Patent Publication Number: US-10761041-B2

Title: Multi-parallel sensor array system

Description:
FIELD 
     The present application generally relates to a sensing system. 
     BACKGROUND 
     The statements in this section merely provide background information related to the present disclosure and may not constitute prior art. 
     Conventional sensors that measure a physical parameter based on a change in resistance, such as thermocouples or strain gauges, are used in a variety of systems. For example, a thermal system, like pedestal heaters, use thermocouples to monitor the temperature of a heater surface. However, such sensors typically require completely isolated wiring (e.g., one for power and one for return) or separate power wires for each sensor and a common wire shared by all sensors. Thus, these sensors require significant number of wires that can be challenging to integrate in a system in which space or access is limited, such as thermal systems. These and other issues are addressed by the present disclosure. 
     SUMMARY 
     This section provides a general summary of the disclosure, and is not a comprehensive disclosure of its full scope or all of its features. 
     In one form, the present application provides a sensing system. The sensing system includes a plurality of resistive elements coupled to a plurality of nodes and a control system configured to index through a plurality of modes to measure an electrical characteristic for each resistive element. Each mode of the plurality of modes represents a different combination of power, return, or open circuit condition applied to each of the plurality of nodes. The control system is configured to calculate, for each of the modes, a total power consumed by the system and a power consumed by each of the resistive elements based on the measured electrical characteristics, to determine a physical parameter. 
     In one form, the control system is configured to calculate resistance of the resistive elements based on the total power consumed, the power consumed by each of the resistive elements, and pre-stored algorithms. 
     In another form, the control system is configured to determine at least one of temperature, strain, light intensity, or gas concentration as the physical parameter of the sensing system based on the calculated resistance. 
     In yet another form, the control system is configured to determine the physical parameter based on the resistance of the resistive elements and predetermined correlation information that associates one or more resistance values to one or more values of the physical parameter. 
     In one form, the control system uses Moore-Penrose pseudoinverse to determine the resistance of the resistive elements. 
     In another form, the control system is configured to test for an open or short circuit condition. 
     In yet another form the control system is configured to compute derivative sensor information such as gradients and rates of change. 
     In yet another form, the control system is configured to determine exceeding temperature ranges of the sensing system. 
     In another form, the electrical characteristic includes voltage and current. 
     In yet another form, the control system is configured to calculate a total conductance of the plurality of resistive elements based on the total power consumed by the sensing system and the power consumed by each of the resistive elements. 
     In another form, the number of plurality of modes is greater than or equal to the number of resistive elements. 
     In yet another form, each of the resistive elements is connected between a pair of nodes from the plurality of nodes. 
     In another form, the resistive elements are comprised of an electrically conductive material with a temperature dependent electrical resistance. 
     In one form, each mode has a set of voltages that are linearly independent of each other. 
     In another form, the control system communicates at least one of the electrical characteristics or the physical parameter to an external device by way of a network controller. 
     In one form, a method for measuring temperature of a sensing system having a plurality of resistive elements coupled to a plurality of nodes is provided. The method includes indexing through a plurality of modes to measure an electrical characteristic for each resistive element. Each mode of the plurality of modes represents a different combination of power, return, or open circuit condition applied to each of the plurality of nodes. The method includes calculating, for each of the modes, a total power consumed by the sensing system and a power consumed by each of the resistive elements based on the measured electrical characteristics, to determine a physical parameter of the sensing system. 
     In another form, the method includes calculating the resistance of the resistive elements based on the total power consumed by the sensing system and the power consumed by each of the resistive elements. The method includes determining the physical parameter based on the resistance of the resistive elements and predetermined correlation information that associates one or more resistance values to one or more values of the physical parameter. 
     In yet another form, resistance of the resistive elements is calculated using Moore-Penrose pseudoinverse. 
     In another form, the physical parameter is at least one of temperature, strain, light intensity, or gas concentration. 
     In one form, the electrical characteristics include voltage and current. 
     In another form, the method includes calculating a total conductance of the plurality of thermal elements based on the total power consumed by the sensing system and the power consumed by each of the resistive elements. 
     In another form, each of the resistive elements is connected between a pair of nodes from the plurality of nodes. 
     In yet another form, the method determines exceeding temperature ranges of the sensing system. 
     In another form, the method includes testing for an open or short circuit condition. 
     In one form, the method includes computing derivative sensor information such as gradients and rates of change. 
     In another form, each mode has a set of voltages that are linearly independent of each other. 
     Further objects, features and advantages of this application will become readily apparent to persons skilled in the art after a review of the following description, with reference to the drawings and claims that are appended to and form a part of this specification. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In order that the disclosure may be well understood, there will now be described various forms thereof, given by way of example, reference being made to the accompanying drawings, in which: 
         FIG. 1  is a block diagram of a sensing system in accordance with teachings of the present disclosure; 
         FIG. 2  is a partial side view of a heater comprising a routing layer, a base heater layer, and a sensor array constructed in accordance with principles of the present disclosure; 
         FIG. 3  is a schematic of a sensing system in accordance with principles of the present disclosure; 
         FIG. 4  is a block diagram of a control system in accordance with principles of the present disclosure; 
         FIG. 5  is a network diagram of the multi-parallel sensor array of  FIG. 3 ; 
         FIG. 6  is an example schematic of a three-wire multi-parallel sensor array in accordance with the principles of the present disclosure; 
         FIG. 7  is a flowchart illustrating a method for calculating the total power for the multi-parallel sensor array in accordance with principles of the present disclosure; and 
         FIG. 8A-8C  are schematics exemplifying calculations for the three-wire multi-parallel thermal array of  FIG. 6  with varied sensing modes in accordance with the principles of the present disclosure. 
     
