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The tradeoff between noise, data rate, and power consumption of transimpedance amplifiers for optical receivers

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Abstract

The inverter-based shunt-feedback transimpedance amplifier (TIA) has become an essential building block for high-speed receivers for optical interconnects in advanced technologies due to its low operating voltage and high efficiency. Previously, the design of TIA’s optimal noise is based on the relationship between the photodetector’s capacitance, \(C_{{\mathrm {D}}}\), and the TIA’s capacitance, \(C_{{\mathrm {I}}}\). In this paper, we present a method to calculate the accurate size of the inverter-based amplifier, feedback resistance \(R_F\), and load capacitance \(C_o\) for the optimal noise. Next, we further discuss the impact of the quality factor, channel length, and input parasitics on these parameters. Our analysis is applied to 65 nm CMOS technology based on MATLAB calculations. The predicted results agree well with the simulation results, offering valuable interpretations and conclusions that reveal the inherent tradeoffs among noise, data rate, and power consumption in the broadband optical TIA.

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Acknowledgements

The authors would like to thank Kadaba R. (Kumar) Lakshmikumar from Cisco Systems for technical discussions. The data, models or code generated during and analyzed during the current study are available from the corresponding author on reasonable request but can be subject to non-disclosure agreements with semiconductor foundries.

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  1. Columbia University, 500 W. 120th St., Mudd 1310, New York, NY, 10027, USA

    Yudong Zhang & Peter R. Kinget

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  1. Yudong Zhang
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Correspondence to Yudong Zhang.

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This work is supported by U.S. Department of Energy Advanced Research Projects Agency Energy under ENLITENED Grant DE-AR0000843.

Appendix

Appendix

1.1 Derivation of transfer function

In Fig. 3, we use Kirchhoff’s laws to derive

$$\begin{aligned} i_{{\mathrm {in}}}&= s C_{{\mathrm {T}}} v_{{\mathrm {in}}} \!+\! (s C_{{\mathrm {F}}} \!+\! g_{{\mathrm {F}}}) \!(v_{{\mathrm {in}}} \!-\! v_{{\mathrm {o}}}) \end{aligned}$$
(10a)
$$\begin{aligned} (s C_{{\mathrm {F}}} \!+\! g_{{\mathrm {F}}}) \!(v_{{\mathrm {in}}} \!-\! v_{{\mathrm {o}}})&= g_m v_{{\mathrm {in}}} + (s C_{{\mathrm {o}}} + g_{{\mathrm {o}}}) v_{{\mathrm {o}}}, \end{aligned}$$
(10b)

where \(g_{{\mathrm {F}}} = 1/R_{{\mathrm {F}}}\) and \(g_{{\mathrm {o}}} = 1/R_{{\mathrm {o}}}\). (1) follows from (10).

We next explain the approximation of the first equation in (1). First, it is easy to understand \(A_0 R_{{\mathrm {F}}} \gg R_{{\mathrm {o}}}\), so we can omit the term \(R_{{\mathrm {o}}}\) in the numerator. Second, the zero from the numerator is located at

$$\begin{aligned} f_z = \frac{A_0}{2 \pi C_{{\mathrm {F}}} R_{{\mathrm {o}}}} = \frac{g_m}{2 \pi C_{{\mathrm {F}}}} > \frac{g_m}{2 \pi C_{{\mathrm {gs}}}}, \end{aligned}$$
(11)

which is even higher than the CMOS transit frequency, \(f_{{\mathrm {T}}}\). As a result, the approximation is completely reasonable.

1.2 Derivation of input-referred noise current

Fig. 11
figure 11

Schematic for the calculation of input-referred noise

Full size image

First, we calculate the input-referred noise current density and derive (2). Second, we derive the total input-referred noise current through integral in (3).

