Transistor mismatch effect on common-mode gain of cross-coupled amplifie

In this paper, the analytical approach of MOS transistor mismatch effect on common-mode gain of cross-coupled amplifier is presented. Transconductance (MOS transistor parameter) mismatch effect on common-mode gain of cross-coupled amplifier was analyzed. This study was started with mathematical derivation for representing the mismatch effect of transconductance between 2 differential pairs of crosscoupled amplifier due to common-mode voltage. The derivation result was simulated based on Monte Carlo simulation with random transconductance mismatch rate from 0.05% until 1%. The common-mode gain increases 36.9 dB and average common-mode gain is -81.1 dB. The transconductance mismatch rate increases followed by increase in common-mode gain. The results can be used by circuit designers to design analog circuits, especially operational amplifier used for biosignals processing to minimize the common-mode gain of their circuits. This research presents aid to circuit designers to improve their circuits performance.


Introduction
CMOS cross-coupled amplifier is frequently used in various application. For example, Fully Balanced Differential Difference Amplifier (FBDDA) is mainly constructed by using Cross-Coupled Amplifier (CCA) [1][2][3][4]. MOS transistor mismatch often happens after fabrication until 3%, depends on the fabrication technology [5]. There are some MOS transistor parameters which can be considered, such as mismatch in mobility, W/L ratio, oxide capacitance and threshold voltage. Furthermore, the W/L ratio corresponds to transconductance (g m ) [6].
As shown in Figure 1, biosignals, such as Electroencephalogram (EEG), Electrooculogram (EOG), Electrocardiogram (ECG), Electromyogram (EMG) are widely used for medical applications and have small amplitude and low frequency. The biosignals amplitude and frequency are in the order of µV to mV and DC to a few kHz, respectively [7][8][9][10][11][12]. Common-Mode Rejection Ratio (CMRR) is ratio of differential gain and common-mode gain (A cm ). As parameter of CMRR, A cm is important to be reduced when main circuit like CCA is applied for biosignals. The smaller the A cm the better the CMRR [13][14][15]. Furthermore, common-mode voltage (V cm ) can be caused by power-line interference which normally has frequency of 50Hz or 60 Hz, depends on local power-line frequency. Therefore, the V cm becomes problem for biosignal processing [16][17][18][19][20]. In this paper, study of g m mismatch effect on A cm of CCA by large signal analysis (mathematical derivation) and simulation is presented. Section 2 discusses the mathematical derivation of circuit analysis of CCA based on the effect of g m mismatch on its A cm . Section 3 describes simulation results representing g m mismatch effect on A cm of CCA. Finally, section 4 concludes this study.

Transconductance Mismatch Effect on Cross-coupled Amplifier
In some text books and literatures, CCA can be realized as shown in Figure 2 [2], [21,22]. It consists of 2 differential pairs which matching between MOS transistors is very important to achieve low common-mode gain. In order to simplified the circuit analysis, R s and R d are used with well-matched condition. MOS transistor mismatch often occurs due to W/L ratio which is designed by circuit designer [23]. Furthermore, W/L ratio has relationship with g m as mentioned in (1) [24,25]. Therefore, the effect of g m mismatch on 2 differential pairs of CCA is analyzed in this section.
current flows through R s can be defined as follows: on the other hand, V op and V on can be derived as follow: output voltage ( out V  ) can be calculated as follows: Where and stand for 2 (2 + Δ 2 ) and 4 (2 + Δ 4 ) , respectively. From the above derivation, the A cm of CCA is proportional to mismatch of g m . Based on (11), the A cm is finite, so that some amount of V cm may pass through the CCA. If the CCA is applied to enhanced amplifier, the V cm will be amplified and deteriorate performance of the enhanced amplifier.

Simulation Result
In this section, the g m mismatch effect was evaluated using simulations. Referring to (11), the simulations were done based on the condition on Table 1. Well-matched resistors R s and R d were given with the value of 100 kΩ and 2 kΩ, respectively. Mismatch rate of both g m1 and g m2 are set in the range of 0,05% -1%.

Effect of Increasing Mismatch Rate to A cm
In this subsection the effect of increasing ∆ is considered and in the next subsection the ∆ is extended to random condition. Assuming the ∆ of both g m1 and g m2 increase from 0,05% until 1%. Figure 3 shows simulation result based on the increasing ∆. From the simulation result, the A cm increases from -100 dB until -74 dB. It means the A cm increases 26 dB by the increase of ∆ about 1%. This means when ∆ is getting bigger, so is the A cm .

Effect of Random Mismatch Rate to A cm
In order to realize the effect of random mismatch rate, Monte Carlo simulation was used. It was done by 500 times with random and different value of ∆ 2 and ∆ 4 to get data of A cm . Figure 4 shows histogram of the A cm . Because of random ∆ 2 and ∆ 4 from 0.05% until 1%, the A cm varies from -111.2 dB to -74.3 dB. From 500 random configuration of ∆ 2 and ∆ 4 values, most of A cm is in the range from -75 dB to -85 dB, therefore average A cm is -81.1 dB. The random mismatch rate from 0.05% until 1% increases A cm 36.9 dB.

Conclusion
The effect of transistor mismatch on common-mode gain of cross-coupled amplifier has been presented and identified based on mathematical derivation and simulation. The common-mode gain increases along with the increase in transconductance mismatch rate. The results can be used by analog circuit designer to design enhanced operational amplifier with low transistor mismatch effect on common-mode gain.
In the future, mathematical derivation and simulation based on multiple mismatch of MOS transistor parameters such as R s , R d , threshold voltage, mobility, and oxide capacitance will be considered in order to represent actual condition. Furthermore, development of common-mode gain reducer and layout technique are also necessary.