Patent Publication Number: US-2023146711-A1

Title: Ultra-Low Power Readout Circuit With High-Voltage Bias Generation For MEMS Accelerometer

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Application No. 63/263,778, filed on Nov. 9, 2021. The entire disclosure of the above application is incorporated herein by reference. 
    
    
     FIELD 
     The present disclosure relates to an ultra-low power readout circuit with high-voltage bias generation and mismatch compensation for MEMS accelerometers. 
     BACKGROUND 
     MEMS (microelectromechanical systems) accelerometers with on-chip CMOS readout circuits (ROIC) are becoming increasingly attractive for IoT monitoring of objects or gestures due to their miniaturized volume and low noise operation. Consisting of a micro-mechanical spring-mass system, MEMS accelerometers are capable of high acceleration sensitivity while maintaining good linearity, low Brownian (mechanical) noise, good temperature consistency and miniaturized volume. 
     However, typical MEMS accelerometers have a fundamental trade-off between the resolution and power consumption, limiting their use to applications that require both high resolution and a high-power budget, or applications where resolution can be sacrificed to accommodate a low power budget. The reason behind this is the trade-off between readout circuit noise and power. High-resolution accelerometers require an ultra-low noise floor for their readout circuit so that the signal-to-noise ratio does not limit the overall resolution. As a result, low-noise amplifiers and signal chopping techniques are typically adopted to reduce thermal noise and flicker noise, respectively, creating a trade-off between the benefit they provide and the circuit power required for their operation. 
     This section provides background information related to the present disclosure which is not necessarily prior art. 
     SUMMARY 
     This section provides a general summary of the disclosure, and is not a comprehensive disclosure of its full scope or all of its features. 
     A motion sensing system is presented. The motion sensing system includes: a motion sensor having a proof mass and cantilever beams, producing acceleration signals with its capacitance change, a bias circuit interfaced with the motion sensor and operable to supply a bias voltage to the motion sensor, where the magnitude of the bias voltage is such that the motion sensor signal is larger than flicker noise associated with the readout circuit, and a readout circuit having a low-power input amplifier coupled to output terminals of the motion sensor without signal chopping. The bias circuit includes at least one Dickson charge pump. The bias voltage supplied by the bias circuit includes a positive bias voltage and a negative bias voltage, such that magnitude of the positive bias voltage differs from the magnitude of the negative bias voltage. The bias circuit is further configured to supply a positive bias voltage and a negative bias voltage to the motion sensor, wherein the difference between the magnitude of the positive bias voltage and the magnitude of the negative bias voltage is set to compensate for the process mismatch in the motion sensor. 
     Further areas of applicability will become apparent from the description provided herein. The description and specific examples in this summary are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure. 
    
    
     
       DRAWINGS 
       The drawings described herein are for illustrative purposes only of selected embodiments and not all possible implementations, and are not intended to limit the scope of the present disclosure. 
         FIG.  1    is a diagram of a proposed architecture for a low power motion sensing system. 
         FIG.  2 A  is a simplified diagram of a fully differential MEMS capacitive accelerometer. 
         FIG.  2 B  is zoomed-in diagram showing the coupling capacitance between the MEMS proof-masses and electrodes. 
         FIG.  3    is a diagram of the proposed high-voltage biasing scheme. 
         FIG.  4 A  is a stress analysis of the proof-mass considering both mechanical force, F m , and electrostatic force, F e . 
         FIG.  4 B  is a chart showing the change of the MEMS/accelerometer sensitivity as the bias voltage, V B , increases under a fixed acceleration. 
         FIG.  5 A  is a stress analysis of the proof-mass in a differential MEMS structure. 
         FIG.  5 B  is a chart showing the change in MEMS/accelerometer sensitivity with bias voltages, V B , for systems with and without electrostatic mismatch compensation (EMC). 
         FIG.  6    is a circuit diagram showing an example embodiment of a high-voltage bias circuit. 
         FIG.  7 A  shows the structure of a voltage divider with serial switched capacitors. 
         FIG.  7 B  is a diagram of a V B+  sampling and division circuit with a separated dirty V B+  to pre-charge the sampling nodes and reduce V B+  ripples. 
         FIG.  7 C  is a diagram of the V B+  and V B−  average circuit 
         FIG.  7 D  is a conceptual waveform showing the transient voltages of  7 B and  7 C. 
         FIG.  8    is a diagram of the pull-in detection and protection circuit. 
         FIG.  9    is a top-level diagram showing the MEMS/accelerometer circuit and the readout circuit. 
         FIG.  10    is a schematic of the combined low noise amplifier and programmable gain amplifier and the auxiliary amplifier shown in  FIG.  9   . 
     
