# On Noise Modeling of Capacitive Feedback Transimpedance Amplifiers in CMOS
## 3 Mathematical and Computer Simulation Results  
Below, we present the results for the noise modeling only, while the corresponding results for the gain modeling can be found in the previous work [22]. The results are demonstrated for five gain-bandwidth configurations of the programmable-gain TIA, as developed according to the OTDR specification. Here, we shall bear in mind that we did not target the equal bandwidth design and, as expected, bandwidth decreases with the increasing gain. The obtained results also confirm that the achieved bandwidth of around 1 GHz for the base gain of 10 k\(\Omega\) is close to the limit for the proposed implementation for the given gain and technological node. As we saw in the previous section, an exact gain model \(Z_{\text{T}}\) is important for the accuracy of the complete noise model due to its coupling in the flicker noise component of \(M_{5}\) and thermal noise component due to \(R_{2}\). Before we proceed to the noise results, let us first elaborate on how the noise model components sum up to the final noise performance and what impact each of these components has over the complete frequency range. To make the analysis more intuitive, we plot the results for all five gain configurations at once in order to observe the impact of different target gains. Let us start with the thermal noise component at the gate of the transistor \(M_{1}\):  
\[\frac{\overline{i_{\text{n}t}^{2}}}{\Delta f}\approx\frac{1}{Z_{\text{IN,eff} }^{2}}\frac{8}{3}\frac{\text{k}_{\text{B}}T}{g_{\text{m},1}}, \tag{18}\]  
and shown in Figure 3 for \(T=298.15\) K. The contribution is mostly responsible for the high-frequency noise, while its contribution to the noise figures in the mid- and low-frequency range is, obviously, insufficient to describe the behavior of a real circuit. Next, we consider the proposed noise model for both \(M_{1}\) thermal noise and the noise component due to \(R_{\text{bias,1}}\):  
\[\frac{\overline{i_{\text{n}t}^{2}}}{\Delta f}+\frac{\overline{i_{\text{n}t}^{ 2}}}{\Delta f}\approx\frac{1}{Z_{\text{IN,eff}}^{2}}\frac{8}{3}\frac{\text{k}_ {\text{B}}T}{g_{\text{m},1}}+\frac{4\text{k}_{\text{B}}T}{R_{\text{bias,1}}}. \tag{19}\]  
The results are shown in Figure 4, where noise due to the bias resistor dominates in the mid- and low-frequency range and significantly pulls up the noise to the level of approximately 1 pA/\(\sqrt{\text{Hz}}\) for the base 10 k\(\Omega\) gain. Further, we extend the noise model by adding the noise contribution due to \(R_{2}\):  
\[\frac{\overline{i_{\text{n}t}^{2}}}{\Delta f}+\frac{\overline{i_{\text{n}t}^{ 2}}}{\Delta f}+\frac{\overline{i_{\text{n}t}^{2}}}{\Delta f}\approx\frac{1}{Z _{\text{in,eff}}^{2}}\frac{8}{3}\frac{\text{k}_{\text{B}}T}{g_{\text{m},1}}+ \frac{4\text{k}_{\text{B}}T}{R_{\text{bias,1}}}+\frac{4\text{k}_{\text{B}}Tk_ {2}}{Z_{\text{T}}^{2}}, \tag{20}\]  
Figure 3: Analytically calculated input-referred noise model for programmable-gain configuration with \(M_{1}\) thermal noise component only (example case for TSMC 0.18 \(\upmu\)m CMOS). with the results shown in Figure 5. Although one may expect to again see the white resistor noise added to the previous results, this is not the case. As we elaborated in the previous section, the noise component \(\overline{\overline{i_{\text{nr2}}^{2}}}\) is typically approximated in the literature as  
\[\frac{\overline{\overline{i_{\text{nr2}}^{2}}}}{\Delta f}\approx\frac{4\text{ k}_{\text{B}}T}{R_{2}\Big{(}1+\frac{\text{C}_{1}}{\text{C}_{2}}\Big{)}^{2}}. \tag{21}\]  
The term represents the white noise (constant power spectral density) and its value is negligible when compared to the noise floor set by the noise component caused by \(R_{\text{bias,1}}\). However, this approximate expression is only valid if we assume a constant gain \(Z_{T}\approx R_{T}\) over the complete frequency range. As this assumption is not valid due to the bandpass behavior of the proposed circuit, the noise due to \(R_{2}\) is amplified for very low frequencies.