# Transimpedance Amplifier for Noise Measurements in Low-Resistance IR Photodetectors
## 3 Materials and Methods
The component list used to build the proposed amplifier is presented in Table 1.  
The most important parameter for any amplifier designed for low-frequency noise measurements is the level of background noise, particularly at low frequencies, where the flicker noise of the active devices in the amplifier may mask the flicker noise generated by the DUT. Since we are mainly interested in the performances at low frequencies, and since it can be easily proven that the compensation network does not contribute significantly to the background noise at frequencies up to a few kHz, in discussing the noise performances of the system, we will neglect the presence of \(R_{C1}\), \(R_{C2}\), \(C_{C1}\) and \(C_{C2}\). The equivalent circuit for noise calculation is shown in Figure 3. The ratio between useful noise (the noise coming from the DUT \(Z_{D}\)) and background noise will be estimated with reference to the input of the last voltage amplifier in the noise measurement chain (\(v_{OB}\), corresponding to the non-inverting input of \(OA_{2}\) in Figure 2). Besides omitting the compensation network, other simplifications are made: (a) the BJT-based current source is assumed to be characterized by very high equivalent impedance; (b) the current noise source representing the noise coming from the BJT-based circuit is omitted, since it is a common mode source whose effects are rejected by \(O_{A2}\); (c) the equivalent input current noise source at the input of \(OA_{3}\) is omitted, since it is shorted by the very low impedance at the output of \(OA_{2}\). Because of the high open-loop gain of the SOA and the fact that we are mainly interested in the noise at low frequencies, we can perform noise estimation under the assumption of a virtual short circuit between the inverting \(G_{2}(-)\) and non-inverting \(G_{1}(+)\) inputs of the SOA. We will also assume that we are working above the cut-off frequency of the AC coupling filter, with the minimum frequency of interest being 1 Hz. Note, however, that the capacitors \(C_{A1}\) and \(C_{A2}\) are not replaced with short circuits, because, due to the large amount of thermal noise introduced by the resistances \(R_{OE}\) and \(R_{A2}\), their finite impedance may result in a non-negligible contribution to the BN of the system even above the cut-in frequency [23]. We will also take advantage of the reasonable assumption that all noise sources are uncorrelated, so that we can estimate the total noise at \(v_{OB}\) by adding the noise contribution from each single source. We can start by estimating the contribution \(S_{OB\_iD}\) to the PSD of the noise at \(v_{OB}\) due to the DUT noise source \(i_{D}\) with PSD \(S_{iD}\). We have:  
\[S_{OB\_iD}=R_{R}^{2}S_{iD} \tag{11}\]  
It can be easily demonstrated that, assuming the PSD \(S_{\_1}\) of \(e_{\_1}\) and \(S_{\_2}\) of \(e_{\_2}\) to be the same, and equal to \(S_{\_}I\), their total contribution \(S_{OB\_J}\) is given by:  
\[S_{OB\_J}=2\bigg{|}1+\frac{R_{\_}R}{Z_{D}}\bigg{|}^{2}S_{\_}I \tag{12}\]  
As far as the contribution of the offset compensation network is concerned, with the component values in Table 1, the equivalent resistance \(R_{\_OE}\) is essentially reduced to \(R_{\_}{Q}\|R_{\_A1}\)  
Figure 3: Simplified equivalent circuit for noise calculation. which, at the minimum frequency of interest (1 Hz), is much greater than the impedance \(X_{CA1}\) of the capacitor \(C_{A1}\). Therefore, the contribution \(S_{OB\_OE}\) due to the thermal noise \(i_{OE}\) of the resistor \(R_{OE}\) is reduced to:  
\[S_{OB\_OE}=\frac{kT}{\pi^{2}f^{2}C_{A1}^{2}R_{OE}}\bigg{|}1+\frac{R_{R}}{Z_{D}} \bigg{|}^{2} \tag{13}\]  
where \(f\) is the frequency, \(k\) is the Boltzmann constant, and \(T\) is the absolute temperature. With respect to the contribution from the thermal noise of the resistances \(R_{D1}\) and \(R_{D2}\) and the equivalent input voltage (\(e_{O1}\)) and current noise sources (\(i_{A1}\) and \(i_{1B}\)), assuming a virtual short circuit at the inputs of the SOA, there is no contribution to the output noise.