# Transimpedance Amplifier for Noise Measurements in Low-Resistance IR Photodetectors
## 3 Materials and Methods
This result, however, is only an approximation, and depends on the magnitude of the loop gain and the magnitude of the PSD associated with the noise sources. Proper calculations show that these contributions can be neglected at low frequencies as long as the loop gain of the SOA in the shunt-shunt configuration in Figure 2 is high and \(g_{m}R_{D}>>1\). In terms of the contribution \(S_{OB\_RR}\) of the thermal noise generated by the feedback resistance \(R_{R}\), it is given by:  
\[S_{OB\_RR}=4 kTR_{R} \tag{14}\]  
Since we are assuming that we are operating well above the cut-in frequency of the AC filter \(R_{A2}C_{A2}\), the reactance of the capacitor \(C_{A2}\) is much lower than the resistance \(R_{A2}\), and therefore, for the contributions \(S_{OB\_A2}\) of the resistance \(R_{A2}\) and \(S_{OB\_OT2}\) of the noise source \(i_{I2}\), we have:  
\[S_{OB\_A2}=\frac{kT}{\pi^{2}f^{2}C_{A2}^{2}R_{A2}};\ \ S_{OB\_II2}=\frac{S_{II2}}{4 \pi^{2}f^{2}C_{A2}^{2}} \tag{15}\]  
where \(S_{II2}\) is the PSD of the current noise source \(i_{I2}\). Finally, it is necessary to add the contribution \(S_{OB\_EI2}\) due to the equivalent noise source \(e_{I2}\), that is:  
\[S_{OB\_EI2}=S_{el2} \tag{16}\]  
where \(S_{el2}\) is the PSD of the current noise source \(e_{I2}\). In order to more easily understand the relative weight of the different noise contributions, we can proceed with the same simplifying assumption used in the previous section to obtain the equivalent input background noise, that is, the DUT impedance is assumed to be essentially a resistance with a value \(R_{D}\), and the condition \(R_{D}/R_{R}<<1\) is satisfied. Starting from the PSD \(S_{OB}\) of the overall noise at the output \(v_{OB}\) in the form:  
Table 1.0  
we can calculate the parameter \(Q_{n}\) as before. We have:  
\[ \begin{array}{l}Q_{n}=\frac{R_{D}S_{iBN}}{4 kT}=Q_{n1}+Q_{n2},\\ Q_{n1}=\frac{2S_{D}}{4 kT}+\frac{R_{D}}{R_{R}}\bigg{(}1+\frac{S_{el2}}{4 kTR_{R} }\bigg{)},\\ Q_{n2}=\frac{1}{4\pi^{2}f^{2}}\bigg{[}\frac{1}{\tau_{A1}^{2}}\frac{R_{OE}}{R }+\frac{1}{\tau_{A2}^{2}}\frac{R_{D}R_{A2}}{R_{R}^{2}}\bigg{(}1+\frac{S_{II2} R_{A2}}{4 kT}\bigg{)}\bigg{]},\tau_{A1}=C_{A1}R_{OE},\tau_{A2}=C_{A2}R_{A2}\end{array}  \tag{18}\]  
In Equation (18), \(Q_{n}\) is expressed as the sum of two contributions to stress the fact that the contribution \(Q_{n2}\) can be made as small as desired by increasing the values of the coupling capacitors \(C_{A1}\) and \(C_{A2}\), at the cost, however, of increasing the time constants \(\tau_{A1}\) and \(\tau_{A2}\); the fact that these time constants increase means that the settling time of the circuit increases as well, and this, besides resulting in a waste of time when connecting a new DUT or setting a different bias voltage, can make offset correction a much more challenging task. When looking at the relative weight of the terms contributing to \(Q_{n2}\), we can start by evaluating the term that contains the PSD \(S_{II2}\). The ADA4625 is a JFET input operational amplifier. The PSD for the equivalent input current noise source reported in the datasheet is 4.5 fA/\(\surd\)Hz, and this means that the contribution of the fraction containing \(S_{II2}\) in Equation (18) is much lower than 1 for values of \(R_{A2}\) on the order of a M\(\Omega\) or more (\(R_{A2}=1\) M\(\Omega\) in our prototype). This, together with the fact that the two time constants \(\tau_{A1}\) and \(\tau_{A2}\) have similar values, and the fact that:  
\[R_{D}\ll R_{R}\ll R_{OE}\approx R_{A2} \tag{19}\]  
clearly indicates that \(Q_{n2}\) is essentially set by the first term in Equation (18), that is, by the thermal noise generated by the equivalent resistance \(R_{OE}\) that is not completely filtered out by the capacitance \(C_{A1}\). At the minimum frequency of interest (1 Hz), assuming a typical value for \(R_{D}\) of 1 k\(\Omega\), we have \(Q_{n2}\approx 0.06\). Note that, because of the proportionality to the inverse of the frequency squared, \(Q_{n2}\) rapidly decreases with increasing frequency.