To calculate the input-referred noise current density, we need to find the input-referred noise for noise sources \(\overline{i_{{\mathrm {n.res}}}^2}\) and \(\overline{i_{{\mathrm {n.D}}}^2}\), and they can be calculated through output noise. As shown in Fig. 11, let us calculate the output voltages \(v_{{\mathrm {o.a}}}\), \(v_{{\mathrm {o.b}}}\), and \(v_{{\mathrm {o.c}}}\) respectively. The admittance \(Y_{{\mathrm {F}}} = 1/Z_{{\mathrm {F}}}\) consists of the parallel of \(R_{{\mathrm {F}}}\) and \(C_{{\mathrm {F}}}\). The admittance \(Y_{{\mathrm {o}}} = 1/Z_{{\mathrm {o}}}\) consists of the parallel of \(R_{{\mathrm {o}}}\) and \(C_{{\mathrm {o}}}\).

From Fig. 11(a), Kirchhoff’s laws yield

$$\begin{aligned} - i_{{\mathrm {n.TIA}}}&= s C_{{\mathrm {T}}} v_{{\mathrm {in.a}}} + Y_{{\mathrm {F}}} (v_{{\mathrm {in.a}}} - v_{{\mathrm {o.a}}}) \end{aligned}$$
(12a)
$$\begin{aligned} 0&= Y_{{\mathrm {F}}} (v_{{\mathrm {in.a}}} - v_{{\mathrm {o.a}}}) + g_m v_{{\mathrm {in.a}}} + Y_{{\mathrm {o}}} v_{{\mathrm {o.a}}}. \end{aligned}$$
(12b)

Solving (12), we get

$$\begin{aligned} i_{{\mathrm {n.TIA}}} = \frac{sC_{{\mathrm {T}}}(Y_{{\mathrm {o}}} + Y_{{\mathrm {F}}}) + (Y_{{\mathrm {o}}} + g_m) Y_{{\mathrm {F}}}}{g_m - Y_{{\mathrm {F}}}} v_{{\mathrm {o.a}}} = H_{{\mathrm {a}}}(s) v_{{\mathrm {o.a}}} \end{aligned}$$
(13)

Similarly, from Fig. 11(b), Kirchhoff’s laws yield

$$\begin{aligned} - i_{{\mathrm {n.res}}}&= s C_{{\mathrm {T}}} v_{{\mathrm {in.b}}} + Y_{{\mathrm {F}}} (v_{{\mathrm {in.b}}} - v_{{\mathrm {o.b}}}) \end{aligned}$$
(14a)
$$\begin{aligned} i_{{\mathrm {n.res}}}&= Y_{{\mathrm {F}}} (v_{{\mathrm {in.b}}} - v_{{\mathrm {o.b}}}) + g_m v_{{\mathrm {in.b}}} + Y_{{\mathrm {o}}} v_{{\mathrm {o.b}}}, \end{aligned}$$
(14b)

and the solution is

$$\begin{aligned} i_{{\mathrm {n.res}}} = \frac{sC_{{\mathrm {T}}}(Y_{{\mathrm {o}}} + Y_{{\mathrm {F}}}) + (Y_{{\mathrm {o}}} + g_m) Y_{{\mathrm {F}}}}{g_m + sC_{{\mathrm {T}}}} v_{{\mathrm {o.b}}} = H_{{\mathrm {b}}}(s) v_{{\mathrm {o.b}}}. \end{aligned}$$
(15)

From Fig. 11(c), Kirchhoff’s laws yield

$$\begin{aligned} 0&= s C_{{\mathrm {T}}} v_{{\mathrm {in.c}}} + Y_{{\mathrm {F}}} (v_{{\mathrm {in.c}}} - v_{{\mathrm {o.c}}}) \end{aligned}$$
(16a)
$$\begin{aligned} i_{{\mathrm {n.D}}}&= Y_{{\mathrm {F}}} (v_{{\mathrm {in.c}}} - v_{{\mathrm {o.c}}}) + g_m v_{{\mathrm {in.c}}} + Y_{{\mathrm {o}}} v_{{\mathrm {o.c}}}, \end{aligned}$$
(16b)

and the solution is

$$\begin{aligned} i_{{\mathrm {n.D}}} = \frac{sC_{{\mathrm {T}}}(Y_{{\mathrm {o}}} + Y_{{\mathrm {F}}}) + (Y_{{\mathrm {o}}} + g_m) Y_{{\mathrm {F}}}}{Y_{{\mathrm {F}}} + sC_T} v_{{\mathrm {o.c}}} = H_{{\mathrm {c}}}(s) v_{{\mathrm {o.c}}}. \end{aligned}$$
(17)

