Abstract:
A method for detecting displacements of a micro-electromechanical sensor including a fixed body and a mobile mass, and forming a first sensing capacitor and a second sensing capacitor having a common capacitance at rest. The first and second sensing capacitors being connected to a first input terminal and, respectively, to a first output terminal and to a second output terminal of the sensing circuit. The method includes the steps of closing a first negative-feedback loop, which is formed by the first and second sensing capacitors and by a differential amplifier, feeding an input of the differential amplifier with a staircase sensing voltage through driving capacitors so as to produce variations of an electrical driving quantity which are inversely proportional to the common sensing capacitance, and driving the sensor with the electrical driving quantity.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a method and a circuit for detecting displacements using micro-electromechanical sensors with compensation of parasitic capacitances and spurious displacements. 
     2. Description of the Related Art 
     As is known, the use of micro-electric-mechanical sensors, or MEMS sensors, with differential capacitive unbalance has been proposed for forming, for example, linear or rotational accelerometers and pressure sensors. 
     In particular, MEMS sensors of the indicated type comprise a fixed body (stator) and a moving mass, generally of suitably doped semiconductor material, connected to each other through elastic elements (springs) and restrained so that, with respect to the stator, the moving mass has predetermined translational and rotational degrees of freedom. Moreover, the stator and the moving mass have a plurality of fixed and, respectively, moving arms, interleaved to each other. In practice, each fixed arm is arranged between a pair of moving arms, so as to form a pair of capacitors having a common terminal and a capacitance which is a function of the relative position of the arms, that is of the relative position of the moving mass with respect to the stator. When the sensor is stressed, the moving mass moves and the capacitance of the capacitors is unbalanced. 
     Depending on the type of structure and relative movement allowed between the moving mass and the stator, it is possible to manufacture MEMS sensors of a linear or rotational type, with variable interspace (distance between each moving arm and the respective fixed arms) and/or with variable facing area (variation of the reciprocal facing area between the moving arms and the respective fixed arms). 
     In all mentioned cases, reading by the sensor (that is detection of an electric quantity representing the variation of the capacitance of the capacitors) leads to problems due to the presence of parasitic capacitors (pad and substrate capacitances). The reading precision is also limited by another drawback, which is caused by spurious displacements, i.e., displacements not according with the designed degrees of freedom and due to non-ideality of mechanical constraints. 
     For the sake of clarity, reference will be made to FIGS. 1 and 2, where a linear MEMS sensor  1  is shown; however, what will be explained hereinafter applies to MEMS sensors of any type. 
     In detail, the sensor  1  comprises a stator  2  and a moving mass  3 , connected to each other by springs  4  so that the moving mass  3  can translate parallel to a first reference axis X, while it is substantially fixed with respect to a second and a third reference axes Y, Z. The sensor  1  is also symmetrical with respect to a longitudinal axis parallel to the first reference axis X. 
     The stator  2  and the moving mass  3  are provided with a plurality of first and second fixed arms  5 ′,  5 ″ and, respectively, with a plurality of moving arms  6 , extending substantially parallel to the plane Y-Z. 
     As shown in detail in FIG. 2, each moving arm  6  is arranged between two respective fixed arms  5 ′,  5 ″, partially facing them. Consequently, the moving arm  6  forms, with the two fixed arms  5 ′,  5 ″, a first and, respectively, a second sensing capacitor  8 ,  9  with paraliel flat faces. In particular, the area of the plates of the sensing capacitors  8 ,  9  is equal to the facing area A of the moving arms  6  and of the fixed arms  5 ′,  5 ″. In particular, the facing area A is substantially a rectangle with sides Ly, Lz. 
     The first and the second sensing capacitor  8 ,  9  have a first and a second sensing capacitance Ca, Cb, respectively, given by the equations:              Ca   =     ɛ        A   X1               (   1   )               Cb   =     ɛ        A   X2               (   2   )                                
     where X1, X2 are the distances between the moving arm  6  and the first and, respectively, the second fixed arms  5 ′,  5 ″ of FIG.  2  and ∈ is the dielectric constant of the air. 
     In the sensor  1 , all the sensing capacitances Ca formed between the moving arms  6  and the first fixed arms  5 ′ are parallel-connected; similarly all the sensing capacitances Cb formed between the moving arms  6  and the second fixed arms  5 ″ are parallel-connected. Consequently, altogether two capacitances are present between the stator  3  and the moving mass  4 , equal to Cl=N*Ca and, respectively, to C2=N*Cb, with N number of moving arms  6  of the sensor  1 . If we define as a common sensing capacitance Cs of the sensor  1  the value of the capacitances C1, C2 at rest, we have: 
     
