Abstract:
A system for obtaining incremental and absolute displacement measurements using systems of electrodes that interact to form variable capacitors and systems that facilitate implementation of the method along with exemplary embodiments of these systems. The capacitors created by the disclosed method have known physical properties and corresponding known mathematical relationships. These laws are exploited in such a way by our method as to overcome inadequacies in existing systems and create superior systems. These superior systems improve upon the existing art by including economically and reliably made sensors based on the area varying principle which eliminate dead zone issues and increase accuracy through a reduction of the influence of gap variations on capacitive systems through the use of compensatory geometrical arrangements of multiple capacitive systems.

Description:
BACKGROUND 
       [0001]    The use of capacitors is known for the effective storing of electrical charges. The capacitance of a capacitor with two parallel electrodes is known to be approximately proportional to the dielectric constant of the medium between the two electrodes, proportional to the overlapped area of the two electrodes, and inversely proportional to the gap distance between the two electrodes. The capacitance changes when a displacement of one electrode with respect to another changes the overlapped area and/or changes the gap distance. Those skilled in the art applied these scientific principles (or relationships) and fabricated various capacitive sensors including displacement (position) measuring ones. 
         [0002]    Based on the above scientific principles, a displacement can be measured in two different ways. One method is detecting capacitance variation due to a gap variation between two parallel electrodes, where the motion is perpendicular to the electrode surfaces. Because a capacitance of a capacitor is extremely sensitive to its gap variation, particularly when the gap is small, this kind of method is found in nano-positioning applications. Those sensors can sense a displacement as small as a few picometers, while they are limited to a measurement range of a few hundred micrometers. Another issue for this kind of sensor is nonlinearity due to the inversely proportional relationship between the capacitance and the gap distance of a capacitor, although this problem can be resolved with a digital linearization technique. 
         [0003]    Another method is detecting a capacitance variation due to a size variation of an overlapped area between two parallel electrodes, where the motion is parallel to the surfaces of the electrodes. A commercially available sensor based on the area varying principle is desirable, but hard to find. Implementing the area varying principle perfectly in practice has proven to be a significant challenge. It is very difficult to measure the size variation of an overlapped area by having a motion in a parallel direction to the electrode surface without any variation of the gap distance due to motion perpendicular to the electrode surfaces. As mentioned above, the capacitance of a capacitor is extremely sensitive to its gap distance. It can be estimated, for example, that increasing or decreasing a capacitor gap of 100 micrometers by one micrometer, creates a capacitance variation of one percent. A 100 micrometer gap is in the practical range while a one micrometer tolerance for mechanical moving parts is quite a challenge for economical mass production. The one percent uncertainty error is unacceptable in terms of accuracy for most applications. As a result of difficult gap variation control, a sensor based on area variation is impractical unless useful compensation techniques are developed to reduce influence of gap variation. 
         [0004]    There is another challenging problem for an area varying capacitive sensor to have a both highly sensitive and long range measurement. An area varying capacitive sensor cannot have a long range measurement because of the cyclic nature of its output signal. Transition points exist where a signal changes from increase to decrease or vice versa. The measurement uncertainties near the transition points are usually too big to be acceptable in terms of accuracy for most applications. This makes a sensor based on area variation impractical for long range measurement unless a solution is found to eliminate the uncertainties. 
       SUMMARY 
       [0005]    A composite capacitor measurement system which may be used for incremental and absolute displacement measurement. Each system can have one or more subsystems (STATORs) which themselves may have an electrically insulated substrate having capacitive electrodes located a predetermined distance from one another. These STATOR subsystems can work in conjunction with one or more subsystems (MOVERs) which can have an electrically insulated substrate with one or more ground electrodes. One or more of the subsystems may be mobile and may move in a controlled, predetermined manner with respect to one or more predetermined subsystems which may be stationary relative to the global system. 
         [0006]    One exemplary embodiment described herein may be a composite capacitive displacement measurement system. The composite capacitive displacement system can have one or more composite components (STATORs) that are stationary with respect to a predetermined globe system. Each of the composite components can include an electrically insulated substrate having at least a pair of electrically conductive capacitive electrodes located a predetermined distance from one another, where the electrodes are alternatively electrically connected together to form two capacitive series. Also, the system can have one or more composite components (MOVERs) that are mobile and move in a controlled manner with respect to the predetermined corresponding components that are stationary relative to the globe system. Each of these components can have an electrically insulated substrate having at least one ground series of electrically conductive ground electrodes located a predetermined distance from one another, where the ground electrodes are electrically connected together to form a ground series. Further, the system can have one or more circuits with an electronic signal processing unit; a system ground; conducting components connecting the ground series to the system ground; and conducting components connecting the capacitive series to the electronic signal processing unit. 
         [0007]    In another exemplary embodiment, a system for providing for substantially simultaneous creation and destruction of variable capacitors in respectively equal amounts may be described. The system can have one or more stationary composite components (STATORs) that are stationary with respect to a predetermined globe system. Each of the components can include an electrically insulated substrate having one pair of two capacitive series of m electrically conductive elementary electrodes located a predetermined distance from one another, where the elementary electrodes are electrically connected together within their own series. The system can also have one or more mobile composite components (MOVERs) that move in a controlled manner with respect to the predetermined corresponding components that are stationary relative to the globe system. Additionally, the system may have an electrically insulated substrate having one ground series of n electrically conductive elementary electrodes located a predetermined distance from one another where the elementary electrodes are electrically connected together within their own series. Also, the system may be such wherein there are more of the capacitive electrodes than the ground electrodes (m&gt;n) such that as at least one MOVER moves with respect to at least one STATOR, capacitive electrodes on the STATOR that previously formed capacitors with ground electrodes of the MOVER are no longer formed capacitors and, simultaneously, capacitive electrodes on STATOR which had not formed capacitors now form capacitors because of the movement of the ground electrodes of the MOVER. 
     
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         [0008]    FIG.  1 - a  is an exemplary side view of one possible embodiment containing two STATORs and one double sided MOVER and showing the geometric configuration of their components. 
           [0009]    FIG.  1 - b  is an exemplary top down view of one STATOR and one side of a double sided MOVER interacting and showing the geometric configuration of their components. 
           [0010]      FIG. 2  is an exemplary electrical drawing that demonstrates how capacitors are created and destroyed using the embodiment shown in FIG.  1 - a  and how the resulting capacitors are numbered. 
           [0011]      FIG. 3  is an exemplary diagram of an electrical configuration useable for connecting the capacitors, processing unit and output for an embodiment of the invention. 
