# On Noise Modeling of Capacitive Feedback Transimpedance Amplifiers in CMOS
## 2 Analytical Methods
The simplified model from above can be used for a very rough analysis of this promising architecture; it may not be extremely helpful for predicting the performance of a real circuit in CMOS and, therefore, more accurate models are needed. In order to develop an accurate noise current model, one shall shortly recall the gain model for the very same circuit. The capacitive feedback TIA topology was originally suggested in the seminal work of [PERSON] [29] and extended in the series of works of [PERSON] [2; 24; 30; 31]. The modeling of the gain for this TIA architecture was revisited in our recent work [22] where we developed accurate gain expressions for circuits implemented under realistic CMOS and power supply constraints. The discussion below complements the original approach of [22] by developing the corresponding noise current model for the very same circuit following an approach similar to the gain model, where an elaborated model is developed incrementally bottom-up from the noise contribution of the circuit's basic components. Clearly, where relevant, for the noise components shaped through some of the blocks in the gain model, one shall recall the original gain model elaborated in [22]. The generic capacitive feedback TIA design along with the major noise sources is shown in Figure 1 and employs the voltage amplifier, the second stage with \(M_{2}\), and a capacitive feedback network formed by two capacitors, \(C_{1}\) and \(C_{2}\). A corresponding circuit-level design of the proposed amplifier, including the relevant noise sources, is shown in Figure 2. Here, we followed a classical approach of using a modification of the single-stage common-source amplifier, although alternative solutions for the core voltage amplifier can also be used [15; 32]. A more detailed discussion on the gain model and motivation behind particular design decisions can be found in our previous work [22]. The input-referred noise current \(i_{\mathrm{n,TIA}}\) is an important parameter when characterizing any given TIA design, as it determines the overall sensitivity of the circuit. The proper characterization of the noise of the TIA for our scenario is even more important, as the PD's current in the case of OTDR applications is very small. Below, we show how the noise model can be derived for the developed circuit and which noise sources have the most significant contribution. As we have argued before, one of the major advantages of the capacitive feedback TIA is that the feedback network itself is noiseless and contributes no noise, as opposed to the traditional resistive-feedback network. Furthermore, the current thermal noise due to \(R_{2}\) will be suppressed when divided by the current gain of the circuit. Unfortunately, this only holds for the circuit operating under benign conditions, and additional significant noise sources shall be taken into account due to imperfections in biasing circuits, current mirrors, etc. We will start the noise analysis from the classical resistor's \(R\) thermal noise. This can be modeled as a voltage source in series with the resistor or, equivalently, as a current source in parallel with it. The noise is generated by the thermal agitation of the charge carriers that occurs inside any electrical conductor, and for the noise representation in the form of the voltage source, one obtains the voltage noise spectral density [33]:  
\[\frac{\overline{v_{\mathrm{nr}}^{2}}}{\Delta f}=4\mathrm{k_{B}}TR, \tag{2}\]  
where \(T\) is the absolute temperature, \(\mathrm{k_{B}}\) is the Boltzmann constant, and \(f\) is frequency. The noise is white and is proportional to the product of the resistance and the temperature, but does not depend on the actual current flowing through the resistance. One can obtain an equivalent current noise spectral density as follows:  
\[\frac{\overline{i_{\mathrm{nr}}^{2}}}{\Delta f}=\frac{4\mathrm{k_{B}}T}{R}. \tag{3}\]  
For the given circuit, there are several sources for the resistor's thermal noise which we may consider for the complete noise model.