With \(v_{{\mathrm {o.a}}} = v_{{\mathrm {o.b}}} = v_{{\mathrm {o.b}}}\) and \(Y_{{\mathrm {F}}} = sC_{{\mathrm {F}}} +1/R_{{\mathrm {F}}}\), we can get the ratio

$$\begin{aligned} \begin{aligned} \,{\mathrm {d}}\overline{i_{{\mathrm {n.TIA}}}^2}&= \frac{|H_{{\mathrm {a}}}(s)|^2}{|H_{{\mathrm {b}}}(s)|^2} \,{\mathrm {d}}\overline{i_{{\mathrm {n.res}}}^2} + \frac{|H_{{\mathrm {a}}}(s)|^2}{|H_{{\mathrm {c}}}(s)|^2} \,{\mathrm {d}}\overline{i_{{\mathrm {n.D}}}^2} \\&= \frac{|g_m + sC_{{\mathrm {T}}}|^2}{|g_m - Y_{{\mathrm {F}}}|^2} \,{\mathrm {d}}\overline{i_{{\mathrm {n.res}}}^2} + \frac{|Y_{{\mathrm {F}}} + sC_{{\mathrm {T}}}|^2}{|g_m - Y_{{\mathrm {F}}}|^2} \,{\mathrm {d}}\overline{i_{{\mathrm {n.D}}}^2} \\&= \frac{|A_{{\mathrm {o}}} R_{{\mathrm {F}}} + sC_{{\mathrm {T}}}R_{{\mathrm {F}}} R_{{\mathrm {o}}}|^2}{|A_{{\mathrm {o}}} R_{{\mathrm {F}}} - R_{{\mathrm {o}}} - sC_{{\mathrm {F}}} R_{{\mathrm {F}}} R_{{\mathrm {o}}}|^2} \,{\mathrm {d}}\overline{i_{{\mathrm {n.res}}}^2} \\& {} + \frac{|R_{{\mathrm {o}}} + s(C_{{\mathrm {T}}}+C_{{\mathrm {F}}})R_{{\mathrm {F}}} R_{{\mathrm {o}}}|^2}{|A_{{\mathrm {o}}} R_{{\mathrm {F}}} - R_{{\mathrm {o}}} - sC_{{\mathrm {F}}} R_{{\mathrm {F}}} R_{{\mathrm {o}}}|^2} \,{\mathrm {d}}\overline{i_{{\mathrm {n.D}}}^2}. \\ \end{aligned} \end{aligned}$$
(18)

To better understand the result, if we apply \(C_{{\mathrm {F}}} = 0\) and \(R_{{\mathrm {F}}} \gg R_{{\mathrm {o}}}\) to (18), we reach the same equation as in [31].

Now we derive the total input-referred noise current through integral.

$$\begin{aligned} \overline{i_{{\mathrm {n.TIA}}}^2} = \int \frac{|Z_{{\mathrm {T}}}(s)|^2}{|Z_{{\mathrm {T}}}(0)|^2} \,{\mathrm {d}}\overline{i_{{\mathrm {n.TIA}}}^2} \end{aligned}$$
(19)

Using the first equation in (1) without any approximation, we can get (3).

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Zhang, Y., Kinget, P.R. The tradeoff between noise, data rate, and power consumption of transimpedance amplifiers for optical receivers. Analog Integr Circ Sig Process 108, 437–446 (2021). https://doi.org/10.1007/s10470-021-01895-y

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