           [0012]      FIG. 4  is an exemplary alternative electrical configuration useable for the capacitors, processing unit and output and is the configuration use for the embodiment shown in FIG.  1 - a.    
           [0013]      FIG. 5  is an example of the digital output curve of the embodiment shown in FIG.  1 - a  and shows the dead zones in output. 
           [0014]      FIG. 6  is an exemplary top down view of one possible embodiment containing one STATOR and one MOVER and showing the geometric configuration of their components. 
           [0015]      FIG. 7  is an exemplary example of the digital output curves of the embodiment shown in  FIG. 6  and how the dead zones have been compensated for by the embodiment. 
           [0016]      FIG. 8  is an exemplary top down view of one possible embodiment containing one STATOR and one MOVER and showing the geometric configuration of their components. 
           [0017]      FIG. 9  is an exemplary side view of one possible embodiment containing two STATORs and one double sided MOVER and showing the geometric configuration of their components. 
           [0018]      FIG. 10  is an exemplary side view of one possible embodiment containing two STATORs and one double sided MOVER and showing the geometric configuration of their components. 
           [0019]      FIG. 11  is an exemplary side by side comparison of two possible embodiments of MOVERs and their components. 
           [0020]    FIG.  12 - a  is an exemplary top down view of just the STATOR and its components in one possible embodiment, the same embodiment that the MOVER in FIG.  12 - b  would be a part of, and the same embodiment shown in  FIG. 12 . 
           [0021]    FIG.  12 - b  is an exemplary top down view of just the MOVER and its components in one possible embodiment, the same embodiment that the STATOR in FIG.  12 - a  would be a part of, and the same embodiment shown in  FIG. 12 . 
           [0022]      FIG. 12   c  is an exemplary top down view of one possible embodiment containing one STATOR and one MOVER and showing the geometric configuration of their components; it is also the embodiment whose STATOR was shown in FIG.  12 - a  and MOVER was shown in FIG.  12 - b.    
           [0023]    FIG.  13 - a  is an exemplary isometric view of one possible embodiment of a completely assembled sensor system for an absolute linear displacement measurement. 
           [0024]    FIG.  13 - b  is an exemplary STATOR from FIG.  13 - a  as viewed from the top down and also the MOVER from FIG.  13 - a  as viewed from the bottom up. 
           [0025]    FIG.  14 - a  is an exemplary isometric view of one possible embodiment of a completely assembled sensor system for an absolute linear displacement measurement. 
           [0026]    FIG.  14 - b  is an exemplary isometric view of the STATOR from FIG.  14 - a  and also the MOVER from FIG.  14 - a.    
           [0027]    FIG.  15 - a  is an exemplary isometric view of one possible embodiment of a completely assembled sensor system for an absolute angular displacement measurement. 
           [0028]    FIG.  15 - b  is an exemplary top down view of the STATORs of the embodiment shown in FIG.  15 - a.    
           [0029]    FIG.  15 - c  is an exemplary top down view of the MOVER of the embodiment shown in FIG.  15 - a.    
           [0030]    FIG.  16 - a  is an exemplary isometric view of a cross section of a complete angular displacement sensor system for an absolute angular displacement measurement of a full 360-degree measurement range. 
           [0031]    FIG.  16 - b  is an exemplary view of the STATORs and MOVER in an unassembled configuration. 
       
    
    
     DETAILED DESCRIPTION 
       [0032]    Aspects of the invention are disclosed in the following description and related drawings directed to specific embodiments of the invention. Alternate embodiments may be devised without departing from the spirit or the scope of the invention. Additionally, well-known elements of exemplary embodiments of the invention will not be described in detail or will be omitted so as not to obscure the relevant details of the invention. Further, to facilitate an understanding of the description discussion of several terms used herein follows. 
         [0033]    As used herein, the word “exemplary” means “serving as an example, instance or illustration.” The embodiments described herein are not limiting, but rather are exemplary only. It should be understood that the described embodiment are not necessarily to be construed as preferred or advantageous over other embodiments. Moreover, the terms “embodiments of the invention”, “embodiments” or “invention” do not require that all embodiments of the invention include the discussed feature(s), advantage(s) or mode(s) of operation(s). 
         [0034]    Generally referring to  FIGS. 1-16 , exemplary embodiments described herein may measure linear and/or angular displacement (or position) based on overlapped area variations of multiple capacitive systems. Exemplary 2D and 3D geometrical compensational arrangements of capacitive systems described herein can reduce the capacitive influence of gap variations that may cause an accuracy problem with measurements and can also overcome a known measurement “dead zone” problem. The “dead zone” solution can make long range measurements with nanometer resolution possible. Additionally, certain exemplary embodiments can reduce the extremely high accuracy requirements of supporting components. 
         [0035]    In a first exemplary embodiment shown in FIG.  1 - a , a capacitive sensor system  100  can have a variety of sub-systems. These sub-systems can include a capacitive series  11  having m elements of substantially equally spaced electrically conductive electrodes  11 - 1 ,  11 - 2 , . . . ,  11 - m ; a capacitive series  12  of m elements of substantially equally spaced electrically conductive electrodes  12 - 1 ,  12 - 2 , . . . ,  12 - m ; a capacitive series  13  of m elements of substantially equally spaced electrically-conductive electrodes  13 - 1 ,  13 - 2 , . . . ,  13 - m ; a capacitive series  14  of m elements of substantially equally spaced electrically conductive electrodes  14 - 1 ,  14 - 2 , . . . ,  14 - m ; a ground series  15  of n elements of substantially equally spaced electrically conductive electrodes  15 - 1 ,  15 - 2 , . . . ,  15 - n ; and a ground series  16  of n elements of substantially equally spaced electrically conductive electrodes  16 - 1 ,  16 - 2 , . . . ,  16 - n . The numbers m of the electrodes of capacitive series  11 ,  12 ,  13 , and  14  may all be equal to each other. The number m can be any positive integer. The numbers n of grounded electrodes  15  and  16  may be equal to each other. The number n can be any positive integer as well. 
