# Portable L-Band Radiometer (PoLRa): Design and Characterization
## 3 Radiometer Characterization
### Active Cold Source Characterization  
Using an active cold source (ACS) with the radiometer hardware that is not temperature-stabilized requires determination of the temperature dependence of the ACS noise temperature. L-band brightness temperature \(T_{\text{sky}}\) of the sky is on the order of several Kelvin [24], depending on the Zenith angle, and in the absence of galactic background radiation. Galactic radiation has been shown to have up to 5 K or more impact on sky brightness temperatures [25], but this is reduced to less than 2 K by the relatively large \(37^{\circ}\) antenna beamwidth, compared to the \(10^{\circ}\) antenna assumed in [25]. Figure 4: (a) Antenna return loss from finite element simulation and VNA measurement; (b) normalized antenna power sensitivity pattern from finite element simulation and solar overpass measurement. The noise temperature at the switch inputs of the two polarizations, \(p~{}=~{}\{H,V\}\), \(T_{\rm in}^{p}\) can be expressed as the following expression:  
\[T_{\rm in}^{p}~{}=~{}(1-a^{p})\cdot T_{\rm sky}+a^{p}\cdot\overline{T}_{\rm phy },~{}{\rm with}~{}a^{p}~{}=~{}10^{L^{p}/10} \tag{2}\]  
where \(a^{p}\) is the absorption of the total transmission path (TP) at the mean antenna/cable physical temperature \(\overline{T}_{\rm phy}\) (assuming that all antenna elements and cables are at uniform temperature). Note that the bar accent on temperature symbols refers to physical temperatures throughout the following discussion. \(L^{p}\) in decibels (dB) is the cumulated loss between the antenna and the radiometer input (the mentioned TP) that accounts for losses due to the non-ideal antenna efficiency, cable loss, adapter and connector losses, and mismatch errors. We consider different losses \(L^{p}\) in each polarization \(p~{}=~{}\{H,V\}\), due to respective variability in the cable and antenna losses of the two transmission paths (TPs). We use the sky and ambient matched resistive source (RS) measurements to perform a two-point calibration of the radiometer with the switch inputs as the reference plane. The radiometer gain \(G^{p}\) and the radiometer inherent offset (off) noise temperature \(T_{\rm off}^{p}\) are given by:  
\[G^{p}~{}=~{}\frac{T_{\rm RS}-T_{\rm in}^{p}}{u_{\rm RS}-u_{\rm sky}^{p}}, \tag{3}\]  
\[T_{\rm off}^{p}~{}=~{}-G^{p}\cdot u_{\rm RS}+T_{\rm RS}, \tag{4}\]  
where \(T_{\rm RS}~{}=~{}\overline{T}_{\rm RS}\) is the noise temperature of the RS, which is equivalent to its physical temperature \(\overline{T}_{\rm RS}\) if the RS is perfectly matched. \(u_{\rm RS}\) is the measured detector voltage for the RS switch position, and \(u_{\rm sky}^{p}\) is the measured detector voltage for the switch positions at the antenna polarizations \(p~{}=~{}\{\rm H,V\}\), while the antenna is directed towards the sky. The calibrated noise temperature \(T_{\rm ACS}^{p}\) of the ACS at the input of the switch is thus:  
\[T_{\rm ACS}^{p}~{}=~{}G^{p}\cdot u_{\rm ACS}+T_{\rm off}^{p} \tag{5}\]  
The noise temperature \(T_{\rm ACS}^{p}\) of an ACS reference is expected to be linearly increasing with its physical temperature, as demonstrated in [26; 27]. Accordingly, the following linear model is applicable to express the ACS noise temperature \(T_{\rm ACS,mod}^{p}\) as a function of its measured physical temperature \(\overline{T}_{\rm ACS}\),  
\[T_{\rm ACS,mod}^{p}~{}=~{}m^{p}\cdot\overline{T}_{\rm ACS}+b^{p}, \tag{6}\]  
where \(m^{p}\) and \(b^{p}\) are the slope (units of K/K) and offset (units of K) of a linear least-squares regression, respectively. Given an ideal switch, because all values are referenced to the switch inputs, there is no polarization dependence on ACS noise temperature, meaning that \(T_{\rm ACS}^{\rm H}~{}=~{}T_{\rm ACS}^{\rm V}\).