## 3 Instrument Characteristics and Tests
### Characteristics of Radiometer Sub-Systems
#### 3.1.3 Linearity of the Power Detector Assembly  
The PDA response is measured with the Micronetics noise module SNM 7114-C2A. Its output is band-pass filtered to cover the frequency range of 1,400 MHz-1,700 MHz, and then amplified by 30 dB yielding, a constant power level of \(P_{\mathrm{0}}\approx\) -25 dBm. Subsequently, \(P_{\mathrm{0}}\) is passed through Agilent 9496B attenuators with the total attenuation variable in the range of 0 dB \(\leq\) \(A_{\mathrm{tot}}\) \(\leq\) 20 Db with a step size of 1 dB.  
Figure 5: (a) Frequency transfer functions of the two frequency channels (LSB = blue, USB = green) of the microwave assembly. The borders of the protected part of the L-band at 1,400 MHz and 1,427 MHz are indicated in red. (b) Frequency response of the MA front-end. The loss measured at the radiometer center frequency of 1,413.5 MHz is \(L_{\mathrm{front}}\) = 1.09 dB.  
For the attenuators set to \(A_{\rm tot}\) = 0 dB, the power \(P_{\rm PDA}\) \(\approx\) -25 dBm injected to the PDA is measured with an Agilent power meter. This reference value and the selected \(A_{\rm tot}\) is used to inject the power levels -25 dBm \(\geq\)\(P_{\rm PDA}\)\(\geq\) -45 dBm (corresponding to 3.16 \(\mu\)W to 0.03 \(\mu\)W) into the detector diode of the PDA. The resulting output voltages \(U_{\rm PDA}\) of the PDA are measured with a multimeter.  
Figure 6 shows the response of the PDA with respect to input power levels -38 dBm \(\leq\)\(P_{\rm PDA}\)\(\leq\) -25 dBm (0.17 \(\mu\)W \(\leq\)\(P_{\rm PDA}\)\(\leq\) 3.33 \(\mu\)W), resulting in the range of output voltages 0.167 V \(\leq\)\(U_{\rm PDA}\)\(\leq\) 3.334 V (black dots). The red circles are the \(P_{\rm PDA}\) from Table 1 estimated for the MA input noise temperatures expected. Fitting the model \(P_{\rm PDA}\) = \(P_{\rm PDA\,0}\) + \(dP_{\rm PDA}\)/\(dU_{\rm PDA}\). \(U_{\rm PDA}\) to the measured relation between the power \(P_{\rm PDA}\) [\(\mu\)W] and the voltages \(U_{\rm PDA}\) [V] yields the small negative offset \(P_{\rm PDA\,0}\)\(\approx\) -0.00352 \(\mu\)W with the standard error of \(\pm\)0.00564 \(\mu\)W. Hence, an apparently negative offset \(P_{\rm PDA\,0}\) is not significant but explained by the errors in \(P_{\rm PDA}\) caused by the known uncertainty (\(\pm\)0.05 dB) of the variable attenuator used in the test setup. Consequently, the PDA is virtually offset-free, implying that the simple linear model depicted in Figure 6 (solid line) is adequate:  
\[P_{\rm PDA}\left[\mu\mbox{W}\right]\mbox{=}\frac{dP_{\rm PDA}}{dU_{\rm PDA}} \cdot U_{\rm PDA}=\mbox{1.00939}\cdot U_{\rm PDA}\left[\mbox{V}\right] \tag{7}\]  
The asymptotic standard error of the power sensitivity \(dP_{\rm PDA}\)/\(dU_{\rm PDA}\) = 1.00939 \(\mu\)W V\({}^{-1}\) is \(\pm\)0.00247 \(\mu\)W V\({}^{-1}\). This and the correlation coefficient \(R\) = 0.99991 confirms the highly linear response of the PDA within a power range that includes the power levels estimated for the operative mode of the instrument.  
Figure 6: Measured voltage response \(U_{\rm PDA}\)(\(P_{\rm PDA}\)) of the PDA with respect to injected power in the range -38 dBm \(\leq\)\(P_{\rm PDA}\)\(\leq\)–25 dBm (black dots). The solid line is the linear fit (7) with the gradient \(dP_{\rm PDA}\)/\(dU_{\rm PDA}\) = 1.00939 \(\mu\)W V\({}^{-1}\). Red circles are the estimates of \(P_{\rm PDA}\) for expected MA input noise temperatures (Table 1).