         [0036]    Still referring to exemplary FIG.  1 - a , the number n of grounded electrodes may be smaller than or equal to the number m of capacitive series. The capacitive series  11  and  12  can be on the same side of an electrically-insulated substrate  17  with electrical shielding  18  on the other side of the substrate. The two capacitive series  11  and  12 , the insulation substrate  17 , and the shielding layer  18  together can form a composite component, known, for example, as STATOR-1  104  of the sensor system. The capacitive series  13  and  14  may be on the same side of an electrically-insulated substrate  19  with electrical shielding  20  on the other side of the substrate. The two capacitive series  13  and  14 , the insulation substrate  19 , and the shielding layer  20  together may form another composite component, known, for example, as STATOR-2  105  of the sensor system. The ground series  15  and  16  can be on two different sides of a common electrically-insulated substrate  21 . The ground series  15  and  16 , together with the insulation substrate  21 , can form a composite component known, for example, as MOVER  106  of the sensor system. It may further be appreciated that any of the STATORs or MOVERs described herein may be composite components. 
         [0037]    The STATOR-1  104  and STATOR-2  105  may be substantially parallel to each other. The MOVER  106  can displace in a substantially parallel manner with respect to the STATOR-1  104  and STATOR-2  105 . It may be noted, however, that STATOR and MOVER are just assigned names for the purpose of describing aspects of the exemplary embodiments. A STATOR can be either stationary or mobile with respect to a global stationary coordinate or a moving coordinate as desired. 
         [0038]    Still referring to exemplary FIG.  1 - a , the gap between the first electrode  11 - 1  of the capacitive series  11  and the first electrode  15 - 1  of the ground series  15  is represented as d1-1. The gap between the first electrode  12 - 1  of the capacitive series  12  and the first electrode  15 - 1  of the ground series  15  is represented as d2-1. The gap between the first electrode  13 - 1  of the capacitive series  13  and the first electrode  16 - 1  of the ground series  16  is represented as d3-1. The gap between the first electrode  14 - 1  of the capacitive series  14  and the first electrode  16 - 1  of the ground series  16  is represented as d4-1. To form a combination of direct compensation and differential compensation as described below, the gaps d1-1, d2-1, d3-1, and d4-1 may all be substantially equal to each other. In the exemplary embodiment, with an arrangement as shown in  FIG. 1 , the gaps d1-1, d2-1, d3-1, and d4-1 can be slightly different within practical fabrication and assembly tolerances without significantly reducing measurement accuracy. 
         [0039]    In exemplary FIG.  1 - b  the widths and lengths of the electrodes of capacitive series  11 ,  12 ,  13  and  14  are represented as Wc and Lc, respectively. The spaces between electrodes of capacitive series  11  and adjacent electrodes of capacitive series  12  are represented as Wcs. The spaces between electrodes of capacitive series  13  and adjacent electrodes of capacitive series  14  are also Wcs. The width and length of the ground series  15  and  16  are represented as Wg and Lg. The spaces between adjacent electrodes of ground series  15  are Wgs. The spaces between adjacent electrodes of ground series  16  are also Wgs. In some exemplary embodiments, Lg can be either larger or smaller than Lc but not equal to Lc as this size difference can eliminate a misalignment error. 
         [0040]    Continuing with exemplary FIG.  1 - b , the width of the overlap between capacitive electrode  11 - 1  and ground electrode  15 - 1  is X1-1, and the overlap area A1-1 may be substantially equal to the multiplication of width X1-1 and the length Lc. The overlap width between capacitive electrode  12 - 1  and ground electrode  15 - 1  is X2-1, and the overlap area A2-1 may be substantially equal to the multiplication of width X2-1 and the length Lc. The overlap width between capacitive electrode  13 - 1  and ground electrode  16 - 1  may be substantially equal to X3-1, and the overlap area A3-1 may be substantially equal to multiplication of width X3-1 and the length Lc. The overlap width between capacitive electrode  14 - 1  and ground electrode  16 - 1  may be substantially X4-1, and the overlap area A4-1 may be substantially equal to multiplication of width X4-1 and the length Lc. These overlaps may be linear functions of displacement X of the MOVER  106 . For example, for the first electrode  11 - 1  of capacitive series  11 , A1-1 may be determined as: 
         [0000]        A 1-1= X 1-1* Lc =( Wc−X )* Lc   [equation 1]
 
         [0000]    and, for the first electrode  12 - 1  of capacitive series  12 , A2-1 may be determined as: 
         [0000]        A 2-1= X 2-1* Lc =( Wg+X−Wc−Wcs )* Lc   [equation 2]
 
         [0000]    The area variations resulting capacitance change as described below may represent an exemplary method of displacement measurement. 
         [0041]    With the above exemplary geometrical arrangement, an electrode K-i of a capacitive series and an electrode G-j of a ground series can form a variable elementary capacitor CK-j, where K may represent the K th  capacitive series and can be  11 ,  12 ,  13  or  14 ; and i may represent the i th  electrode in the capacitive series, and can be any number of 1, 2, . . . , m; G may represent the G th  ground series and can be  15  or  16 ; and j can represent the j th  electrode in the ground series and can be any number of 1, 2, . . . , n; CK-j can represent the elementary capacitor formed by an electrode of the K th  capacitive series and j th  electrode of a ground series. For example, electrode  11 - 1  of capacitive series  11  and electrode  15 - 1  of ground series  15  can form elementary capacitor C 11 - 1  with a capacitance proportional to the value of (A1-1/d1-1). Furthermore,  11 - 2  of capacitive series  11  and  15 - 2  of ground series  15  can form elementary capacitor C 11 - 2 ;  12 - 1  and  15 - 1  can form C 12 - 1 ;  13 - 1  and  16 - 1  can form C 13 - 1 ;  14 - 1  and  16 - 1  can form C 14 - 1 ;  11 - n  and  15 - n  can form C 11 - n ;  12 - n  and  15 - n  can form C 12 - n ;  13 - n  and  16 - n  can form C 13 - n ;  14 - n  and  16 - n  can form C 14 - n ; and so on. The capacitance of capacitor C 11 - 2  may be proportional to the value of (A1-2/d1-2), the C 12 - 1  the value of (A2-1/d2-1), the C 13 - 1  the value of (A3-1/d3-1), the C 14 - 1  the value of (A4-1/d4-1), and so on. 
         [0042]    There are numbers (m-n) of electrodes of the capacitive series  11 ,  12 ,  13 , and  14 , which may not be able to form elementary capacitors with an electrode of ground series  15  or  16  at the illustrated position in exemplary FIG.  1 - a . But as the MOVER  106  displaces to the right, there may be new elementary capacitors created at the same time when some existing capacitors are destroyed. The number of newly created and destroyed elementary capacitors may be substantially equal to each other. Therefore the number of elementary capacitors may be substantially the same throughout any movement, while the total capacitances of the systems may vary as the overlapped areas vary. 