    
    
     Corresponding reference numerals indicate corresponding parts throughout the several views of the drawings. 
     DETAILED DESCRIPTION 
     Example embodiments will now be described more fully with reference to the accompanying drawings. 
       FIG.  1    depicts a proposed architecture for a low power motion sensing system  100 . The motion sensing system  100  is comprised of a motion sensor  104 , a bias circuit  102 , and a readout circuit  106 . The bias circuit  102  is interfaced with the motion sensor  104  and operates to supply a bias voltage to the motion sensor  104 . More specifically, the bias circuit  102  is configured to supply a positive bias voltage and a negative bias voltage to the motion sensor  104 . 
     The motion sensing system  100  has two operating modes. In Full Function mode  108 , the readout circuit generates a rail-to-rail analog voltage output that covers a ±1.5 g measurement range for accelerations. Alternatively, in the absence of acceleration, the motion sensing system  100  may switch to an ultra-low power Motion Detection mode  110 . In Motion Detection mode  110 , power usage is significantly reduced and the measurement range is ±3 g. Operating modes are discussed in greater detail below. 
     Of note, the magnitude of the bias voltage is such that output signal of the motion sensor  104  is larger than the flicker noise associated with the readout circuit  106  as further described below. The magnitudes of the positive bias voltage and the negative bias voltage can preferably be set to compensate for process mismatch in the motion sensor  104 . 
     In the example embodiment, the bias circuit  102  is further defined as a high-voltage bias circuit although other types of bias circuits are contemplated by this disclosure. 
     In the example embodiment, the readout circuit  106  is further defined as a complementary metal-oxide-semiconductor (CMOS) analog front end (AFE) circuit although other types of readout circuits are contemplated by this disclosure. While reference is made to an AFE circuit, this disclosure also encompasses other types of readout circuits for motion sensors. 
     In the example embodiment, the motion sensor  104  is further defined as a triaxial MEMS capacitive accelerometer although other types of accelerometers are contemplated by this disclosure. While reference is made to an accelerometer, this disclosure also encompasses other types of motion sensors having a proof mass and cantilever beams, producing acceleration signals with its capacitance change. 
     During operation, the motion sensing system  100  utilizes a non-chopping open loop sensing scheme to eliminate the switching loss while keeping the low noise floor (dominated by the flicker noise in low frequency domain) with delicate amplifier structure design and sizing. With no chopping at the accelerometer&#39;s bias nodes (which are purely capacitive), high-voltages are generated by the bias circuit  102  and applied as bias voltages to increase the accelerometer sensitivity with little power overhead. The bias circuit  102  compresses its output ripple by splitting its sampling and output electrodes, and it produces programmable positive/negative voltages to compensate for the process mismatch in MEMS devices and avoid mechanical interferences (e.g., electrostatic pull-in). 
       FIGS.  2 A and  2 B  depict the simplified diagram of a fully differential MEMS capacitive accelerometer  200  of the example embodiment. The MEMS capacitive accelerometer  200  is a micro-mechanical structure comprising fixed electrodes  202  and movable proof-masses  204 . Both the electrode  202  and the proof-masses  204  have multiple “fingers” or beams  202 ,  208  that cross-couple together, forming a coupling capacitance between the fingers  202 ,  208 . When an acceleration  210  occurs, the proof-mass fingers  208  deflect from their initial position while the electrode fingers  202  stay stationary (relative to the substrate  212 ), changing the gap distance between the proof-mass fingers  208  and the electrode fingers  202  causing a capacitance change that can be detected to reflect the acceleration  210 . 
       FIG.  2 A  shows the simplified diagram of the fully differential MEMS capacitive accelerometer  200  that comprises two proof-masses  204  and two electrodes  202  in the example embodiment. The proof-masses  204  are anchored to the substrate  212  via suspension beams  214 , and their displacement, x, under the acceleration  210 , a, can be expressed as 
     
       
         
           
             
               
                 
                   
                     dx 
                     da 
                   
                   = 
                   
                     
                       m 
                       
                         k 
                         m 
                       
                     
                     = 
                     
                       1 
                       
                         ω 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where m represents the proof mass, k m  is the spring constant of the suspension beam  214 , and ω is the fundamental frequency of this mechanical system, which determines the bandwidth of the accelerometer  200 . The proof-mass  204  displacement causes the capacitance change of C 1    216  and C 2    218  between itself and two neighboring electrodes  202 , as shown in  FIG.  2 B . The values of C 1    216  and C 2    218  are expressed as follows: 
     
