capacitance neutralization is a procedure used in differential circuits to cancel unwanted parasitic capacitances by adding neutralization capacitances of opposite polarity using the differential voltages . fig3 illustrates an exemplary differential circuit that includes a first capacitive core c a and a second capacitive core c b . the first capacitive core c a includes variable capacitors c 1a and c 2a . the second capacitive core c b includes variable capacitors c 1b and c 2b . fig3 also shows the unwanted feed - through capacitances c ft1 , c ft2 , c ft3 , c ft4 that are parallel to the desired sense capacitors c 1a , c 2a , c 1b , c 2b , respectively . fig4 illustrates the exemplary differential circuit of fig3 with on - chip neutralization capacitors added to cancel the unwanted feed - through capacitances . the neutralization capacitors c n1 , c n2 , c n3 , c n4 are added in a way to cancel the unwanted feed - through capacitances c ft1 , c ft2 , c ft3 , c ft4 , respectively . fig5 illustrates the impact of adding too little or too much neutralization capacitance in a case where the feed - through capacitances are 50 ff . if we have the ideal case where the neutralization capacitors perfectly cancel the feed - through capacitances , then the measurement voltage vo is proportional to the displacement x of the proof mass as shown in eq . ( 2 ). if the neutralization capacitance is less than the feed - through capacitance ( c n = 0 ° f . or c n = 40 ff ), then the feed - through capacitances are not completely canceled and the measurement voltage vo is greater than it would be in the ideal case . if the neutralization capacitance is greater than the feed - through capacitance ( c n = 60 ff or c n = 100 ff ), then the feed - through capacitance is canceled and we have excess neutralization capacitance which causes the measurement voltage vo to be less than it would be in the ideal case . fig5 also shows that the deviation of the measurement voltage δvo from ideal case increases as the displacement of the proof mass δx increases . thus , it is desirable to determine the necessary amount of neutralization capacitance to add to cancel the feed - through capacitance without adding excess neutralization capacitance . one method is simply to estimate the mean feed - through capacitance for several parts and put in a nominal neutralization capacitance to cancel the mean feed - through capacitance . however both the feed - through and neutralization capacitances vary from part to part due to process variation and mismatch which leads to improper cancellation . this method can be relatively inexpensive and provides some benefit but is not necessarily accurate for a particular part . a more accurate but expensive method to cancel the feed - through capacitance is to put each chip inside a centrifuge . then as the acceleration is increased using the centrifuge , the output voltage can be read for various on - chip neutralization capacitor settings . the on - chip neutralization capacitance can be set using trim bits . the neutralization capacitance that corresponds to the most linear output measurement versus acceleration curve can then be programmed into the chip . this procedure is expensive in terms of test cost and time since it requires use of a centrifuge and time to load the chips and sweep the measurement versus acceleration curves for each of the chips . an alternative method for determining the necessary amount of neutralization capacitance is to measure the linearity of the measured output for two or more precisely known forces . a force f 0 and twice that force , 2 * f 0 , can be applied to the proof mass and the change in the output can be measured to determine the linearity between the input forces and the output measurements . fig6 shows three scenarios for this method when the forces f 0 and 2 * f 0 are applied . other ratios of forces or more forces can be used as long as the relationships between the forces are known . fig6 a shows the ideal situation where the neutralization capacitance perfectly cancels the feed - through capacitance , c n = c ft . in this case , the change in output voltage δv 1 caused by the force f 0 equals the change in output voltage δv 2 caused by the force 2 * f 0 which produces a linear relationship between displacement and output voltage . fig6 b shows the situation where too much neutralization capacitance is added , c n & gt ; c ft . in this case , the change in output voltage δv 1 caused by the force f 0 is greater than the change in output voltage δv 2 caused by the force 2 * f 0 which produces a decreasing non - linear relationship between displacement and output voltage . fig6 c shows the situation where too little neutralization capacitance is added , c n & lt ; c ft . in this case , the change in output voltage δv 1 caused by the force f 0 is less than the change in output voltage δv 2 caused by the force 2 * f 0 which produces an increasing non - linear relationship between displacement and output voltage . it can be seen from fig6 a - 6c that if the relationship between the applied forces is known , then the non - linearity of the resulting output measurements will indicate whether the neutralization capacitance should be increased or decreased to better model the linearity of the ideal case . a force can be applied to the proof mass using electrostatic actuation . fig7 shows an exemplary automatic calibration procedure 700 for tuning out the feed - through capacitance using on - chip neutralization capacitance , c neut . the calibration procedure 700 uses a binary search method between a lower limit for on - chip neutralization capacitance , c min , and an upper limit for on - chip neutralization capacitance , c max . other search schemes known in the art can also be used for finding a value of neutralization capacitance to cancel the feed - through capacitance . at block 702 , the process is initialized by setting the lower binary search limit to the lower limit for on - chip neutralization capacitance , c low = c min , setting the upper binary search limit to the upper limit for on - chip neutralization