PATENT DOCUMENT

Publication Number: US-10859407-B2
Application Number: US-201916581661-A
Country: US
Kind Code: B2

Title: Motion sensing using hall voltage signals

Abstract:
An integrated circuit (IC) chip including an array of asymmetrically distributed magnetic field sensing elements. Additionally, an integrated circuit (IC) chip includes a substrate, a sensing coil supported by the substrate and enclosing a portion of substrate, and a Hall effect sensor supported by the portion of the substrate enclosed by the sensing coil.

Claims:
What is claimed is: 
     
       1. A method comprising:
 vibrating a magnet relative a first coil in response to driving a current through the first coil; 
 concurrently measuring a back electromotive force (bEMF) signal using a second coil affixed to the first coil and a Hall voltage signal using a Hall effect sensor (HES) disposed within the second coil; and 
 determining a velocity of the vibrating magnet by using a sensing matrix and the concurrently measured bEMF signal and Hall voltage signal, 
 wherein an inverse of the sensing matrix maps the concurrently measured bEMF signal and Hall voltage signal to the velocity of the vibrating magnet and a rate of the current driven through the first coil, and 
 wherein the method further comprises determining the rate of the current concurrently with the determining of the velocity. 
 
     
     
       2. The method of  claim 1 , further comprising:
 verifying that the determined rate of the current exceeds a threshold; and 
 adjusting one or more elements of the sensing matrix in response to the verifying. 
 
     
     
       3. The method of  claim 1 , further comprising:
 measuring the current concurrently with the measuring of the bEMF signal and the Hall voltage signal; 
 determining a rate of the measured driving current by differentiating the measured driving current; 
 verifying that the determined rate of the current is different from the rate of the measured driving current by a threshold; and 
 adjusting one or more elements of the sensing matrix in response to the verifying. 
 
     
     
       4. The method of  claim 3 , wherein the adjusting comprises applying a filter on the one or more elements of the sensing matrix. 
     
     
       5. The method of  claim 4 , wherein the applying of a filter comprises performing a running average of the one or more elements of the sensing matrix. 
     
     
       6. A displacement measurement system comprising:
 (i) a haptic engine comprising
 a) a first coil, 
 b) a mass supporting a magnet, the mass being movable relative to the first coil, and 
 c) a hybrid sensor affixed to the first coil, wherein the hybrid sensor comprises 
 (I) a substrate, 
 (II) a second coil disposed on the substrate and enclosing a portion of substrate, and 
 (III) a Hall effect sensor (HES) element disposed on the portion of the substrate enclosed by the second coil; and 
 
 (ii) a digital signal processor (DSP) configured to determine displacements of the mass based on a back electromotive force (bEMF) signal and a Hall voltage signal induced in the second coil and in the HES, respectively, due to magnetic field changes caused by motion of the mass when a driving current is being driven through the first coil. 
 
     
     
       7. The displacement measurement system of  claim 6 , wherein the hybrid sensor is an IC chip. 
     
     
       8. The displacement measurement system of  claim 7 , wherein the hybrid sensor is an ASIC with the DSP formed on the IC chip. 
     
     
       9. The displacement measurement system of  claim 6 , wherein
 the substrate is a PCB board, and 
 the HES is formed on a chip, the chip being connected to a socket of the PCB board disposed on the portion of the PCB board enclosed by the second coil. 
 
     
     
       10. The displacement measurement system of  claim 9  wherein the DSP is disposed on the PCB board. 
     
     
       11. The displacement measurement system of  claim 6 , comprising
 a plurality of the hybrid sensors c) affixed to the first coil, 
 wherein the DSP is configured to determine the displacement of the mass based on the bEMF signal and the Hall voltage signal output by each of the plurality of the hybrid sensors c). 
 
     
     
       12. The displacement measurement system of  claim 11 , wherein the plurality of the hybrid sensors c) forms a 1D array. 
     
     
       13. The displacement measurement system of  claim 11 , wherein the plurality of the hybrid sensors c) forms a 2D array.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This disclosure is a divisional of and claims priority to U.S. patent application Ser. No. 15/674,227, filed Aug. 10, 2017, which claims priority to U.S. Provisional Application Ser. No. 62/396,036, filed Sep. 16, 2016, and claims priority to U.S. Provisional Application Ser. No. 62/482,166, filed Apr. 5, 2017, the entire contents of which are incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present disclosure is generally related to motion sensing. For example, aspects of the present disclosure are related to localizing a moving magnet using an array of asymmetrically distributed Hall-effect sensing elements. As another example, aspects of the present disclosure are related to localizing a moving magnet based on Hall voltage signals and back electromotive force (bEMF) signals that are concurrently acquired using a hybrid sensor that includes a Hall effect sensor (HES) and a sensing coil. 
     BACKGROUND 
     A haptic engine (also referred to as a vibration module) is a linear resonant actuator that determines one of acceleration, velocity and displacement of a moving mass.  FIGS. 12A-12B  show aspects of a conventional haptic engine (HE) in which position of a magnet (M), that is moving relative to a fixed coil (C), is encoded in the intensity of magnetic field flux and sensed by Hall-effect sensing elements (HSEs), also referred to interchangeably as HESs or simply Hall sensors, disposed on a top side of the coil, and on a bottom side of the coil. For example, a displacement ΔX along the x-axis and a displacement ΔZ along the z-axis of the magnet, that is moving relative to the fixed coil, is determined as: 
                       Δ   ⁢           ⁢   X     ∝       V     H   top       +     V     H   btm           ,           (   1   )                   Δ   ⁢           ⁢   Z     ∝       (       V     H   top       +     V     H   btm         )       Δ   ⁢           ⁢   X         ,           (   2   )               
where the magnetic field flux induces a Hall voltage V Htop  in the Hall sensor disposed on the top side of the coil, and a Hall voltage V Hbtm  in the Hall sensor disposed on the bottom side of the coil.
 
     As another example, the displacement ΔX of the magnet along the x-axis can be obtained as:
 
Δ X =LUT( V   H   −ηI )  (3),
 
where V H  is voltage output by an HSE, I is a current through the driving coil C, and η is an EM coupling factor. V H  is proportional to a magnitude B of a total field caused by the magnet and induced by the driving coil, while ηI is proportional to a magnitude of a field induced by the driving coil. In EQ. (3), the difference represents the Hall voltage due to the field caused only by the magnet. LUT is a look-up table mapping the measured value of the field caused only by the magnet and a corresponding value of the displacement.
 
       FIG. 12C  shows examples of errors of the displacement measurements for various frequencies of driving currents. For driving currents having frequencies in the mid-frequency range, a sensing error can be caused by the inaccuracy of η. As such, a conventional HE needs an external reference displacement sensor (e.g., laser or bEMF model fitting, etc.) for calibration, as shown in  FIG. 12C . 
     For instance, bEMF can be determined conventionally using the driving coil itself, in the following manner: 
                     bEMF   =     V   -   RI   -     L   ⁢     dI   dt       -     R   ⁢           ⁢     τ   ADC     ⁢     dI   dt           ,           (   4   )               
where R and L are the resistance and inductance, respectively, of the driving coil. Here, the first term is the voltage across the driving coil, the second term is a driver term, and the third term is an inductance term. However, as shown in EQ. (4), accuracy of bEMF-based motion sensing is prone to a number of error sources. The coil resistance R is very sensitive to temperature changes and quantization error associated with analog-to-digital conversion (ADC). Typically, copper&#39;s temperature coefficient of resistance is approximately 0.4%/deg C. This can represent a large error source when the engine is operating in power-limited regime (away from resonance frequency) where bEMF can be approximately 10% of the RI term. Similarly, when R is estimated in real-time with very small signal magnitude (typically a calibration tone in kHz range), the estimation itself is also prone to errors of 1 to 10%. Another error source in Eq. (4) is the timing synchronization between driving coil voltage V and driving coil current I when an ADC delay τ ADC  between the measured driving coil voltage V and measured driving coil current I is finite (i.e., non-zero). As such, for driving currents having frequencies in a high-frequency range, a finite false inductance term, given by the fourth term in EQ. (4), can be sensed as part of bEMF, as shown in  FIG. 12C . Such timing synchronization can be expressed as
 
                           ⁢           τ   D     +           ⁢     τ   ADC       ≈       L   +     L   E       R       ,             (   5   )               
where τ D  is the group delay between voltage and current caused by the inductance, and L E  is the false inductance term caused by the ADC group delay.
 