       
         
           
             
               
                 
                   
                     C 
                     1 
                   
                   = 
                   
                     
                       ε 
                       ⁢ 
                       A 
                     
                     
                       
                         g 
                         ⁢ 
                         0 
                       
                       - 
                       x 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               C 
               2 
             
             = 
             
               
                 ε 
                 ⁢ 
                 A 
               
               
                 
                   g 
                   ⁢ 
                   0 
                 
                 + 
                 x 
               
             
           
         
       
     
     where ε is the permittivity, A is the area of parallel plates, and g 0  is the initial gap distance between the centered proof mass  204  and the electrodes  202 . Taking C 1    216  as an example, the capacitance sensitivity of C 1    216  to displacement can be derived as 
     
       
         
           
             
               
                 
                   
                     
                       dC 
                       1 
                     
                     dx 
                   
                   = 
                   
                     
                       ε 
                       ⁢ 
                       A 
                     
                     
                       
                         ( 
                         
                           
                             g 
                             0 
                           
                           - 
                           x 
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Combining (1) and (3), the MEMS capacitance sensitivity to the acceleration  210  is 
     
       
         
           
             
               
                 
                   
                     
                       dC 
                       1 
                     
                     da 
                   
                   = 
                   
                     
                       
                         
                           dC 
                           1 
                         
                         dx 
                       
                       ⁢ 
                       
                         dx 
                         da 
                       
                     
                     = 
                     
                       
                         m 
                         ⁢ 
                         ε 
                         ⁢ 
                         A 
                       
                       
                         
                           
                             k 
                             m 
                           
                           ( 
                           
                             
                               g 
                               0 
                             
                             - 
                             x 
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     To maintain good linearity in sensing accelerations, the MEMS is usually designed with a large mechanical stiffness k m  to make proof-mass displacement x&lt;&lt;g 0 , and both C 1    216  and C 2    218  will have constant sensitivities within the accelerometer measurement range: 
     
       
         
           
             
               
                 
                   
                     
                       dC 
                       1 
                     
                     da 
                   
                   = 
                   
                     
                       
                         dC 
                         2 
                       
                       da 
                     
                     = 
                     
                       
                         m 
                         ⁢ 
                         ε 
                         ⁢ 
                         A 
                       
                       
                         
                           k 
                           m 
                         
                         ⁢ 
                         
                           g 
                           0 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     The fully differential MEMS capacitive accelerometer  200  shown in  FIG.  2 B  produces two pairs of C 1    216  and C 2    218  with the opposite sensitivity for accelerations. C 1    216  and C 2    218  are configured as a capacitive Wheatstone bridge that has twice the sensitivity compared with a single-ended accelerometer and possesses superior common-mode rejection to noise, offset, etc. 
       FIG.  3    depicts the high-voltage biasing scheme for the MEMS capacitive accelerometer  200  of the example embodiment. To convert the capacitance change in the accelerometer  200  to a more convenient readout, the accelerometer  200  is biased with a voltage V B  so it can produce electrical signals that reflect the incoming accelerations. In conventional MEMS capacitive accelerometers, there is a large trade-off between the accelerometer resolution and power consumption, which is consistent with the fundamental trade-off between the AFE noise and power. In power-constrained applications such as Internet-of-Things (IoT) devices, it remains challenging for MEMS capacitive accelerometers to achieve a sub-mg sensitivity with μW-level power consumption. 
     To overcome this power-noise dilemma, the accelerometer  200  signal is increased to reduce the noise and equivalently improve the signal-to-noise ratio (SNR). By applying a significantly higher bias voltage (e.g., 10× compared to conventional MEMS capacitive accelerometers), the accelerometer  200  signal is raised much higher than the noise floor of the AFE circuit  106 . This eliminates the need for power-hungry linear noise amplifiers (LNA) and signal chopping to suppress thermal noise and flicker noise, respectively, and one can design the AFE circuit  106  with a nA-level supply current while still maintaining a good SNR. 
     Compared to conventional MEMS capacitive accelerometers, the high-voltage biased MEMS accelerometer  200  is not subject to a trade-off between AFE  106  power and AFE  106  noise. Instead, power-resolution performance of the accelerometer  200  is determined by what voltage levels can be applied to the accelerometer  200  for a given power budget. 
       FIG.  4 A  depicts a stress analysis of the proof mass  204  when considering both mechanical force F m  and electrostatic force F e  of the example embodiment. 
     The benefit of high-voltage bias is not obvious when considering the impact of bias voltage on the accelerometer&#39;s  200  mechanical movement. The large voltage stress across the proof mass  204  and electrodes  202  generates electrostatic force between them and results in an additional movement of the proof mass  204 . To quantitatively analyze the impact of the electrostatic force, C 1    216  is taken as an example to calculate the force between a proof mass  204  and electrode  202 , as shown in  FIG.  4 A . When a bias voltage V B    402  is applied across the proof mass  204  and electrode  202 , the total energy stored in C 1    216  is expressed by 
         E=C   1   V   B   2   (6)
 