    
    
     The drawings described herein are for illustration purposes only and are not intended to limit the scope of the present disclosure in any way. 
     DETAILED DESCRIPTION 
     The following description is merely exemplary in nature and is not intended to limit the present disclosure, application, or uses. 
     Referring to  FIG. 1 , a sensor array system  100 , which may also be referred to as a sensing system, measures a physical parameter based on a resistance of an element whose resistance value varies with a change in the physical parameter. In one form, the sensing system  100  includes a control system  102  and a multi-parallel sensor array  104  (i.e., a sensor array) having a plurality of resistive elements (not shown). As described further herein, the control system  102  operates the sensor array  104  in accordance with one or more sensing modes in which power is applied to the sensor array  104  from a power supply  106 . The control system  102  is configured to determine a value of the physical parameter based on a resistance of the sensor array  104 . More particularly, the control system  102  calculates the resistance of the resistive elements of the sensor array  104  based on measured electrical characteristics and on the total power consumed by the sensor array  104 . Using the calculated resistance, the control system  102  determines a value of the physical parameter. 
     The sensing system  100  can be applied to a variety of systems to measure physical parameters, such as temperature, physical deformation (i.e., strain), light intensity, or gas concentration. In one example, the sensing system of the present disclosure is used to measure a temperature profile across a surface of a heater used for semiconductor processing. Such a heater system is described in pending U.S. application Ser. No. 13/598,995 filed on Aug. 30, 2012, which is commonly assigned with the present application and the disclosures of which is incorporated herein by reference in their entirety. 
     As an example,  FIG. 2 , illustrates a heater  200  for a semiconductor processor. The heater  200  includes a sensor array  202  disposed above a base heater layer  204  with a routing layer  206  disposed between the base heater layer  204  and the sensor array  202 . The sensor array  202  includes multiple resistive elements  208 , and the base heater layer  204  includes one or more heater circuits  210  that are operable to generate heat. The routing layer  206  is configured to route power lines that extend from the base heater layer  204  to the sensor array  202 . For example, the base heater layer  204  defines one or more apertures (not shown) that extend through the base heater layer  204 , and the routing layer  206  defines a cavity (not shown) that is in communication with the apertures. Power lines are routed through the apertures of the base heater layer  204  to the internal cavity of the routing layer  206 . From the internal cavity, the power lines are connected to the sensor array  202 . In one form, the sensor array  202  is used to monitor the temperature of the heater  200  using the teachings of the present disclosure. 
     The sensing system of the present disclosure can be used with other systems and should not be limited to heaters for semiconductor processing, i.e. mechanical systems. In addition, the sensor array can be used to measure other physical parameters, and should not be limited to temperature. For example, an array of strain gauges may be coupled to structural components of bridges, buildings, and other mechanical systems to take strain measurements, thereby reducing the amount of wires required for taking the measurements. 
     A sensor array of the sensing system includes a plurality of resistive elements that are coupled to a plurality of power nodes. Each node may then be coupled to a power line for receiving power, return, or being placed in an open condition. For example, referring to  FIG. 3 , a sensor array  300  includes six resistive elements  310   1  to  310   6 , which are collectively referred to as resistive elements  310 , and four power nodes  320   1  to  320   4 , which are collectively referred to as power nodes  320 . The resistive elements  310  are arranged in a multi-parallel fashion across pairs of power nodes  320 . As such, each power node  320  has one resistive element  310  connected between itself and each of the other power nodes  320 , and each resistive element  310  is connected between a pair of power nodes from the plurality of power nodes  320 . 