         [0043]    Elementary capacitors formed by electrodes of capacitive series K and electrodes of ground series G may be in parallel connections. They can form capacitive systems CK. In the exemplary embodiment shown in FIG.  1 - a , there are four capacitive systems, C 11 , C 12 , C 13  and C 14 . Equivalent circuit diagrams of the elementary capacitors and the capacitive systems are shown in exemplary  FIG. 2  as a further embodiment. Exemplary  FIG. 2  can also illustrate that as the MOVER  106  displaces to the right, there may be new elementary capacitors created at the same time that existing capacitors are destroyed. The numbers of newly created and destroyed elementary capacitors may be substantially equal to each other. The number of elementary capacitors may be the same throughout any movement. 
         [0044]    The capacitive systems C 11  and C 13  can be arranged in a superposition configuration so that values of (C 11 +C 13 ) or (C 11 *C 13 ) can be obtained. The capacitive systems C 12  and C 14  can be arranged in superposition configuration so that values of (C 12 +C 14 ) or (C 12 *C 14 ) may be obtained. 
         [0045]    In the above exemplary embodiment as shown in FIG.  1 - a , the elementary electrodes of capacitive series  11  and ones of capacitive series  13  may be on different planes, but can be in alignment with respect to each other. There can also be an allowable degree of error due to fabrication and assembly. Similarly, the elementary electrodes of capacitive series  12  and those of capacitive series  14  can be on different planes, but may be in alignment with each other, while also possibly having an allowable small degree of error due to fabrication and assembly. The elementary electrodes of ground series  15  and  16  can be on different planes, but may be in alignment with each other, while also possibly having an allowable degree of error due to fabrication. The elementary capacitors with capacitive electrode pairs, in alignment, can directly compensate each other against undesired motion in the gap direction, which may be perpendicular to the designed motion direction. In the above exemplary embodiment as shown in FIG.  1 - a , for example, the elementary capacitor C 11 - 1  formed by the capacitive electrode  11 - 1  and the ground electrode  15 - 1  can form a direct compensation with the elementary capacitor C 13 - 1  composed of the capacitive electrode  13 - 1  and the ground electrode  16 - 1 . As a result of elementary capacitor direct compensation, the capacitive systems CI and C 13  can form direct compensation, and the capacitive systems C 12  and C 14  can form direct compensation against the undesired motion in the gap direction, which may be perpendicular to the designed motion direction. 
         [0046]    The capacitive systems C 11  and C 12  can be arranged in a differential configuration so that values of (C 11 -C 12 ) or (C 11 /C 12 ) can be obtained. The capacitive systems C 13  and C 14  can be arranged in a differential configuration so that values of (C 13 -C 14 ) or (C 13 /C 14 ) can be obtained. 
         [0047]    In the above exemplary embodiment, and still referring to exemplary FIG.  1 - a , the elementary electrodes of capacitive series  11  and series  12  can all be in the same plane with an allowable small degree of error due to fabrication. Similarly, the elementary electrodes of capacitive series  13  and series  14  can all be in a same plane with an allowable small degree of error due to fabrication and assembly. The elementary electrodes of ground series  15  can all be in the same plane with an allowable small degree of error due to fabrication. Adjacent elementary capacitors can be differentially compensating each other against undesired motion perpendicular to the designed motion direction. In the above exemplary embodiment of FIG.  1 - a , the elementary capacitor C 11 - 1  may be constructed by capacitive electrode  11 - 1  and ground electrode  15 - 1  to form differential compensation with elementary capacitor C 12 - 1  which in turn can be constructed by capacitive electrode  12 - 1  and ground electrode  15 - 1 . When the MOVER  106  displaces to the right at the one half point of the repeat cycle for example, the elementary capacitor C 12 - 1  composed of capacitive electrode  12 - 1  and ground electrode  15 - 1  could form a differential compensation with elementary capacitor C 11 - 2  which itself is composed of capacitive electrode  11 - 2  and ground electrode  15 - 1 . As a result of adjacent capacitor compensations the capacitive systems C 11  and C 12  can form a differential compensation, and the capacitive systems C 13  and C 14  can form a differential compensation against, for example, undesired motion in the gap direction, that may be perpendicular to the designed motion direction. 
         [0048]    In the above exemplary embodiment as shown in FIG.  1 - a , multiple capacitive systems can be arranged to form a combination of direct compensation and differential compensation interactions that may optimally reduce the influence of the motion in the gap direction. Capacitive system C 11  can simultaneously form direct compensation with capacitive system C 13  and differential compensation with capacitive system C 12 . Capacitive system C 12  can simultaneously form direct compensation with capacitive system C 14  and differential compensation with capacitive system C 11 . Capacitive system C 13  can simultaneously form direct compensation with capacitive system C 11  and differential compensation with capacitive system C 14 . Capacitive system C 14  can simultaneously form direct compensation with capacitive system C 12  and differential compensation with capacitive system C 13 . 
         [0049]    The repeating patterns of electrically-conductive electrodes of both the capacitive series and the ground series can be fabricated onto insulated substrates by several techniques. Fabrication techniques include, but are not limited to, printed circuit board (PCB) methods, various thin-film deposition methods, printed electronics methods, or any other known or desired method known to a person of ordinary skill in the art. Exemplary conductive materials for the electrodes include, but are not limited to, copper, silver, gold, aluminum and their alloys, their coatings and inks, or any other desired material or combination of materials known to a person of ordinary skill in the art. Exemplary insulating substrate materials include, but are not limited to, various resin laminates, glasses, ceramics and plastic sheets and tubes, or any other desired material or combination of materials known to a person of ordinary skill in the art. 