     The electrostatic force F e    404  between the proof mass  204  and electrode  202  can be derived by 
     
       
         
           
             
               
                 
                   
                     F 
                     e 
                   
                   = 
                   
                     
                       dU 
                       dx 
                     
                     = 
                     
                       
                         
                           
                             dC 
                             1 
                           
                           dx 
                         
                         ⁢ 
                         
                           V 
                           B 
                           2 
                         
                       
                       = 
                       
                         
                           ε 
                           ⁢ 
                           
                             AV 
                             B 
                             2 
                           
                         
                         
                           
                             ( 
                             
                               
                                 g 
                                 0 
                               
                               - 
                               x 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Note that F e    404  increases nonlinearly with the proof-mass displacement, and it is always a destabilizing (positive feedback) force that fights against the mechanical recovery force F m    406  by the accelerometer suspension beam  214 . In a stable accelerometer  200 , F e    404  always needs to stay lower than F m    406  or the electrostatic force will keep moving the proof mass towards the electrode and eventually result in an electrostatic pull-in. Taking the expression of F m    406  and F e    404 , the equation becomes 
     
       
         
           
             
               
                 
                   
                     
                       k 
                       m 
                     
                     ( 
                     
                       
                         g 
                         0 
                       
                       - 
                       x 
                     
                     ) 
                   
                   &gt; 
                   
                     
                       
                         ε 
                         ⁢ 
                         
                           AV 
                           B 
                           2 
                         
                       
                       
                         
                           ( 
                           
                             
                               g 
                               0 
                             
                             - 
                             x 
                           
                           ) 
                         
                         2 
                       
                     
                     ⁢ 
                         
                     or 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     V 
                     B 
                   
                   &lt; 
                   
                     
                       
                         
                           
                             k 
                             m 
                           
                           ( 
                           
                             
                               g 
                               0 
                             
                             - 
                             x 
                           
                           ) 
                         
                         3 
                       
                       
                         ε 
                         ⁢ 
                         A 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Equations (8) and (9) reveal an important trade-off between the accelerometer bias voltage  402  and the proof-mass displacement range. With a larger V B    402  applied to the accelerometer  200 , its proof-mass displacement needs to be more constrained to maintain F m &gt;F e  and avoid pull-in. When V B    402  exceeds 
     
       
         
           
             
               
                 
                   
                     k 
                     m 
                   
                   ⁢ 
                   
                     g 
                     0 
                     3 
                   
                 
                 
                   ε 
                   ⁢ 
                   A 
                 
               
             
             , 
           
         
       
     
     the proof mass  204  will destabilize and pull-in even without any displacement (acceleration) applied, so it implies a theoretically maximum V B    402  that can be used to bias the accelerometer  200 . 
     Another way to understand the impact of V B    402  is through the change in the capacitance sensitivity of the accelerometer that was derived in Equation (4). Intuitively, if the proof mass  204  initially moves a distance x 1  with input acceleration, it moves closer to the electrode and experiences a greater attraction force from the electrode. The attraction force will move the proof mass  204  an additional distance x 2  so that its overall displacement becomes x 1 +x 2  under the same acceleration. The proof mass  204  behaves as if it has a ‘reduced stiffness’ from the suspension beam  214 , so Equation (4) is rewritten as 
     
       
         
           
             
               
                 
                   
                     
                       dC 
                       1 
                     
                     da 
                   
                   = 
                   
                     
                       m 
                       ⁢ 
                       ε 
                       ⁢ 
                       A 
                     
                     
                       
                         ( 
                         
                           
                             k 
                             m 
                           
                           + 
                           
                             k 
                             e 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               g 
                               0 
                             
                             - 
                             x 
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     where k m  and k e  represent the mechanical stiffness (by suspension beams  214 ) and electrostatic stiffness (by high-voltage bias V B    402 ), respectively, and their values are expressed by 
     
       
         
           
             
               
                 
                   
                     k 
                     m 
                   
                   = 
                   
                     
                       
