Referring to Figure 1, we denote the voltage gain of amplifier 1 as and the voltage gain of amplifier 2 as . The output, is then given by:
Here, is the noise voltage at the sky horn, is the noise voltage of the reference load, and are the noise voltages contributed by the amplifiers, is the ratio of the DC gains after the two detector diodes, and the diodes are assumed to be perfect square law detectors with a constant of proportionality of .
We will now denote the power gain of amplifier 1 as () and of amplifier 2 as () and calculate the time average of , denoted by . Terms like vanish because the signal and amplifier noise are uncorrelated. Under the assumption that the noise signals in the two amplifiers are uncorrelated, the terms also vanish. The term is equal to , where is the effective bandwidth, is the noise temperature of the signal entering horn 1, and is Boltzmann's constant. We have not specified where the bandpass of the signal gets defined, and we have assumed that it is the same in both legs of the radiometer. Using similar formulae for , , and , becomes:
We now see that in order to null the output (i.e. make ), we must adjust to the proper value. In the simple case that and , this value is
Equation 2 identifies some potential systematic effects. If the reference load temperature, , changed slightly, or if the noise temperature, , of one the amplifiers fluctuated for example, there is the potential that these changes could be confused with the sky signal variations that we are interested in measuring. In the following section of this paper, we will calculate the magnitude of these effects under the assumption that the fluctuations in the various parameters are uncorrelated.
Before calculating these effects we briefly examine the expected magnitude of gain and noise temperature fluctuations. We can infer that cryogenic HEMT amplifiers have noise temperature fluctuations with a type spectrum because we know that the amplifiers have type gain fluctuations (Wollak (1995), Jarosik (1996), Seiffert et al. (1996)). We can estimate the magnitude of noise temperature fluctuations from the following argument. Assuming that each stage of the amplifier has the same level of fluctuation, we can conclude that the transconductance of an individual HEMT device also fluctuates according to:
where is the number of stages of the amplifier, typically . An optimal low noise amplifier design will have equal noise contributions from the gate and drain of the HEMT, which mean the changes in will lead to changes in (Pospieszalsky (1989)). This can be expressed as
We can write the spectrum of the gain fluctuations as:
Putting this together we get:
We can therefore write the noise temperature fluctuations as
with ; a normalisation of (relying on the references above) is appropriate for the 30 and 45 GHz radiometers. Throughout, we will use units of for so that we will not need to refer to the sampling frequency of the radiometer. In these units then, has units of Hz and is dimensionless. We also note that the value of will generally depend on the physical temperature of the amplifier. The values for given here should be regarded as estimates rather than precise values. For the radiometers at higher frequencies, it will be necessary to use HEMT devices with a smaller gate width to achieve the lowest amplifier noise figure. We expect that the gate widths will be roughly that of the devices used for the lower frequency radiometers and this will lead to fluctuations that are roughly a factor of higher (Gaier (1997), Weinreb (1997)). We will therefore adopt a normalisation of for the 70 and 100 GHz radiometers.