# Fast Calculation of Transimpedance Gain and Equivalent Input Noise Current Density for High-speed Optical Preamplifier Design

#### Analytical expressions for the relationships between the transimpedance gain and S-parameters, equivalent input noise current density and noise figure for high-speed optical transimpedance amplifier design are proposed in this article. This technique i...

Optoelectronic integrated circuits (OEIC) have attracted the interest of many researchers because of their important role in the hardware for information technology. The optical receivers have key roles in high-speed optical fiber communications, in high-speed chip-to-chip interconnections in computers, efficient networking between computers and in other diverse areas such as medical imaging. One of the most critical building blocks in an optical link system is the transimpedance amplifier (TIA), which converts the photodiode (PD) current into an amplified voltage. The requirements for a typical TIA are high bandwidth, high transimpedance gain, adequate power gain, low noise, low input impedance, small area and low power consumption for array applications.^{1-4}

The major design goals of the TIAs are the transimpedance gain and equivalent input noise current density. The transimpedance gain of the TIAs must be large enough to overcome the noise of the subsequent stage, typically a 50 Ω driver or a limiting amplifier. The equivalent input noise current density determines the minimum input current that yields a given bit error rate, directly impacting the link budget. Unfortunately, the transimpedance gain and equivalent input noise current density cannot be measured directly from microwave and noise equipment, while the S-parameters and noise figure of the TIAs can be measured with a vector network analyzer (VNA) and a noise figure meter in a straightforward manner. Therefore, a fast transformation between S-parameters/noise figure and transimpedance gain/equivalent input noise current density is needed.

Analytical expressions for trans-impedance gain and equivalent input noise current of optical receivers have been derived^{5} and simple expressions for the relationship between the transimpedance gain and Z-parameters are given.^{6-8} However, these expressions are not always valid and the transimpedance gain and equivalent input noise current density cannot be directly and accurately calculated from S-parameters and noise figure measurement data. Based on the author’s knowledge, a comprehensive analysis for fast transformation between S-parameters/noise figure and transimpedance gain/equivalent input noise current density has not yet been published.

In this article, a simple but efficient transformation technique for TIAs is proposed and the analytical expressions for the relationships between the transimpedance gain and S-parameters, the equivalent input noise current density and the noise figure for high-speed optical transimpedance preamplifier design are derived. This technique is based on the signal and noise equivalent circuit model of the optical receiver front-end.

In contrast with previous publications,^{5-8} this method has the following advantages:

- The transimpedance gain can be directly derived from S-parameters for arbitrary source and load impedances and simplified expressions for two special cases (source impedances are zero and 50 Ω) are also given.
- The equivalent input noise current density can be determined from noise figure measurements without four noise parameters (minimum noise figure, noise resistance and optimum source reflection coefficient) of TIA in 50 Ω and non-50 Ω systems.

**Figure 1** Simulated model of the optical receiver front-end.

### Theoretical Analysis

**Transimpedance Gain**

The schematic of the optical receiver front-end circuit is shown in **Figure 1**, where Y_{S} is the photodiode (PD) input admittance. Typically this will be that of the PIN/APD and it is almost totally capacitive (that is Y_{S} = jωC_{pd}). Y_{L} is the load admittance, which is generated by the input admittance of the next stage (typically Y_{L} = Y_{0} = 0.02 mS).

The transimpedance is defined as the magnitude of the ratio of the output voltage V_{2} at a load impedance and the photocurrent through the photodiode i_{s}. Based on the small-signal circuit model analysis for an optical receiver frond-end and applying Kirchhoff’s current law, the transimpedance gain of the optical receiver front-end can be expressed as:

The relationship between the S-parameters and Y-parameters can be expressed as:

where Y_{0} ( = 0.02 mS) is the characteristic admittance of the system.