     Further, large offsets of the magnet&#39;s cage relative to one side of the HE&#39;s housing can produce dead-zones in displacement sensitivity, as shown in  FIG. 12D . Furthermore, the conventional HE can be sensitive to temperature change. At least for the above reasons, the conventional HE shown in  FIGS. 12A-12B  requires external calibration, however, module to system test correlations can be elusive. 
     SUMMARY 
     In this disclosure, technologies are described for measuring displacements of a moving mass, which is part of a magnetic field source, by using an array of asymmetrically distributed magnetic field sensing elements. Other technologies described in this disclosure are directed to localizing a moving magnet based on Hall voltage signals and bEMF signals concurrently acquired using a hybrid sensor that includes a HES and a sensing coil. Here, the magnetic field sensing elements can be Hall-effect sensing elements fabricated, using CMOS technologies, as part of an integrated circuit chip. As such, the array of asymmetrically distributed Hall-effect sensing elements or the hybrid sensor along with driving circuitry, conditioning circuitry and processing circuitry can be integrated in an application specific integrated circuit (ASIC). 
     One aspect of the disclosure can be implemented as an integrated circuit (IC) chip that includes a first magnetic field sensing element disposed at a first location of the IC chip; a second magnetic field sensing element disposed at a second location of the IC chip; and a third magnetic field sensing element disposed at a third location of the IC chip, the first location, the second location and the third location being distributed along a first direction, such that the second location is between the first location and the third location, and the second location is separated from the first location by a first distance and from the third location by a second distance different from the first distance; and signal processing circuit configured to determine displacements of a mass, when the mass is in motion along a direction of motion having a first component along the first direction and a second component along a second direction normal to the IC chip, where the mass supports at least a portion of a magnetic field source, and where the displacements of the mass are determined, at least in part, based on location-specific changes of a magnetic field emitted by the magnetic field source, the magnetic field changes being caused by the motion of the mass. 
     Implementations can include one or more of the following features. In some implementations, the location specific magnetic field changes can include magnetic field changes sensed at the first location by the first magnetic field sensing element, magnetic field changes sensed at the second location by the second magnetic field sensing element, and magnetic field changes sensed at the third location by the third magnetic field sensing element. In some implementations, the signal processing circuit can be disposed in an area of the IC chip that is between the second location and third location along the first direction, and the second distance is larger than the first distance. 
     In some implementations, the determined displacements of the mass can include a displacement ΔX along the first direction (e.g., the x-axis) and a displacement ΔZ along the second direction (e.g., the z-axis). In some cases, each of the first, second and third magnetic field sensing elements can be configured to sense a component of the magnetic field along the second direction. For example, each of the first, second and third magnetic field sensing elements can include a uniaxial Hall-effect sensor element. In some cases, each of the first, second and third magnetic field sensing elements is configured to (i) sense components of the magnetic field along each of the first direction, the second direction and a third direction orthogonal on both the first direction and the second direction, and (ii) provide to the signal processing circuit an angle of the magnetic field relative to the second direction and a magnitude of the magnetic field. For example, each of the first, second and third magnetic field sensing elements can include a tri-axial Hall-effect sensor element. 
     In some cases, the IC chip can include a fourth magnetic field sensing element disposed at a fourth location of the IC chip, the fourth location being separated from the first location by a third distance along a third direction orthogonal on both the first direction and the second direction. Here, the direction of motion has a third component along the third direction, and the determined displacements of the mass further include a displacement ΔY along the third direction (e.g., the y-axis). Further, the IC chip can include a fifth magnetic field sensing element disposed at a fifth location of the IC chip, the fifth location being separated from the third location by the third distance along the third direction and from the fourth location by a sum of the first distance and second distance along the first direction. 
     Furthermore, the IC chip can include a sixth magnetic field sensing element disposed at a sixth location of the IC chip, the sixth location being (i) between the fourth location and the fifth location, (ii) separated from the fourth location by the first distance along the first direction, and (iii) separated from the second location by the third distance along the third direction; a seventh magnetic field sensing element disposed at a seventh location of the IC chip, such that the third location is between the second location and the seventh location, where the seventh location is separated from the third location by the first distance along the first direction; and an eighth magnetic field sensing element disposed at an eighth location of the IC chip such that the fifth location is between the sixth location and the seventh eighth, where the eighth location is separated from the fifth location by the first distance along the first direction, and separated from the seventh location by the third distance along the third direction. Here, the magnetic field sensing elements provide to the signal processing circuit a gradient along the first direction of the magnetic field, e.g., δ 2 B Z /δX 2 . 
     In some implementations, the IC chip can include one or more driving circuits each of which includes a programmable current source; two or more of the magnetic field sensing elements; and a dummy load connected to each other in series. Additionally, the IC chip can include a band-gap reference circuit; sample and hold circuits; and for each of the driving circuits, (i) a multiplexer circuit including input ports and an output port, where an output of each of the magnetic field sensing elements, the dummies and the band-gap reference circuit is coupled with a respective input port of the multiplexer through a respective sample and hold circuit; (ii) a programmable gain amplifier circuit coupled with the output port of the multiplexer circuit; and (iii) an analog to digital converter (ADC) circuit coupled between the programmable gain amplifier circuit and the signal processing circuit. In some cases, the IC chip can include a chopping multiplexer pair coupled between the output port of the multiplexer circuit and the input of the programmable gain amplifier circuit and between the output of the programmable gain amplifier circuit and the input of the ADC circuit. In some cases, the signal processing circuit and one or more of the dummy loads, the band-gap reference circuit, sample and hold circuits, multiplexer circuits, programmable gain amplifier circuits and ADCs are disposed in a central area of the IC chip that is between the second location and third location along the first direction, and the second distance is larger than the first distance. 
     In some implementations, the signal processing circuit can determine, when the mass is at rest, a gradient of the magnetic field along the first direction based on (i) a value of the magnetic field sensed by the first magnetic field sensing element at the first location, (ii) a value of the magnetic field sensed by the second magnetic field sensing element at the second location, and (iii) the first distance between the first location and the second location. As such, the displacements of the mass are determined based on a combination of the location-specific changes of the magnetic field and the gradient of the magnetic field along the first direction. 
     In some implementations, the signal processing circuit can include a microcontroller unit. In some implementations, the signal processing circuit can include a Field-Programmable Gate Array. In some implementations, the IC chip can be an application specific IC (ASIC). 
     Another aspect of the disclosure can be implemented as a haptic engine that includes the mass and the above-summarized IC chip. 
     Another aspect of the disclosure can be implemented as a computing device that includes the haptic engine. 
     Another aspect of the disclosure can be implemented as a displacement measurement system that includes a substrate; a first magnetic field sensing element disposed at a first location of the substrate; a second magnetic field sensing element disposed at a second location of the substrate; and a third magnetic field sensing element disposed at a third location of the substrate, the first location, the second location and the third location being distributed along a first direction, such that the second location is between the first location and the third location, and the second location is separated from the first location by a first distance and from the third location by a second distance larger than the first distance by a predetermined factor; and signal processing circuit configured to determine displacements of a mass, when the mass is in motion along a direction of motion having a first component along the first direction and a second component along a second direction normal to the IC chip, where the mass supports at least a portion of a magnetic field source, and where the displacements of the mass are determined, at least in part, based on location-specific changes of a magnetic field emitted by the magnetic field source, the magnetic field changes being caused by the motion of the mass. 
     Implementations can include one or more of the following features. In some implementations, the predetermined factor can be a range of 1.1 to 10. In some implementations, the signal processing circuit can be disposed in an area of the substrate that is between the second location and third location along the first direction. In some implementations, the substrate, the first magnetic field sensing element, the second magnetic field sensing element, and the third magnetic field sensing element can be included in an integrated circuit chip. In some implementations, the substrate is a PCB board; and the first magnetic field sensing element is formed on a first chip, the second magnetic field sensing element is formed on a second chip, and the third magnetic field sensing element is formed on a third chip. 
     The above-disclosed technologies can result in one or more of the following potential advantages. For example, the array of asymmetrically distributed HSEs can measure ΔX, ΔY and ΔZ at the same time, can measure rotation around X, Y, and Z axes, and can differentiate rotational motion from translational motion. 
     As another example, the array of asymmetrically distributed magnetic field sensing elements can be placed closer to the magnetic field source of the haptic engine and away from potential dead-zones, compared to the HSEs used in the conventional haptic engine in  FIGS. 12A-12B . As yet another example, unlike the HSEs used in the conventional haptic engine in  FIGS. 12A-12B  that require calibration based on an external reference displacement sensor, the array of asymmetrically distributed magnetic field sensing elements can be self-calibrated. Also, the array of asymmetrically distributed magnetic field sensing elements is integrated in a single CMOS chip or SoC along with necessary driver circuitry, unlike the HSEs used in the conventional haptic engine in  FIGS. 12A-12B  that are driven with external circuitry. 
     Furthermore, as the array of asymmetrically distributed magnetic field sensing elements is integrated in a single CMOS chip or SoC, the chip can be disposed, as part of a haptic engine, on a single side of a magnetic field source. In contrast, the HSEs used in the conventional haptic engine in  FIGS. 12A-12B  are disposed, inside a conventional haptic engine, above and below the magnet, and for this reason, results of the displacement measurements taken with the two conventionally placed HSEs are sensitive to alignment of the HSEs. 
     The reason for the HSEs used in the conventional haptic engine in  FIGS. 12A-2B  to be placed above and below the magnet is to allow for results of ΔZ measurements to be insensitive to temperature. In contrast, ΔZ measurements can be performed with the array of asymmetrically distributed magnetic field sensing elements in a ratiometric manner, in which case results of the ΔZ measurements are intrinsically insensitive to temperature change. 
     Another aspect of the disclosure can be implemented as an integrated circuit (IC) chip that includes a substrate; a sensing coil supported by the substrate and enclosing a portion of substrate; and a Hall effect sensor (HES) supported by the portion of the substrate enclosed by the sensing coil. 
     Implementations can include one or more of the following features. In some implementations, when the IC chip is disposed within a varying magnetic field, the sensing coil outputs a bEMF signal proportional to a change of the magnetic field, and, concurrently, the HES outputs a Hall voltage signal proportional to a magnitude of the magnetic field. 
     In some implementations, the sensing coil can be shaped like a polygon. In some implementations, the sensing coil can be shaped like an oval. In some implementations, the HES can be supported at a center of symmetry of the sensing coil. 
     In some implementations, the IC chip can include a plurality of layers stacked on the substrate. In some cases, the sensing coil and the HES can be formed in the same layer of the plurality of layers. In some cases, the sensing coil and the HES can be formed on respective different layers of the plurality of layers. 
     In some implementations, the IC chip can include signal conditioning circuitry coupled with output terminals of the sensing coil and output terminals of the HES. In some implementations, the IC chip can further include analog-to-digital converter (ADC) circuitry. Here, input terminals of the ADC circuitry are coupled with output terminals of the signal conditioning circuitry. In some implementations, the IC chip can additionally include MUX circuitry coupled between the output terminals of the signal conditioning circuitry and the input terminals of the ADC circuitry. In some implementations, the IC chip can additionally include signal processing circuitry coupled with output terminals of the ADC circuitry. In some cases, the signal processing circuit can include a microcontroller unit. In some cases, the signal processing circuit can include a Field-Programmable Gate Array. In some cases, the IC chip can be an application specific IC (ASIC). 
     In some implementations, a haptic engine can include a driving coil; a mass supporting a magnet, the mass being movable relative to the driving coil; and any one of the foregoing implementations of the disclosed IC chip affixed to the driving coil. In some implementations, a computing device can include the foregoing haptic engine. 
     Another aspect of the disclosure can be implemented as a method that includes vibrating a magnet relative a first coil in response to driving a current through the first coil; concurrently measuring a bEMF signal using a second coil affixed to the first coil and a Hall voltage signal using a Hall effect sensor (HES) disposed within the second coil; and determining a velocity of the vibrating magnet by using a sensing matrix and the concurrently measured bEMF signal and Hall voltage signal. 
     Implementations can include one or more of the following features. In some implementations, an inverse of the sensing matrix maps the concurrently measured bEMF signal and Hall voltage signal to the velocity of the vibrating magnet and a rate of the driving current through the first coil. Here, the method can further include determining the rate of the driving current concurrently with the determining of the velocity. 
     In some implementations, the method can further include verifying that the determined rate of the driving current exceeds a threshold; and adjusting one or more elements of the sensing matrix in response to the verifying. In some implementations, the method can further include measuring the driving current concurrently with the measuring of the bEMF signal and the Hall voltage signal; determining a rate of the measured driving current by differentiating the measured driving current; verifying that the determined rate of the driving current is different from the rate of the measured driving current by a threshold; and adjusting one or more elements of the sensing matrix in response to the verifying. In either of these implementations, the adjusting can include applying a filter on the one or more elements of the sensing matrix. For example, the applying of a filter can include performing a running average of the one or more elements of the sensing matrix. 
     Yet another aspect of the disclosure can be implemented as a displacement measurement system that includes (i) a haptic engine that includes a) a first coil, b) a mass supporting a magnet, the mass being movable relative to the first coil, and c) a hybrid sensor affixed to the first coil. The hybrid sensor includes (I) a substrate, (II) a second coil disposed on the substrate and enclosing a portion of substrate, and (III) a Hall effect sensor (HES) element disposed on the portion of the substrate enclosed by the second coil. Additionally, the displacement measurement system includes (ii) a digital signal processor (DSP) configured to determine displacements of the mass based on a bEMF signal and a Hall voltage signal induced in the second coil and in the HES, respectively, due to magnetic field changes caused by motion of the mass when a driving current is being driven through the first coil. 
     Implementations can include one or more of the following features. In some implementations, the hybrid sensor can be an IC chip. For example, the hybrid sensor can be an ASIC with the DSP formed on the IC chip. 
     In some implementations, the substrate can be a PCB board, and the HES can be formed on a chip. Here, the chip is connected to a socket of the PCB board disposed on the portion of the PCB board enclosed by the second coil. Further, the DSP can be disposed on the PCB board. 
     In some implementations, the displacement measurement system can include a plurality of the hybrid sensors c) affixed to the first coil, Here, the DSP can be configured to determine the displacement of the mass based on the bEMF signal and the Hall voltage signal output by each of the plurality of the hybrid sensors c). In some cases, the plurality of the hybrid sensors c) can form a 1D array. In some cases, the plurality of the hybrid sensors c) can form a 2D array. 
     The above-disclosed technologies can result in one or more of the following potential advantages. For example, by using the disclosed hybrid sensors, the above noted EM coupling factor η-accuracy errors in the Hall voltage measurements and the errors in bEMF measurements caused by resistance estimation and the finite ADC delay τ ADC  can be addressed concurrently. As such, when the disclosed technologies are used in combination with a closed loop controller of HE, the improvements in the foregoing error sources can improve the controller&#39;s performance, as shown in  FIGS. 11B-11C . For instance, command-to-displacement transfer function (CDTF) bandwidth increases when accuracy of the EM coupling factor η increases. CDTF describes the transfer function between an input displacement command to a close-loop controlled HE and an output displacement (i.e., the actual movement of HE). A wider bandwidth in CDTF means HE can support a richer set of haptic vocabularies with greater fidelity. 
     Furthermore, bEMF is prone to timing error (τADC) between driving coil voltage and driving coil current measurements, as these are typically a small calibration tone on the order of 100 mV at 2 kHz superimposed on top of an engine drive signal on the order of 6.6 V at frequencies lower than 400 Hz. As shown in Table 1, the disclosed motion sensing technologies can be rendered accurate because the timing errors are constrained to values of 1 μs or less. 
                                         TABLE 1                       τ ADC     −2 μs   0   +2 μs                                                            R   8.29   8.24   8.18           L   93.33   109.87   126.3                        
Here, R and L are the resistance and inductance, respectively, of the driving coil.
 
     As such, measuring, based on the disclosed technologies, displacement of a moving mass inside a haptic engine can be used to avoid crash of the mass, minimize saliency variation over population, and improve reliability of the haptic engine. 
     Details of one or more implementations of the disclosed technologies are set forth in the accompanying drawings and the description below. Other features, aspects, descriptions and potential advantages will become apparent from the description, the drawings and the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  shows an example of an IC chip including an array of asymmetrically distributed magnetic field sensing elements. 
         FIG. 1B  shows an example of a haptic system that has an IC chip including an array of asymmetrically distributed magnetic field sensing elements. 
         FIG. 1C  shows another example of a haptic system that has an IC chip including an array of asymmetrically distributed magnetic field sensing elements. 
         FIG. 2A  shows an example of a spatial distribution of a magnetic field emitted by a magnetic field source of a haptic engine. 
         FIG. 2B  shows components of a magnetic field sensed by a tri-axial Hall-effect sensing element. 
         FIGS. 3A-3C  show aspects of another example of a spatial distribution of a magnetic field emitted by a magnetic field source of a haptic engine. 
         FIGS. 4A-4C  show examples of changes of a magnetic field, that are caused by various displacements of a magnetic field source, as measured by asymmetrically distributed magnetic field sensing elements. 
         FIG. 5A  shows an example of a process for equalizing sensitivity of magnetic field sensing elements. 
         5 B- 5 D show aspects of a calibration process applied to an array of asymmetrically distributed magnetic field sensing elements. 
         FIG. 5E  shows an example of a process for determining a displacement of a mass using an array of asymmetrically distributed magnetic field sensing elements. 
         FIG. 5F  shows an example of a system for determining a displacement of a mass using an array of asymmetrically distributed magnetic field sensing elements. 
         FIG. 6  shows an example of another IC chip including an array of asymmetrically distributed magnetic field sensing elements. 
         FIG. 7  shows an example of an IC chip including an array of symmetrically distributed magnetic field sensing elements. 
         FIG. 8  is circuit diagram of an ASIC including an array of magnetic field sensing elements. 
         FIG. 9  shows an example of a haptic engine (HE) that includes a hybrid sensor for concurrently measuring Hall voltage and bEMF. 
         FIG. 10  shows an example of a hybrid sensor including a HES and a sensing coil for concurrently measuring Hall voltage and bEMF. 
         FIGS. 11A-11C  show aspects of a measurement technique for concurrently measuring Hall voltage and bEMF. 
         FIGS. 12A-12D  show aspects of displacement sensing using a conventional HE. 
     
