# On Noise Modeling of Capacitive Feedback Transimpedance Amplifiers in CMOS
## 4 Discussion  
Apart from the low-frequency \(M_{5}\) noise, the nonideal behavior of a practical biasing (resistive) circuit was also often neglected by other authors while deriving the noise models. This was exactly followed by the author in [5], where the the noise component due to \(R_{2}\) was kept while assuming a perfect biasing circuit. However, for a realistic circuit operating under voltage headroom constraint with resistive biasing, such a model would clearly result in an overoptimistic estimation for the noise level in mid-range frequencies. Note that none of the components of resistive noise was considered in the series of works of [PERSON] (see his latest work [24]). While biasing noise was similarly ignored due to transistor-based implementation and the assumption of effectively infinite resistance, the noise from \(R_{2}\) was neglected due to an assumption of its minor impact. According to the author, the noise contribution was solely due to the noise of the core amplifier (our first noise component \(\overline{i_{\text{nr}}^{2}}\) presented above). Figure 10: Proposed analytical model for the input-referred noise current density and Cadence-simulated noise for programmable-gain TIA configuration in IHP 0.25 \(\upmu\)m BiCMOS. Solid lines are due to Cadence simulation; dashed lines are due to the developed analytical noise model. Figure 9: Proposed analytical model for the input-referred noise current density and Cadence-simulated noise for programmable-gain TIA configuration in TSMC 0.18 \(\upmu\)m CMOS. Solid lines are due to Cadence simulation; dashed lines are due to the developed analytical noise model. The assumption of a perfect biasing for \(M_{1}\) results in yet another difference with the models so far presented in the literature. Authors typically assumed a pure reactive input impedance (see the series of works of [PERSON] and work [PERSON] (2018)), resulting in  
\[\overline{i_{\text{n,TIA}}^{2}}\approx\overline{v_{\text{n,OpAmp}}^{2}}\bigg{[} C_{\text{T}}+\frac{C_{1}}{1+C_{1}/C_{2}}\bigg{]}^{2}s^{2}. \tag{23}\]  
Technically, the model can be derived from our noise model when one assumes perfect biasing \(R_{\text{bias},1}\rightarrow\infty\). Then, the model for the input impedance becomes  
\[\lim_{R_{\text{bias},1}\rightarrow\infty}Z_{\text{in}}=\frac{1}{1+s\Big{(}C_{ \text{T}}+\frac{C_{1}C_{2}}{C_{1}+C_{2}}\Big{)}}\approx\frac{1}{1+s(C_{\text{ T}}+C_{2})}\approx\frac{1}{1+sC_{\text{T}}}. \tag{24}\]  
In the capacitive network, the input impedance is formed by \(C_{\text{T}}\) in parallel with the series connection of \(C_{1}\) and \(C_{2}\). As the \(C_{\text{T}}\) dominates the input impedance, one can rearrange the noise model, and since in capacitive feedback TIA holds \(C_{1}\gg C_{2}\), we obtain  
\[\overline{i_{\text{n,TIA}}^{2}} \approx \overline{v_{\text{n,OpAmp}}^{2}}\bigg{[}C_{\text{T}}+\frac{C_{ 1}C_{2}}{C_{1}+C_{2}}\bigg{]}^{2}s^{2}\] \[\approx \overline{v_{\text{n,OpAmp}}^{2}}\bigg{[}C_{\text{T}}+\frac{C_{ 1}C_{2}}{C_{1}}\bigg{]}^{2}s^{2}=\overline{v_{\text{n,OpAmp}}^{2}}[C_{\text{T }}+C_{2}]^{2}s^{2},\]  
and the latter expression is yet another simplified noise model, as reported by [PERSON] (2018). How the previously known noise model of [PERSON] compares to the proposed noise model can be found in Figure 11, where both models are shown for the very same set of parameters in CMOS. The comparison employs the very same set of true circuit parameters, where the simplified model of [PERSON] is fed with the required parameter values of the true operating CMOS circuit without any additional tuning or inflation of the parameter values. As the noise model proposed by [PERSON] is even simpler when compared to the one of [PERSON] (it even ignores the noise component due to \(R_{2}\)), it is not discussed any further. Based on Figure 11, we see that for the given valid CMOS TIA design we cannot confirm the simplified noise model to represent the dominating noise behavior as well as to correctly represent even the base noise level (compare Figures 9 and 11).