       
           Cs=C 1 =C 2  (3) 
       
     
     After a movement of the moving arm  4  purely along the axis X, the sensing capacitances C1, C2 present variations with an opposite sign and with a same absolute value, and equal to a capacitive unbalance ΔCs. 
     In greater detail, supposing for simplicity&#39;s sake that the distances X1, X2 are initially the same and equal to a rest distance X0, from equations (1)-(3) it results that the component ΔCsx of the capacitive unbalance ΔCs according to the first reference axis X is given by the equation:                  Δ                 CSx     =         -          Cs          X            Δ                 X     =           ɛ                 A       X0   2          Δ                 X     =       Cs   X0        Δ                 X                
            Δ                 CSx     =         -          Cs          X            Δ                 X     =           ɛ                 A       X0   2          Δ                 X     =       Cs   X0        Δ                 X                   (   4   )                                
     where ΔX is the movement of the moving mass  4  long the first reference axis X. 
     In presence of a spurious movement ΔY parallel to the second reference axis Y, the capacitive unbalance ΔCs has a component ΔCsy given by the equation:                Δ                 CSy     =         -          Cs          Y            Δ                 Y     =         -       ɛ                 Ly     X0          Δ                 Y     =       -     CS   Ly          Δ                 Y                 (   5   )                                
     Any spurious movements ΔZ along the third reference axis Z are instead compensated by virtue of the axial symmetry of the sensor MEMS  1 . 
     While the unbalance introduced by the movement ΔX is of a differential type and is itself suitable to be detected by a fully differential sensing operational amplifier (see, for example, the article “A Three-Axis Micromachined Accelerometer with a CMOS Position-Sense Interface and Digital Offset-Trim Electronics” by M. Lemkin, B. Boser, IEEE Journal of Solid-State Circuits, Vol. 34, N. 4, Pages 456-468), the movement ΔY introduces a notable common mode variation of the common sensing capacitance Cs, as it causes a variation of the facing area A (FIG.  2 ). 
     Since the sensing operational amplifier allows detection of a voltage that is directly proportional to the capacitive unbalance ΔCs, which in turn is directly proportional to the common sensing capacitance Cs, the common mode variation due to the movement ΔY introduces a significant sensing error. 
     BRIEF SUMMARY OF THE INVENTION 
     An embodiment of the present invention overcomes the above-mentioned drawbacks. 
     According to an embodiment of the present invention, a method and a circuit are provided for detection of displacements through a micro-electromechanical sensor. The sensor includes a fixed body and a mobile mass, and forms a first sensing capacitor and a second sensing capacitor having a common capacitance at rest. The first and second sensing capacitors are connected to a first input terminal and, respectively, to a first output terminal and to a second output terminal of the sensing circuit. 
     According to an embodiment of the invention, the method includes the steps of closing a first negative-feedback loop, which is formed by the first and second sensing capacitors and by a differential amplifier, feeding an input of the differential amplifier with a staircase sensing voltage through driving capacitors so as to produce variations of an electrical driving quantity which are inversely proportional to the common sensing capacitance, and driving the sensor with the electrical driving quantity. 
     According to another embodiment of the invention, a circuit for detecting displacements in the sensor is provided, including a first negative-feedback loop, which can be closed selectively and which includes the first and second sensing capacitors and first amplifier means. The circuit also includes voltage-source means connected to the first amplifier means via capacitive driving means and supplying a staircase sensing voltage when the first negative-feedback loop is closed, so as to produce variations of an electrical driving quantity of the sensor, which are inversely proportional to the common sensing capacitance. 
    