    
    
     Certain illustrative aspects of the systems, apparatuses, and methods according to the disclosed technologies are described herein in connection with the following description and the accompanying figures. These aspects are, however, indicative of but a few of the various ways in which the principles of the disclosed technologies may be employed and the disclosed technologies are intended to include all such aspects and their equivalents. Other advantages and novel features of the disclosed technologies may become apparent from the following detailed description when considered in conjunction with the figures. 
     DETAILED DESCRIPTION 
       FIG. 1A  is a plan view, e.g., in the (x,y) plane, of an example of an integrated circuit (IC) chip  100  that includes an array  110  of magnetic field sensing elements and mix-signal circuitry  120  formed on a die  101 . The magnetic field sensing elements of the array  110  are distributed on the die  101  along a first direction, e.g., along the x-axis, at locations  102 A,  102 B,  102 C, for instance. In this example, the mix-signal circuitry  120  is disposed in a central area  104  of the die  101  that separates, along the x-axis, the second location  102 B from the third location  102 C. As such, the second distance d 2  is larger than the first distance d 1 , and the magnetic field sensing elements of the array  110  are said to be asymmetrically distributed along the x-axis. 
       FIG. 1B  is a side view, e.g. in the (x,z) plane, of a portion of an example implementation of a haptic engine  150  that has a frame  152 . The haptic engine  150  includes, encapsulated inside the frame  152 , a mass  154 , a magnetic field source  156  and the IC chip  100  shown in  FIG. 1A . Here, the mass  154  can move relative to the frame  152  along the x-axis (e.g., through vibration left-and-right on page), along the z-axis (e.g., through vibration up-and-down on page), along the y-axis (e.g., through vibration in-and-out of page), or combinations of two or all three of these motions. The magnetic field source  156  includes portions  156 F that are disposed on (i.e., are “fixed” to) the frame  152 , e.g., coils, etc. As such, these are referred to as the fixed portions  156 F of the magnetic field source. In some implementations, coil sections  156 FA and  156 FB can be made from one winding (looped in and out of the page) and have continuous current flow. The magnetic field source  156  also includes portions  156 M that are part of the mass  154 , e.g., permanent magnets, etc. As these move along with the mass  154 , they are referred to as the mobile (or moving) portions  156 M of the magnetic field source. For example, the mass  154  can be formed from a stainless steel cage with enclosures in which the mobile portions  156 M of the magnetic field source are held. In this example, a value of the mass  154  is a sum of a mass of all the mobile portions  156 M of the magnetic field source (e.g., the total mass of the mobile permanent magnets) and a mass of the stainless steel cage that holds them. 
     In the example illustrated in  FIG. 1B , a left magnet  156 MA is oriented with its north pole towards the top of the page and its south pole towards the bottom of the page, and a right magnet  156 MB is oriented with its north pole towards the bottom of the page and its south pole towards the top of the page. Coil sections  156 FA have the same electrical current flow (e.g., current IL), while coil sections  156 FB have the opposite electrical current flow (e.g., current IL). For instance, when coil sections  156 FA have current flowing out of the page and coil sections  156 FB have current flowing into the page (as shown in  FIG. 1B ), the coil  156 F experiences a Lorentz force to the left of the page and the magnets  156 M, along with the mass  154 , will move towards the right of the page. In this manner, an alternating current IL that is provided through the coil  156 F causes a periodic Lorentz force that drives, along the x-axis, the mass  154  which includes the magnets  156 M. An amplitude and frequency of the displacement ΔX of the mass  154  is proportional to respective amplitude and frequency of the coil current IL. 
     The fixed portions  156 F and the mobile portions  156 M of the magnetic field source are configured and arranged relative to each other, such that, when the mass  154  is at rest, the magnetic field source  156  as a whole emits a magnetic field B. Only two lines of a spatial distribution of the magnetic field B(X,Y,Z) emitted by the magnetic field source  156  is shown in  FIG. 1B , however, more details of the spatial distribution of the magnetic field B(X,Y,Z) emitted by the magnetic field source  156  will be illustrated in  FIGS. 2A, 3A-3B . Moreover, the IC chip  100  is disposed on (i.e., is “fixed” to) the frame  152  of the haptic engine  150  in a sensing plane  105  of the spatial distribution of the magnetic field B(X,Y,Z) emitted by the magnetic field source  156 . In this manner, each of the magnetic field sensing elements of the array  110  can determine one or more components of the magnetic field B, in the sensing plane  105 , at the respective locations  102 A,  102 B,  102 C. Note that in this configuration of the haptic engine  150 , the fixed portions  156 F and the mobile portions  156 M of the magnetic field source are used both for driving the mass  154  along the x-axis and for sensing the mass&#39; displacement ΔX along the x-axis, ΔZ along the z-axis, and ΔY along the y-axis, as explained below. 
       FIG. 1C  is a side view, e.g. in the (x,z) plane, of a portion of another example implementation of a haptic engine  150 ′ that has a frame  152 ′. The haptic engine  150 ′ includes, encapsulated inside the frame  152 ′, a mass  154 ′, a magnetic field source  156 ′ and the IC chip  100  shown in  FIG. 1A . Here, the mass  154 ′ can move relative to the frame  152 ′ along the x-axis (e.g., through vibration left-and-right on page), along the z-axis (e.g., through vibration up-and-down on page), along the y-axis (e.g., through vibration in-and-out of page), or combinations of two or all three of these motions. The magnetic field source  156 ′ includes portions  156 ′F that are disposed on (i.e., are “fixed” to) the frame  152 ′, in this case, permanent magnets. As such, these are referred to as the fixed portions  156 ′F of the magnetic field source. The magnetic field source  156 ′ also includes portions  156 ′M that are part of the mass  154 ′, in this case, portions of a coil. As these move along with the mass  154 ′, they are referred to as the mobile (or moving) portions  156 ′M of the magnetic field source. In some implementations, coil sections  156 ′MA and  156 ′MB can be made from one winding (looped in and out of the page) and have continuous current flow. For example, the mass  154 ′ can be formed from a stainless steel cage with enclosures in which the mobile portions  156 ′M of the magnetic field source are held. Additionally, the magnetic field source  156 ′ also includes a marker magnet set  157  that can be attached to the stainless steel cage, for instance. The marker magnet set  157  can include one or more permanent magnets arranged in accordance with a predetermined pattern in the (x,y) plane. In the example shown in  FIG. 1C , the permanent magnets of the marker magnet set  157  are arranged in a row in which adjacent ones have opposing polarities. In this manner, a value of the mass  154 ′ is a sum of a mass of all the mobile portions  156 ′M of the magnetic field source (e.g., the mass of the portions of the mobile coil together with the mass of the marker magnet set  157 ) and a mass of the stainless steel cage that holds them. 
     In the example illustrated in  FIG. 1C , magnets  156 ′FA on the left side are oriented with their north pole towards the top of the page and their south pole towards the bottom of the page, and magnets  156 ′FB on the right side are oriented with their north pole towards the bottom of the page and their south pole towards the top of the page. Coil section  156 ′MA and coil section  156 ′MB have opposite respective electrical current flows (e.g., currents IL). For instance, when coil section  156 ′MA has current IL flowing out of the page and coil section  156 ′MB has current IL flowing into the page (as shown in  FIG. 1C ), such that the coil  156 ′M experiences a Lorentz force to the left of the page and will move, along with the mass  154 ′, towards the left of the page. In this manner, an alternating current IL that is provided through the coil  156 ′M causes a periodic Lorentz force that drives, along the x-axis, the mass  154 ′ which includes the coil  156 ′M and the marker magnet set  157 . An amplitude and frequency of the displacement ΔX of the mass  154 ′ is proportional to respective amplitude and frequency of the coil current IL. 
     The fixed portions  156 ′F and the mobile portions  156 ′M of the magnetic field source are configured and arranged relative to each other to emit, when the mass  154 ′ is at rest, a magnetic field B′. Additionally, the marker magnet set  157  are configured to emit a marker magnetic field B M  having a known spatial period δ X  between zero crossings thereof. Moreover, the IC chip  100  is disposed on (i.e., is “fixed” to) the frame  152 ′ of the haptic engine  150 ′ in a sensing plane  105 ′ of the spatial distribution of the combined magnetic fields B′(X,Y,Z) and B M (X,Y,Z) emitted by the magnetic field source  156  as a whole. Note that the marker magnet set  157  is spaced apart from the sensing plane  105 ′ such that, at the sensing plane, a strength of the marker magnetic field B M  emitted by the marker magnet set  157  is 10×, 100× or 100× stronger than a strength of the magnetic field B′ emitted by the combination of permanent magnets  156 ′F and coil  156 ′M of the magnetic field source. In this manner, each of the magnetic field sensing elements of the array  110  can determine one or more components of the marker magnetic field B M , in the sensing plane  105 ′, at the respective locations  102 A,  102 B,  102 C. Note that in this configuration of the haptic engine  150 ′, the magnets  156 ′F and the coil  156 ′M of the magnetic field source are used for driving the mass  154 ′ along the x-axis, and the marker magnet set  157  is used for sensing the mass&#39; displacement ΔX along the x-axis, ΔZ along the z-axis, and ΔY along the y-axis, as explained below. 
     Referring again to  FIG. 1A , in some implementations, each magnetic field sensing element  110 ZA,  110 ZB,  110 ZC is a single-axial magnetic field sensing element configured to measure, at its respective location  102 A,  102 B,  102 C, a single component of a magnetic field B that is normal to the plane of the die  101 , e.g., component B Z  along the z-axis. In these cases, each of the single-axial magnetic field sensing elements  110 ZA,  110 ZB,  110 ZC can be a Hall-effect sensing element configured to measure the B Z  component of a magnetic field B. In some implementations, each of the single-axial magnetic field sensing elements  110 ZA,  110 ZB,  110 ZC can be either a magneto-resistor or a flux-gate. In some implementations, each magnetic field sensing element  110 A,  110 B,  110 C is a tri-axial magnetic field sensing element configured to measure, at its respective location  102 A,  102 B,  102 C, all three components B Z , B X  and B Y  of a magnetic field B. For example, the tri-axial magnetic field sensing element  110 A includes a combination of (i) a single-axial magnetic field sensing element  110 ZA configured to measure, at its location  102 A, a component B Z  along the z-axis, (ii) a single-axial magnetic field sensing element  110 XA configured to measure, at its location  102 A, a component B X  along the x-axis, and (iii) a single-axial magnetic field sensing element  110 YA configured to measure component B Y  along the y-axis. Here, each of the tri-axial magnetic field sensing elements  110 A,  110 B,  110 C can be a tri-axial Hall-effect sensing element configured to measure the B Z , B X  and B Y  components of a magnetic field B. 
       FIG. 2A  is a cross-section  158  in the plane (x,z) of a portion of an example of the spatial distribution of the magnetic field B(X,Y,Z) emitted by the magnetic field source  156 . The sensing plane  105  also is overlaid, in  FIG. 2A , on the (x,z) cross-section  158  of the spatial distribution of the magnetic field B(X,Y,Z). Note that, at a location  102 A of the sensing plane  105 , where one of the magnetic field sensing elements of the array  110  is disposed, the magnetic field B has non-zero B Z , B X  and B Y  components.  FIG. 2B  shows components B Z ( 102 A), B X ( 102 A), B Y ( 102 A) of the magnetic field B measured at location  102 A where the tri-axial Hall-effect sensing element  110 A is located. 
       FIG. 3A  is a representation of a spatial distribution  160 Z of the axial component B Z (X,Y) along the z-axis of the magnetic field B, measured in the sensing plane  105  (e.g., parallel to the (x,y) plane).  FIG. 3B  is a representation of a spatial distribution  160 XY of the transverse component B XY (X,Y) within the (x,y) plane of the magnetic field B, determined, e.g., by a digital signal processor  122  of the mix-signal circuitry  120 , by using
 
 B   XY ( X,Y )=√{square root over ( B   X   2 ( X,Y )+ B   Y   2 ( X,Y ))}  (6).
 
In EQ. 6, a component B X (X,Y) along the x-axis and a component B Y (X,Y) along the y-axis of the magnetic field B are measured concurrently with the component B Z (X,Y), in the same sensing plane  105 .
 