                         ma 
                         x 
                       
                       ⁢ 
                            
                       
                         k 
                         e 
                       
                     
                     = 
                     
                       - 
                       
                         
                           2 
                           ⁢ 
                           ε 
                           ⁢ 
                           
                             AV 
                             B 
                             2 
                           
                         
                         
                           
                             ( 
                             
                               
                                 g 
                                 0 
                               
                               - 
                               x 
                             
                             ) 
                           
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     With a larger V B    402 , the accelerometer&#39;s  200  overall stiffness (k m +k e ) decreases, resulting in a higher capacitance sensitivity to acceleration  210 . This further transfers into a non-linear increase in the accelerometer  200  signal V IN  at given accelerations  210  as shown in  FIG.  4 B . When V B    402  is small, the electrostatic feedback is negligible, and V IN  increases linearly with V B    402 . When V B  becomes large and generates enough strong electrostatic force to the proof mass  204 , a super-linear increase in V IN  will occur. 
       FIG.  5 A  depicts a stress analysis of the proof-mass  204  in a differential MEMS structure. 
     To take advantage of the high-voltage bias while mitigating its side effect due to electrostatic feedback, the MEMS&#39;s differential structure is utilized and a balanced positive and negative voltage is applied on the two electrodes neighboring a proof mass. As shown in  FIG.  5 A , a first electrode  502  is biased with a positive high-voltage, V B+   506 , while a second electrode  504  is biased with a negative voltage, V B−   508 , of the same magnitude. When the proof-mass  204  is DC coupled to ground/substrate  21 , it will experience equal electrostatic forces F e1  and F e2  from the first electrode  502  and the second electrode  504 , respectively, but in opposite directions, so forces F e1    510  and F e2    512  cancel each other out. Then the proof-mass  204  will no longer suffer from electrostatic feedback regardless of the value of V B+   506  and V B−   508 . 
     However, maintaining a balanced electrostatic force on both proof-masses  204  is tricky in practical applications, and electrostatic feedback still exists due to electrostatic mismatch, defined as F mis =F e1 +F e2 . There are two reasons for a non-zero electrostatic mismatch: 
     1) During MEMS fabrication, process variation can cause a mismatch in the MEMS&#39;s mechanical parameters (e.g., area A or gap distance g 0  in Equation (7)). Circuit non-idealities also induce electrical mismatch, such as voltage errors and ripples, making it difficult to generate exactly equalized V B+  and V B− . 
     2) The values of F e1  and F e2  diverge with input acceleration regardless of their equilibrium condition in the stationary state. When acceleration occurs, F mis  can be written as 
     
       
         
           
             
               
                 
                   
                     F 
                     mis 
                   
                   = 
                   
                     
                       
                         F 
                         
                           e 
                           ⁢ 
                           1 
                         
                       
                       + 
                       
                         F 
                         
                           e 
                           ⁢ 
                           2 
                         
                       
                     
                     = 
                     
                       
                         
                           ε 
                           ⁢ 
                           
                             AV 
                             
                               B 
                               + 
                             
                             2 
                           
                         
                         
                           
                             ( 
                             
                               
                                 g 
                                 0 
                               
                               - 
                               x 
                             
                             ) 
                           
                           2 
                         
                       
                       - 
                       
                         
                           ε 
                           ⁢ 
                           
                             AV 
                             
                               B 
                               - 
                             
                             2 
                           
                         
                         
                           
                             ( 
                             
                               
                                 g 
                                 0 
                               
                               + 
                               x 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     which is simplified as 
     
       
         
           
             
               
                 
                   
                     F 
                     mis 
                   
                   = 
                   
                     ε 
                     ⁢ 
                     
                       AV 
                       B 
                       2 
                     
                     ⁢ 
                     
                       
                         4 
                         ⁢ 
                         
                           g 
                           0 
                         
                         ⁢ 
                         x 
                       
                       
                         
