Waveguide and the Klystron

Establish the loss that forces the change.

1. Measure the insertion loss of a known length of coaxial cable at 1 MHz.
2. Repeat at 100 MHz, then as high as your equipment reaches.
3. Plot loss per metre against frequency.

Loss climbs steadily and then steeply. Two mechanisms are at work and both worsen with frequency: the skin effect confines current to an ever-thinner layer on the conductor surface, raising effective resistance; and the dielectric between the conductors absorbs energy directly, an effect that grows roughly in proportion to frequency.

The inner conductor is the real casualty. It is thin, it carries the full current, it is wrapped in insulation, and it therefore has nowhere to send its heat.

A hollow pipe is a high-pass filter whose corner frequency is set by its width.

1. Make a rectangular waveguide section from sheet brass or copper — a good size for experiment is roughly 22 × 10 mm, which works above about 6.5 GHz.
2. Fit a coupling probe at each end: a short quarter-wave pin projecting into the guide.
3. Sweep frequency and measure transmission from end to end.
4. Find the frequency below which nothing gets through.

Below the cutoff frequency the guide transmits nothing at all; above it, transmission is efficient. Cutoff for the dominant mode occurs when the free-space wavelength is twice the guide’s WIDE dimension — so the wide side alone sets the low-frequency limit, and the narrow side does not enter into it.

The reason a pipe does not short the signal is worth stating plainly: there is no inner conductor, so there is no circuit to short. The wave travels as a pattern of electric and magnetic fields that must satisfy the boundary condition that electric field is perpendicular to a conducting wall. Only certain patterns can do that, and only above a certain frequency does any pattern fit.

Measure inside the guide, and discover the wavelength is not what you expect.

1. Cut a narrow slot along the CENTRE of the guide’s broad wall, parallel to the axis.
2. Insert a small probe on a sliding carriage and feed its output to a detector.
3. Terminate the far end with a short circuit and slide the probe along, recording detected level against position.
4. Measure the distance between adjacent minima and double it.

The guide wavelength is LONGER than the free-space wavelength at the same frequency, and it stretches toward infinity as you approach cutoff. The wave is effectively zig-zagging down the pipe rather than travelling straight, so its progress along the axis is slower than its progress through space.

The slot position is not arbitrary and this is the detail worth remembering: a longitudinal slot in the centre of the broad wall cuts no current lines, so it does not disturb the fields and does not radiate. The same slot cut across the guide would interrupt the wall currents and turn the instrument into an antenna. Slotted lines are cut this way on purpose, and slotted-waveguide antennas are cut the opposite way on purpose.

Velocity modulation converts a steady beam into a pulsating one without any grid switching it.

1. Sketch the two-cavity klystron: an electron gun, a buncher cavity, a drift space, a catcher cavity and a collector.
2. Follow one electron through: the buncher cavity’s RF field either speeds it up or slows it down depending on when it arrives.
3. Reason about what happens to a stream of such electrons after they travel down the drift space.

Electrons that were speeded up catch those that were slowed down, so by the end of the drift space the uniform stream has sorted itself into dense bunches. No electron has been created or removed — they have merely been redistributed in time — and those bunches, arriving periodically at the catcher cavity, drive it hard and deliver useful RF power.

Notice what has happened to the transit-time problem. In a triode, transit time destroys control. Here the drift time IS the mechanism, and it is chosen deliberately: too short and the bunches have not formed, too long and they have passed through each other and dispersed.

The wave has to leave the pipe and enter free space, and a sudden opening does that badly.

1. Terminate a waveguide with a plain open end and measure the reflected power.
2. Now fit a flared horn — a gradual expansion from guide dimensions to a much larger aperture.
3. Measure reflected power again.
4. Map the radiation pattern of both, as you did for the dipole earlier in this batch.

The abrupt opening reflects a substantial fraction back down the guide; the flared horn reflects very little and produces a clean, narrow beam. The horn is an impedance transformer — it changes the wave’s impedance gradually from that of the guide to that of free space, and gradual is the entire point.

That principle has appeared repeatedly in this batch under different names: the balun matching a dipole to coax, the IF transformer’s coupling, the tapered cable joint in the power batch. Wherever two media meet, an abrupt transition reflects and a gradual one does not.