     Accordingly, resistive element  310   1  is connected between power nodes  320   1  and  320   2 , resistive element  310   2  is connected between power nodes  320   1  and  320   3 , resistive element  310   3  is connected between power nodes  320   1  and  320   4 , resistive element  310   4  is connected between power nodes  320   2  and  320   3 , resistive element  310   5  is connected between power nodes  320   2  and  320   4 , and resistive element  310   6  is connected between power nodes  320   3  and  320   4 . While  FIG. 3  illustrates a sensor array having six resistive elements and four power nodes, the sensor array may be configured in other suitable ways to have two or more resistive elements and two or more power nodes. 
     As discussed above, the resistive elements  310  are sensors or devices whose resistance is dependent on a physical property. For example, the resistive elements  310  are any one of resistance temperature detectors (RTDs), thermistors, strain gauges, photocells, and/or gas sensors, among others. The resistances of such devices vary due to one or more of the following physical properties: temperature; physical deformation; light intensity; and gas concentration, among others. By calculating the resistance of the resistive elements  310 , the value of the physical property may also be determined as set forth in greater detail below. 
     In one form, the system  300  further includes a plurality of switches that are operable to electrically couple the plurality of power nodes  320  to one of return (V−), power (V−), or open circuit condition. For example, in  FIG. 3 , four switches  330   1  to  330   4 , which are collectively referred to as switches  330 , are coupled to the power nodes  320 , such that each power node  320  is coupled to one switch  330  to selectively apply one of return (V−), power (V−), or open circuit condition to the power node. The switches  330  may be a circuit of discreet elements including, but not limited to, transistors, comparators and SCR&#39;s or integrated devices for example, microprocessors, field-programmable gate arrays (FPGA&#39;s), or application specific integrated circuits (ASIC&#39;s). 
     A control system  340  is configured to operate the sensor array  300 , and is implemented as a computer system. For example,  FIG. 4  illustrates the control system  340  as a computer system that includes a processor  410  for executing instructions such as those described in the routing described below. The instructions may be stored in a computer readable medium such as memory  412  or storage devices  414 , for example a disk drive, CD, or DVD. The computer may include a display controller  416  responsive to instructions to generate a textual or graphical display on a display device  418 , for example a computer monitor. In addition, the processor  410  may communicate with a network controller  420  having a data port to communicate data or instructions to other systems, for example other general computer systems. The network controller  420  may communicate over Ethernet or other known protocols to distribute processing or provide remote access to information over a variety of network topologies, including local area networks, wide area networks, the Internet, or other commonly used network topologies. 
     In one form, the control system  340  is configured to determine the physical property detected by the resistive element  310  based on the resistance of the resistive elements  310  and on predetermined information that correlates one or more resistance values with values of the physical property. As described further herein, the control system  340  determines the resistance by measuring electrical characteristics of the resistance elements  310  at different modes. That is, in one form, the control system  340  operates the switches  330  to selectively apply power to the power nodes  320 , and index through a plurality of sensing modes to measure electrical characteristics of the resistive elements  310 . Sensing modes are an application of voltages and/or current to the power nodes that result in some distribution of power through the network. The amount of power applied to the power nodes can be selected based on the system, but is generally low enough to measure voltage and/or current through the resistive elements, such as 2-5 Vs. 
     In one form. the resistance of the resistive elements  310  is determined based on the power consumed by the sensor array. By way of explanation, the sensor array of  FIG. 3  is represented as the network diagram shown in  FIG. 5  in which six resistors (i.e., g 1 , g 2 , g 3 , g 4 , g 5 , and g 6 ) are coupled to four nodes (a, b, c, d). From the network the following variables and relationships are established.
 