    
     BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S) 
     For a better understanding of the present invention, two embodiments thereof are now described, purely by way of non-limiting example, with reference to the attached drawings, wherein: 
     FIG. 1 is a perspective view of a micro-electromechanical sensor of a known type; 
     FIG. 2 is a perspective view of an enlarged detail of the sensor of FIG. 1; 
     FIG. 3 is a simplified circuit diagram of a sensing circuit for a micro-electromechanical sensor object of a previous patent application; 
     FIG. 4 is a graph of quantities present in the circuit of FIG. 3; 
     FIGS. 5-7 show circuit diagrams of a sensing circuit for a micro-electromechanical sensor according to a first embodiment of the present invention, in different operating configurations; and 
     FIGS. 8 and 9 show circuit diagrams of a sensing circuit for a micro-electromechanical sensor according to a second embodiment of the present invention, in different operating configurations. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     To overcome the drawbacks described above, a method and a circuit for reading a MEMS sensor have been proposed in U.S. patent application Ser. No. 10/081,134, filed on Feb. 20, 2002 in the name of the same Applicants, which is incorporated by reference in its entirety. 
     This patent application will be briefly described with reference to FIG. 3, where the MEMS sensor  1  is schematized through a first and a second equivalent sensing capacitor  11 ,  12 , having capacitances equal to the first and, respectively, to the second sensing capacitance C1, C2, and further having first terminals connected to a first and, respectively, a second sensing node  13 ,  14  and second terminals connected to a common node  15 . In FIG. 3, the parasitic capacitances of the sensor MEMS  1  are schematized by parasitic capacitors  17 ,  18  connected between the sensing nodes  13 , respectively  14 , and ground. 
     A sensing circuit  30  comprises a sensing operational amplifier  20 , a feedback stage  21 , a compensation stage  31  and a signal generator  60 , which is connectable to the common node  15  through a first input of a selector  61  and supplies a sensing voltage Vs. 
     Briefly, the sensing operational amplifier  20 , having fully differential topology, has its inputs connected to the first and, respectively, to the second sensing node  13 ,  14 , is connected in a charge-integrator configuration, and supplies an output voltage Vo. 
     The feedback stage  21  comprises an amplifying circuit  25  and a first and a second feedback capacitor  26 ,  27 , having first terminals connected to an output  25   a  of the amplifying circuit  25  and second terminals connected to the first and to the second sensing node  13 ,  14 , respectively. The amplifying circuit  25 , has differential inputs connected to the first and to the second sensing node  13 ,  14 , respectively, and receives a reference voltage V REF  on a reference input  25   b  and supplies a feedback voltage V FB  on its output  25   a.    
     The compensation stage  31  has an input, connected to the output  25   a  of the amplifying circuit  25 , and an output  31   a , connectable to the common node  15  through a second input of the selector  61 . Moreover the compensation stage  31  supplies a compensation voltage Vc, linked to the common sensing capacitance Cs of the sensor MEMS  1  approximately by an inverse proportional function, as explained in detail below. 
     The compensation stage  31  comprises a memory capacitor  32 , a decoupling stage  33 , preferably an operational amplifier in follower configuration, and an inverting amplifier  35 . 
     The memory capacitor  32  has a first terminal connected to ground and a second terminal alternatively connectable to the output  25   a  of the amplifying circuit  25  and to the decoupling stage  33  through respective first and second switches  36 ,  37 , controlled in counterphase. 
     The inverting amplifier  35  has an input terminal  35   a  connected to the output of the decoupling stage  33 ; and an output terminal forming the output  31   a  of the compensation stage  31  and supplying the compensation voltage Vc. 
     The sensing circuit  30  exploits the fact that the common sensing capacitance Cs is, to a first approximation, linked to the compensation voltage Vc through an inverse proportionality relation. 
     In fact, when the sensing voltage Vs (which has a constant value) is supplied to the mobile mass  4 , the feedback voltage V FB  supplied by the amplifier circuit  25  assumes a value directly proportional to the overall sensing capacitance Cs, as also does the output voltage Vo. In two subsequent steps, the feedback voltage V FB  is stored and then transferred to the compensation operational amplifier  35 . Given that the gain G of the inverting amplifier  35  is negative and the variations in the common sensing capacitance Cs that are due to spurious displacements ΔY are of the order of femtoFarads, the pattern of the compensation voltage Vc with respect to the variations in the common sensing capacitance Cs is a first-order approximation of a relation of inverse proportionality (see FIG. 4, which also shows a curve Vc(inversely proportional to the common sensing capacitance Cs). In other words, in a preset neighborhood I of a rest value Cs 0  of the common sensing capacitance Cs, it is correct to assume that 
     