         [0050]    In the above exemplary embodiment as shown in FIG.  1 - b , the widths and lengths of the electrodes of capacitive series  11 ,  12 ,  13  and  14  are represented as Wc and Lc. The applicable range of width for Wc may be between 0.001 mm approximately and 200 mm approximately, with an exemplary range of between 0.1 mm approximately and 200 mm approximately. An applicable range of length for Lc is between 0.005 mm approximately and 200 mm approximately, with an exemplary range of between 5 mm approximately and 200 mm approximately. The spaces between electrodes of capacitive series  11  and adjacent electrodes of capacitive series  12  are represented as Wcs. The spaces between electrodes of capacitive series  13  and adjacent electrodes of capacitive series  14  are also Wcs. The applicable range of width for Wcs can be between 0.001 mm approximately and 200 mm approximately. The width and length of the ground series  15  and  16  are represented as Wg and Lg. The spaces between adjacent electrodes of ground series  15  are Wgs. The spaces between adjacent electrodes of ground series  16  are also Wgs. The sum of the width Wg and the space Wgs can be equal to two times of the sum of width Wc and the space Wcs, that is: 
         [0000]        Wg+Wgs= 2*( Wc+Wcs ).  [equation 3]
 
         [0000]    Additionally, in some exemplary embodiments, the width Wg can be equal to or larger than the width Wc, and the width Wg may be equal to or smaller than the sum of width Wc and two times of the space Wcs, that is 
         [0000]        Wc≦Wg ≦( Wc+ 2* Wcs ).  [equation 4]
 
         [0051]    In a further exemplary embodiment, the length Lg of the electrodes of ground series  15  and  16  may be smaller than or larger than, but not equal to the length Lc of electrodes of the capacitive series  11 ,  12 ,  13 , and  14 . For example, the length Lg can be about 0.002 mm to about 10 mm larger than or smaller than the length Lc. 
         [0052]    In the above exemplary embodiment as shown in FIG.  1 - a , the electrodes of both the capacitive series and the ground series may be rectangular shapes. It may be appreciated that, in other embodiments, other shapes may be suitable for real applications. As such the electrodes of both the capacitive series and the ground series are not limited to being planar or flat as shown in exemplary FIG.  1 - a . Exemplary shapes of electrodes illustrated in the following sections are cylindrical or are sections of a cylinder. However, it may be appreciated that the shapes of electrodes can be sections of any of a variety of shapes, for example flat rectangular, flat circular, cylindrical rectangular, flat triangular, flat circular triangular, cylindrical triangular, or any other shape or combination of shapes as desired. 
         [0053]    Turning now to the circuitry of a measurement system, in an exemplary embodiment, the capacitive systems C 11 , C 12 , C 13  and C 14  can be connected to an Electronic Signal Processing Unit  800  as shown in exemplary  FIG. 3 . The grounded electrodes may all be connected to the Processor Ground  801  through various conducting components, as desired. The Electronic Signal Processing Unit  800  may be any desired type of signal processor, including, but not limited to, a signal conditioner, an analog to digital converter, a digital signal processor and the like. The signal processor may further produce a digital output. In the exemplary embodiments, capacitances of capacitive systems C 11 , C 12 , C 13  and C 14  can be independently measured, and there may not be a bridge arrangement. For example, there may be one side of a capacitor, which may be connected directly to ground, as opposed to an excitation source. There may be no bridge excitation circuit applied in such an exemplary embodiment, although the application of such a circuit may be utilized in other embodiments, as desired. 
         [0054]    In a direct compensation between C 11  and C 13 , the two capacitive systems may be in a parallel configuration and they can be measured together. So are capacitive systems C 12  and C 14 . Taking the advantage of the parallel configurations, only two independent measurements need be utilized. The measurement circuit can be simplified as shown in exemplary  FIG. 4 . 
         [0055]    In a further exemplary embodiment, a curve of digital outputs of a sensor system with the geometrical arrangement shown in exemplary FIG.  1 - a  and with the electrical configuration shown in exemplary  FIG. 4  is provided in exemplary  FIG. 5 . The curve can indicate that the output signal has a cyclic behavior, as may be expected. The value of a measured signal can change from increasing to decreasing, and then can change back to increasing. This cyclic behavior can make a long range measurement possible and, in some exemplary embodiments, more sensitive. However, it may also introduce “dead zones”  110  as marked around the transition locations, where the uncertainty of measurement can be unacceptably large. The dead zones can be unavoidable because of the existence of spaces (Wcs) between adjacent elementary electrodes of capacitive series for the differential compensation. For direct compensation, the dead zones may be unavoidable because the edges of electrodes are not able to be made with nanometer precision. Without resolving the dead zone problem, measurement may be limited to less than a half cycles distance. 
         [0056]    Thus, using exemplary embodiments described herein, the dead zone problem may be resolved and may allow for making a long range measurement with nanometer accuracy a reality. 
         [0057]    Referring now to  FIG. 6 , in one exemplary embodiment that resolves the dead zone problem a Composite Measurement System  201  may be constructed with two capacitive sensor systems, System-1  30  and System-2  40 . The System-1  30  and System-2  40  may be substantially similar to the capacitive sensor system  100  described in an exemplary embodiment above or a part of that capacitive sensor system  100 . For example, to demonstrate the method of resolving the dead zone problem a simplified version or a part of the capacitive sensor system  100  can be repeatedly applied in the Composite Measurement System  201 , and also in the following paragraphs for Composite Measurement Systems  202 ,  203  and  204 . 
         [0058]    Referring still to exemplary  FIG. 6  and in a further embodiment, the System-1  30  can include: a capacitive series  31  of m elements of substantially equally spaced electrically conductive electrodes  31 - 1 ,  31 - 2 , . . . ,  31 - m ; a capacitive series  32  of m elements of substantially equally spaced electrically conductive electrodes  32 - 1 ,  32 - 2 , . . . ,  32 - m ; and a ground series  35  of n elements of equally spaced electrically conductive electrodes  35 - 1 ,  35 - 2 , . . . ,  35 - n . For demonstration simplicity the number m of the electrodes of capacitive series  31  and  32  is 6, although it can be any positive integer. Similarly, for demonstration simplicity the number n of the electrodes of ground series  35  is 3, although it can be any positive integer. Capacitive series  31  and  32  can be on the same side of an electrically insulated substrate  37 , forming part of a STATOR. Ground series  35  can be on a different electrically insulated substrate, forming part of the MOVER. The MOVER can displace in a substantially parallel manner with respect to the STATOR, which can result in variations of overlap areas between capacitive electrodes and ground electrodes, and variations of capacitances of capacitors C 31 - 1 , C 31 - 2 , C 31 - 3 , C 32 - 1 , C 32 - 2 , and C 32 - 3 . With the kind of geometrical arrangement as described above, capacitive systems C 31  and C 32  can together form a differential compensation reducing the influence of undesired motion in the gap direction. The elements as described with respect to this exemplary embodiment may also be true for the System-2  40 , which can include the capacitive system C 41  and C 42 . The relative geometrical arrangement of the System-1  30  and System-2  40  is illustrated in exemplary  FIG. 6 . While the ground electrodes of the two systems may be in alignment, the capacitive electrodes of C 41  and C 42  can be shifted a distance of Wss with respect to the capacitive electrodes of C 31  and C 32 . The value of Wss can be equal to (Wc+Wcs)/2, whereas Wc can be the widths of the electrodes of the capacitive series and Wcs can be the space between adjacent electrodes of the capacitive series. 