       FIG. 3C  shows a first cross-section  162 X of the representation of the spatial distribution  160 Z of the axial component B Z (X,Y) and a second cross-section  163 X of the representation of the spatial distribution  160 XY of the transverse component B XY (X,Y). Each of the cross-sections  162 X,  163 X corresponds to the segment AB parallel to the x-axis and spans a first region (denoted I) adjacent to first end point A, a second region (denoted II) between intermediate points P 1 , P 2 , and a third region (denoted III) adjacent to second end point B. The first cross-section  162 X is linear in Region II, where it has large slope δB Z /δX, and non-linear in each of Regions I, III, where it has nearly zero slope δB Z /δX but large curvature δ 2 B Z /δX 2 . For these reasons, a measurement of the displacement ΔX, performed based on the spatial distribution  160 Z of the axial component B Z (X,Y), has highest sensitivity, that is proportional to δB Z /δX, in Region II. In contrast, the sensitivity of the measurement of the displacement ΔX, performed based on the same spatial distribution  160 Z of the axial component B Z (X,Y), is proportional to δ 2 B Z /δX 2  in Region I and Region III. Moreover, the second cross-section  163 X is linear in each of Regions I, III, where it has large slope δB XY /δX, and non-linear in Region II, where it has nearly zero slope δB XY /δX but large curvature δ 2 B XY /δX 2 . For these reasons, a measurement of the displacement ΔX, performed based on the spatial distribution  160 XY of the transverse component B XY (X,Y), has highest sensitivity, that is proportional to δB XY /δX, in Region I and Region III. In contrast, the sensitivity of the measurement of the displacement ΔX, performed based on the same spatial distribution  160 XY of the transverse component B XY (X,Y), is proportional to δ 2 B XY /δX 2  in Region II. 
     Referring again to  FIG. 3A , note that a third cross-section of the representation of the spatial distribution  160 Z of the axial component B Z (X,Y), that corresponds to the segment CD parallel to the y-axis, is linear between intermediate point P 3  and end point D, where it has large slope δB Z /δY, and non-linear in between intermediate point P 3  and end point C, where it has nearly zero slope δB Z /δY but large curvature δ 2 B Z /δY 2 . For this reason, a measurement of the displacement ΔY, performed based on the spatial distribution  160 Z of the axial component B Z (X,Y), has highest sensitivity, that is proportional to δB Z /δY, point P 3  and end point D. 
     Referring now to  FIG. 1B , motion of the mass  154 , that includes at least one of a displacement ΔX along the x-axis, a displacement ΔZ along the z-axis, and a displacement ΔY along the y-axis, causes rearrangement of the fixed portions  156 FA,  156 FB and mobile portions  156 MA,  156 MB of the magnetic field source with respect to each other. As such, the magnetic field B emitted by the magnetic field source  156  as a whole will change as its mobile portions  156 MA,  156 MB are moving, along with the mass  154 , relative to its fixed portions  156 FA,  156 MB. In this manner, the magnetic field sensing elements of the array  110  will measure, at their respective asymmetrically distributed locations  102 A,  102 B,  102 C in the sensing plane  105 , the changes of the magnetic field B caused by the motion of the mass  154 . 
     Additionally, the DSP  122  of the mix-signal circuitry  120  uses the changes of the magnetic field B, measured by the magnetic field sensing elements of the array  110  at their respective asymmetrically distributed locations  102 A,  102 B,  102 C in the sensing plane  105 , to determine one or more of the displacements ΔX, ΔZ, and ΔY of the mass  154 . In this manner, the displacements ΔX, ΔZ, and ΔY of the mass  154  are determined by the DSP  122  with an accuracy corresponding to the accuracy of lithography processes used to establish/form the separations d 1  and d 2 . Note that it is the asymmetric arrangement of the magnetic field sensing elements, in which the separations between adjacent magnetic field sensing elements are different, d 2 ≠d 1 , that enables the DSP  122  to differentiate translational motion of the mass  154  along x, y, and z-axes, as described below. For instance, for the array  110 , the separation d 2  between the magnetic field sensing elements  110 B and  110 C has to be larger than the separation d 1  between the magnetic field sensing elements  110 A and  110 B by at least 2%, 5% or 10%, so the displacements ΔX, ΔZ, and ΔY are measurable in accordance with the techniques described herein. Typically, for the array  110 , the separation d 1  between the magnetic field sensing elements  110 A and  110 B is in a range of 10-100 μm, and the separation d 2  between the magnetic field sensing elements  110 B and  110 C is suitably 2×-10× larger than d 1 . 
     For example, the first magnetic field sensing element  110 ZA measures the component B Z ( 102 A) of the magnetic field at the first location  102 A, and the second magnetic field sensing element  110 ZB measures the component B Z ( 102 B) of the magnetic field at the second location  102 B. Here, d 1  is the separation along the x-axis between the first location  102 A, where the first magnetic field sensing element  110 ZA is located, and the second location  102 B, where the second magnetic field sensing element  110 ZB is located. Because the magnetic field B decays approximately linearly along the z-axis (given a separation along the z-axis between the sensing plane  105  and the adjacent surface of the stainless steel cage that encompasses mass  154  is at least 10× smaller than a separation along the x-axis between the mobile portions  156 MA,  156 MB of the magnetic field source), a displacement ΔZ along the z-axis and a displacement ΔX along the x-axis of the mass  154  induce a change of the component ΔB Z ( 102 A) at the first location  102 A and a change of the component ΔB Z ( 102 B) at the second location  102 A in accordance with the following system of two linear equations: 
                     [           Δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   A     )                   Δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   B     )               ]     =         [           α   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   A     )                 δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢           ⁢   A     )           δ   ⁢           ⁢   X                 α   ⁡     (         B   Z     ⁡     (     102   ⁢   A     )       +       d   1     ⁢       δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   A     )           δ   ⁢           ⁢   X           )               δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   B     )           δ   ⁢           ⁢   X             ]     ⁡     [           Δ   ⁢           ⁢   Z               Δ   ⁢           ⁢   X           ]       .             (   7   )               
The DSP  122  is configured to solve the above system of linear equations for the two unknown displacements ΔZ, ΔX and unknown parameter a in terms of the measured changes ΔB Z ( 102 A) and ΔB Z ( 102 B). Note that, in view of the above noted linear approximation, the magnetic flux gradient satisfies the following first condition:
 
                       δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢           ⁢   A     )           δ   ⁢           ⁢   X       =         δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢           ⁢   B     )           δ   ⁢           ⁢   X       .             (   8   )               
Moreover, given that the first magnetic field sensing element  110 ZA and the second magnetic field sensing element  110 ZB are both calibrated in terms of a ΔZ correction, if the second magnetic field sensing element  110 ZB measures, at the second location  102 B at time t, a component B Z ( 102 B;t) equal to the component B Z ( 102 A;t−τ) measured, at the first location  102 A at an earlier time t−τ, by the first magnetic field sensing element  110 ZA, then the DSP  122  can infer, without having to perform a calibration based on an external displacement reference, that the mass  154  has traveled, over the time z, exactly the separation d 1  between the first and second magnetic field sensing elements. In such cases, the following second condition is satisfied:
 
If  B   Z (102 A;t −τ)= B   Z (102 B;t ),then Δ X=d   1   (9).
 
Note that, because the first magnetic field sensing element  110 ZA and the second magnetic field sensing element  110 ZB are typically at the same temperature, the determination of the displacement ΔX in accordance with EQ. 9 is temperature insensitive. Graph  164 X* shown in  FIG. 4A  illustrates that the second condition from EQ. 9 can be used for self-calibration of the array  110  of magnetic field sensing elements by measuring the magnetic field component B Z (t−τ) at a location  102   n , and then measuring it again, as B Z (t), after a time τ at a different location  102 ( n +1) that is separated by a known separation d n .
 
     As such, the DSP  122  is configured to substitute EQs. 8-9 into the system of linear equations of EQ. 7 to determine the following displacement ΔZ along the z-axis: 
                     Δ   ⁢           ⁢   Z     ≅           Δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   B     )         -     Δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   A     )             α   ⁢           ⁢       d   1     ⁡     (       δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   A     )           δ   ⁢           ⁢   X       )           .             (   10   )               
Note that equations EQs. 7-10 can be generalized to account for the remaining magnetic field sensing elements of the array  110 , e.g., the third magnetic field sensing element  110 ZC that is separated from the second magnetic field sensing element  110 ZB by separation d 2 .
 
     Referring now to a system of linear equations similar to the one in EQ. 7, if both the first condition given by EQ. 8 is satisfied and the mass  154  moves only along the x-axis (i.e., a displacement ΔX≠0 and a displacement ΔZ→0), then the displacement ΔX of the mass causes a uniform change along the x-axis of the component B Z  of the magnetic field. As such, graph  164 X shown in  FIG. 4B  illustrates that ΔB Z ( 102 A)=ΔB Z ( 102 B)=ΔB Z ( 102 C), although B Z ( 102 A)&gt;B Z ( 102 B)&gt;B Z ( 102 C). In other words, for changes ΔB Z  of a component B Z  of the magnetic field that are caused by the same displacement ΔX of the mass  154  but are measured at different locations, a change ΔB Z  at a location where the component B Z  has a small magnitude is equal to another change ΔB Z  at another location where the component B Z  has a large magnitude. 
     However, if both the first condition given by EQ. 8 is satisfied and the mass  154  moves only along the z-axis (i.e., a displacement ΔZ≠0 and a displacement ΔX→0), then the displacement ΔZ of the mass causes a scaled change along the x-axis of the component B Z  of the magnetic field. As such, graph  164 Z shown in  FIG. 4C  illustrates that ΔB Z ( 102 A)&gt;ΔB Z ( 102 B)&gt;ΔB Z ( 102 C) because ΔB Z ( 102 A)=(1+αΔZ)B Z ( 102 A) and ΔB Z ( 102 B)=(1+αΔZ)B Z ( 102 B) and ΔB Z ( 102 C)=(1+αΔZ)B Z ( 102 C), where B Z ( 102 A)&gt;B Z ( 102 B)&gt;B Z ( 102 C). In other words, for changes ΔB Z  of a component B Z  of the magnetic field that are caused by the same displacement ΔZ of the mass  154  but are measured at different locations, a change ΔB Z  at a location where the component B Z  has a small magnitude is smaller than another change ΔB Z  at another location where the component B Z  has a large magnitude. 
     As illustrated in  FIGS. 4B-4C , if a separation d n  between locations of magnetic field sensing elements is small (e.g., like the separation d 1  between the second magnetic field sensing element  110 ZB and the first magnetic field sensing element  110 ZA), then it cannot be distinguished whether the measured changes ΔB Z ( 102   n ) and ΔB Z ( 102 ( n +1)) were caused by a displacement ΔZ along the z-axis or a displacement ΔX along the x-axis, because it is likely that the magnitudes of B Z ( 102   n ) and B Z ( 102 ( n +1)) are about the same at nearby locations. As such, magnetic field sensing elements that are separated by a large separation d n (e.g., like separation d 2  between the second magnetic field sensing element  110 ZB and the third magnetic field sensing element  110 ZC or like separation d 1 +d 2  between the first magnetic field sensing element  110 ZA and the third magnetic field sensing element  110 ZC) will be used to determine a displacement ΔZ of the mass  154 . In such cases, given a large d n  and a known gradient δB Z /δX (or a pre-calibrated magnitude of B Z  at the locations separated by d n ), a displacement ΔZ of the mass  154  can be estimated with fine resolution using EQ. 10. Further, as a small magnitude B Z  changes by the same ΔB Z  as a large magnitude B Z  when caused by a displacement ΔX along the x-axis, magnetic field sensing elements that are separated by a small separation d n (e.g., like separation d 1  between the adjacent first and second magnetic field sensing elements  110 ZA,  110 ZB) will be used to determine a displacement ΔX of the mass  154 . 
     The DSP  122  can perform determinations of a displacement ΔY of the mass  154  along the y-axis by generalizing EQ. 6-10. Referring again to  FIG. 3A , the nonlinearity of a cross-section along the segment CD parallel to the y-axis of the representation of the spatial distribution  160 Z of the axial component B Z (X,Y) suggests that (i) determination of a displacement ΔY along the y-axis is sensitive to the curvature of the cross-section along the segment CD between end point C and point P 3 , and (ii) determination of ΔY is sensitive to the slope of the cross-section along the segment CD between point P 3  and end point D. Moreover, as a large δB Z /δY is needed to obtain an accurate ΔY displacement measurement (using an equation similar to EQ. 8), a magnetic field sensing element to be used for determining the ΔY displacement will be disposed between the center and the edge of the die  101 , where a value of δB Z /δY is expected to be largest. This is shown in  FIG. 1A , where the magnetic field sensing elements  110 ZA,  110 ZB,  110 ZC are off-centered along the width of the die  101 . 
     Referring again to EQ. 8, note that values of the gradient δB Z /δX at various locations of the die  101  where the magnetic field sensing elements of the array  110  are disposed are measured as part of a calibration procedure described below. A procedure to equalize sensitivity of the magnetic field sensing elements of the array  110  is described first. 
       FIG. 5A  is a flow chart of an example of a process  500  used to equalize sensitivity across the magnetic field sensing elements of the array  110  of the IC chip  100  when the latter is part of the haptic engine  150 . In this example, each magnetic field sensing element  110 Zn, where n is {A, B, C, . . . }, is a Hall-effect sensing element that transduces a magnitude and direction of a component B Z ( 102   n ) of the magnetic field along the z-axis, at the Hall-effect sensing element&#39;s location  102   n , to a Hall voltage V( 102   n ). The process  500  is performed by at least one of the circuits of the mix-signal circuitry  120 , and by controller circuitry of the haptic engine  150 . 
     At  510 , while the mass  154  is at rest and no current is provided to the fixed portions  156 FA,  156 FB of the magnetic field source (e.g., the coils A and B), i.e., for I L =0, Hall voltages {v( 102   n ;t 1 )} at respective locations { 102   n } are measured, at time t 1 , across all n Hall-effect sensing elements of the array  110 . 
     At  520 , the fixed portions  156 FA,  156 FB of the magnetic field source (e.g., the coils A and B) of the haptic engine  150  are driven at high frequency with known coil current I L0 ≠0, to avoid moving the mobile portions  156 MA,  156 MB of the magnetic field source (e.g., the permanent magnets A and B). 
     While performing  520 , the following operations are being performed. At  522 , a respective Hall voltage v( 102   n ;t 2 ) at a corresponding location  102   n  is measured, at time t 2 , on each of the n Hall-effect sensing elements of the array  110 . At  524 , sensitivity across all n Hall-effect sensing elements of the array  110  is equalized, by adjusting their gain, so v( 102   n ;t 2 )−v( 102   n ;t 1 ) is the same for all n Hall-effect sensing elements. This ensures that each Hall-effect sensing element  110 Zn will output the same value of the Hall voltage V( 102   n ) when the Hall-effect sensing element senses the same field. At  526 , a relationship between the Hall voltages v( 102   n ;t 2 ) and the known coil current I L0  is determined. For instance, a linear approximation can be used for this purpose, v( 102   n ;t 2 )=η( 110 Zn)I L0 , where η( 110 Zn) is referred to as the electromagnetic (EM) coupling factor for the Hall-effect sensing element  110 Zn. 
     At  528 , and at subsequent operations performed by any of the circuits of the mix-signal circuitry  120 , and by the controller circuitry of the haptic engine  150 , if at time t a Hall-effect sensing element  110 Zn measures a voltage V( 102   n ;t) while the fixed portions  156 FA,  156 FB of the magnetic source are driven with a coil current IL, then the contribution ΔV( 102   n ;t) to the measured voltage caused by the mobile portions  156 MA,  156 MB of the magnetic source is determined as ΔV( 102   n ;t)=V( 102   n ;t)−η( 110 Zn)IL. In other words, a contribution of the coil current-induced Hall voltage η( 110 Zn)IL is subtracted from the sensed Hall voltage V( 102   n ;t), and only an effective Hall voltage ΔV( 102   n ;t) caused by the moving permanent magnets is used for all subsequent calculations. 
       FIG. 5B  is a flow chart of a calibration process  530  used for determining a gradient δB Z /δX at the sensing plane  105  of the haptic engine  150 . In some implementations, each magnetic field sensing element  110 Zn, where n is {A, B, C, . . . }, is a Hall-effect sensing element that measures a magnitude and direction of a component B Z ( 102   n ) of the magnetic field along the z-axis at the Hall-effect sensing element&#39;s location  102   n . The process  500  is performed by at least one of the circuits of the mix-signal circuitry  120 , and by controller circuitry of the haptic engine  150 . 
     At  535 , the haptic engine  150 , which is a linear resonant actuator (LRA), is driven with sinusoidal input voltage at resonance. While 535 is in progress, the following operations are performed. 
     At  540 , the gradient δB Z /δX at the sensing plane  105  of the haptic engine  150  is estimated for a set of values of B Z , in the range from 0 to B: B Z ={0, B Z1 , B Z2 , . . . , B} Here, the gradient δBZ/δX at a location  102 ( n +0.5) between locations  102   n  and  102 ( n +1) that are separated by a separation d n  is estimated geometrically, in accordance with  FIG. 5C , in the following manner: 
                       δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢     (     n   +   0.5     )       )           δ   ⁢           ⁢   X       ≅             B   Z     ⁡     (     102   ⁢     (     n   +   1     )       )       -       B   Z     ⁡     (     102   ⁢   n     )           d   n       .             (   11   )               
The value of δB Z /δX at the exact locations  102   n  and  102 ( n +1) is later obtained by interpolation between a large set of δB Z ( 102 ( n +0.5))/δX values measured at different values of B Z ( 102   n ). Note that while the foregoing set of large values is obtained by sweeping over X, in order to change the value of B Z ( 102   n ), the actual value of x at location  102 ( n +0.5) is not needed.
 