       
           Vc=K/Cs   (6) 
       
     
     where K is a constant of proportionality. 
     The compensation voltage thus obtained is supplied to the common node  15 . In this step, the output voltage Vo is given by the following relation:              Vo   =     2        K2   Ci            Δ                 Cs     Cs               (   7   )                                
     Since, according to equations (4) and (5), the capacitive unbalancing ΔCs is given by                Δ                 Cs     =         Δ                 Csx     +     Δ                 Csy       =       Cs        (         Δ                 X     X0     -       Δ                 Y     Ly       )       ≅     Cs          Δ                 X     X0                   (   8   )                                
     the output voltage Vo is found to be substantially independent of the common sensing capacitance Cs. In fact, combining equations (7) and (8) we obtain              Vo   =     2        K2   Ci            Δ                 X     X0               (   9   )                                
     The sensing circuit  30  has, however, some limitations, which are mainly due to the fact that a first-order approximation is made. Following upon this approximation, in fact, linearity errors may occur, especially when the spurious displacements of the mobile mass  4  are of a considerable amount. In this case, the compensation of the spurious displacements may be imprecise, and, moreover, distortions are introduced that degrade the performance of the sensing circuit. 
     Secondly, it is necessary to introduce a special compensation stage, which involves a considerable increase in the overall dimensions of the circuit. 
     An further improvement is therefore described below, which overcomes the limitations outlined above. 
     With reference to FIGS. 5-7, number  100  designates a sensing circuit for a MEMS sensor  101 , here of the differential input type. The MEMS sensor  101 , per se known and having a stator  2  and a mobile mass  3  as shown in FIGS. 1 and 2, has a first input terminal  102  and a second input terminal  103 , which are connected to the mobile mass  3 , and a first output terminal  104  and a second output terminal  105 , which are connected to the stator  2  and can be represented schematically by four equivalent sensing capacitors  107 - 110 . In detail, a first equivalent sensing capacitor  107 , having capacitance C11, is connected between the first input terminal  102  and the first output terminal  104 ; a second equivalent sensing capacitor  108 , having capacitance C12, is connected between the first input terminal  102  and the second output terminal  105 ; a third equivalent sensing capacitor  109 , having capacitance C21, is connected between the second input terminal  103  and the first output terminal  104 ; and a fourth equivalent sensing capacitor  110 , having capacitance C22, is connected between the second input terminal  103  and the second output terminal  105 . 
     In addition, at rest, the capacitances of the sensors are all equal to a common sensing capacitance Cs. When, instead, the MEMS sensor  101  is excited, a capacitive unbalancing ΔCs is generated, defined by equations (4) and (5), and the capacitances of the equivalent sensing capacitors  107 - 110  are given by the relations              C11   =     C22   =     Cs   +     Δ                 Cs                 (   10   )               C12   =     C21   =     Cs   -     Δ                 Cs                 (   11   )                                
     The sensing circuit  100  comprises a sensing operational amplifier  111 , having fully differential topology, a driving stage  112 , a feedback stage  114 , and at least one reference line  115 , which supplies a reference voltage Vref. 
     The sensing operational amplifier  111  has a non-inverting input  111   c  and an inverting input  111   d , which are respectively connected to the first output terminal  104  and to the second output terminal  105  of the MEMS sensor  101 , and an inverting output  111   a  and a non-inverting output  111   b , between which an output voltage Vo is supplied. In addition, a first feedback switch  116  is connected between the inputs of the operational amplifier  111 . A first integration capacitor  117  is connected between the non-inverting input  111   c  and the inverting output  111   a , and a second integration capacitor  118  is connected between the inverting input  111   d  and the non-inverting output  111   b  of the sensing operational amplifier  111 , which is consequently in a charge-integrator configuration. Both of the integration capacitors  117 ,  118  have integration capacitance Ci. 