         [0059]    In a further embodiment, and referring now to exemplary  FIG. 7 , the Composite Measurement System  201  can output two digital cyclic curves corresponding to the System-1  30  (displaying Curve-1 U 30 ) and System-2  40  (displaying Curve-2 U 40 ). Dead zones may still exist on each of the individual curves, however, as can be seen in exemplary  FIG. 7  at the dead zone locations of Curve-1 U 30  from System-1  30 ; Curve-2 U 40  from System-2  40  shows good or desired behavior for measurement. Similarly, at the dead zone locations of Curve-2 U 40  from the System-2  40 ; the Curve-1 U 30  from System-1  30  shows good or desired behavior for measurement. Therefore a complete measurement may be accomplished by taking good or desired results from System-1  30  and System-2  40  alternatively. In addition, the good behavior sections of both curves can extend in both ends and overlap each other. The overlapped good sections of the two curves can provide a cross-checking function of measurement and make measurements much more reliable. 
         [0060]    As an exemplary embodiment  FIG. 8  shows a Composite Measurement System  202 . In the Composite Measurement System  202 , ground electrode series  65  of the System-2  60  may be shifted a distance of Wss with respect to the ground electrode series  55  of System-1  50 , while the capacitive electrodes of the systems may be substantially aligned. The value of Wss may be equal to (Wc+Wcs)/2. The Composite Measurement System  202  can output two digital cyclic curves substantially similar to the ones in exemplary  FIG. 7 . 
         [0061]    Similar to the composite system  201 , as an exemplary embodiment,  FIG. 9  shows Composite Measurement System  203 . An exemplary difference between the Composite Measurement System  201  and  203  may be the geometrical arrangement. The System-1  78  of the composite system  203  can include: a capacitive series  71  of m elements of equally spaced electrically conductive electrodes  71 - 1 ,  71 - 2 , . . . ,  71 - m ; a capacitive series  72  of m elements of substantially equally spaced electrically conductive electrodes  72 - 1 ,  72 - 2 , . . . ,  72 - m ; and a ground series  75  of n elements of substantially equally spaced electrically conductive electrodes  75 - 1 ,  75 - 2 , . . . ,  75 - n . For the purposes of simplicity, in this exemplary embodiment, the number m of the electrodes of capacitive series  71  and  72  is 4, although it may be appreciated by one having ordinary skill in the art that it can be any positive integer. Similarly, the number n of the electrodes of the ground series  75  for the sake of simplicity during the demonstration is 2 although it can be any positive integer. The capacitive series  71  and  72  may be on the same side of an electrically-insulated substrate  701  with an electrical shielding  702  on the other side of the substrate, forming STATOR-1. The ground series  75  can be on one side of a different electrically-insulated substrate  705 , forming a part of a MOVER. The MOVER can displace in a substantially parallel manner with respect to the STATOR-1, resulting in variations of overlap areas between capacitive electrodes and ground electrodes, and furthermore, the variations of capacitances of capacitors C 71 - 1 , C 71 - 2 , C 72 - 1 , and C 72 - 2 . As described earlier with this kind of geometrical arrangement the capacitive systems together may form a differential compensation that reduces the influence of undesired motion in the gap direction. The description of this exemplary embodiment may also be true for System-2  79 , that can include capacitive system C 73  and C 74 . Capacitive series  73  and  74  may be on the same side of an electrically-insulated substrate  703  with an electrical shielding  704  on the other side of the substrate forming STATOR-2. STATOR-2 may be fixed substantially parallel to STATOR-1. The ground series  76  can be on the opposite side of the ground series  75  of the electrically-insulated substrate  705  forming another part of the MOVER. The relative geometrical arrangement of the System-1  78  and System-2  79  is illustrated in exemplary  FIG. 9 . While the electrodes  75 - 1  and  75 - 2  of the ground series  75  may be in alignment with the electrodes  76 - 1  and  76 - 2  of the ground series  76  respectively; the capacitive electrodes of series C 73  and C 74  can be shifted correspondingly a distance of Wss with respect to the capacitive electrodes of series C 71  and C 72 . The value of Wss is equal to (Wc+Wcs)/2. Composite Measurement System  203  can output two digital cyclic curves substantially similar to the ones in exemplary  FIG. 7 . 
         [0062]    Similar to the Composite Measurement System  203 , as an exemplary embodiment,  FIG. 10  shows a Composite Measurement System  204  which utilizes the features of Capacitor Sensor System  100  and adds to them the Wss shift as described above in the Composite Measurement System  202 . The value of Wss may be equal to (Wc+Wcs)/2. The composite measurement system  204  can output two digital cyclic curves substantially similar to the ones in exemplary  FIG. 7 . 
         [0063]    For the geometrical arrangements shown in exemplary FIG.  1 - a  and exemplary  FIG. 9 , the MOVER can be made simplify from a single conductive sheet of metals, for example, by a wire cutting technique instead of the ones described earlier, although it may be appreciated by a person with ordinary skill in the art that other techniques can be utilized. To illustrate the difference, exemplary  FIG. 11  can provide a comparison of these MOVERs with only two electrodes included for each of two ground series. On the left is the kind of MOVER  100  described in earlier exemplary embodiments. Two electrodes  15 - 1  and  15 - 2  of a ground series  15  may be on the top side of an electrically-insulated substrate  21 , and can be connected through wires to a ground terminal of a measurement system. Additionally it may be noted that there can exist two electrodes  16 - 1  and  16 - 2  of another ground series  16  that may be on the bottom side of the substrate  21  which are not shown in exemplary  FIG. 11 . This kind of MOVER  100  can be fabricated by a PCB technique for example, although any desired technique may be utilized. On the right side of  FIG. 11  is the kind of MOVER that can be fabricated from a single conductive sheet of metal by removing a rectangular piece with a wire cutting method or any other desired method known to a person of ordinary skill in the art. The surfaces of the single piece of the conductor can serve as multiple electrodes of ground series  15  and  16 . The electrodes may be connected to a ground terminal though other conductive components of the measurement system as desired. Such an exemplary single sheet MOVER may be made very thin, very light and very small so it can make the measurement system have a faster dynamic response behavior, also, it may be useful for mini or micro devices if desired. 