Substituting Equations 2 to 5 into Equation 1:

If the source impedance is infinite (that is Y_{S} = 0), and the output end of the TIA is connected to a matched load (that is Y_{L} = Y_{0} = 0.02 mS), the corresponding transimpedance gain of the receiver front-end can be simplified as follows:

when the TIA is operated in a 50 Ω system (Y_{L} = Y_{S} = Y_{0} = 20 mS), the transimpedance gain can be written as:

The transimpedance gain and bandwidth versus Y_{S} and Y_{L} are summarized in **Table 1**. It can be observed that the transimpedance 3 dB bandwidth can be determined from the forward transmission coefficient S_{21} only when the input and output ports are terminated in matched loads. The physical meanings of the three transimpedance gains mentioned above are as follows: Z_{T} is the transimpedance gain of whole optical receiver front-end (PD+TIA); is the transimpedance gain of TIA; and is proportional to the power gain S_{21} of the TIA.

**Figure 2** Noise model of the 50 Ω noise figure measurement system for an optical preamplifier.

**Equivalent Input Noise Current Density of the TIA**

**Figure 2** shows the noise model of the noise figure measurement system for an optical preamplifier. It is noted that 50 Ω standard resistances have been used for source and load impedances (Z_{S}=Z_{L}=Z_{0}=50 Ω). and are the input and output impedances of the TIA, respectively. is the total output noise voltage density, and is the equivalent input noise current density of the TIA.

The noise figure of the TIA can be expressed as follows:^{10}

Where A_{v} is the voltage gain and can be expressed as:

Where is the transimpedance of the TIA . The corresponding equivalent input noise current density of the TIA can be derived as follows:

**Figure 3** Noise model of the optical receiver front-end.

**Equivalent Input Noise Current Density of the Optical Receiver Front-end**

**Figure 3** shows the noise model of a typical optical receiver front-end in a non-50 Ω system, where is the total equivalent input noise current density and is the total output noise voltage density of the receiver front-end. It is noted that the input port of the TIA is connected to the PD, not the matched load. Therefore, Equation 13 is only valid for TIA design, not for the whole receiver front-end.

Assuming that the total output noise voltage density is generated mainly by the TIA (here the noise contribution of the PD is neglected), the equivalent input noise current density of the receiver front-end can be expressed as follows:

Traditionally, the output end of TIA is connected to a matched load (Y_{l} =Y_{o}=0.02 mS), then can be written as:

when the TIA is operated in a 50 Ω system (Y_{L} = Y_{S} = Y_{0} = 20 mS), the equivalent input noise current density can be simplified:

It is noted that Equation 16 is the conventional formula for predicting the equivalent input noise current density for a TIA IC. The is dependent on the noise figure only, and independent on the S-parameters of the TIA. The equivalent input noise current density versus Y_{S} and Y_{L} is summarized in **Table 2**.

**Figure 4** Schematic of the 10 Gb/s HEMT-based TIA IC.

### Experimental Analysis

In order to demonstrate the expressions derived in the Theoretical Analysis section for the TIA, a HEMT-based TIA, which operates at 10 Gb/s, has been designed using a 0.2 µm PHEMT process.^{11} **Figure 4** shows a schematic of the developed TIA IC, using both enhancement- and depletion-mode (E-D) transistors. This IC consists of three parts: a parallel-feedback amplifier core, a source-follower buffer and an output match stage. The source-follower buffer improves the flatness of the gain-frequency characteristics by separating the parallel-feedback loop from the large input capacitance of the output buffer (that is eliminating the Miller capacitance loading to the previous stages). The output stage is designed for a 50 Ω output impedance match. **Figure 5** shows the experimental setup. All measurements were carried out on-wafer, using Air-Coplanar Probes. The wafer probes were calibrated using the Line-Reflect-Match (LRM) calibration method for S-parameter measurement. The noise parameter measurement method proposed here has been tested on wafer up to 26 GHz. The corresponding chip photograph of the 10 Gb/s HEMT TIA is shown in **Figure 6**.^{11}

**Figure 5** Experimental setup.

**Figure 6** Photograph of the 10 Gb/s HEMT TIA chip.

**Figure 7** shows the measured magnitudes and phases of the S-parameters of the TIA IC. The high gain |S_{21}| of 25 dB and the broad 3 dB bandwidth over 10.8 GHz have been obtained. Good matching is also achieved, |S_{11}| less than -10 dB and |S_{22}| less than -7 dB for the whole frequency range. The corresponding noise figure versus frequency is shown in **Figure 8**.