capacitance , c high = c max , and setting the search counter to 1 . at block 704 , the system sets the value of neutralization capacitance to be checked , c neut , to the center of the working binary search range ( c low + c high )/ 2 . at block 706 , no electrostatic actuation is applied to the proof mass and the output value is recorded . this output will include the sensor offset and a fraction of earth &# 39 ; s gravitational force depending on the angle of mounting of the proof mass . at block 708 , an electrostatic force of f 0 is applied to the proof mass and the change in the output value ( δv 1 ) is recorded . at block 710 , an electrostatic force of 2 * f 0 is applied to the proof mass and the change in the output value ( δv 2 ) is recorded . under some conditions , a force of 2 * f 0 can be applied by making the duty cycle of actuation twice as long as the duty cycle for the force of f 0 . other multiples of f 0 can also be used to determine linearity . the method then determines whether the neutralization capacitance should be increased or decreased to better cancel the feed - through capacitance . these steps can be implemented by increasing and decreasing a trim setting for neutralization capacitance . at block 712 , the system tests whether δv 2 is less than δv 1 . in this case there is too much neutralization capacitance compared to the feed - through capacitance . if δv 2 is less than δv 1 then control passes to block 714 , otherwise control passes to block 716 . at block 714 , the system decreases the neutralization capacitance by setting the upper limit for the binary search to the current value , c high = c neut , and control passes to block 720 . at block 716 , the system tests whether δv 2 is greater than δv 1 . in this case there is less neutralization capacitance as compared to the feed - through capacitance . if δv 2 is greater than δv 1 then control passes to block 718 , otherwise control passes to block 722 . at block 718 , the system increases the neutralization capacitance by setting the lower limit for the binary search to the current value , c low = c neut , and control passes to block 720 . at block 720 , the system checks whether the binary search is complete by checking if the binary search counter equals n . larger values of n indicate finer resolution in determining the neutralization capacitance that cancels the feed - through capacitance , which should provide more accurate cancellation . if the binary search is not complete ( cnt & lt ; n ), then control passes back to block 704 and the output readings are taken for a new value of neutralization capacitance . if the binary search is complete ( cnt = n ), then control passes to block 722 . block 722 is reached because either : ( a ) the system determined that δv 2 is not less than or greater than δv 1 , which leaves that δv 2 equals δv 1 ; or ( b ) the binary search is complete . in either case the final value of neutralization capacitance c neut will be used . the system can also include a threshold linearity check to ensure that the selected value of neutralization capacitance c neut meets the threshold . for example , the threshold check can determine whether the absolute value of the difference between δv 2 and δv 1 is less than the threshold linearity for the final value of neutralization capacitance c neut . one way of applying electrostatic force for sensor calibration is by voltage actuation . fig8 shows an exemplary voltage actuation circuit 800 . under voltage actuation , a fixed voltage v is applied to sensor capacitor c 1 while sensor capacitor c 2 is shorted . the relationship between the applied electrostatic force f e and the mechanical force f m can be expressed as : eq . ( 5 ) shows that for voltage actuation , the displacement x is not a linear function of duty cycle , i . e ., doubling the duty cycle of actuation does not result in doubling the displacement of the proof mass . thus , this method of voltage actuation does not provide the desired linear actuation as a function of duty cycle . another way of applying electrostatic force for sensor calibration is by charge control actuation . fig9 shows an exemplary charge control actuation circuit 900 comprising a sensor core with variable sensor capacitors c 1 and c 2 , an amplifier 902 , a capacitor ca and an input voltage source va . under charge control actuation , the sensor core is placed in a first feedback path of the amplifier 902 , and the sensor capacitor c 2 is shorted so that charge is placed on only the sensor capacitor c 1 of the sensor core . the amplifier 902 also includes a second parallel feedback path with a switch that is open during a phase φ a1 . during phase φ a1 , the sensor core is reset and a constant charge va is gathered on capacitor ca . then during phase φ a2 , the charge is dumped from capacitor ca onto the sensor capacitor c 1 . in this case , the relationship between the applied electrostatic force f e and the mechanical force f m can be expressed as : eq . ( 7 ) shows that for charge control actuation , the displacement x is a linear function of duty cycle , i . e ., doubling the duty cycle of actuation results in doubling the displacement of the proof mass . this method of charge control actuation appears to provide the desired linear actuation as a function of duty cycle . the above analysis shows charge control actuation without the feed - through capacitances . fig1 shows an exemplary charge control actuation circuit 1000 with the feed - through and neutralization capacitances . unfortunately , when the feed - through capacitance cft is included , the charge q 0 is not only dumped on capacitor c 1 but is also dumped on capacitor cft , and the charge is not linearly split between capacitors c 1 and cft based on the displacement , x . a similar neutralization principle can be used for the charge control circuit 1000 . ideally , if cft = cneut then there is perfect cancellation and the charge control is perfectly linear . however , if cft ≠ cneut then there is non - linearity . fortunately , this cancellation non - linearity is in the same direction as the