## 3 Instrument Characteristics and Tests
### Characteristics of Radiometer Sub-Systems
#### 3.1.4 Characteristics of the Active Cold Source  
In Section 2.1.3, the ACS noise temperature is estimated to \(T_{\mathrm{ACS}}\approx 48\) K, based on the specifications (Appendix) of the components involved (Figure 2, ISO2 and AMP5) at the physical temperature \(T_{0}=313\) K. The low \(T_{\mathrm{ACS}}\) makes it challenging to calibrate the ACS absolutely in a lab experiment. On the one hand, the impact of losses is strong and difficult to control and, on the other, it is difficult to find a highly accurate noise standard with an even lower noise temperature. Nevertheless, such lab measurements are performed using the resistive source (RS) at \(T_{\mathrm{RS}}=300\) K and the calibrated noise diode at \(T_{\mathrm{ND}}=1575\) K as standards to be compared with \(T_{\mathrm{ACS}}\), which is to be determined. After amplifying these noise temperatures with the two amplifiers of the MA, their frequency responses are measured with a Agilent E4408B spectrum analyzer. The associated power levels for the frequency range (1,413 \(\pm\) 500) MHz are determined to be \(P_{\mathrm{ACS}}=0.778\)\(\upmu\)W, \(P_{\mathrm{RS}}=2.748\)\(\upmu\)W, and \(P_{\mathrm{HS}}=11.888\)\(\upmu\)W. Finally, the known reference noise temperatures \(T_{\mathrm{RS}}=300\) K and \(T_{\mathrm{ND}}=1,\!575\) K are used to determine \(T_{\mathrm{ACS}}\approx 39\) K by considering a linear relation between power and injected noise. This calibration of the ACS is error-prone due to the applied extrapolation, which multiplies the measurement uncertainties of the reference sources. Hence, a calibration procedure using the RS and the cold sky as a reference source is applied to determine \(T_{\mathrm{ACS}}\) more accurately. The noise temperature of the RS is \(T_{\mathrm{RS}}=T_{0}\), which is significantly higher than \(T_{\mathrm{ACS}}\). In contrast, the noise standard \(T_{\mathrm{sky,\,in}}=T_{\mathrm{sky}}+\Delta T_{\mathrm{B,\,FC}}\approx 10\) K (see Section 2) at the input of the radiometer looking towards the sky is smaller than \(T_{\mathrm{ACS}}\), which allows the ACS to be calibrated using linear interpolation instead of extrapolation:  
\[T_{\mathrm{ACS}}=\frac{T_{\mathrm{RS}}-T_{\mathrm{sky,\,in}}}{U_{\mathrm{RS }}-U_{\mathrm{sky,\,in}}}\Bigl{(}U_{\mathrm{ACS}}-U_{\mathrm{sky,\,in}}\Bigr{)} +T_{\mathrm{sky,\,in}} \tag{8}\]  
The output voltage \(U_{\mathrm{RS}}\) is measured for the resistive source (RS) switched to the radiometer input port, and \(U_{\mathrm{sky,\,in}}\) is measured with the instrument oriented towards the sky. As described in Section 2, \(T_{\mathrm{sky,\,in}}=T_{\mathrm{sky}}+\Delta T_{\mathrm{B,\,FC}}\) is the received sky brightness \(T_{\mathrm{sky}}\), complemented with \(\Delta T_{\mathrm{B,\,FC}}\) due to the loss of the FC. According to (6), the latter is particularly significant for low antenna brightness such as \(T_{\mathrm{sky}}\). If the radiometer is not pointing exactly towards the galaxy, the sun, or the moon, \(T_{\mathrm{sky}}\) received varies marginally over the sky hemisphere. In this case \(T_{\mathrm{sky}}\) can be computed as the sum of the down-welling atmospheric radiance plus the cosmic background emission (assumed to be 2.7 K), attenuated by the atmosphere [31]. Evaluating the model [31] for the radiometer set-up at WSL (zenith angle \(\theta=30\), elevation 554 m a.s.l.) and air temperatures between 0 \({}^{\circ}\)C and 30 \({}^{\circ}\)C yielded 4.44 K \(\leq T_{\mathrm{sky}}\leq 4.48\) K.  
These theoretical values are used in the calibration procedure to determine \(T_{\mathrm{ACS}}\) for 7 different set point temperatures \(T_{0}\) of the assembled ELBARA II system (Figure 1). The data to determine \(T_{\mathrm{ACS}}\) consist of records of single radiometer voltages \(U_{\mathrm{sky\,in}}\), \(U_{\mathrm{RS}}\), and \(U_{\mathrm{ACS}}\) measured every 10 minutes between 10 p.m. and 2 a.m. on seven successive days in April 2009.