                           ( 
                           
                             
                               g 
                               0 
                               2 
                             
                             + 
                             
                               x 
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     When x 2 &lt;&lt;g 0   2 , F mis  increases proportionally with the proof-mass displacement. But if x grows large enough under strong accelerations, the increase of F mis  becomes dramatic and eventually converges into the single-ended electrostatic force described in Equation (7). 
     For both of the reasons listed above, the electrostatic mismatch is exacerbated quadratically with the V B  increase, and thus both issues need to be carefully considered in the high-voltage bias scheme for MEMS capacitive accelerometers. Electrostatic mismatch can be overcome using Electrostatic Mismatch Compensation (EMC), which strategically manipulates the bias voltages to improve upon the challenges raised by F mis . The EMC technique has two goals: 
     1) to extend the linear region of MEMS sensitivity, and thus accelerometer sensitivity, to higher V B  levels. As discussed above, the electrostatic mismatch due to MEMS process variation and circuit nonideality becomes more obvious with a larger V B . To compensate for the mismatch, EMC directly equalizes F e1  and F e2  by inducing an intended voltage skew ΔV B  between V B+  and V B−  and maintaining ultra-low voltage errors and ripples for ΔV B . As a result, the sensitivity of the MEMS, and thus the accelerometer, stays linear until a higher V B  threshold, as shown by curve  516  in  FIG.  5 B  and as compared to curve  514  in  FIG.  5 B , and it guarantees a wider region of clean bias without electrostatic feedback on the MEMS/accelerometer. 
     2) to optimize the trade-off between MEMS sensitivity, and thus accelerometer sensitivity, and full scale. At very large V B , non-linearity appears in the MEMS/accelerometer sensitivity, and electrostatic mismatch is primarily caused by the proof-mass displacement a s a result of input acceleration. It is beneficial to achieve higher MEMS sensitivity at the cost of losing its dynamic range, but the process must be properly controlled to guarantee the MEMS/accelerometer linearity to accelerations and avoid electrostatic pull-in. EMC achieves this by carefully choosing the values of V B+  and V B−  so that sufficient dynamic range is achieved, and the pull-in point is pushed into a higher bias voltage. EMC also determines the safe margin on the bias voltages when considering the variation across MEMS chips/wafers. 
     EMC guarantees a more stable, predictable, and variation-robust MEMS/accelerometer operation when using a high-voltage bias, resulting in a better accelerometer SNR. EMC is implemented on the high-voltage bias circuit described below. 
       FIG.  6    depicts the top-level diagram of the high-voltage bias circuit  600  of the example embodiment. The EMC technique relies on generating precisely controlled V B+   614  and V B−   616  with proper values to compensate for various process variations (e.g. MEMS process variations) and various circuit non-idealities (e.g. CMOS circuit non-ideality). In the example embodiment, the high-voltage biases are up-converted from V DD    612  using Dickson charge pumps  602 ,  604  for the large conversion ratio, chip integration, and high efficiency with low load current. In the example embodiment, V B+   614  and V B−   616  are DC voltages. 
       FIG.  6    shows the positive charge pump  602  and negative charge pump  604  on the high-voltage bias circuit  600  used to generate V B+   614  and V B−   616 , respectively. The charge pump outputs  614 ,  616  are sampled and compared with the respective positive reference voltage, V CM    618 , and negative reference voltage, V DM    620 , and the comparison results  622 ,  624  modulate the charge pumps&#39; operations in a delta-sigma manner to form a closed-loop control on the bias voltages  614 ,  616 . In the example embodiment, V B+   614  and V B−   616  are in the range of 20-30 V while V CM    618  and V DM    620  are in the range of 0-2 V, so the bias voltages  614 ,  616  must be divided before they can be compared with the reference voltages  618 ,  620 . However, any voltage errors from the reference are amplified by the large division ratio (e.g., 20:1) when the voltage errors appear in the bias voltages. In the example embodiment, the programmable reference voltages  618 ,  620  are multiplexed  610  from a resistive voltage divider  626  that divides 2V with 128 poly-resistors, and the quantization error is 2 V/128≈15 mV. The resulting error on V B+   614  and V B−   616  will then become 15 mV×20=300 mV, making it difficult to achieve EMC with the required voltage precision. 
     To avoid this quantization error, voltage sampling and division (20×)  606  is only performed for V B+   614  to control the positive charge pump  602 . For V B−   616 , the arithmetic mean of V B−   616  is sampled with V B+   614  at  608  and the mean value is directly compared with the negative reference voltage V DM    620  to determine the negative charge pump  604  operation. As a result, V B−   616  will follow the change of V B+   614  while keeping a programmable voltage skew ΔV B =(V B+ +V B− ) that is determined by the second comparison. Expressed as an equation, the bias voltages are refactored into a “common-mode” part and a “differential-mode” part: 
         V   B+ =20 V   CM   (14)
 
         V   B− =−20 V   CM +2 V   DM   (15)
 
     where V CM    618  and V DM    620  are the reference voltages that are used by the comparison for the positive and negative charge pumps  602 ,  604 , respectively. While the voltage error of V CM    618  is multiplied by 20 on both V B+   614  and V B−   616 , the V DM    620  error only has a 2× effect on (V B+ +V B− ). As x&lt;&lt;g 0  remains true within the measurement range of the accelerometer of the example embodiment, Equation (12) can be rewritten with the condition x&lt;&lt;g 0 : 
     