     The driving stage  112  comprises a signal-generator circuit  120 , which has an output  120   a  supplying a staircase sensing voltage Vs with steps of predetermined amplitude and duration, and a pair of driving capacitors  121 ,  122 , which have driving capacitance Cd and present first terminals in common, connected to the output  120   a  of the signal-generator circuit  120 , and second terminals, connected, respectively, to the non-inverting input and to the inverting input of the sensing operational amplifier  111 . 
     The feedback stage  114  comprises a feedback operational amplifier  124  and a holding capacitor  125  having holding capacitance Ch. 
     The feedback operational amplifier  124  has a non-inverting input  124   a  connected to the reference line  115 ; an inverting input  124   b  connected to the second output terminal  105  of the MEMS sensor  101  via a second feedback switch  128  and to the reference line  115  via a first supply switch  126 ; and an output  124   c  which supplies a compensation voltage Vc. A second initialization switch  127  is connected between the reference line  115  and the second output terminal  105  of the MEMS sensor  101 . In addition, the output  124   c  of the feedback operational amplifier  124  is connected to the reference line  115  via a third initialization switch  140 , and is also connected to the first input terminal  102  and to the second input terminal  103  of the MEMS sensor  101  via a first driving switch  129  and via a second driving switch  130 , respectively. The first input terminal  102  and the second input terminal  103  of the MEMS sensor  101  are in turn connected to the reference line  115  via a third driving switch  131  and a fourth driving switch  132 , respectively. 
     The holding capacitor  125  has a first terminal connected to the output  124   c  of the feedback operational amplifier  124  and a second terminal which can be selectively connected to the reference line  115 , via a first holding switch  134 , and to the inverting input  124   b  of the feedback operational amplifier  124 , via a second holding switch  135 . 
     Operation of the sensing circuit  100  is described hereinafter. First, an initialization step is performed, during which the initialization switches  126 ,  127 ,  140 , the feedback switches  116 ,  128 , the third and fourth driving switches  131 ,  132 , and the first holding switch  134  are closed, while the first and second driving switches  129 ,  130  and the second holding switch  135  are open (FIG.  5 ). Consequently, the input terminals  102 ,  103  and output terminals  104 ,  105  of the MEMS sensor  101  and the inputs  124   a ,  124   b  of the feedback operational amplifier  124  are set at the reference voltage Vref. Also the compensation voltage Vc at the output  124   c  of the feedback operational amplifier  124  is initially equal to the reference voltage Vref. 
     Next, the common sensing capacitance Cs is detected and stored. In detail (FIG.  6 ), the initialization switches  126 ,  127 ,  140  and the third driving switch  131  are open, while the first driving switch  129  is closed. In this way, a negative-feedback loop  136  formed by the feedback operational amplifier  124  and by the first and second equivalent sensing capacitors  107 ,  108  is closed. In addition, the feedback operational amplifier  124  is in an inverting-amplifier configuration; in particular, the driving capacitors  121 ,  122  and the sensing capacitors  107 ,  108  respectively form input elements and feedback elements of the inverting amplifier. 
     Immediately after switching of the switches  126 ,  127 ,  140 ,  129 , and  131 , the signal-generator circuit  120  generates a voltage step having an amplitude Vs, said step being supplied to the input  124   b  of the feedback operational amplifier  124  through the driving capacitors  121 ,  122 , and, on account of the presence of the negative-feedback loop  136 , determines a variation in the compensation voltage ΔVc given by the relation                Δ                 Vc     =         -       2      Cd       2      Cs            Vs     =       -     Cd   Cs          Vs               (   12   )                                
     In this step, in fact, the first equivalent sensing capacitor  107  and the second equivalent sensing capacitor  108  are parallel connected, and likewise the first driving capacitor  121  and the second driving capacitor  122  are parallel connected (the first feedback switch  116  is still closed). Consequently, the capacitance altogether present between the output  120   a  of the signal-generator circuit  120  and the inverting input  124   b  of the feedback operational amplifier  124  is equal to 2Cd, while the capacitance between the inverting input  124   b  and the output  124   c  of the feedback operational amplifier  124 , according to equations (10) and (11), is given by the relation 
     