         [0064]    The Composite Measurement Systems  201 ,  202 ,  203  and  204 , as illustrated in exemplary FIGS.  6  and  8 - 10 , respectively, can provide incremental measurements for a long range measurement. A measurement system may lose track of its position each time power is turned off. When power is turned on again the measurement system may need to find or define its home or zero position. Additionally measurement can be relative to a home position as a result of the cyclic nature of the measured signal. 
         [0065]    Exemplary  FIGS. 12   a - c  can provide further embodiments of a Composite Measurement System  205  that can provide an absolute measurement, where exemplary  FIG. 12   a  is a stator, exemplary  FIG. 12   b  is a mover and exemplary  FIG. 12   c  is an assembly. Composite Measurement System  205  can be constructed with three capacitive sensor systems, System-1  101 , System-2  102 , and System-3  103 . The pair of System-1  101  and System-2  102  may be utilized for fine measurement. The System-3  103  may be utilized for a coarse measurement. The System-1  101 , System-2  102 , and System-3  103  may be substantially similar to the ones of Composite Measurement System  202  described above. For the purposes of demonstrating the method of obtaining an absolute measurement only three capacitive sensor systems are used here. Another three more capacitive sensor systems can be added to the Composite Measurement System  205  to optimally form a combination of direct compensation and differential compensation if desired. 
         [0066]    The System-1  101  of the composite system  205  can include: 1) a capacitive series  111  of m elements of substantially equally spaced electrically conductive electrodes  111 - 1 ,  111 - 2 , . . . ,  111 - m;  2) a capacitive series  112  of m elements of substantially equally spaced electrically conductive electrodes  112 - 1 ,  112 - 2 , . . . ,  112 - m ; and 3) a ground series  115  of n elements of substantially equally spaced electrically conductive electrodes  115 - 1 ,  115 - 2 , . . . ,  115 - n . For the purposes of this exemplary embodiment the number m of the electrodes of capacitive series  111  and  112  is 8, although it can be any positive integer as desired. The number n of the electrodes of the ground series  115  for demonstrative purposes is 4, although it can be any positive integer as desired. The capacitive series  111  and  112  may be on the same side of an electrically insulated substrate  120 , forming STATOR-1  107  a part of System-1. The ground series  115  can be fabricated from a single sheet of metal by wire cutting, or any other desired manner, and can form a part of the MOVER  108 . The MOVER  108  can displace in a substantially parallel manner with respect to STATOR-1  107 , which can result in variations of overlap areas between capacitive electrodes and ground electrodes, and furthermore variations of the capacitances of capacitors C 111 - 1 , C 112 - 1 , . . . , C 111 - 8  and C 112 - 8 . Further it may be noted that the description of this exemplary embodiment may also be applied for System-2  102  that can include the capacitive systems C 113  and C 114 , while the electrodes of ground series  116  are shifted a distance of Wss with respect to the electrodes of ground series  115 . The value of Wss may be equal to (Wc+Wcs)/2, where Wc is equal to the width of the electrodes of capacitive series  111 , and Wcs may be equal to the space between electrodes of the adjacent capacitive series  111  and  112 . System-1  101  and System-2  102  together can output two digital cyclic curves substantially similar to those in exemplary  FIG. 7 . 
         [0067]    The System-3  103  can be substantially similar to the capacitive sensor system  101  with the numbers of electrodes of each capacitive series and ground series equal to one. System-3  103  can include two capacitive electrodes  117  and  118  and a single ground electrode  119 . The two capacitive electrodes  117  and  118  may be on the electrically-insulated substrate  120 , forming a part of STATOR-1  107 . The ground electrode  119  may be fabricated together with electrodes of ground series  115  and  116  from a single sheet of metal by wire cutting, or in any other desired manner known by a person of ordinary skill in the art, and can form a part of a MOVER  108 . The width Wcl of the capacitive electrodes  117  and  118  and the width Wsl of ground electrode  119  may be designed as: 
         [0000]        Wcl≦Wc*m+Wcs *( m− 1)  [equation 5]
 
         [0000]        Wsl ≦( Wc+Wcs )* m   [equation 6]
 
         [0000]    where Wc is the widths of the electrodes of capacitive series  111  and Wcs is the space between electrodes of the adjacent capacitive series  111  and  112 . The System-3  103  may provide a linear curve of reading signal as a function of position. 
         [0068]    Additional examples are provided as follows to further illustrate exemplary principles and uses of the embodiments described herein. Such examples may show: 1) direct and differential compensation of capacitive systems to reduce influence of gap variation for area variation sensor, 2) multiple uses of capacitive systems to overcome dead zone problem for long range measurement; and 3) multiple uses of capacitive systems to upgrade measurement from incremental to absolute. 
         [0069]    Exemplary FIG.  13 - a  can demonstrate how a complete sensor system  300  for an absolute linear displacement measurement might be assembled. The STATOR  301  and MOVER  302  may form variable capacitors due to area variation, the system  300  can have a STATOR base  303 , a MOVER base  304 , two linear bearings  305  and  306 , two motion guiders  307  and  308 , and four guider supports  311 ,  312 ,  313 , and  314 , although it is envisioned that other embodiments may utilize different quantities of these elements. The MOVER  302  may displace with the MOVER base  304 , the two linear bearings  305  and  306 , while the STATOR  301  may be fixed on to the stator base  303 . The STATOR  301  can have three sets of differential capacitive measuring systems: System-1, System-2 and System-3. 
         [0070]    To further illustrate the electrode arrangement of the measuring systems, exemplary FIG.  13 - b  can show the top view of STATOR  301  and the bottom view of MOVER  302 . Each of the three measuring systems can be constructed with electrodes of a pair of two capacitive series and one ground series as described above. Additionally, for the System-3  103  of the coarse measurement, the ground series may be a single electrode of rectangular in shape, and each of the two capacitive series may be triangular in shape. The STATOR and MOVER with electrodes can be fabricated with a PCB technique, or any other desired manner known to a person of ordinary skill in the art. 
