5.7 Mixer

A sketch of a mixer is shown on figure 5.6, again grossly over-simplified. The junction is mounted across the waveguide, in the direction of the electric field. One side of the junction is connected to the outside of the mixer block, both to bring out the IF beat signal, and to provide the DC bias. That connection is made through a low-pass filter to avoid losing precious RF energy.

Figure 5.6: Rough sketch of the main elements of a mixer

One end of the waveguide is the input of the mixer; the other end must be terminated somehow. At the zero-order approximation, one would like the junction to ``see'' an open circuit when ``looking into'' the rear end of the waveguide. More generally, the junction should see a pure imaginary impedance, so that no energy is wasted. A simple calculation shows that a transmission line having a length $l$, and terminated into a short-circuit, has an apparent impedance:

$$Z_{BS} = j Z_0 \tan(2\pi l/\lambda)$$

where $Z_0$ and $\lambda$ are respectively the propagation impedance and wavelength in the waveguide, and $l$ is the distance to the short-circuit. In particular, for $l = (\frac{n}{2}+\frac{1}{4})\lambda$, the apparent impedance is an open circuit. More generally, by adjusting $l$, an arbitrary imaginary impedance can be placed in parallel with the junction. Together with the tuning structures mentioned in the previous section, such an adjustable backshort contributes to achieve the best possible match of the junction impedance.

For various reasons (one of which is reducing the noise contribution from the atmosphere) it is desirable that the mixer should operate in single-sideband mode. We explain how this is achieved with a crude zero-order model. Assume that the best impedance match of the junction is obtained when the apparent impedance of the backshort seen from the junction is an open circuit. Assume we observe in the lower sideband at a frequency $\nu_L=\nu_{LO}-\nu_{IF}$, and want to reject the upper sideband $\nu_U=\nu_{LO}+\nu_{IF}$. That condition can be achieved if, at the frequency $\nu_U$, the junction is short-circuited. So, we must meet the two conditions:

$$l = (\frac{n}{2}+\frac{1}{4}) \lambda_L$$

$$l = (\frac{n}{2}+\frac{1}{2}) \lambda_U$$

for some integer $n$; we gloss over the distinction between free-space and waveguide wavelengths. The two conditions (one unknown) can be approximately met for some $l$ close to

$$l_\mathrm{reject} = \frac{1}{8} \frac{c}{\nu_{IF}}$$

Because for current mixers in the 100GHz band, the IF frequency is relatively low (1.5GHz), single-sideband operation requires additional tricks...

Returning to practicalities, tuning a receiver requires several steps (which used to make astronomers a bit nervous at the 30-m telescope when all was done manually). First the local oscillator must be tuned and locked at the desired frequency. Then the backshort is set at the appropriate position, and the junction DC bias voltage is set. Finally the LO power is adjusted to reach a prescribed junction DC current (of the order of $20\mu A$). These adjustments are made by a combination of table lookup and optimization algorithms under computer control. Altogether this involves between 11 and 13 adjustments, mechanical or electrical, yet this process takes only a few minutes with the current systems.