     At  545 , a value of B Z  as a function, X(B), over the position along the x-axis in a range from B 1  to B 2  is estimated by integrating over the inverse of the determined gradient 1/(δB Z /δX) in the following manner: 
                     X   ⁡     (   B   )       =       ∫     B   ⁢           ⁢   1       B   ⁢           ⁢   2       ⁢       1     (       δ   ⁢           ⁢     B   Z         δ   ⁢           ⁢   X       )       ⁢       dB   Z     .                 (   12   )               
Based on the assumption that X(B) is a single-valued function of B Z , the determined function X(B) can be used to map every measured value of B Z  (for instance B Z ( 102   n ) at location  102   n ) to a unique value of X.  FIG. 5D  shows a look up table (LUT)  546 , shown here as a graph of X vs. B Z , that stores an example of a set of pairs of estimated positions X(B Z ) of the moving mass  154 , between X MIN  and X MAX , and corresponding measured values of the magnetic field component B Z , between B ZMIN  and B ZMAX , obtained based on EQ. 12. Thus, by simply measuring B Z (m), at a sensor  110 Zn, the position of the moving mass  154  can be estimated by simply reading from LUT  546  a value X(m) corresponding to the measured value B Z (m).
 
     In this manner, a displacement of the mass  154  is referenced by a combination of geometries of the IC chip  100  and of the haptic engine  150  (e.g., separations d 1 , d 2 , etc. between magnetic field sensing element  110 ZA,  110 ZB, etc. of the array  110 ) rather than geometry of an external tester. Further note that the positions of the mass  154  estimated from LUT  546  based on measurements of the magnetic field component B Z  corresponds to a “macroscopic motion” of the mass, e.g., over 10s or 100s of microns. 
     Moreover, the DSP  122  can determine displacements ΔZ and ΔX of the moving mass  154  that correspond to its “microscopic motion”, e.g., at a sub-micron or micron scale, by using the gradient δBZ/δX that has been determined in accordance with the process  530 , as described below. 
       FIG. 5E  is a flow chart of a process  550  used during run-time of the IC chip  100  to determine displacements ΔZ and ΔX of a mass  154 , where the mass and the IC chip are part of the haptic engine  150 , and the IC chip is disposed at a sensing plane  105  of the haptic engine. Here, the displacements ΔZ and ΔX correspond to microscopic motion riding on macroscopic motion, the latter causing the mass  154  to sweep through locations X(B), in accordance with EQ. 12 and LUT  546 . In some implementations, each magnetic field sensing element  110 Zn, where n is {A, B, C, . . . }, is a Hall-effect sensing element that measures a magnitude and direction of a component B Z ( 102   n ) of the magnetic field along the z-axis at the Hall-effect sensing element&#39;s location  102   n . The process  550  is performed by the DSP  122  of the mix-signal circuitry  120 . The process  550  uses sensors at more than 2 locations  102   n  with different separations, d n  (and hence different sensitivities of the measurements of displacement ΔZ and displacement ΔX, respectively, according to EQs. 7-10), to simultaneously solve for the displacements ΔZ and ΔX. 
     At  555 , a displacement ΔZ(α) along the z-axis is estimated using two magnetic field sensing elements of the array  110  disposed at locations spaced apart by a large separation, in accordance with EQ. 10, and using the gradient δBZ/δX determined based on the process  530 . For example, changes ΔB Z  measured by the second magnetic field sensing element  110 ZB and the third magnetic field sensing element  110 ZC at the respective second location  102 B and third location  102 C that are separated by a separation d 2  can be used for the estimation performed at  555 . As another example, changes ΔB Z  measured by the first magnetic field sensing element  110 ZA and the third magnetic field sensing element  110 ZC at the respective first location  102 A and third location  102 C that are separated by a separation d 1 +d 2  can be used for the estimation performed at  555 . 
     At  560 , parameter a and a displacement ΔX along the x-axis of the mass  154  are determined using two magnetic field sensing elements of the array  110  disposed at locations spaced apart by a small separation and the gradient δBZ/δX determined based on the process  530 . For example, changes ΔB Z  measured by the first magnetic field sensing element  110 ZA and the second magnetic field sensing element  110 ZB at the respective first location  102 A and second location  102 B that are separated by a separation d 1  can be used for the determination performed at  560 . This determination is performed by inverting EQ. 7 in the following manner: 
                     [         α             Δ   ⁢           ⁢   X           ]     =           [           Δ   ⁢           ⁢       ZB   Z     ⁡     (     102   ⁢   A     )                 δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   A     )           δ   ⁢           ⁢   X                 Δ   ⁢           ⁢     Z   ⁡     (         B   Z     ⁡     (     102   ⁢   A     )       +       d   1     ⁢       δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   A     )           δ   ⁢           ⁢   X           )                 δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   B     )           δ   ⁢           ⁢   X             ]       -   1       ⁡     [             Δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   A     )         ⁢                       Δ   ⁢           ⁢       B   Z     ⁡     (     102   ⁢   B     )               ]       .             (   13   )               
Note that EQs. 10 and 13 constitute a system of three linear equations with three unknowns a, ΔZ, ΔX. Such system of linear equations can have a unique solution.
 
     Note that when the magnetic field sensing elements of the array  110  are tri-axial Hall-effect sensing elements  110 A,  110 B,  110 C, etc., the motion of the mass  154  of the haptic engine  150  can be sensed based on the angle ang(B) formed by the magnetic field with the z axis, instead of based on the component B Z  along the z-axis of the magnetic field. The angle ang(B) is determined by the DSP  122 , based on the spatial distribution  160 Z of the axial component B Z (X,Y) shown in  FIG. 3A  and the spatial distribution  160 XY of the transverse component B XY (X,Y) shown in  FIG. 3B , as 
                       ang   ⁡     (   B   )       ⁢     (     X   ,   Y     )       =         tan     -   1       ⁡     (         B   Z     ⁡     (     X   ,   Y     )           B   XY     ⁡     (     X   ,   Y     )         )       .             (   14   )               
In EQ. 14, the transverse component B XY (X,Y) is obtained by the DSP  122  using EQ. 6. In the example shown in  FIG. 2A , orientation of ang(B) at the sensing plane  105  corresponds to orientation of the magnetic field lines thereat.
 
     Note that the displacements ΔX and ΔZ of the mass  154 , in terms of changes Δ[ang(B)] of the magnetic field angle ang(B), can be determined by solving EQs. 7-13, where Δ[ang(B)] is substituted for ΔB Z . In this manner, the displacements ΔX and ΔZ of the mass  154  can now be encoded in angular domain, so they can be determined based on a ratio-metric measurement of the magnetic field angle ang(B), in accordance with EQ. 14. As determinations of the displacements ΔX and ΔZ of the mass  154  are based on a ratio-metric measurement, results of these determinations are insensitive to temperature change. Further, this approach also enables the DSP  122  to differentiate rotational motion from translational motion. Furthermore, processes  530  and  550  can be used, in a manner similar to the one described above in connection with  FIGS. 5B-5E , to determine the displacements ΔX and ΔZ of the mass  154  based on the changes Δ[ang(B)] measured by the tri-axial Hall-effect sensing elements  110 A,  110 B,  110 C. For instance, the DSP  122  uses first the process  530  to determine the gradient δ[ang(B)]/δX, and then the process  550  to determine the displacements ΔX and ΔZ of the mass  154  based on the determined gradient δ[ang(B)]/δX. 
     Note that technologies described above in connection with  FIGS. 5A-5E  use deterministic EQs. 10-12 to transform an input set of values of measured magnetic field B Z  or measured change ΔB Z  of the magnetic field into an output set of values of a position X of a moving mass  154 , or variations (also referred to as displacements) ΔX and ΔZ around its position X. In this manner, each instance of an input set of values of measured magnetic field B Z  or measured change ΔB Z  of the magnetic field results in an output set of values of the position X and displacements ΔX and ΔZ that is independent of other output sets of values of the position X and displacements ΔX and ΔZ that have been previously obtained from respective sets of previously measured magnetic field B Z  or measured change ΔB Z  of the magnetic field. 
     In an embodiment alternative to the one shown in  FIGS. 5A-5E , the disclosed technologies can be used in conjunction with a state-observe feedback control system in which it is not necessary to perform a self-calibration.  FIG. 5F  shows a state-observer feedback control system  570  that includes uses the array  110  of magnetic field sensing elements, e.g., described above in connection with  FIG. 1A , look-up-table (LUT)  572 , a state-observer  574 , a controller  576  and the haptic engine  150  (that includes the mass  154 , the magnetic field source  156 , etc.) 
     The state-observer feedback control system  570  uses the array  110  of magnetic field sensing elements to measure various magnetic field components B X (k), B Y (k), B Z (k) at a respective position  102 A,  102 B, etc. of each sensor  110 A,  110 B, etc. These measured values are used to interpolate LUT  572  to generate a measured plant output y[k]. Note that, in this example, the plant is the haptic engine  150 , and x[k] is a state of the haptic engine. Here, the state x[k] of the haptic engine includes a current X position of the mass  154 . As such, in this example, the plant output y[k] represents a measured X position of the mass  154  given its current position x[k] and a driving signal u[k] acting on the mass. For instance, the driving signal u[k] can be a coil current I L . 
     The LUT  572  can be calibrated using a reference sensor. For example, a laser profiler can be used to measure the X position (y[k] state is the X position, in this case) of the engine  150 &#39;s moving mass  154  while the engine is actuated. A set of B field measurements is collected by the sensor array  110  over this set of known y[k] states and the LUT  572  is generated to lookup y[k] for a given set of B field measurements. 
     The state observer  574  receives the measured engine output y[k] (e.g., the measured X position) from LUT  572  and uses it as a first input. Moreover, the state observer  574  estimates the engine state {circumflex over (x)}[k] and an estimated engine output ŷ[k], using the following equations:
 
 {circumflex over (x)} [ k+ 1]= A{circumflex over (x)} [ k ]+ Bu [ k ]+ L ( y [ k ]− ŷ [ k ])  (15),
 
 ŷ [ k ]= C{circumflex over (x)} [ k ]+ Du [ k ]  (16).
 
Note that by using a parametrized model of the engine  150  in accordance with EQs. 15-16, the state observer  574  obtains an estimate of the next engine state {circumflex over (x)}[k+1] based on a current estimate of the same, a current estimate of the engine output ŷ[k], and a discrepancy between the output measurements y[k] and the output estimate ŷ[k].
 