 i   w =[ i   a   i   b   i   c   i   d ] T   Wire currents:
 
 v   w =[ v   a   v   b   v   c   v   d ] T   Wire voltages:
 
 g =[ g   1   g   2   g   3   g   4   g   5   g   6 ] T   Conductances:
 
 i   G =[ i   1   i   2   i   3   i   4   i   5   i   6 ] T   Currents through conductances:
 
 v   G =[ v   1   v   2   v   3   v   4   v   5   v   6 ] T   Voltages across conductances:
 
     
       
         
           
             
                 
             
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     In one form, the power consumed by the entire array for any given mode is determined by Equation 1 in which the “∘” operator designates an element-by-element multiplication (i.e., a Hadamard product) and the row vector “s” is defined to be the squares of the leg voltages (i.e., s=(v g ∘v g ) T ).
 
 p=i   w   T   v   w =( v   g   ∘v   g ) T   g =(( Dv   w )∘( Dv   w )) T   g=sg   Equation 1
 
     More particularly, in one form, the total network power is determined using Equations 2 or 3 which use measured wire voltages V i  and measured wire currents I i . In Equations 2 and 3, 
                 g   ij     =     1     R     ij   ⁢                   ;         
w is the number of wires. By setting the equations to each other, as shown in Equation 4, the conductance (g) of a resistive element, and thus, the resistance (R=1/g) is determined.
 
     
       
         
           
             
               
                 
                   
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     To determine the resistance of each of the resistive elements, multiple measurements may be taken. For example, if there are “n” number of resistive elements, at least “n” number of measurements should be taken to obtain “n” number of measurements of wire voltages V i  and currents I i . These measurements are taken during the application of the sensing modes, where each sensing mode has voltages that are linearly independent of each other. In one form, the number of sensing modes is greater than or equal to the number of resistive elements. 
     Using vector-matrix notation, Equation 1 is rewritten as Equation 5 for the k&#39;th mode, and Equation 6 represents the matrix for all the mode equations. From Equation 6, the resistance of the resistive elements is calculated by solving for g using Equation 7 and taking the reciprocal of the conductance. 
                           p   k     =       s   k     ⁢     g   k         ,   where     ⁢     
     ⁢       p   k     =     ∑       V   i     ⁢     I   i           ⁢     
     ⁢     s   k     =       [             (       V   1     -     V   2       )     2             ⋮               (       V     w   -   1       -     V   w       )     2           ]     T       ⁢     
     ⁢       g   k     =     [           g   12             ⋮             g       w   -   1     ,   w             ]               Equation   ⁢           ⁢   5                   [           p   1             ⋮             p   m           ]     =         [           s   1             ⋮             s   m           ]     ⁢   g   ⁢           ⁢   or   ⁢           ⁢   p     =   Sg       ,           Equation   ⁢           ⁢   6               
where m is the number of modes
 