       
           C 11 +C 12 =Cs+ΔCs+Cs−ΔCs= 2 Cs   (13) 
       
     
     The compensation voltage Vc (initially equal to Vref) goes to, and is held, at a driving voltage Vcd given by the equation 
     
       
           Vcd=Vref+ΔVc   (14) 
       
     
     In practice, the variation in the compensation voltage ΔVc, which is inversely proportional to the common sensing capacitance Cs, as shown by relation (12), is stored by the holding capacitor  125 . In addition, this value of the driving voltage Vcd is also fed on the first input terminal  102  of the MEMS sensor  101 , which is directly connected to the output  124   c  of the feedback operational amplifier  124 . 
     Next, the capacitive unbalancing ΔCs is detected. In particular, the step of the sensing-voltage Vs terminates, and the feedback switches  116 ,  128 , the first holding switch  129 , and the first and fourth driving switches  129 ,  132  are opened, while the second holding switch  135 , and the second driving switch  130  and the fourth driving switch  131  are closed (FIG.  7 ). Given that the second feedback switch  128  is open, the negative-feedback loop  136  is open. In addition, switching of the holding switches  134 ,  135  enables feedback connection of the holding capacitor  125  between the inverting input  124   b  and the output  124   c  of the feedback operational amplifier  124 . In this way, the charge on the holding capacitor  125  is conserved, and hence the compensation voltage Vc on the output  124   c  of the feedback operational amplifier  124  is kept at the driving voltage Vcd. 
     In addition, the compensation voltage Vc is used for driving the MEMS sensor  101 . In detail, the output  124   c  of the feedback operational amplifier  124  is disconnected from the first input terminal  102  and connected to the second input terminal  103  of the MEMS sensor  101 . Consequently, the voltage on the first input terminal  102  of the MEMS sensor  101  switches from the driving value Vcd to the value of the reference voltage Vref. Instead, the voltage on the second input terminal  103  switches from the value of the reference voltage Vref to the driving value Vcd. In other words, voltage steps of opposite sign and of an amplitude equal to the variation in the compensation voltage ΔVC, and hence inversely proportional to the common sensing capacitance Cs, are applied simultaneously to the input terminals  102 , 103  of the MEMS sensor  101 . 
     Thanks to the above relation of inverse proportionality, the output voltage Vo in this step is given by the following relation, which is analogous to equation (7):              Vo   =       2      Δ                 Vc          2      Δ                 Cs     Ci       =       A   Cs            Δ                 Cs     Ci                 (   15   )                                
     where A is a constant. 
     As already discussed previously, in particular with reference to equations (8) and (9), the output voltage Vo is independent of the common sensing capacitance Cs, given that the capacitive unbalancing ΔCs is directly proportional to the common sensing capacitance, and hence the errors due to spurious displacements of the mobile mass of the MEMS sensor  101  are substantially eliminated. 
     In addition, the use of staircase voltages with steps of equal amplitude and opposite sign for driving the MEMS sensor  101  makes it possible to maintain constant the common-mode voltages at the output terminals  104 ,  105  of the MEMS sensor  101 , thus the parasitic capacitances (not illustrated) associated to these output terminals have no effect. 
     FIGS. 8 and 9, in which parts in common with those already shown are designated by the same reference numbers, illustrate a different embodiment of the invention, according to which a sensing circuit  150  for a MEMS sensor  151  comprises the sensing operational amplifier  111 , in a charge-integrator configuration, the reference line  115 , the signal-generator circuit  120 , a biasing line  165 , which supplies a biasing voltage Vb, and a feedback stage  152 . 
     The MEMS sensor  151 , in the present case of the single-input type described in FIGS. 1 and 2, has an input terminal  153  and a first output terminal  154  and a second output terminal  155 , and may be schematically represented by a first equivalent sensing capacitor  156  and a second equivalent sensing capacitor  157 . In particular, the first equivalent sensing capacitor  156  is connected between the input terminal  153  and the first output terminal  154  and has a capacitance equal to the common sensing capacitance Cs when the MEMS sensor  151  is at rest, and a capacitance equal to Cs+ΔCs when the MEMS sensor  151  is excited and a capacitive unbalancing ΔCs occurs. The second equivalent sensing capacitor  157  is connected between the input terminal  153  and the second output terminal  155  and has a capacitance equal to the common sensing capacitance Cs when the MEMS sensor  151  is at rest, and a capacitance equal to Cs−ΔCs when the MEMS sensor  151  is excited. 
     The feedback stage  152  comprises an amplifier circuit and a first feedback capacitor  159  and a second feedback capacitor  160 . 
     The amplifier circuit  158 , per se known and described in detail in the aforementioned article, has a pair of differential inputs  158   a ,  158   b , respectively connected to the first output terminal  154  and to the second output terminal  155  of the MEMS sensor  151 , a reference input connected to the reference line  115 , and an output  158   c , which supplies a feedback voltage Vfb and is connected to a feedback node  161  via a feedback switch  162  and to the input terminal  153  of the MEMS sensor  151  via a driving switch  164 . The input terminal  153  of the MEMS sensor  151  is moreover connected to the driving line  165  via a biasing switch  166 . 
     The first feedback capacitor  159  and the second feedback capacitor  160 , both having feedback capacitance Cfb, have first terminals connected to the feedback node  161  and second terminals respectively connected to the first output terminal  154  and to the second output terminal  155  of the MEMS sensor  151 . 
     The signal-generator circuit  120 , which supplies a voltage step having amplitude Vs, has an output  120   a , which is connected to the feedback node  161 . 
     In an initial operating step, the sensing circuit  150  is initialized by closing the biasing switch  166  and the driving switch  164  and opening the feedback switch  162 . In this way, the input terminal  153  of the MEMS sensor  151  and the output  158   c  of the amplifier circuit  158  go to the biasing voltage Vb (namely, Vfb=Vb). 
     Next (FIG.  8 ), the biasing switch  166  is opened, and the signal-generator circuit  120  supplies a voltage step having amplitude Vs. In this phase, the driving switch  164  closes a first negative-feedback loop  168 , which is formed by the amplifier circuit  158  and by the equivalent sensing capacitors  156 ,  157  of the MEMS sensor  151 . Consequently, the voltage step of amplitude Vs brings about a variation ΔVfb in the feedback voltage Vfb generated by the amplifier circuit  158 , the operation of which is described in detail in the aforementioned article. In practice, the amplitude of the variation ΔVfb of the feedback voltage Vfb is given by the expression 
     