feed - through nonlinearity due to the feed - through capacitance , cft . therefore , this cancellation non - linearity can actually help with neutralization because if there is an error in cft cancellation , this cancellation error makes that feed - through error appear larger , and when cft = cneut there is no cancellation or feed - through error . exemplary timing diagrams for an auto - calibration procedure are shown in fig1 . an exemplary differential capacitive sensor system is shown in fig1 . during auto calibration , time is divided between actuation phases φ a and measurement phases φ m . the circuit of fig1 is referred to during the actuation phase description , and the circuit of fig1 is referred to during the measurement phase description . the circuit of fig1 is not shown in the circuit of fig1 for clarity . those of skill in the art will understand that there will be other switches in series with the sensor switches that enable actuation . these switches are not shown in fig1 . fig1 only shows the circuit during the measurement phase . including all of the switches in fig1 would make fig1 extremely complicated . thus , fig1 shows the sensor and the capacitance - to - voltage circuit during the measurement phase , and fig1 shows the sensor and actuation circuits during the actuation phase . some of the details regarding the switches that connect the sensor to the actuation amplifier 1002 or the capacitance - to - voltage amplifier are not shown , for example the sensor is disconnected from the capacitor ca during the measurement phase . during the actuation phase , the sensor is connected as shown in fig1 . for a differential sensor system , each of the capacitor cores c a and c b are connected as shown in fig1 . the actuation phase φ a is further divided into two subphases : φ a1 and φ a2 . during the actuation phase , the sensor is disconnected from the front end , so φ 2 equals 0 . during subphase φ a1 , the sensor is reset and a constant charge is gathered on capacitor ca . then during subphase φ a2 , the charge is dumped onto the mems capacitor c 1 . the actuation force is linearly proportional to tw , the duration or duty cycle of the actuation subphase φ a2 . then φ a1 is high during the last part of the actuation phase to reset the capacitors to provide zero force and get them ready for the measurement phase . in a differential system , the actuation force can be applied to both cores at substantially simultaneously to simulate the application of a force in operation . during the measurement phase , the sensor is connected as shown in fig1 . the measurement phase φ m is divided into two subphases : φ 1 and φ 2 . during the first subphase φ 1 , the mems capacitors are connected to the front end of the capacitive sensor system and the displacement is read out by the system as a voltage v 0 . during the second subphase φ 2 , the voltages are applied to the sensor so that there is no net electrostatic force on the proof masses of the cores c a and c b to cause erroneous actuation . by alternating between the actuation and measurement phases , at a very high speed ( for example , 100 khz ), an accurate , low noise measurement can be made . charge control actuation has not previously been reported for linear actuation . charge - control can be used for linear actuation to obtain immunity to unwanted vibrations that cause the proof mass of the sensor to shake around . continuous self - test is a technique to check that a sensor continues to function properly during operation . in continuous self test , a background signal ( force ) is applied continuously on the sensor . this background signal is usually a pseudo - random signal , for example a square wave or a sine wave . by measuring the displacement of the proof mass , an output signal caused by this background signal can be read out by the transducer system . if the output signal is within some operational limits of what is expected for the background signal , then it can be assumed that the sensor is working properly . if the output signal is beyond some warning limits , then it can be assumed that the sensor is malfunctioning and it cannot be used to make safety - critical decisions . in an automotive application , an example of a safety critical decision would be detecting a skid and applying brakes selectively on the wheels according to the sensor signals . linear actuation can be important in a self - test application . for example , in a car there are many vibrations that are several tens of gravitational force ( g ) that are within the sensor bandwidth . an exemplary sensor bandwidth for an accelerometer is usually less than 5 khz . these multi - g vibrations can cause the proof mass to shake around . if the actuation of the proof mass during self test is non - linear ( i . e ., the actuation force depends on the proof mass displacement ), these vibrations can mix with the actuation signal and change the dc value for the sensor making the subsequent readings inaccurate . an example of non - linear actuation is the voltage actuation shown in eq . ( 5 ). the electrostatic force due to voltage actuation is a function of the proof mass displacement squared x 2 , so these vibrations causing the proof mass to shake around during self test can cause inaccurate results . linear actuation can avoid this change to the dc value for the sensor during continuous self - test . an example of linear actuation is the charge - control actuation shown in eq . ( 7 ). the electrostatic force due to charge - control actuation is not a function of the proof mass displacement x . so vibrations causing the proof mass to shake around during self test do not effect the self - test . thus , the self - test can provide accurate results during operation even if there are multi - g vibrations on the sensor . while this invention has been described as having an exemplary design , the present invention may be further modified within the spirit and scope of this disclosure . this application is therefore intended to cover any variations , uses , or adaptations of the invention using its general principles .