 g=S   −1   p,R= 1/ g   Equation 7
 
     The control system  340  is configured to index through “m” number of modes which is greater than or equal to the number of resistive elements, to obtain m measurements. When the number of modes is equal to the number of a resistive elements, or in other words, when the S matrix is square and full rank, the conductance (g) is solved using Equation 7. Alternatively, when the number of sensing modes is greater than the number of resistive elements (i.e., is not square and has full rank), then Moore-Penrose pseudoinverse is used to obtain to g=S + p. Since the pseudoinverse is equal to the inverse when S is invertible, the latter equation is used to determine g as long as S is full column rank. 
     In one form, the sensing system includes measurement noise in i w  and v w  as well as numerical errors in computing S + p. Assuming that g is constant, or approximately so, for the whole set of measurements, in one form, the noise and error be reduced by taking additional power samples and using linear regression (i.e., g=(S T S) −1 S T p) to estimate g. If g is not constant, that is, g=(f(x)) for some parameter vector x, then a numerical nonlinear least squares method is used to estimate g. 
     Once the control system  340  calculates the resistance of the resistive elements, a value of the physical parameter is determined using, for example, predetermined information, such as a look-up table, that associates resistance values to values of the physical parameters. For example, if the resistive elements are thermistors, a look up table or algorithm is used to determine the temperature associated with the calculated resistance. 
     Various configurations of the control system  340  may include determining exceeding physical parameter ranges of the system, testing for an open or short circuit condition, and/or computing derivative sensor information such as gradients and rates of change. For example, the control system  340  is configured to determine exceeding ranges of temperature, pressure, light intensity, gas concentration, etc., by comparing the measured values of the physical parameter to predetermined limit values. In the event that the measured values are greater than the predetermined limit value, the control system  340  can be configured to issue an alert notification. 
     The control system  340  can be configured in various suitable ways to test for an open or short circuit conditions. For example, in one form, an open circuit condition is detected when the measured resistances are at high orders of magnitude and are approaching infinity. Short circuit conditions on the other hand is detected when the power node currents exceed predetermined values. 
     As mentioned above, the control system  340  may also be configured to compute derivative sensor information such as gradients. For example, in one form, the control system  340  computes gradients by taking a consecutive series of at least two of the measured resistances and applying gradient computational algorithms such as finite difference, exact curve fit, and/or least-squares curve fit, just to name a few, and comparing them with the derivative. 
     The control system  340  of the sensor array may be configured in various suitable ways to index through multiple sense modes to calculate the resistance of the resistive elements. An example operation of the system having the multi-parallel sensor array is described with reference to  FIGS. 6, 7, and 8A to 8C . 
       FIG. 6 , illustrates a multi-parallel sensor array  600  having three resistive elements  610   1 ,  610   2 , and  610   3 , which are collectively referred to as resistive elements  610 , and three power nodes  620   1 ,  620   2 , and  620   3 , which are collectively referred to as power nodes  620 . Like the multi-sensor array described above, each resistive element  610  is coupled to a pair of nodes  620 , and each power node  620  is operable to apply power, return, or set in an open circuit condition by way of, for example, a switch  630  (i.e., switches  630   1 ,  630   2 , and  630   3  in the figure). In the following, resistive elements  610   1 ,  610   2 , and  610   3  may also be referenced as R 12 , R 23 , and R 13 , respectively, where the numbers identify the power nodes to which the respective resistive element is connected between. 
     In one form, a control system, which is similar to control system  340 , operates the switches  630  based on a plurality of sensing modes. For example, the control system is configured to operate the sensor array  600  based on three sensing modes (K 1 , K 2 , and K 3 ), which are defined in Table 1 below, for determining the resistances of the three resistive elements  610 . In Table 1, power nodes  620   1 ,  620   2 , and  620   3  are represented by PN 1 , PN 2 , and PN 3 , respectively. The values 0 and 1 represent return and power, respectively, and for each sensing mode, a different combination of power and return is applied to the power nodes  620 . In another form, the control system is configured to apply more than three sensing modes that include different combinations of power, return, and/or open circuit condition, and should not be limited to the three sensing modes provided below. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Sensing Modes 
               
            
           
           
               
               
            
               
                   
                 Node Operation 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Modes 
                 PN1 
                 PN2 
                 PN3 
               
               
                   
                   
               
               
                   
                 K1 
                 0 
                 0 
                 1 
               
               
                   
                 K2 
                 0 
                 1 
                 0 
               
               
                   
                 K3 
                 1 
                 0 
                 0 
               
               
                   
                   
               
            
           
         
       
     
       FIG. 7  illustrates an example sensing routine  700  to be performed by the control system to calculate the resistance of each resistive element  610  based on electrical characteristics measured during the three sensing modes. More particularly, to demonstrate the resistance measurement feature of the sensing system, for the following example, it is assumed that the resistances of the resistive elements  610   1 ,  610   2 , and  610   3  are 1Ω, 3Ω, and 2Ω, respectively. 
     At  710 , the control system sets the kth mode to 1 and applies mode K 1  to the sensor array  600 . Accordingly, the nodes PN 1  and PN 2  are coupled to return, and PN 3  is coupled to power. For brevity, power is provided as 1V. 
     In operation, at  720 , the control system measures and stores the electrical characteristics of the sensor array  600  for the applied mode. For example, the control system measures the electric current flowing through each of the power nodes  620   1 ,  620   2 ,  620   3  as i 1 , i 2 , and i 3 , respectively, and the voltage applied to the nodes. Here, for explanation purposes only, the electric current through the power nodes  620  are calculated using the known resistance values of the resistive elements  610  and the voltages applied to the nodes  620 . For example, the current through node  620   1  is i 1 = 
                       v   1     -     v   2         R   12       +         v   1     -     v   3         R   13         =       -   0.500     ⁢   A       ,         
in which R 12  and R 13  are the resistance of resistive elements  610   1  and  610   3 , respectively. Using similar calculations, the current through power nodes  620   2  and  620   3  is determined as i 2 =−0.333 A and i 3 =0.833 A, respectively.
 
