7.3.8 Applications

A radar transmitter, with up to megawatts of peak power, feeds its antenna through a rigid waveguide run with bends, a rotary joint and a feed horn (Figure 7.3.11). Satellite earth stations do the same with the uplink amplifier. A coaxial cable of comparable size would break down at a fraction of the power and lose far more, because its small inner conductor carries the whole current in a thin skin. The two numbers that matter follow from Eqs. (7.3.37) and (7.3.42):

with the breakdown field $E_{\mathrm{bd}}\approx 3MV/m$ of dry air. Table 7.3.2 gives typical values for air-filled copper guides.

Table 7.3.2. Typical TE${}_{10}$ feed lines (air, copper, $E_{\mathrm{bd}}=3MV/m$).

• Air-traffic-control radars, with about $1\mathrm{MW}$ of peak power, need the large S-band guide. Higher bands handle less power, because the cross-section scales as $ab\propto 1/f^2$.
• Guides are kept dry and slightly pressurised: humidity lowers $E_{\mathrm{bd}}$, pressure raises it.
• The feed run also sets the noise of the system. A loss $L$ in front of a receiver at the physical temperature $290\mathrm{K}$ adds the noise temperature $(L-1)\times 290\mathrm{K}$, so $0.5\mathrm{dB}$ ($L=1.122$) adds about $35\mathrm{K}$. Worked example 7.3.1 puts numbers on a complete S-band feed.

The slotted line. A narrow slot on the centre line of the broad wall cuts no current (Figure 7.3.7) and hardly disturbs the TE${}_{10}$ field. A probe sliding in it samples $|E_y|$ along $z$. In front of a mismatched load the incident and reflected waves form a standing wave whose adjacent minima are ${\lambda }_g/2$ apart and whose depth gives the standing-wave ratio. This was the classic way to measure ${\lambda }_g$, and with it the frequency and the load impedance (Home exercise 7.3.2).

The slotted array. A longitudinal slot offset by $\delta$ from the centre line cuts the crosswise current $J_x\propto \cos (\pi x/a)$, which has opposite signs on the two sides of the centre line: it radiates, more strongly for a larger offset (Figure 7.3.12a). Slots spaced ${\lambda }_g/2$ apart see the guide wave $180^{\circ }$ apart in phase, and putting them on alternate sides of the centre line adds another $180^{\circ }$. All slots then radiate in phase and the beam is normal to the wall (Figure 7.3.12b):

For WR-90 at $9.4\mathrm{GHz}$, ${\lambda }_g=4.46\mathrm{cm}$ and $d=2.23\mathrm{cm}$. A $1.2\mathrm{m}$ array holds $53$ slots and gives, with uniform excitation, a beam about $1.4^{\circ }$ wide. Ship, coastal and weather radars use such arrays: they are rugged, have low loss and need no feed network. Away from $f_0$, $\tau d\ne \pi$ and the beam tilts (Worked example 7.3.2).

The magnetron of a domestic oven delivers about $1\mathrm{kW}$ at $2.45\mathrm{GHz}$. It ends in a short antenna that sticks into a rectangular guide like a probe, parallel to the $E$ of TE${}_{10}$ (Figure 7.3.13). One end of the guide is shorted, the other opens into the cavity through an aperture.

• The short is a node of $E_y$. A probe ${\lambda }_g/4$ in front of it sits at a maximum of the standing wave, where the wave reflected by the short adds in phase to the wave sent towards the cavity.
• A guide of about WR-340 size ($86\times 43\mathrm{mm}$) is single-mode at $2.45\mathrm{GHz}$: $f_{co,10}=1.74\mathrm{GHz}$ and $f_{co,20}=f_{co,01}=3.47\mathrm{GHz}$.

The guide wavelength and the probe position are

Typical numbers:

• $1\mathrm{kW}$ in TE${}_{10}$: $Z_{TE}=534\Omega$ and $E_0=\sqrt{4Z_{TE}P/ab}=24kV/m$, far below breakdown.
• The wall loss is about $0.015dB/m$: under $1\mathrm{W}$ is lost in a $20\mathrm{cm}$ feed.
• Behind the aperture the cavity is a resonator with many modes (Section 7.8). A stirrer or a turntable evens out the heating.
• The door screen and the vents are waveguides far below cutoff: they give tens of dB of decay in a few millimetres (Home exercise 7.3.1).

Worked example 7.3.3 goes through the design.