, that guides EM energy down its length. No center conductor. The walls are the only metal. Inside is air or vacuum. At microwave frequencies a properly sized waveguide has dramatically lower loss than coax of comparable cross-section because there is no inner conductor for current to dissipate $I^2R$ on a small surface.

Pipe-versus-trumpet analogy. The bore of a trumpet shapes sound; the bell focuses it forward and the metal walls keep most energy traveling out the open end rather than leaking through the brass. A waveguide does the same for microwaves. The wave is constrained by walls, geometry sets which "modes" can ring inside, and energy travels down the pipe. The big difference: a trumpet works at any frequency you blow; a waveguide has a sharp cutoff frequency below which it refuses to propagate at all. Below cutoff the wave is evanescent and decays exponentially; above cutoff it propagates almost losslessly.

Real numbers: WR-90 X-band waveguide (22.86 × 10.16 mm internal) running 10 GHz signals has attenuation around 0.1 dB/m in copper. High-quality coax at 10 GHz drops over 1 dB/m, ten times worse. Above 18 GHz the gap widens; by 60 GHz coax is unusable and waveguide is the only option for runs longer than a few centimeters.

The cost is mechanical. Waveguide is heavy, rigid, bulky. You cannot snake it around corners. Connectors are precision flanges with bolts. So waveguide is used where you need it: between a klystron and an antenna, in the feed of a parabolic dish, in the high-power signal chain of a radar. For everything else there is microstrip or short coax jumpers.

2.3 Why no TEM in a hollow waveguide

Coax supports a TEM mode where both $\vec{E}$ and $\vec{H}$ are perpendicular to propagation. TEM requires two conductors so the current down the inner can return on the outer. A hollow waveguide has only one conductor (the walls). There is no return path for an axial current, so pure TEM is impossible in a single-conductor waveguide. The waves that do exist must have either axial $\vec{E}$ (TM, transverse magnetic) or axial $\vec{H}$ (TE, transverse electric), never both zero.

This is the key consequence: in a hollow waveguide, something must be longitudinal. Either $E_z \neq 0$ (TM) or $H_z \neq 0$ (TE). Coax, with two conductors, supports TEM with no longitudinal component. That is the deep reason coax behaves like a "transmission line" while waveguide behaves like a "mode-resonator-pipe."