     Referring to  FIG. 7 , at  730  the control system increments k, and determines if k is greater than the total number of modes (i.e., k&gt;m), at  740 . That is, the control system determines whether the sensor array has been indexed through all the modes. If k is less than the total number modes, the control system applies mode k to the sensor array at  750  and returns to  720  to measure the electrical characteristics. As it relates to the sensor array  600 , from mode K 1 , the control system applies modes K 2  and K 3 , and measures and stores the electrical characteristics of the sensor array  600 . Table 2 below summarizes the current through each power node for each of the modes. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Electric Current 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Modes 
                 i 1   
                 i 2   
                 i 3   
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 K1 
                 −0.500 
                 −0.333 
                 0.833 
               
               
                   
                 K2 
                 −1.000 
                 1.333 
                 −0.333 
               
               
                   
                 K3 
                 1.500 
                 −1.000 
                 −0.500 
               
               
                   
                   
               
            
           
         
       
     
     When the control system has indexed through all of the modes, the control system, at  760 , calculates the total power generated by the sensor array  600  for each of the modes K 1 , K 2 , and K 3  using Equation 2. For example, for mode K 1  the total power is p 1 =i 1 v 1 +i 2  v 2 +i 3  v 3 =0.833 W. Similarly, the total power for modes K 2  and K 3  are equal to p 2 =1.333 W and p 3 =1.500 W. Here, since the resistance of the resistive elements  610  are known, the total power can be verified by taking the sum of the power applied to each resistive element  610  during each mode. For example,  FIGS. 8A, 8B, and 8C  illustrate the power applied to each resistive element  610  for modes K 1 , K 2 , and K 3 , respectively. As illustrated, the total power for mode K 1  is p 1 =p R12 +p R13 +p R23 =0.000+0.500+0.333=0.833 W, which is the same as p 1 =i 1 v 1 +i 2  v 2 +i 3  v 3 =0.833 W. Accordingly, the total power (p) for modes K 1 , K 2 , and K 3  are p 1 , p 2 , and p 3  and represented in the following matrix. 
     
       
         
           
             p 
             = 
             
               
                 [ 
                 
                   
                     
                       
                         p 
                         1 
                       
                     
                   
                   
                     
                       
                         p 
                         2 
                       
                     
                   
                   
                     
                       
                         p 
                         3 
                       
                     
                   
                 
                 ] 
               
               = 
               
                 [ 
                 
                   
                     
                       0.833 
                     
                   
                   
                     
                       1.333 
                     
                   
                   
                     
                       1.500 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     From  760 , the control system solves for conductance using Equations 6 and 7, at  770 . That is, the control system calculates conductance of the resistive elements based on the power determined and the voltages applied to the resistive elements for each mode. For example, with respect to sensor array  600 , for each mode, s i =[v 12     i     2  v 13     i     2  v 23     i     2 ], and the S matrix encompassing all of the modes is a full-square matrix and is provided below. Furthermore, to determine the conductance as provided in Equation 7, the inverse of the S matrix is determined and multiplied by the power matrix (p) resulting in the conductance for each resistive element  610 . 
     
       
         
           
             S 
             = 
             
               [ 
               
                 
                   
                     0 
                   
                   
                     1 
                   
                   
                     1 
                   
                 
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
                 
                   
                     1 
                   
                   
                     1 
                   
                   
                     0 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               S 
               
                 - 
                 1 
               
             
             = 
             
               0.5 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         - 
                         1 
                       
                     
                     
                       1 
                     
                     
                       1 
                     
                   
                   
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       1 
                     
                   
                   
                     
                       1 
                     
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             g 
             = 
             
               
                 
                   S 
                   
                     - 
                     1 
                   
                 
                 ⁢ 
                 p 
               
               = 
               
                 
                   
                     0.5 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               - 
                               1 
                             
                           
                           
                             1 
                           
                           
                             1 
                           
                         
                         
                           
                             1 
                           
                           
                             
                               - 
                               1 
                             
                           
                           
                             1 
                           
                         
                         
                           
                             1 
                           
                           
                             1 
                           
                           
                             