       
         Δ Vfb=Vs ( Cfb/Cs )  (16) 
       
     
     and is thus inversely proportional to the common sensing capacitance Cs. 
     The value of the feedback voltage Vfb on the output  158   c  of the amplifier circuit  158  and on the input terminal  153  of the MEMS sensor  151  is therefore 
     
       
           Vfb=Vb+ΔVfb   (17) 
       
     
     Next, all the switches  162 ,  164 ,  166  switch (FIG.  9 ). In this way, the first negative-feedback loop  168  is opened, and a second negative-feedback loop  170  formed by the amplifier circuit  158  and by the feedback capacitors  159 ,  160  is closed. Thanks to the second negative-feedback loop  170 , the amplifier circuit  158 , in a per se known manner, maintains at a constant value the common-mode voltage between the first output terminal and the second output terminal of the MEMS sensor  151 , and hence between the inputs of the sensing operational amplifier  111 , which thus operates correctly. 
     In addition, in this step the input terminal  153  of the MEMS sensor  151 , which is again connected to the biasing line  165 , goes to the biasing voltage Vb. In practice, then, the voltage at the input terminal  153  undergoes an amplitude variation equal to the variation ΔVfb of the feedback voltage Vfb, and hence a variation that is inversely proportional to the common sensing capacitance Cs. 
     Consequently, the output voltage Vo is given by the equation              Vo   =       2      Δ                 Vfb          Δ                 Cs     Ci       =       B   Cs            Δ                 Cs     Ci                 (   18   )                                
     where B is a constant. 
     Also in this case, the contributions due to spurious displacements are thus cancelled out by applying to the input terminal of the MEMS sensor  151  a voltage inversely proportional to the common sensing capacitance Cs. 
     The advantages of the present invention emerge clearly from the foregoing description. 
     First, the sensing circuit described herein has not any problems of linearity, since no approximations are made, but rather the properties of the feedback amplifiers are exploited for generating a voltage which is in itself inversely proportional to the common sensing capacitance Cs. Consequently, the spurious displacements of the mobile mass of the MEMS sensor are effectively eliminated without introducing any distortions, and hence the precision of the sensing circuit is markedly improved. 
     Second, it is not necessary to provide additional compensation stages, and consequently the sensing circuit is simpler to build and has small overall dimensions. 
     Finally, it is clear that modifications and variations may be made to the circuit and method described herein, without thereby departing from the scope of the present invention. 
     All of the above U.S. patents, U.S. patent application publications, U.S. patent applications, foreign patents, foreign patent applications and non-patent publications referred to in this specification and/or listed in the Application Data Sheet, are incorporated herein by reference, in their entirety.