     The estimated engine state {circumflex over (x)}[k] along with an input command are used by the controller  576  to provide a drive signal u[k]. The drive signal u[k] is returned to the state observer  574  through a feedback loop, where it is used by the state observer  574  as a second input along with the first input noted above. Additionally, the drive signal u[k] is provided to the engine  150  having an engine state x[k]. 
     Moreover, the IC chip  100  can be modified in various ways and then used in the haptic engine  150  to determine a displacement ΔY along the y-axis of the mass  154  concurrently with determining the displacements ΔX and ΔZ thereof. 
       FIG. 6  is a plan view, e.g., in the (x,y) plane, of an example of a modified IC chip  600  that includes an array  610  of magnetic field sensing elements, an array of contact pads  666 , and mix-signal circuitry  620  formed on a die  601 . The magnetic field sensing elements of the array  610  are distributed on the die  601  along two rows extending along a first direction. e.g., along the x-axis. Here, some of the magnetic field sensing elements of the array  610  are disposed at locations  602 A,  602 B,  602 C along the first row, and at locations  602 E,  602 G along the second row, for instance. The second location  602 B is between the first location  602 A and the third location  602 C, and the second location is separated from the first location by a first distance d 1  and from the third location by a second distance d 2  different from the first distance. Moreover, the first and second rows are separated from each other by a third distance d 3 . In this example, the mix-signal circuitry  620  is disposed in a central area  604  of the die  601  that separates, along the x-axis, the second location  602 B and the third location  602 C on the first row, as well as the fourth location  602 E and the fifth location  602 G on the second row. As such, the second distance d 2  is larger than the first distance d 1 , and the magnetic field sensing elements of the array  610  are said to be asymmetrically distributed along the x-axis. 
     In some implementations, each magnetic field sensing element  610 ZA,  610 ZB,  610 ZC,  610 ZE,  610 ZG is a single-axial magnetic field sensing element configured to measure, when the IC chip  600  is disposed in the sensing plane  105  of the haptic engine  150 , at its respective location  602 A,  602 B,  602 C,  602 E,  602 G, the B Z  component of the magnetic field B. In other implementations, each magnetic field sensing element  610 A,  610 B,  610 C,  610 E,  610 G is a tri-axial magnetic field sensing element configured to measure, when the IC chip  600  is disposed in the sensing plane  105  of the haptic engine  150 , at its respective location  602 A,  602 B,  602 C,  602 E,  602 G, all three components B Z , B X  and B Y  of a magnetic field B. In this case, for example, the tri-axial magnetic field sensing element  610 A includes a combination of (i) a single-axial magnetic field sensing element  610 ZA configured to measure, at its location  602 A, a component B Z  along the z-axis, (ii) a single-axial magnetic field sensing element  610 XA configured to measure, at its location  602 A, a component B X  along the x-axis, and (iii) a single-axial magnetic field sensing element  610 YA configured to measure component B Y  along the y-axis. 
     The array  610  of magnetic field sensing elements of the IC chip  600  is used to measure changes in the magnetic field, e.g., ΔB Z  or Δ[ang(B)], at respective locations of the magnetic field sensing elements, such that determinations of both displacement ΔX along the x-axis and displacement ΔY along the y-axis can be performed concurrently with determinations of displacement ΔZ along the z-axis of the mass  154 . Here, magnetic field sensing elements, at the locations  102 A,  102 C separated by d 1 +d 2  on the first row, are used by the DSP  622  of the mix-signal circuitry  620  to determine displacement ΔZ of the mass  154  using EQ. 8; magnetic field sensing elements, at the locations  102 A,  102 B separated by d 1  on the first row, are used by the DSP  622  to determine displacement ΔX of the mass  154  using EQs. 8 and 11; and magnetic field sensing elements, at the locations  102 A,  102 E separated by d 3  between the first and second rows, are used by the DSP  622  to determine displacement ΔY of the mass  154  using equations similar to EQs. 8 and 11. 
       FIG. 7  is a plan view, e.g., in the (x,y) plane, of another example of a modified IC chip  700  that includes an array  710  of magnetic field sensing elements, an array of contact pads  766 , and mix-signal circuitry  720  formed on a die  701 . The magnetic field sensing elements of the array  710  are distributed on the die  701  along two rows extending along a first direction. e.g., along the x-axis. Here, some of the magnetic field sensing elements of the array  710  are disposed at locations  702 A,  702 B,  702 C,  702 D along the first row, and at locations  702 E,  702 F,  702 G,  702 H along the second row, for instance. The second location  702 B (the sixth location  702 F) is between the first location  702 A (the fifth location  702 E) and the third location  702 C (the seventh location  702 G), and the second location (the sixth location) is separated from the first location (the fifth location) by a first distance d 1  and from the third location (the seventh location) by a second distance d 2  different from the first distance. Also, the third location (the seventh location) is separated from the fourth location (the eighth location) by the same first distance d 1 . Moreover, the first and second rows are separated from each other by a third distance d 3 . In this example, the mix-signal circuitry  720  is disposed in a central area  704  of the die  701  that separates, along the x-axis, the second location  702 B and the third location  702 C on the first row, as well as the sixth location  702 F and the seventh location  702 G on the second row. As such, the magnetic field sensing elements of the array  710  are said to be symmetrically distributed along the x-axis. 
     In some implementations, each magnetic field sensing element  710 ZA,  710 ZB,  710 ZC,  710 ZD,  710 ZE,  710 ZF,  710 ZG,  710 ZH is a single-axial magnetic field sensing element configured to measure, when the IC chip  700  is disposed in the sensing plane  105  of the haptic engine  150 , at its respective location  702 A,  702 B,  702 C,  702 D,  702 E,  702 F,  702 G,  702 H the B Z  component of the magnetic field B. In other implementations, each magnetic field sensing element  710 A,  710 B,  710 C,  710 D,  710 E,  710 F,  710 G,  710 H is a tri-axial magnetic field sensing element configured to measure, when the IC chip  700  is disposed in the sensing plane  105  of the haptic engine  150 , at its respective location  702 A,  702 B,  702 C,  702 D,  702 E,  702 F,  702 G,  702 H, all three components B Z , B X  and B Y  of a magnetic field B. In this case, for example, the tri-axial magnetic field sensing element  710 A includes a combination of (i) a single-axial magnetic field sensing element  710 ZA configured to measure, at its location  702 A, a component B Z  along the z-axis, (ii) a single-axial magnetic field sensing element  710 XA configured to measure, at its location  702 A, a component B X  along the x-axis, and (iii) a single-axial magnetic field sensing element  710 YA configured to measure component B Y  along the y-axis. 
     The array  710  of magnetic field sensing elements of the IC chip  700  is used to measure, at respective locations of the magnetic field sensing elements, changes ΔB Z  (or Δ[ang(B)]) in the magnetic field. As such, the DSP  722  can determine curvature of the spatial distribution  160 XY of the transverse component B XY (X,Y) of the magnetic field, e.g., in terms of δ 2 B Z /δX 2 . For instance, changes ΔB Z  measured by magnetic field sensing elements disposed on the first row, at locations  702 A,  702 B separated by d 1 , on one side of the central area  704 , are used by the DSP  722  to determine a first instance of δB Z /δX; and changes ΔB Z  measured by magnetic field sensing elements disposed on the first row, at locations  702 C,  702 D also separated by d 1 , on the opposing side of the central area  704 , are used by the DSP  722  to determine a second instance of δB Z /δX. However, a difference of the first and second instances of δB Z /δX—over the separation (d1+d2) between the magnetic field sensing elements disposed on the first row, at locations  702 A,  702 C—is δ 2 B Z /δX 2 . 
     Note that the foregoing capability of the IC chip  700  can be used when a displacement ΔX extends outside of the linear region II into region I or region III, or both, as shown in  FIG. 3C . In such cases EQ. 2 can be modified to include quadratic changes in the magnetic field, e.g., δ 2 B Z /δX 2 , which are measured by the IC chip  700  as explained above. Then, the DSP  722  can determine displacement ΔZ along the z-axis and displacement ΔX along the x-axis from EQs. 2 and 6-11 appropriately modified to include quadratic terms δ 2 B Z /δX 2 . 
     Note that, for either of the implementations of the IC chip  100 ,  600  and  700 , by placing the magnetic field sensing elements of their respective arrays  110 ,  610 ,  710  on both sides of respective mixed-signal circuitry  120 ,  620 ,  720  makes efficient use of the silicon real-estate while producing a large value of the separation d n  between at least some of the magnetic field sensing elements. In addition, the magnetic field sensing elements that are asymmetrically disposed in the X or Y directions are beneficially used for measuring changes in the magnetic field, e.g., ΔB Z  or Δ[ang(B)], so the measured changes in the magnetic field can then be used for accurately determining the displacement ΔZ. 
       FIG. 8  is a circuit diagram of an example of mix-signal circuitry  820  integrated together with multiple magnetic field sensing elements on an IC chip that is implemented as an ASIC. In this example, each of the magnetic field sensing elements is implemented as a uniaxial Hall-effect sensing element. Any one of the IC chips  100 ,  600 ,  700  described above in connection with  FIGS. 1A, 6, 7 , respectively, can be implemented as the ASIC shown in  FIG. 8 . The ASIC shown in  FIG. 8  can be placed in the sensing plane  105  of the haptic engine  150  that includes mass  154 . 
     In the example shown in  FIG. 8 , the ASIC chip includes one or more driving circuits  826 P,  826 R, etc. Each driving circuit  826 P (or  826 R, etc.) includes a programmable current source  826 P (or  826 R, etc.), two or more Hall-effect sensing elements, and a dummy load  828 P (or  828 R, etc.) connected to each other in series. For example, the programmable current source  826 P (or  826 R, etc.) can be implemented as a single current steered DAC. Using chip  100  as an example, a single current source  826 P (or  826 R, etc.) can be shared between 2 or more sensors. One example would be one current source for sensors  110 XA and  110 XB, another current source for sensors  110 YA and  110 YB, another current source for sensors  110 ZA and  110 ZB, another current source for sensors  110 XC and  110 YC, and another current source for  110 ZC. In yet another example, a single current source can be shared by  110 XA,  110 YA, and  110 ZA. Other combinations are also possible. This grouping in series of the driving circuit  826 P (or  826 R, etc.) can cause power savings, as Hall-effect sensing elements are conventionally connected in parallel, each with its own current source, which consumes a lot of power. The dummy load  828 P (or  828 R, etc.) is a reference resistor with known resistance for calibrating a respective current source  826 P (or  826 R, etc.) For example, such a dummy load resistor can be fabricated in modern complementary-metal-oxide-semiconductor (CMOS) technologies using gate-poly-silicon, doped silicon, biased transistor (in linear region), etc. 
     In the example shown in  FIG. 8 , the mix-signal circuitry  820  further includes a band-gap reference circuit  830 . A temperature insensitive background calibration can be performed by comparing the voltage across the dummy loads  828 P,  828 R, etc., against the single band-gap reference circuit  830 . 
     In the example shown in  FIG. 8 , the mix-signal circuitry  820  further includes sample-and-hold (S/H) circuits  834 , each of the S/H circuit including a capacitor. The S/H circuitry  834  allows global shutter capture which reduces motion blur. 
     In the example shown in  FIG. 8 , the mix-signal circuitry  820  further includes, for each of the driving circuits, an analog multiplexer circuit  832 P (or  832 R, etc.), a programmable gain amplifier (PGA) circuit  836 P (or  836 R, etc.), and an analog to digital converter (ADC) circuit  840 P (or  840 R, etc.) The analog multiplexer circuit  832  includes input ports and an output port, wherein an output of each of the corresponding magnetic field sensing elements, the respective dummy  828 P (or  828 R, etc.), and the band-gap reference circuit  830  is coupled with a respective input port of the analog multiplexer circuit through a respective sample and hold circuit  834 . Note that the analog multiplexer circuit  832 P (or  832 R, etc.) is bootstrapped to (1) provide random-access from the Hall-effect sensing elements to the parallel PGA circuits  836 P,  836 R, etc., and (2) reduce parasitic charge transfer in large (W/L) high-speed field-effect transistors (FETs). This multiplexer is functionally defined to have multiple inputs and multiple outputs, and routing between a specified input and a specified output is selected by a corresponding address input. In terms of circuit realization, such a multiplexer can be realized by a plurality of multiplexer modules  832 P,  832 R, etc., one for each of the programmable amplifiers  836 P,  836 R, etc. Each one of the multiplexer modules  832 P,  832 R, etc. will have only one output that feeds into one of the PGA/ADC channels  836 P/ 840 P,  836 R/ 840 R, etc. Each one of the sample and hold circuits  834  will feed into one of the inputs on one of these multiplexer modules  832 P,  832 R, etc. The address input for each multiplexer module  832 P,  832 R, etc. can be shared or individually addressed. 
     The PGA circuit  836 P (or  836 R, etc.) is coupled with the output port of the analog multiplexer circuit  832 P (or  832 R, etc.) Note that the S/H circuitry  834  also level shifts the Hall voltage output by each of the Hall-effect sensing elements to input common-mode (CM) level of the PGA circuit  836 P (or  836 R, etc.) 
     Offset and low-frequency noise of the PGA circuit  836 P (or  836 R, etc.) can be removed by chopping. As such, in some implementations, the mix-signal circuitry  820  further includes a pair of multiplexers  838 P (or  838 R, etc.), also referred to as a chopping multiplexer pair, that includes a first multiplexer coupled between the output port of the analog multiplexer circuit  832 P (or  832 R, etc.) and the input port of PGA circuit  836 P (or  836 R, etc.) and a second multiplexer coupled between the output port of the PGA circuit  836 P (or  836 R, etc.) and the input port of ADC circuit  840 P (or  840 R, etc.). When the pair of multiplexers  838 P (or  838 R, etc.) is switched simultaneously, the polarity of the input and output signals of the PGA circuit  836 P (or  836 R, etc.) is inverted. This action has no impact on the analog output of the multiplexer  832 , but it inverts the sign of any offset or low-frequency noise from the PGA circuit  836 P (or  836 R, etc.) which can be canceled out later by addition or averaging. 
     Further, the ADC circuit  840 P (or  840 R, etc.) is coupled between the PGA circuit  836 P (or  836 R, etc.) and the DSP  822 . Note that the multiple PGA circuit  836 P (or  836 R, etc.) and ADC circuit  840 P (or  840 R, etc.) channels are being parallelized to increase sampling speed and relax analog design requirement. 
     The mix-signal circuitry  820  includes a DSP  822  coupled with the ADC circuits  840 P,  840 R, etc. Here, the DSP  822  can be implemented as a FPGA, or MCU, or any digital state-machine, for instance, and configured to perform any one of the techniques described above (e.g., processes  530 ,  550 ) for determining displacements ΔZ, ΔX and ΔY of the mass  154  based on changes of a magnetic field measured by the Hall-effect sensing elements included in the ASIC. The determined displacements ΔZ, ΔX and ΔY can be output to an off-chip computing system  842  for further processing, storing, etc. 
     Note that another advantage of the technologies described above is that only digital information is communicated between the DSP  822  and the off-chip computing system  842 . This interface between the DSP  822  and the off-chip computing system  842  can be implemented using the pads  666  shown in  FIG. 6  or the pads  766  shown in  FIG. 7 . In these examples, only a small number of pads is provided on IC chip  600  or  700  for high speed digital data transmission, which can save cost relative to some conventional implementations of magnetic field sensing systems in which an analog signal is output by each sensor to an off-chip computing system, such that a pair of pads needs to be provided for each sensor. 
     Note that the architecture of the mix-signal circuitry  820  relies on ΔB sensing which inherently cancels out a Hall voltage offset output by the Hall-effect sensing elements of the ASIC shown in  FIG. 8 . Low-frequency noise is minimized when a sampling speed is high enough to ensure time correlation between the two samples that generate ΔB. 
     In summary, processes  530  and  550  can be used to encode one or more displacements ΔX, ΔY, and ΔZ of a mass  154  of a haptic engine  150  in magnetic field intensity along one axis, e.g., B Z  (or alternatively B X  or B Y ), and measuring it via uniaxial Hall-effect sensing elements (or other uniaxial magnetic field sensing elements.) Self-calibration of mass  154 &#39;s displacement can be performed by determining a gradient δB Z /δX based on the known distance, dn, between uniaxial Hall-effect sensing elements. Discriminating the displacements ΔX, ΔY, and ΔZ during runtime can be performed by using (i) changes ΔB Z  measured by different combinations of uniaxial Hall-effect sensing elements with different separation distances, d n , and (ii) self-calibration data relating to the gradient δB Z /δX to solve for each motion component separately. Same processes can be used based on gradient δB X /δX or gradient δB Y /δX. 
     Moreover, processes  530  and  550  can be used to encode one or more displacements ΔX, ΔY, and ΔZ of a mass  154  of a haptic engine  150  in magnetic field angle ang(B), and measuring it via tri-axial Hall-effect sensing elements (or other tri-axial magnetic field sensing elements.) Self-calibration of mass  154 &#39;s displacement can be performed by determining a gradient δ[ang(B)]/δX based on the known distance, dn, between tri-axial Hall-effect sensing elements. Discriminating the displacements ΔX, ΔY, and ΔZ motion during runtime can be performed by using (i) changes Δ[ang(B)] measured by different combinations of tri-axial Hall-effect sensing elements with different separation distances, d n , and (ii) self-calibration data relating to the gradient δ[ang(B)]/δX to solve for each motion component separately. 
     Referring again to  FIG. 1A , in other embodiments, the mixed signal circuitry  120  can be provided as a separate discrete chip located away from the array  110  of magnetic field sensing elements. In yet another embodiment, the array  110  of magnetic field sensing elements and the mixed-signal circuitry  120  can each be fabricated on separate chips and integrated onto a common base substrate, e.g.,  101 , as a system-in-package. The same implementations can be provided for the systems  600  and  700 . Note that in each of the foregoing embodiments, the separation d 1  between the magnetic field sensing elements  110 A and  110 B (or  610 A and  610 B, or  710 A and  710 B) is different from the separation d 2  between the magnetic field sensing elements  110 B and  110 C (or  610 B and  610 C, or  710 B and  710 C). 
     Additionally, in some embodiments, the magnetic field sensing elements of the array  110  of IC chip  100  can also function as temperature sensors. For example, when the magnetic field sensing elements of the array  110  are implemented as Hall sensors, a change in resistance Δρ(ΔΘ) of each Hall sensor is a function of mostly a change in temperature ΔΘ, and a change in Hall voltage ΔV(ΔB;ΔΘ) output by the Hall sensor is a function of both a change in magnetic field intensity ΔB and also a change in temperature ΔΘ. In this manner, a measurement of the change in resistance Δρ(ΔΘ) of a Hall sensor can be used to determine the change in temperature ΔΘ of the Hall sensor, and a measurement of the change in Hall voltage ΔV(ΔB;ΔΘ) output by the Hall sensor can be used to determine the change magnetic field intensity ΔB at the location of the Hall sensor for the determined change in temperature ΔΘ. As such, the temperature change ΔΘ determined in this manner can be useful for improving the accuracy of ΔZ displacement and ΔX displacement measurements performed using the IC chip  100  when various temperature coefficients of the IC chip  100  are modeled and compensated for. 
       FIG. 9  shows a haptic engine (HE)  900  that includes a driving coil  910  and a magnet  920  that are movable relative to each other, and a hybrid sensor  1000  that is static relative one of the magnet or the driving coil. The hybrid sensor  1000  is used for sensing relative movement between the magnet  920  and the driving coil  910  via both bEMF and Hall voltage. In the example illustrated in  FIG. 9 , HE  900  has a frame  905  that encapsulates a mass  930  arranged and configured to move relative the frame, at least, along the x-axis (e.g., through vibration left-and-right on page), and optionally along the z-axis (e.g., through vibration up-and-down on page), or combinations of these motions. Here, the driving coil  910  is disposed on (i.e., are fixed to) the frame  905 . In some implementations, sections of the driving coil  910  can be made from one winding (looped in and out of the page) and have continuous current flow. Further, the mass  930  is formed from a stainless cage with enclosures that hold portions of the magnet  920 . 
     In the example illustrated in  FIG. 9 , a left portion of the magnet  920  is oriented with its north pole towards the top of the page and its south pole towards the bottom of the page, and a right portion of the magnet is oriented with its north pole towards the bottom of the page and its south pole towards the top of the page. Left sections of the driving coil  910  have the same electrical current flow (e.g., current I along the negative y-axis), while right sections of the driving coil have the opposite electrical current flow (e.g., current I along the positive y-axis). As such, the driving coil  910  experiences a Lorentz force to the left of the page and the magnet  920 , along with the mass  930 , will move towards the right of the page. In this manner, an alternating (i.e., driving) current I that is provided through the driving coil  910  causes a periodic Lorentz force that drives, along the x-axis, the mass  930  that includes the magnet  920 . An amplitude and frequency of the displacement ΔX of the mass  930  is proportional to respective amplitude and frequency of the driving current I. 
     The driving coil  910  and the magnet  920  are configured and arranged relative to each other, such that, when the mass  930  is at rest, a combination of the driving coil and the magnet emits a magnetic field B. Only two lines of a spatial distribution of the magnetic field B(X,Y,Z) emitted by the combination of driving coil and magnet is shown in  FIG. 9 . Moreover, the hybrid sensor  1000  is disposed on (i.e., it is “affixed” to) the frame  905  of HE  900  at a “sensing point” P of the spatial distribution of the magnetic field B(X,Y,Z) emitted by the combination of driving coil  910  and magnet  920 . In this manner, hybrid sensor  1000  can determine a magnitude of the magnetic field B and its time dependence, at the sensing point P, as described below. 
       FIG. 10  shows an example implementation of the hybrid sensor  1000  that includes analog front-end circuitry  1005 . Here, the analog front-end circuitry  1005  includes a coil structure  1010  (also referred to as sensing coil) and a Hall sensing element (HES)  1020 . Both the coil structure  1010  and the HES  1020  can be fabricated using various semiconductor technologies (e.g., CMOS technologies, etc.) Optionally, the hybrid sensor  1000  can include, in addition to the analog front-end circuitry  1005 , data conversion circuitry  1030 . Such integration of the analog front-end circuitry  1005  with the data conversion circuitry  1030  can reduce signal path mismatch (non-linearity, timing, frequency response, gain, etc.) resulting in improved measurement accuracy when the hybrid sensor  1000  is used in HE  900  for measuring the displacement ΔX of mass  930 . In this manner, HE  900  suitably has a digital-in interface (e.g., implemented as a source of PWM input signals for activating the driving coil  910 ), and a digital-out interface (e.g., implemented as the data conversion circuitry  1030  of the hybrid sensor  1000 ), both of which are universal across module and system testers, making the tester code simpler and less error prone. 
     The sensing coil  1010  of the analog front-end circuitry  1005  is arranged and configured to sense flux change ΔB/Δt, at the location of the sensing coil, as induced bEMF across the sensing coil. For example, when the hybrid sensor  1000  is disposed inside HE  900  at sensing point P, as shown in  FIG. 9 , the sensing coil  1010  senses flux change ΔB/Δt, at the sensing point P, due to a rate of change ΔX/Δt in the displacement (i.e., the velocity) of the mass  930 . Note that while a current I S  is being induced by the flux change ΔB/Δt in the sensing coil  1010 , no current is driven there through. As such, the sensing coil  1010  outputs the induced bEMF&#39;s value as a voltage value E. The velocity ΔX/Δt of the mass  930  is obtained from 
                     E   =         K   M     ⁢       Δ   ⁢           ⁢   X       Δ   ⁢           ⁢   t         +     M   ⁢       Δ   ⁢           ⁢   I       Δ   ⁢           ⁢   t             ,           (   17   )               
where ΔI/Δt is a change in the driving current through the driving coil  910 , M (measured in Henrys) is a mutual coupling between the driving coil  910  and the sensing coil  1010 , and K M  (measured in N/A) is a motor constant.
 
