# On Noise Modeling of Capacitive Feedback Transimpedance Amplifiers in CMOS
## 3 Mathematical and Computer Simulation Results
By adding the flicker noise component for the input transistor \(M_{1}\), we obtain  
\[\frac{\overline{\overline{i_{\text{nr}}^{2}}}}{\Delta f}=\frac{1}{Z_{\text{ ineff}}^{2}}\frac{K_{\text{f}}}{f}\frac{1}{C_{\text{ox}}WL}. \tag{22}\]  
Figure 4: Analytically calculated input-referred noise model for programmable-gain configuration with \(M_{1}\) thermal noise component and noise due to \(R_{\text{bias,1}}\) (example case for TSMC 0.18 \(\upmu\)m CMOS). Figure 5: Analytically calculated input-referred noise model for programmable-gain configuration with \(M_{1}\) thermal noise component and noises due to \(R_{\text{bias,1}}\) and \(R_{2}\) (example case for TSMC 0.18 \(\upmu\)m CMOS). The parameter \(K_{\mathrm{f}}/C_{\mathrm{ox}}WL\) for \(M_{1}\) gives us around 10 \(\upmu\)A with the model performance containing four noise components shown in Figure 6. The impact of the flicker noise can be clearly seen in the transition region between middle and high frequencies, where the famous \(1/f\) characteristic of the pink noise is hidden behind the shaped noise due to \(R_{2}\). The \(1/f\) noise, due to its square-root dependency in PSD, gives the 10 dB/decade drop which is only partially seen in the transition region. We can complete the noise model by adding the last component due to the flicker noise of \(M_{5}\). The shaping filter for this noise is shown in Figure 7 and is simply the scaled inverse of our designed transimpedance \(Z_{\mathrm{T}}\). It also becomes obvious that the noise has a significant impact on extremely low frequencies only. Figure 8 shows the overall noise model. \[\frac{\overline{i_{\text{n}t}^{2}}}{\Delta f}+\frac{\overline{i_{\text{n}t}^{ 2}}}{\Delta f}\approx\frac{1}{Z_{\text{IN,eff}}^{2}}\frac{8}{3}\frac{\text{k}_ {\text{B}}T}{g_{\text{m},1}}+\frac{4\text{k}_{\text{B}}T}{R_{\text{bias,1}}}. \tag{19}\]0 Analytically calculated input-referred noise model for programmable-gain configuration with \(M_{1}\) thermal noise component, noises due to \(R_{\mathrm{bias},1}\), \(R_{2}\) and flicker noise on \(M_{1}\) (example case for TSMC 0.18 \(\upmu\)m CMOS). \[\frac{\overline{i_{\text{n}t}^{2}}}{\Delta f}+\frac{\overline{i_{\text{n}t}^{ 2}}}{\Delta f}\approx\frac{1}{Z_{\text{IN,eff}}^{2}}\frac{8}{3}\frac{\text{k}_ {\text{B}}T}{g_{\text{m},1}}+\frac{4\text{k}_{\text{B}}T}{R_{\text{bias,1}}}. \tag{19}\]1 Shaping filter due to \(G_{\mathrm{MS}}\) (example case for TSMC 0.18 \(\upmu\)m CMOS). After we demonstrated how the noise model is constructed from contributions of separate components, we are in the position to compare the proposed analytical models with the Cadence-based simulation using exactly the parameter values from the suggested design. The results for 1.8 V TSMC 0.18 \(\upmu\)m CMOS and 2.5 V IHP 0.25 \(\upmu\)m BiCMOS can be found in Figures 9 and 10, correspondingly. The slightly better noise performance of BiCMOS can also be partly explained by the fact that this technology employs a 2.5 V power supply and, therefore, voltage headroom problems are easier to avoid. In the case of CMOS, a relatively good fit of the analytical model and numerical simulation can be seen for both low- and high-frequency ranges, where the mismatch in the noise minimum point is below 10% at the frequency of 500 MHz for the 10 k\(\Omega\) gain. Similarly, a good match can be also seen for the noise model at other gain configurations. A slight mismatch between the developed noise model can be found in the transition from the low- to mid-frequency ranges, where the post-layout simulated noise exceeds the prediction of the model. While the same is true for the BiCMOS-based implementation, the latter is also better suited to the problematic segment from the low- to mid-frequency range. Still, the mismatch in the noise minimum point is a bit larger and reaches 15% at the frequency of 500 MHz for the base gain case. Additionally to the effects discussed before, we also clearly see the famous \(1/f\) behavior of the \(M_{1}\) flicker noise in the low-frequency region.