         [0071]    Exemplary FIG.  14 - a  and FIG.  14 - b  show another embodiment of a sensor system  400  for an absolute linear displacement measurement. Although the underlying scientific principles and methods used are the same, this system exemplifies the variety of shapes the electrodes may take. Here electrodes of both capacitive series  401  and  402  and ground series  403  have a cylindrical shape. The capacitive series  401  and  402  are on a cylindrical electrically insulated substrate  404 , forming the STATOR of the system. The capacitive series  401  and  402  can be fabricated with a flexible PCB glued to the substrate  404  or by direct electronic printing to the substrate  404 , as well as any other method known to a person of ordinary skill in the art. The ground series  403  may be fabricated from a single conductive tube of metal with a wire cutting method or any other method known to a person of ordinary skill in the art, and may be attached to a slider  409 . The ground series  403 , slider  409  and a shaft  408  on the slider form the MOVER of the system. The substantial parallelism of the motion of the ground series with respect to the capacitive series may be ensured by the linear bearing between the shaft  408  and the cylindrical surface  407  of the substrate  404 . In addition, there may be pins  405  that are able to slide in the groove  406  to ensure no rotation occurs during the parallel displacement. The capacitive series  401  and  402  can form capacitors with the ground series  403  with a differential compensation. 
         [0072]    The scientific principles used by these exemplary embodiments can be applied to measure an angular displacement. As my first demonstrative example of an angular displacement sensor, two major parts of a sensor system  500 , STATOR-1  501  and MOVER  503  are shown in exemplary FIG.  15 - a . This example may be used for an absolute angular displacement measurement with a full 360-degree measurement range. You can have a STATOR-2  502  (not shown in FIG.  15 - a ) in the system  500 , which may be similar to STATOR-1  501 . The STATOR-1  501  and STATOR-2  502  may be made of PCB with detailed patterns as illustrated in exemplary FIG.  15 - b . Each STATOR can have two circles of capacitive electrodes. STATOR-1  501  as shown in FIG.  15 - b  has as an outer circle of capacitive electrodes which can be used for fine measurement, and which may be constructed with 180 electrodes. Alternating electrodes in STATOR-1  501  may be connected together to form a capacitive series allowing for two capacitive series. This is similar to the way capacitive series  11  and  12  alternated on STATOR-1  104  in FIG.  1 - a . STATOR-1  501  as shown in FIG.  15 - b  has an inner circle which may be used for coarse measurement, and may be constructed with two electrodes. Each of the electrodes can form one capacitive series. Note the shape of all the electrodes in this exemplary embodiment is a circular section with an angular width Wc and radius length Lc, instead of rectangle. 
         [0073]    The MOVER  503  with detailed patterns as illustrated in exemplary FIG.  15 - c  can be fabricated from a single conductive sheet of metal by a wire cutting method, or any other desired method known to a person of ordinary skill in the art. The electrodes on the STATORs can form variable capacitors with the surfaces of the MOVER. To ensure the MOVER rotates in a manner that ground electrodes keep parallel to capacitive electrodes there could be other components to fix the two STATORs and support the MOVER. 
         [0074]    The main difference between STATOR-1  501  and STATOR-2  502  may be their orientations. STATOR-1  501  is face up as shown in exemplary FIG.  15 - a , while the STATOR-2  502  would be facing down if it were shown in the figure. Geometrical arrangement of the STATOR-2  502  with respect to STATOR-1  501  may be similar to the one shown in exemplary  FIG. 9  for linear displacement in that the capacitive electrodes may be displaced an angular width Wss. The value of the angle Was is equal to (Wc+Wcs)/2 with that Wc may be the angular width of the electrodes of the capacitive series from a perspective point of the center and Wcs is the angular width of the space between adjacent electrodes of the capacitive series. 
         [0075]    Exemplary FIG.  16 - a , for demonstrative purposes, shows a cross section of a complete angular displacement sensor system  600  for an absolute angular displacement measurement of a full 360-degree measurement range. In addition to the cylindrically shaped components STATOR-1  601 , STATOR-2  602 , and MOVER  603 , the system  600  can have a STATOR base  604 , a MOVER base  605 , two bearings  607  and  608 , and a bearing holder  606 . These components may be utilized to keep the rotation in a desired manner so that corresponding electrodes displace in a substantially parallel motion during rotation. The STATOR-1  601  and STATOR-2  602  may be fixed on to the STATOR base  604 . The MOVER  603  can rotate with the MOVER base  605  with respect to the STATOR-1  601  and STATOR-2  602  through the bearings  607  and  608 . 
         [0076]    To further illustrate the electrode arrangement of the measuring systems, exemplary FIG.  16 - b  shows the cylindrical STATOR-1  601 , STATOR-2  602  and MOVER  603  in an unassembled configuration. The STATOR-1  601  can have an electrically insulated cylindrical substrate and four series  611 ,  621 ,  612 , and  622  of 45 elements of equally spaced electrically conductive electrodes connected together alternatively. The shape of the capacitive electrodes may be a sector of a cylinder. The STATOR-2  602  can have an electrically insulated cylindrical substrate and three capacitive electrodes  613 ,  614  and  617 . The shape of these two electrodes  613  and  614  may be a triangular sector of a cylinder. The shape of the electrode  617  may be a sector of a cylinder. 
         [0077]    The MOVER  603  can include an electrically insulated cylindrical substrate, two ground series  615  and  616  of about 45 elements of substantially equally spaced electrical-conducting electrodes on the inner surface, and two ground electrodes  618  and  619  on the outer surface. Upon assembly, about seven variable capacitors or composite capacitors can be formed from capacitive electrodes with ground electrodes. They may be composite capacitors C 611  from the capacitive series  611  with ground series  615 ; C 621  from  621  and  615 ; C 612  from  612  with  616 ; C 622  from  622  with  616 ; capacitors C 613  from the  613  with  618 ; C 614  from  614  with  618 ; and C 617  from  617  with  619 . Differential compensations can be formed between C 611  and C 621 , and between C 612  and C 622  for fine measurements, and between C 613  and C 614  for a coarse measurement. For the relative geometrical arrangement of  611 ,  621 ,  612 , and  622  for the fine measurements, the ground electrodes may be in alignment while the capacitive electrodes are rotationally shifted an angular width Wss. The value of the angle Wss can be equal to (Wc+Wcs)/2 where Wc can be the angular width of the electrodes of capacitive series from the perspective of the center and Wcs can be the angular width between adjacent electrodes of the capacitive series from the perspective of the center. The function of the capacitor C 617  may be to define the position of zero degree. 
         [0078]    The foregoing description and accompanying figures illustrate the preferred embodiments and modes of operation of the invention. However, the invention should not be construed as being limited to the particular embodiments discussed above. Additional variations of the embodiments discussed above will be appreciated by those skilled in the art. 
         [0079]    Therefore, the above-described embodiments should be regarded as illustrative rather than restrictive. Accordingly, it should be appreciated that variations to those embodiments can be made by those skilled in the art without departing from the scope of the invention as defined by the following claims.