**Figure 7** S-parameters of the 10 Gb/s HEMT-based TIA IC: (a) magnitude and (b) phase.

**Figure 8** Measured noise figure of the 10 Gb/s HEMT-based TIA IC.

The transimpedance gain (TG) and equivalent input noise current density (EINCD) can be obtained from AC and noise signal analysis by using commercial circuit design tools (such as SPICE). However, it is difficult to measure them directly using conventional microwave signal and noise measurement systems. Alternatively, they can be calculated from S-parameters and noise figure measurements by using the proposed transformation expressions.

**Figure 9** Transimpedance gain vs. frequency for the 10 Gb/s TIA.

**Figure 10** 3 dB bandwidth of the transimpedance gain vs. capacitance of PD.

**Figure 9** shows the transimpedance gain, which is derived from the measured S-parameters of the 10 Gb/s TIA; the corresponding 3 dB bandwidth versus capacitance of the PD is shown in **Figure 10**. It can be found that the 3 dB bandwidth of the TIA transimpedance gain is approximately 11.5 GHz, and the corresponding optical receiver front-end 3 dB bandwidth decreases with the increase of the capacitance of the PD (Y_{S} = jωC_{pd}). As long as the capacitance of the PD is kept less than 0.6 pF (3 dB bandwidth of 8 GHz), the proposed TIA can be operated at a 10 Gb/s bit rate. **Figure 11** shows the comparison of the predicted transimpedance gain for the 10 Gb/s TIA obtained from Equations 7 and 8. It is obvious that the gain and bandwidth predicted by Equation 7 is better than that of a TIA operating in a matching system. That means the transimpedance gain will be underestimated using the conventional formula.

**Figure 11** Comparison of transimpedance gains for 10 Gb/s TIA.

**Figure 12** Equivalent input noise current density. (EINCD) vs. frequency for the 10 Gb/s TIA

**Figure 12** shows the EINCD, which is derived from the measured noise figure of 10 Gb/s TIA. The corresponding average values versus capacitance of the PD are shown in **Figure 13**. It can be found that equivalent input noise current density increases with the increase of capacitance of the PD (Y_{S} = jωC_{pd}). As long as the capacitance of the PD is kept less than 0.3 pF, the equivalent input noise current density of the proposed TIA can be less than **Figure 14** shows the comparison of predicted equivalent input noise current density for the 10 Gb/s TIA by using the proposed Equation 13 and the conventional Equation 16. It is obvious that the equivalent input noise current density predicted by Equation 13 is better than that predicted by the conventional formula. This means the equivalent input noise current density will be overestimated using the conventional formula.

**Figure 13** Equivalent input noise current density (EINCD) vs. capacitance of PD.

**Figure 14** Comparison of the equivalent input noise current density (EINCD) for the 10 Gb/s TIA.

### Conclusion

Analytical expressions for the relationships between the transimpedance gain from S-parameters, equivalent input noise current density and noise figure for high-speed optical transimpedance preamplifier design is proposed in this article. The validity of the new approach is proven by using a 10 Gb/s high electron mobility transistor (HEMT)-based transimpedance preamplifier.

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**Jianjun Gao** *received his BEng and PhD degrees from Tsinghua University in 1991 and 1999, respectively, and his MEng degree from the Hebei Semiconductor Research Institute in 1994. From 1999 to 2001, he was a post-doctoral research fellow at the Microelectronics R&D Center, Chinese Academy of Sciences, developing PHEMT optical modulator drivers. In 2001, he joined the school of Electrical and Electronic Engineering, Nanyang Technological University (NTU), Singapore, as a research fellow in semiconductor device modeling and on wafer measurement. In 2003, he joined the Institute for High-Frequency and Semiconductor System Technologies, Berlin University of Technology, Germany, as a research associate working on InP HBT modeling and circuit design for high speed optical communication. In 2004, he joined the Electronics Engineering Department, Carleton University, Canada, as a post-doctoral fellow, working on the semiconductor neural network modeling technique. From 2004 to 2007, he was a full professor in the radio engineering department at Southeast University, Nanjing, China. Since 2007, he has been a full professor in the school of information science and technology, East China Normal University, Shanghai, China. His main areas of research include characterization, modeling and on wafer measurement of microwave semiconductor devices, optoelectronics device and high-speed integrated circuit for radio frequency and optical communication.*