                               - 
                               1 
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           0.833 
                         
                       
                       
                         
                           1.333 
                         
                       
                       
                         
                           1.500 
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   [ 
                   
                     
                       
                         1.000 
                       
                     
                     
                       
                         0.500 
                       
                     
                     
                       
                         0.333 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             R 
             = 
             
               
                 
                   1 
                   / 
                   g 
                 
                 -&gt; 
                 
                   [ 
                   
                     
                       
                         
                           R 
                           12 
                         
                       
                     
                     
                       
                         
                           R 
                           13 
                         
                       
                     
                     
                       
                         
                           R 
                           23 
                         
                       
                     
                   
                   ] 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                   
                   
                     
                       2 
                     
                   
                   
                     
                       3 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Based on the above, the resistance of the resistive elements  610   1  (R 12 ),  610   2  (R 23 ), and  610   3  (R 13 ) are calculated to be 1Ω, 3Ω, and 2Ω. Accordingly, as demonstrated herein, by operating the sensor array  600  in accordance with the three sensing modes provided in Table 1, the resistances of resistive elements  610  are calculated based on the electrical characteristics taken during those modes. During operation, the control system is configured to measure the electrical characteristics (i.e., measure the current and voltage applied to each node for each of the modes). This data is then used to determine the total power consumed and then the resistance using the algorithms described herein. 
     With continuing reference to  FIG. 7 , using the resistance, the control system at  780 , determines the physical parameter detectable by the resistive element  610  using predetermined correlation information, which may include but is not limited to algorithms and/or look-up tables. 
     The sensing system of the present disclosure is configured to measure temperature at multiple regions with a reduced number of wires to connect the sensor array to power. Specifically, each resistive element is a sensor for measuring a physical parameter, and with the multi-parallel configuration a sensor array having, for example, six sensors requires four wires. Conversely, conventional systems in which the sensors share a common node, still require 7 wires. Furthermore, the physical parameter is determined based on a calculated resistance, which is further based on the power of the system. 
     In accordance with teachings of the present disclosure, the methods described herein may be implemented by software programs executable by a computer system. Further, in an exemplary, non-limited embodiment, implementations can include distributed processing, component/object distributed processing, and parallel processing. Alternatively, virtual computer system processing can be constructed to implement one or more of the methods or functionality as described herein. 
     Further, the methods described herein may be embodied in a computer-readable medium. The term “computer-readable medium” includes a single medium or multiple media, such as a centralized or distributed database, and/or associated caches and servers that store one or more sets of instructions. The term “computer-readable medium” shall also include any medium that is capable of storing, encoding or carrying a set of instructions for execution by a processor or that cause a computer system to perform any one or more of the methods or operations disclosed herein. 
     In other embodiments, dedicated hardware implementations, such as application specific integrated circuits, programmable logic arrays and other hardware devices, can be constructed to implement one or more of the methods described herein. Applications that may include the apparatus and systems of various embodiments can broadly include a variety of electronic and computer systems. One or more embodiments described herein may implement functions using two or more specific interconnected hardware modules or devices with related control and data signals that can be communicated between and through the modules, or as portions of an application-specific integrated circuit. Accordingly, the present system encompasses software, firmware, and hardware implementations. 
     Further, it is noted that any of the topologies described may be used with any of the processing methods. Additionally, any the features described with respect to one topology or method may be used with the other topologies or methods. 
     In accordance with teachings of the present disclosure, the methods described herein may be implemented by software programs executable by a computer system. Further, in an exemplary, non-limited embodiment, implementations can include distributed processing, component/object distributed processing, and parallel processing. Alternatively, virtual computer system processing can be constructed to implement one or more of the methods or functionality as described herein. 
     Further, the methods described herein may be embodied in a computer-readable medium. The term “computer-readable medium” includes a single medium or multiple media, such as a centralized or distributed database, and/or associated caches and servers that store one or more sets of instructions. The term “computer-readable medium” shall also include any medium that is capable of storing, encoding or carrying a set of instructions for execution by a processor or that cause a computer system to perform any one or more of the methods or operations disclosed herein. 
     As a person skilled in the art will readily appreciate, the above description is meant as an illustration of the principles of the invention. This description is not intended to limit the scope or application of the invention in that the invention is susceptible to modification, variation and change, without departing from spirit of the invention, as defined in the following claims.