# On Noise Modeling of Capacitive Feedback Transimpedance Amplifiers in CMOS
## 2 Analytical Methods
The first noise source is due to the resistor \(R_{\mathrm{bias,1}}\) in the biasing circuit of the input stage. Differently from other works, where the performance of the biasing circuit is considered to be perfect and contributes no noise to TIA, we may not consider this noise source to be negligible in the real circuit. Thus, the noise contribution due to \(R_{\mathrm{bias,1}}\) becomes  
\[\frac{\overline{i_{\mathrm{nr1}}^{2}}}{\Delta f}=\frac{4\mathrm{k_{B}}T}{R_{ \mathrm{bias,1}}}, \tag{4}\]  
and is directly added to the input noise current. Additionally, we also have the noise component due to \(R_{2}\). In this case, however, the component is not directly added to the input but is suppressed by the approximate gain due to capacitive feedback:  
\[\frac{\overline{i_{\mathrm{nr2}}^{2}}}{\Delta f}=\frac{4\mathrm{k_{B}}TR_{2}}{ Z_{\mathrm{T}}^{2}}\approx\frac{4\mathrm{k_{B}}T}{R_{2}\Big{(}1+\frac{C_{1}}{C_{2}} \Big{)}^{2}}. \tag{5}\]  
Figure 2: A circuit-level implementation of a fixed-gain capacitive feedback TIA and contributing noise sources: (1) thermal noise due to \(R_{\mathrm{bias,1}}\), (2) thermal noise due to \(R_{2}\), (3) \(M_{1}\) thermal noise, (4) \(M_{1}\) flicker noise, and (5) \(M_{8}\) flicker noise (adapted from [22]). Even though the approximation above may not be valid in the general case, it is clear that the noise contribution due to \(R_{2}\) is damped by a significant factor for the frequencies where the TIA gain is significant. Thus, the final thermal noise component consists of two contributions: noise current caused by \(R_{\text{bias,1}}\) in the front-end biasing circuit for \(M_{1}\), and scaled noise from \(R_{2}\):  
\[\frac{\overline{i_{\text{nr}}^{2}}}{\Delta f}=\frac{\overline{i_{\text{nr1}}^{ 2}}}{\Delta f}+\frac{\overline{i_{\text{nr2}}^{2}}}{\Delta f}=\frac{4\text{k} _{\text{B}}T}{R_{\text{bias,1}}}+\frac{4\text{k}_{\text{R}}TR_{2}}{Z_{\text{T} }^{2}}. \tag{6}\]  
There are several noise sources to be evaluated when it comes to the contribution of the metal oxide semiconductor (MOS) transistor. The two most important noise sources of the transistor are the thermal noise and the \(1/f\) noise, if we ignore the secondary effects such as the shot noise due to the leakage current of the drain-source reverse diodes. The thermal noise of the channel can be represented by a voltage source in series with a gate and the voltage spectral density:  
\[\frac{\overline{v_{\text{nr}}^{2}}}{\Delta f}=\frac{8}{3}\frac{\text{k}_{ \text{B}}T_{\text{gd0}}}{g_{\text{m}}^{2}}, \tag{7}\]  
where \(g_{\text{d0}}\) is the zero-bias drain conductance. If we approximate \(g_{\text{d0}}\) with the transconductance \(g_{\text{m}}\), a somewhat simpler expression can be obtained [33]:  
\[R_{\mathrm{T}}=\left(1+\frac{C_{1}}{C_{2}}\right)R_{2}. \tag{1}\]0  
For the design shown in Figure 2, we may consider this noise component only for the transistor \(M_{1}\) with \(g_{\text{m,1}}\). As the charge carriers move at the interface, so-called flicker noise is caused by the random trapping of the charges. This noise contribution typically varies from process to process and depends on the impurities at the oxide-silicon interface. The noise is typically modeled by a current source across the drain, with the spectral density as  
\[R_{\mathrm{T}}=\left(1+\frac{C_{1}}{C_{2}}\right)R_{2}. \tag{1}\]1  
where \(W\) and \(L\) are, correspondingly, the width and the length of the channel, \(K_{\text{f}}\) is the flicker noise constant, and \(C_{\text{ox}}\) is the gate oxide capacitance per unit area. When referred as input, the noise is represented by a voltage source between gate and source with  
\[R_{\mathrm{T}}=\left(1+\frac{C_{1}}{C_{2}}\right)R_{2}. \tag{1}\]2  
Recall that the value of \(K_{\text{f}}\) of PMOS devices is an order lower than that of NMOS devices, and the former devices also exhibit less flicker noise. As this noise component has an inverse dependency on frequency, it is also often called \(1/f\) noise (yet another name for pink noise).