     Additionally, the HES  1020  of the hybrid sensor  1000  is arranged and configured to sense absolute field intensity B, at the location of the HES, as a Hall voltage V H  output by the HES. For example, when the hybrid sensor  1000  is disposed inside HE  900  at sensing point P, as shown in  FIG. 9 , the HES  1020  senses the field intensity B, at the sensing point P, due to a displacement ΔX of mass  930 . 
     Note that in the example illustrated in  FIG. 10 , the sensing coil  1010  and the magnetic sensor  1020  of the analog front-end circuitry  1005  are disposed at a common location, e.g. the sensing point P of HE  900 , such that the hybrid sensor  1000  can concurrently sense the absolute field intensity B (via Hall voltage measurements) and its rate of change ΔB/Δt (via bEMF measurements), at the common location. In this manner, for HE  900 , the velocity ΔX/Δt of the magnet-carrying mass  930  relative the frame  905 , on which the hybrid sensor  1000  is disposed, can be sensed via both bEMF and Hall voltage measurements performed concurrently to, but independently from, each other. Beneficially, by sensing movement of the magnet-carrying mass  930  relative the frame  905  via both bEMF signal and Hall voltage signal, the above noted EM coupling factor η-accuracy errors and the errors caused by non-zero ADC delay τ ADC  can be concurrently addressed. Further note that the bEMF measurements are not prone to errors in coil resistance because virtually no current flows in the sensing coil  1010 . 
     Lateral size of each of the sensing coil  1010  and the HES  1020  is of order tens to hundreds of microns. As the HES  1020  is disposed “inside” the sensing coil  1010 , the lateral size of the former can be 90%, 80%, 50%, 20% or 10% smaller than the lateral size of the latter. In some implementations, the HES  1020  can be disposed, in regard to the (x,y) plane, at the center of the sensing coil  1010 . Moreover, note that in implementations in which at least some portions of the hybrid sensor  1000  are formed using CMOS technologies, components and routing of the analog front-end circuitry  1005  can be built vertically, e.g., along the z-axis, by using a planer multi-level fabrication process. In some such implementations, the HES  1020  can be formed in the same layer as the sensing coil  1010 , i.e., in the same (x,y) plane. In other such implementations, the HES  1020  can be formed in a first layer parallel to the (x,y) plane and the sensing coil  1010  can be formed in a second layer parallel to the (x,y) plane that is higher along the z-axis in the stack-up than the first layer. In some cases, the second layer in which the sensing coil  1010  has been formed can be deposited prior to the first layer in which the HES  1020  has been formed, so the second layer is lower along the z-axis in the stack-up. 
     In the example illustrated in  FIG. 10 , the data conversion circuit  1030  of the hybrid sensor  1000  includes pre-amplifier circuits  1032 A,  1032 B (also referred to as signal conditioning circuits) respectively coupled with the sensing coil  1010  and the HES  1020 , a multiplexer (MUX)  1034  coupled with the pre-amplifier circuits, and an analog-to-digital converter (ADC)  1036  coupled with the MUX. Optionally, the data conversion circuit  1030  includes a processing module  1038  coupled with the ADC. 
     The pre-amplifier circuits  1032 A,  1032 B are arranged and configured to receive, filter and amplify the analog bEMF signal E(t) across the sensing coil  1010  and the analog Hall voltage signal V H (t) across the HES  1020 . The ADC  1036  is configured to digitize the analog bEMF signal E(t) and the analog Hall voltage signal V H (t) received through the MUX  1034 , and to provide digitized bEMF signal e(t) and the digitized Hall voltage signal v H (t) to the processing module  1038 . 
     When the hybrid sensor  1000  is used in HE  900 , the processing module  1038  is configured to determine the velocity ΔX/Δt and displacement ΔX of the magnet-carrying mass  930  relative the frame  905  from the digitized bEMF voltage signal e(t) and the Hall voltage signal v H (t). Note that in some implementations, the processing module  1038  is part of a digital signal processor (DSP) that is remote from the hybrid sensor  1000 . In such cases, the digitized bEMF signal e(t) and the digitized Hall voltage signal v H (t) are transmitted from the hybrid sensor  1000  to the DSP through communication channels. 
     For example, the processing module  1038  is configured to determine the speed ΔX/Δt of the magnet-carrying mass  930  relative the frame  905  of HE  900  based on values of the simultaneously measured bEMF signal e(t) and Hall voltage signal v H (t) in accordance with 
                     [             Δ   ⁢           ⁢   X       Δ   ⁢           ⁢   t                   Δ   ⁢           ⁢   I       Δ   ⁢           ⁢   t             ]     =         G     -   1       ⁡     [           e   ⁡     (   t   )                   Δ   ⁢           ⁢       v   H     ⁡     (   t   )           Δ   ⁢           ⁢   t             ]       .             (   18   )               
The sensing matrix G from EQ. (18) is expressed as
 