       
         
           
             
               
                 
                   
                     F 
                     mis 
                   
                   = 
                   
                     
                       ε 
                       ⁢ 
                       
                         A 
                         ⁡ 
                         ( 
                         
                           
                             V 
                             
                               B 
                               + 
                             
                           
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                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             V 
                             
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     which shows that F mis  is reduced proportionally with ΔV B =(V B+ +V B− ). 
     Besides the quantitation errors, that are improved with the above technique, V B+   614  and V B−   616  may also suffer from noise and fluctuation from the supply voltage V DD    612  if V CM    618  and V DM    620  are directly generated from V DD    612 . To overcome this, the supply voltage V DD    612  is divided with a subthreshold voltage reference  628  that has a −41 dB power supply rejection and a &lt;1% error from 0° C. to 100° C. Because the subthreshold voltage reference  628  has a large current variation across temperatures, its output voltage is buffered before applying it to the voltage divider  626  to guarantee a sufficient current that flows through the poly-resistors and generates precise V CM    618  and V DM    620 . 
       FIG.  7 A  depicts a serial-connected switched-capacitor voltage converter of the example embodiment. Several circuit challenges are raised with sampling/dividing the high-voltage V B+   614  and V B−   616 . First, the switched-capacitor voltage divider  702  induces a switching loss approximately equal to 0.5 f CV 2 , where f is the sampling frequency, C is the sampling capacitance and V is the voltage swing. For the sufficiently fast charge pump feedback control required by EMC (e.g., f=1000 Hz, C=100 fF and V=30 V), the resulting power losses on V B+   614  and V B−   620  are in the 100 nW range, and it takes even more power consumption from V DD    612  to replenish the bias voltage losses. To mitigate the power overhead that results from frequently sampling/dividing the high-voltage nodes, a serial-connected switched-capacitor voltage converter  700  is implemented. By implementing the serial switched-capacitor divider, its AC signal division is only related to Φ 2    706 , so Φ 2    706  can be highly duty-cycled to keep it on and update V B+ /20 with any ripples and variations that occur at V B+   614 . Meanwhile, Φ 1    704  is only turned on once after a long period of time (e.g., seconds) so the sampling frequency will be in the sub-Hz range, significantly reducing power consumption. 
       FIGS.  7 B and  7 C  depict the high-voltage sampling circuit of the example embodiment and division/average circuits of the example embodiment, respectively. A second challenge manifests itself when the storage capacitor  708  charge shares with the sampling capacitor  710 , resulting in ripples on V B+   614  and V B−   616 . Although a large capacitor ratio is guaranteed, the ripples can be in the 100 mV range due to the high-voltage scales of V B+   614  and V B−   616 , causing an unpredictable, transient F mis  to the accelerometer  200  and increasing the common-mode noise seen by the readout circuit  106 . To address this issue, V B+   614  and V B−   616  are separated from two ‘dirty’ nodes, DV B+   712  and DV B−   714  (see  FIG.  7 C ), each through a large RC constant (τ=1 GΩ 100 pF=0.1 s). During voltage sampling, DV B+   712  and DV B−   714  first pre-charge the sampling capacitors  710  to near V B+   614  and V B−   616  so that the ripples occur on the dirty nodes instead of the actual accelerometer  200  bias voltages. The dirty nodes&#39; voltage loss will later be replenished by the charge pump  602 , but through the large RC network. As a result, V B+   614  and V B−   616  only see charge pump ripples rather than the much larger sampling ripples. 
       FIG.  7 D  describes the transient waveform during voltage sampling. The pulse width of Φ 1    704  and Φ 2    706  remains short compared to the sampling period, and Φ 3    716  stays high for most of the time to provide an AC throughput as discussed above. Timing switches  718  are implemented with high-voltage transistors  720  with their control signals level-shifted by capacitive level shifters  722 , also mitigating control power with low sampling frequency. Furthermore, in the voltage average circuit in  FIG.  6   , current-limiting resistors  724  are included to reduce voltage spikes on (V B+ +V B− )/2  726  due to the timing difference of the V B+   614  and V B−   616  switches and to prevent the spike from damaging the comparator circuits  630 . 
