# On Noise Modeling of Capacitive Feedback Transimpedance Amplifiers in CMOS
## 2 Analytical Methods  
Although the classical resistive-feedback TIA provides a fair balance of the most important TIA parameters, it has an inherent problem with the noise performance of the feedback resistor \(R_{\text{F}}\) as its noise is directly added to the input-referred noise current and drastically degrades the noise figures of this well-accepted design. Although numerous works have been reported on modifications to this established TIA topology, the major issue with this design is that all major circuit characteristics depend entirely on the value of \(R_{\text{F}}\). Here, a larger value of \(R_{\text{F}}\) is needed for lower noise and an improved gain, while a smaller value is required for designs with a larger bandwidth. Finally, technological constraints are also important, as the large value of \(R_{\text{F}}\) may be limited by the maximum realizable on-chip resistance and parasitic capacitances [25]. Thus, by striving for a better TIA performance, it may be inevitable to introduce some structural changes to this reference TIA design in order to obtain an additional degree of freedom for balancing noise, gain, and bandwidth. On the other hand, the basic feedback structure of the TIA itself has important structural benefits as it ensures almost constant transimpedance gain in the bandwidth of interest while decreasing the sensitivity to process and temperature variations. Therefore, it may be still beneficial to keep the general feedback structure of the amplifier, while substituting the noisy feedback resistor with something which is less noisy. An intuitive candidate for such a noise-free element is a capacitor, but a direct replacement of \(R_{\text{F}}\) with a capacitor will result in a phase shift which requires a subsequent stage for phase correction. Such TIAs formed by an integrator and cascaded differentiator, also often called capacitive feedback TIAs, have been also reported and have their own advantages (see, e.g., [25; 26] for further details). However, in this work, we follow an alternative approach where the feedback network is formed by a dedicated network of two capacitors \(C_{1}\) and \(C_{2}\)[24] and is shown in Figure 1 with separate blocks corresponding to the feedback circuit (a), the forward gain (b), and the source follower (c). An advantage of the proposed approach that it does not require the second-stage differentiator. The development of the analytical noise model is provided for the programmable-gain TIA configuration as required when implementing the front-end amplifier according to OTDR requirements [22; 23]. The necessity for the programmable-gain configuration is  
Figure 1: The general structure of the proposed capacitive feedback TIA with major blocks: (**a**) feedback block, (**b**) forward-gain block; (**c**) source-follower block (adopted with changes from [22]). caused by an inherent relationship between the responsivity of the PD and the specification for the dynamic range and gain of the measurement equipment. With the amplifier, which is able to support a wide range of gain-bandwidth configurations, one can effectively use the same circuit with different PDs enabling varying operating modes for the very same device. This also includes support for operations with both very weak optical signals and short optical pulses. The mid-frequency value of the transimpedance gain \(R_{\mathrm{T}}\) for a generic capacitive feedback architecture can be, in theory, approximated as [1]:  
\[R_{\mathrm{T}}=\left(1+\frac{C_{1}}{C_{2}}\right)R_{2}. \tag{1}\]  
This expression for the transimpedance gain is valid under the assumption of an infinite gain \(A\), and the overall gain of this TIA circuit is defined by the feedback circuit formed solely by the two feedback capacitors \(C_{1}\) and \(C_{2}\). It was also recently proved that the performance of the capacitive feedback TIA in terms of Pareto frontier [27] is always greater than that of the resistive-feedback TIA, reaching four to five times for low output capacitance [28].