## 3 Results and Discussion  
When the measurements were carried out with the attenuator inside, the minimum temperature obtained was 83.6 K. This is due to the thermal gradient produced for the connections of the RF cables (model AM10-SR141 TP-AM10) between the SMA feed-through at the external wall of the cryostat (at room temperature) and the PCB. Figure 9 shows the results of the attenuator at room temperature (RT) and the minimum temperature achieved using the chip testing platform. The \(S_{21}\) shows an attenuation higher than 20 dB for the whole frequency range. An \(S_{11}\) close to 0 dB is obtained for most of the frequency range. In the interest frequency band for our application (0.8-2 GHz), an attenuation between \(-40.5\) and \(-45.5\) dB was obtained for room temperature and one of between \(-38\) and \(-43.5\) dB for a temperature of 83.6 K.  
The circuit was tested and characterised for different temperatures ranging from 296.5 K to 83.6 K (21 temperature steps), as shown in Figure 10. As can be seen, the values of \(S_{11}\) and \(S_{21}\) do not vary significantly and follow a general trend. Figure 11 shows a 3D plot and a colour map, which illustrate that the attenuation does not vary significantly with the temperature in the range of interest. Figure 11a shows a 3D plot where the generation of valleys in the attenuation (from \(-20\) to \(-70\) dB) at high frequencies (above 5 GHz) are appreciated. Below that frequency, the attenuation remains between \(-30\) and \(-50\) dB. Figure 11b depicts a grey colour map of the 3D plot, and it shows a good response against changes in temperature for frequency ranges between 800 MHz and 2 GHz.  
Figure 8: Evolution of pressure and temperature inside the cryostat without PCB or cables.  
Figure 9: Experimental results at room temperature and the lowest temperature (83.6 K): (**a**) \(S_{21}\) and (**b**) \(S_{11}\).  
In order to see the maximum differences that can be obtained for the range of temperatures across the complete bandwidth, two curves with the maximum and minimum values are plotted in Figure 12a. It can be clearly appreciated how there are frequency ranges in which higher differences in the attenuation value occur. Figure 12b,c represent these differences in the attenuation across the frequencies in absolute values and percentages, respectively. Although, in many cases, the variations are less than 5 dB, there are some frequency ranges in which the difference between the maximum and minimum values can reach up to 30 dB. A histogram representing the dispersion of the \(S_{21_{MAX}}-S_{21_{MIN}}\) difference is depicted in Figure 12d. The standard deviation (\(\sigma\)) is a commonly employed parameter for quantifying data dispersion. For the entire frequency range, a standard deviation of 5.84 dB and an arithmetic mean (\(\bar{x}\)) of 6.55 dB were obtained. In contrast, within the range of interest, these values drop to a \(\sigma\) of 0.39 dB and a \(\bar{x}\) of 2.27 dB.  
Figure 11: Experimental results from 296.5 K to 83.6 K: (a) 3D plot and (b) colour map.  
Figure 10: Experimental results from 296.5 K to 83.6 K: (a) \(S_{21}\), (b) \(S_{11}\), (c) \(S_{21}\) (region of interest) and (d) \(S_{11}\) (region of interest).  
Therefore, with the results obtained, it is necessary to be cautious with regard to the working frequency, since although, in general, it has flat attenuation and very good performance, there are some frequency ranges in which the attenuation value can vary considerably. In terms of our application, the frequency range of interest (0.8-2 GHz) has differences in the attenuation smaller than 2 dB produced for the different temperatures. Therefore, for the whole range of interest, the attenuation differences are lower than 5.5 dB at the lowest temperature, which are adequate and do not negatively affect the operation of the system.