       FIG.  8    depicts a diagram of the pull-in detection and protection circuit  800 . When EMC optimizes the trade-off between accelerometer sensitivity and full scale, it applies the highest V B+   802  and V B−   804  with a safe margin for input accelerations and process variation. However, it is still possible during the operation/calibration phase that an improper bias voltage is applied and triggers an electrostatic pull-in for the accelerometer  200 . While the pull-in is mechanically recoverable for the accelerometer, it raises issues for the readout circuit  106  because of the electrical contact between the proof mass  204  and electrode  202 , including the potential breakdown of the transistor&#39;s gate oxide and permanent damage to the readout circuit. 
     To prevent this from occurring, a pull-in detection and protection circuit  800  is implemented on the high-voltage bias circuit  600 . A smaller (5 pF) capacitor  806  is connected to V B+   802  so that when the first proof-mass  906  pulls-in with the first electrode  902 , the voltage drop at the first electrode  902  is large enough to be detected. The first electrode  902  voltage drop is AC-coupled to the high-voltage bias circuit with a high-pass filter consisting of 0.5 pF capacitor  814  and a 100 MΩ resistor  816  to generate a reset signal that grounds V B+   802  via transistor  812 . By controlling the bandwidth of this feedback, the pull-in detection and protection circuit  800  can detect and ground V B+   802  before it generates a high-enough voltage spike to damage the readout circuit. After V B+   802  is grounded to 0, the electrostatic force between the first proof mass  906  and the first electrode  902  disappears, and the first proof mass  906  is recentered by the suspension beam  214 . Meanwhile, in the pull-in detection and protection circuit  800 , V DD    818  will re-charge the 0.5 pF capacitor  814  through the DC path that included the 100 MΩ resistor  820 , and the reset signal is then retracted to enable V B+   802  to rebuild its voltage. 
       FIG.  9    depicts a top-level diagram showing the accelerometer circuit and the readout circuit.  FIG.  10    depicts a schematic of the combined low noise amplifier and programmable gain amplifier and the auxiliary amplifier shown in  FIG.  9   . The high-voltage bias circuit generates a proper pair of V B+   916  and V B−   918  with EMC and applies the bias voltages  916 ,  918  to the first electrode  902  and second electrode  904  of the accelerometer, respectively. When acceleration  210  occurs, differential MEMS signal V IN+   924 , V IN−   926  is generated across the first proof-mass  906  and the second proof-mass  908  due to the MEMS capacitance change, and the signal is amplified by the AFE circuit  106 . Equation (7) shows that the MEMS signal declines with the proof-mass parasitic capacitance C par    920 . To reduce C par    920  due to the MEMS-CMOS interconnect, the MEMS and CMOS AFE circuits are eutectically bonded at the wafer level and then the wafer is diced into 2-layer face-to-face bonded MEMS-CMOS chips. 
     On the AFE circuit  106 , a two-stage capacitive coupled amplifier design is adopted consisting of an LNA  910  followed by a PGA  912 , as shown in  FIG.  10   . The combined LNA  910  and PGA  912  design further contains auxiliary amplifiers  922  to shift their output DC-levels to the input for maximized dynamic range. The detailed schematic of the combined LNA and PGA  1000  and the auxiliary amplifier  922  can be found in  FIG.  10   . Tunable amplifier bias voltages, V BP 1-3    1002 , are generated on chip to cover the temperature range of −40° C. to 80° C. and possible process variations. Both the LNA  910  and PGA  912  consume nW-level power, but their noise floor is far below the significantly increased MEMS signals, so a high SNR is achieved. In the example embodiment, the size of the LNA  910  input pair is enlarged with a width of 187 μm and a length of 0.42 μm, trading off the available chip area for lower flicker noise. 
     As discussed above in  FIG.  1   , the system has two operating modes. In the FF mode, the AFE circuit  106  generates a rail-to-rail analog voltage output that covers a ±1.5 g measurement range for accelerations  210 . In the absence of acceleration  210 , the system can switch to an ultra-low-power MD mode to only output a 1-bit signal when there is an acceleration  210  exceeding the detection threshold. During the MD mode, V DD    612  is reduced from 2V to 1.2V, and bias current across the amplifiers  910 ,  912 ,  922  is further reduced to sub-nA levels to save circuit power. 
     The foregoing description of the embodiments has been provided for purposes of illustration and description. It is not intended to be exhaustive or to limit the disclosure. Individual elements or features of a particular embodiment are generally not limited to that particular embodiment, but, where applicable, are interchangeable and can be used in a selected embodiment, even if not specifically shown or described. The same may also be varied in many ways. Such variations are not to be regarded as a departure from the disclosure, and all such modifications are intended to be included within the scope of the disclosure.