                     G   =     [           K   M         M               S   H     ⁢       δ   ⁢           ⁢     B   ⁡     (   X   )           δ   ⁢           ⁢   X             η         ]       ,           (   19   )               
where matrix element K M  and matrix element M are provided in terms of EQ. (17), and matrix element η is provided in terms of EQ. (3). Additionally, matrix element
 
               S   H     ⁢       δ   ⁢           ⁢     B   ⁡     (   X   )           δ   ⁢           ⁢   X             
represents a field sensitivity obtained from the LUT referenced in EQ. (3). This matrix element, also referred to as the LUT element, is expressed in units of Hall counts per micrometer. The factor
 
               δ   ⁢           ⁢     B   ⁡     (   X   )           δ   ⁢           ⁢   X           
describes a gradient of field intensity B(X) along the x-axis, and encompasses all the nonlinearities of the field intensity. The factor S H  is the Hall sensitivity, expressed in units of mT/count. The LUT element can be obtained in simulation or calibrated by bEMF model fitting. In fact, values of all the elements of sensing matrix G can be are obtained during an initial calibration of the hybrid sensor  1000 , e.g., in the factory. For example, the motor constant K M  can be calibrated by measuring the bEMF around HE&#39;s resonance frequency for a known mass of HE. The EM coupling factor η can be calibrated by measuring Hall voltage of HES  1020  when driving high-frequency currents (e.g. 2 kHz) through the driving coil  910 . Similarly, the mutual coupling M can be determined at the same time when driving high-frequency currents (e.g. 2 kHz) through the driving coil  910  and measuring the induced voltage across the sensing coil  1010 . The foregoing measured values are the “t=0 values” of the elements of the sensing matrix G.
 
     Moreover, the displacement ΔX of the magnet-carrying mass  930  relative the frame  905  can be calculated from its velocity ΔX/Δt determined from EQs. (18)-(19). For instance, the determined velocity ΔX/Δt can be integrated to obtain the displacement ΔX. 
     Note that, by solving EQ. 18 based on the concurrently measured signals e(t) and v H (t), the processing module  1038  determines not only the velocity ΔX/Δt of the magnet-carrying mass  930  relative the frame  905  of HE  900 , but also a rate ΔI/Δt of the driving current through the driving coil  910  of HE  900 . As such, while actuating HE  900 , i.e., as the magnet-carrying mass  930  is vibrated relative to the hybrid sensor  1000  that is affixed to the frame  905 , in the vicinity of a resonant frequency, ΔI/Δt exhibits a resonant current-rate value, and ΔX/Δt exhibits a resonant velocity value; whereas, away from the resonant frequency, ΔI/Δt exhibits a non-resonant current-rate value larger than the resonant current-rate value, while ΔX/Δt exhibits a non-resonant velocity value smaller than the resonant velocity value. As the elements of the sensing matrix G are considered to be parameters that have constant values in time, for bEMF measurements performed in accordance with EQ. (17), the first term will carry a larger weight than the second term near a resonant frequency, and the second term will carry a larger weight that the first term away from the resonant frequency. 
     During operation of HE  900 , the driving current I through the driving coil  910  can also be measured, as I M (t), in addition to the bEMF E(t) signal and the Hall voltage V H (t) signal. In this manner, EQs. (18)-(19) are used first by the processing module  1038  to determine the rate ΔI/Δt of the driving current through the driving coil  910  based on the concurrently measured the bEMF E(t) signal and the Hall voltage V H (t) signal. Moreover, ΔI/Δt determined from EQs. (18)-(19) can be compared with ΔI M /Δt determined by differentiating the measured driving current I M . If the determined ΔI/Δt is different from the measured ΔI M /Δt by more than a threshold value, it is assumed that one or more of the values of the elements of the sensing matrix G (e.g., most likely, M, or 
                 S   H     ⁢       δ   ⁢           ⁢     B   ⁡     (   X   )           δ   ⁢           ⁢   X         )         
have changed. As the values of the elements of the sensing matrix G can, and typically do slowly, vary in time, a filter can be applied to the sensing matrix G to implement a running average of the matrix element values.
 
       FIG. 3A  is a flow chart a process  300  that summarizes the techniques described above for measuring velocity ΔX/Δt of the magnet-carrying mass  930  of HE  900  using the hybrid sensor  1000 . As shown in  FIG. 9 , the hybrid sensor  1000  is affixed to the frame  905  of HE  900  and, thus, it at rest with respect to the driving coil  910 . 
     At  310 , the magnet-carrying mass  930  is vibrated relative the driving coil  910  in response to driving a current I through the driving coil. 
     At  320 , a bEMF signal E(t) and a Hall voltage signal V H (t) are concurrently measured using, respectively, the sensing coil  1010  of the hybrid sensor  1000  and the HES  1020  of the hybrid sensor. As shown in  FIG. 10 , the HES  1020  is disposed within the sensing coil  1010 . 
     At  330 , a velocity ΔX/Δt of the vibrating magnet-carrying mass  930  and a rate ΔI/Δt of the driving current through the driving coil  910  are determined by the processing module  1038 . This determination is performed using EQs. (18)-(19) based on the concurrently measured bEMF signal E(t) and Hall voltage signal V H (t). 
     In some implementations, at  340 A, the processing module  1038  can verify whether the rate ΔI/Δt of the driving current, as determined at  330 , exceeds a threshold. 
     In some implementations, at  335 , the driving current I through the driving coil  910  can be measured directly and concurrently with the measuring of the bEMF signal E(t) and Hall voltage signal V H (t). Here, the measured driving current I M  is differentiated to determine a rate ΔI M /Δt of the measured driving current. At  340 B, the processing module  1038  can verify whether the rate ΔI/Δt of the driving current, as determined at  330 , is different from the rate ΔI M /Δt of the measured driving current, as determined at  335 . For instance, ΔI/Δt is deemed to be different from ΔI M /Δt if 
             rms   ⁡     (       (       Δ   ⁢           ⁢   I       Δ   ⁢           ⁢   t       )     -     (       Δ   ⁢           ⁢     I   M         Δ   ⁢           ⁢   t       )       )           
is larger than a threshold.
 
     If the verification performed at either  340 A or  340 B is false, then the process  300  can be reiterated. However, if the verification performed at either  340 A or  340 B is true, then, at  350 , the processing module  1038  can adjust one or more elements of the sensing matrix G, given by EQ. (19), prior to potentially reiterating the process  300 . For instance, adjustment of the EM coupling factor η can cause improvement in the sensing of the velocity ΔX/Δt of the vibrating magnet-carrying mass  930 , as illustrated in the simulation results presented in  FIGS. 3B-3C . As described above, the adjusting can be performed by applying a filter on the one or more elements of the sensing matrix G, starting from respective factory-calibrated values of these elements. In some implementations, the application of the filter involves performing a running average of the one or more elements of the sensing matrix G. 
     Referring again to  FIGS. 9-10 , as the bEMF is measured on the sensing coil  1010  of the hybrid sensor  1000  and not on the driving coil  910  of HE  900 , errors caused by variations in the driving coil&#39;s resistance can be reduced/avoided. While actuating HE  900 , variations of order 0.48%/deg in the driving coil  910 &#39;s resistance can occur over changes in temperature of order 30°, thus rendering bEMF measurements, that use the driving coil for sensing magnetic field variations ΔB/Δt, inaccurate. However, when the sensing coil  1010  of analog front-end circuitry  1005  is used for sensing magnetic field variations ΔB/Δt, the resistance of the sensing coil stays substantially constant when the HE  900 &#39;s temperature changes, because there is no active (driving) current, but only a small sensing current, that circulates through the sensing coil. 
     Moreover, HES  1020 &#39;s resistance can be used as a temperature sensor for thermal compensation of the sensing coil  1010 &#39;s resistance value and Hall voltage&#39;s value V H  measured over the HES. For instance, in the example shown in  FIG. 10 , a current source  1022  can be used to induce a desired current through HES  1020  between its pair of terminals that are not used for measuring Hall voltage. In this manner, a measurement of the voltage across this pair of terminals is used to determine the resistance of HES  1020 . For example, a change in resistance Δρ(ΔΘ) of HES  1020  is a function of mostly a change in temperature ΔΘ, and a change in Hall voltage ΔV H (ΔB;ΔΘ) output by HES is a function of both a change in magnetic field intensity ΔB and also a change in temperature ΔΘ. In this manner, a measurement of the change in resistance Δρ(ΔΘ) of HES  1020  can be used to determine the change in temperature ΔΘ of HES, and a measurement of the change in Hall voltage ΔV H (ΔB;ΔΘ) output by HES can be used to determine the change magnetic field intensity ΔB at the location of HES for the determined change in temperature ΔΘ. As such, the temperature change ΔΘ determined in this manner can be useful for improving the accuracy of ΔX displacement measurements performed using the hybrid sensor  1000  when various temperature coefficients of the sensing coil  1010  and HES are modeled and compensated for. 
     Note that the analog bEMF signal E(t) across the sensing coil  1010  and the analog Hall voltage signal V H (t) across the HES  1020  are out of phase by 90° (the former signal being proportional to the velocity ΔX/Δt of mass  930  and the latter being proportional to the displacement ΔX of the mass). In this manner, a finite (non-zero) signal is always output by the analog front-end circuitry  1005  to the data conversion circuitry  1030 , and, hence, an rms of a combination of the digitized bEMF signal e(t) and the digitized Hall voltage signal v H (t) is always non-zero. As such, in some implementations, a pair of hybrid sensors  1000  can be used in HE  900 , so the pair of hybrid sensors sandwiches, along the Z-axis, the moving magnet-carrying mass  930 . In this manner, a displacement ΔZ along the Z-axis of the magnet-carrying mass  930  can be determined as 
                       Δ   ⁢           ⁢   Z     ∝       (       V     rms   t       -     V     rms   b         )       (       V     rms   t       +     V     rms   b         )         ,           (   20   )               
where the rms of the digital outputs of the hybrid sensors disposed respectively above and below the magnet-carrying mass  930  are
 
 V   rms     t   =√{square root over ( E   t   2   +V   H     t     2 )}≠0  (21),
 
 V   rms     b   =√{square root over ( E   b   2   +V   H     b     2 )}≠0  (22).
 
Because both terms in the denominator of EQ. (20) are non-zero, in accordance with EQs. (21)-(22), the displacement ΔZ along the Z-axis can be accurately determined.
 
     In other implementations, a plurality of hybrid sensors  1000  can be arranged inside HE  900  to form a 1D-array, e.g., to form a row of hybrid sensors. In yet other implementations, a plurality of hybrid sensors  1000  can be arranged inside HE  900  to form 2D-array, e.g., a rectangular array of hybrid sensors  1000  arranged in rows and columns. This arrangement of the plurality of hybrid sensors  1000  can be referred to as a “magnetic camera.” In either of these implementations, each of the plurality of hybrid sensors  1000  of the array is operated independently from each other to sense the absolute field intensity B and its rate of change ΔB/Δt, at its associated node P(i,j) of the array, where i is a row index and j is a column index. 
     A few embodiments have been described in detail above, and various modifications are possible. The disclosed subject matter, including the functional operations described in this specification, can be implemented in electronic circuitry, computer hardware, firmware, software, or in combinations of them, such as the structural means disclosed in this specification and structural equivalents thereof, including system on chip (SoC) implementations, which can include one or more controllers and embedded code. 
     While this specification contains many specifics, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular embodiments. Certain features that are described in this specification in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination. 
     Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system components in the embodiments described above should not be understood as requiring such separation in all embodiments. 
     Other embodiments fall within the scope of the following claims.

Metadata:
Filing Date: 20190924
Publication Date: 20201208
Grant Date: 20201208
Priority Date: 20160916
Inventors: CHEN, DENIS G.
HAJATI, ARMAN
BHATTACHARYYA, MANOJ K.
Assignee: APPLE INC
CPC Classifications: [{"code": "G01D5/145", "inventive": true, "first": true, "tree": "[]"}, {"code": "G01D5/2013", "inventive": true, "first": true, "tree": "[]"}, {"code": "G01D5/142", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01D5/145", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01D5/145", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01D5/2013", "inventive": true, "first": true, "tree": "[]"}, {"code": "G01D5/142", "inventive": true, "first": false, "tree": "[]"}]
Family ID: 61620199