Abstract:
Disclosed herein is a novel sensing technique, termed Multiple Excitation Capacitance Polling (MECaP), that improves the efficiency of Electrical Capacitance Tomography (ECT). Unlike traditional alternating current techniques, where excitation signal is applied to an electrode one at a time, MECaP involves simultaneously applying multiple excitation signals, in a progressively increasing fashion, to multiple electrodes on an ECT sensor. The received signals are filtered or otherwise decomposed (e.g., Fourier transformed) into different components, and the individual components are used to generate an image of the article or substance disposed between the electrodes. Because multiple capacitances can be simultaneously measured as a consequence, scanning with MECaP can significantly increase the image scanning speed. For example, scanning with MECaP may enable frames rates of tens of kHz for imaging dynamic processes such as engine combustion.

Description:
RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Application No. 61/221,806, filed on Jun. 30, 2009, the entire teachings of which are incorporated herein by reference. 
    
    
     BACKGROUND 
     Electrical capacitance tomography (ECT) is a technique to determine the dielectric permittivity distribution in the interior of an object from external capacitance measurements. ECT&#39;s advantages over conventional tomographic techniques include non-intrusiveness, non-destruction, no radiation hazard, and low cost. In industry, ECT has found many applications, such as measurement of gas/liquid and gas/solid flows in pipelines; analysis of dynamic processes in fluidized beds; monitoring of mixing and separation processes; visualization of combustion flames in engine cylinders; and detection of leakage from buried water pipes. 
     ECT enables insight into the material distribution within a closed vessel, and consequently, into the governing mechanism in processes occurring within the vessel, without disturbing the processes themselves. Research on ECT encompasses sensor design, capacitance measuring circuit design, and image reconstruction. While sensor and circuit design involve hardware, image reconstruction typically involves software. 
       FIG. 1  shows a typical ECT system  100  that includes the following components: (1) an ECT sensor  110  that registers physical responses from the object being measured; (2) excitation and measurement circuitry  120  that drives the sensor  110  and conditions the received signals; and (3) a computer-based data acquisition and channel switching system  130  to provide control signals  134  for the sequential excitation of the electrodes. The computer-based system  130  can also reconstruct tomographic images of the object based on the measured data  132 . 
     As shown in  FIG. 1 , the ECT sensor  110  consists of a total of M electrodes  111   a - 111   m  that are mounted symmetrically outside of a cylindrical container  114 . Traditionally, during each scanning frame, an excitation signal in the form of an alternating current (AC) voltage is applied to one of the electrodes, e.g., electrode  111   a , and the remaining electrodes  111   b - 111   m  are kept at ground potential to act as detector electrodes. Subsequently, the voltage potential at each of the remaining electrodes  111   b - 111   m  is measured, one at a time, by the measurement electronics  120  to determine the inter-electrode capacitance. Specifically, in the first round of the measurements, inter-electrode capacitance is measured between electrodes  111   a  and  111   b . Next, capacitance is measured between electrodes  111   a  and  111   c , then between electrode  111   a  and  111   d , and so on. In the second round of measurements, the capacitances is measured first between electrodes  111   b  and  111   c , then between electrodes  111   b  and  111   d , and so on. Measurement cycles continue with each individual electrode assuming the role of an excitation source and a detector, alternately, until all the inter-electrode capacitances are measured. 
     The measured capacitances can be then represented in a matrix and used to reconstruct the tomographic image of the object within the cylindrical container. Analytically, with M electrodes  111   a - 111   m  in the ECT sensor  110 , a total number of H independent capacitance measurements take place, where H=M(M−1)/2. In ECT systems reported in the literature, the number of electrodes M was typically chosen to be 6, 8, 12, 16, and 32. Thus, the number of independent capacitance measurements H was 15, 28, 66, 120, and 496, respectively. 
     The number of capacitance measurements to be performed by the ECT sensor is a parameter that affects the quality of the reconstructed image. Increasing the number of electrodes improves the resolution of the reconstructed image. For a given measurement area, however, increasing the number of electrodes also causes each electrode to have a smaller surface, thus decreasing the magnitude of the inter-electrode capacitance, which, in turn, leads to a lower signal-to-noise ratio (SNR) given fixed background noise. 
     One solution to this problem is the grouping technique, which involves combining two or more electrodes into one segment to increase the magnitude of the received signal, as shown in  FIG. 2 . However, excitation of the electrodes within each segment is still performed in steps of one electrode at a time only. This grouping configuration is repeated along the sensor circumference by shifting the connection by one electrode in each measurement to form a series of independent measurements. 
       FIG. 2  shows an eight-electrode sensor  110  operated in the four-segment mode. The total surface area covered by electrodes  111   a - 111   m  equals that of a four-electrode ECT sensor without segmentation, so the SNR remains the same as that of the four-segment sensor. After measuring the first six capacitance values for a first configuration  202 , the electrodes  111   a - 111   m  are re-connected in a second configuration  204 . This can be viewed as a rotation of the four electrodes by θ=45°. As a result, the total number of independent measurements performed by the ECT electronics in each scanning frame is twice as high (twelve vs. six) compared to a conventional four-electrode sensor, thus achieving higher resolution for better image reconstruction. 
     Scanning speed is an important parameter of an ECT system, as it determines the usability of ECT for on-line, real-time applications involving fast changing dynamics, such as combustion or explosion within an enclosure. Currently, the maximum scanning speed achieved by ECT is about 300-1,000 frames/second. While satisfactory for general applications, this speed is far lower than what can be achieved with optical methods, although optical methods need direct access to the process to be monitored and thus are subject to various constraints (e.g., blockage of the line of sight) in real-world environments. For engine combustion process imaging, which is not easily monitored using optical methods, it is desirable to resolve the process at every crank angle (i.e., in 1° increments) at crank rotational speeds of up to 6,000 rev/min. This requires an ECT system that can collect data at a speed of up to 36,000 frames/second. 
     SUMMARY 
     Embodiments include an apparatus, corresponding method, and corresponding non-transitory computer-readable medium with code for electronic capacitance tomography that includes measuring multiple excitation signals simultaneously. Source and receiver electrodes capacitively coupled together measure a spatial distribution of a substance within capacitive distance of at least one pair of the source and receiver electrodes. An excitation module coupled to the source electrodes excites simultaneously each source electrode with a different excitation signal. A measurement module coupled to the receiver electrodes simultaneously measures capacitance between pairs of source and receiver electrodes. The measured capacitances represent the spatial distribution of the substance. 
     In some embodiments, the different excitation signals are at different frequencies. Alternatively, the different excitation signals may be modulated with different codes. In any case, each receiver electrode may receive simultaneously excitation signals from more than one source electrode. The measurement module may be configured to measure capacitances between each receiver electrode and every source electrode simultaneously. 
     Further embodiments may include a discrimination module coupled to the measurement module. The discrimination module may be configured to discriminate among signals received simultaneously at a given receiver electrode from more than one source electrode, possibly with a filter bank configured to filter excitation signals at different frequencies. Alternatively, the discrimination module may decompose (e.g., Fourier transform) the signals detected by the receiver electrodes to measure the capacitances. 
     Still other embodiments may include an imaging module coupled to the measurement module. The imaging module may be configured to convert the measured capacitances to an indication, such as a 3D rendering, of the spatial distribution of the substance. Alternatively, a storage module coupled to the measurement module may store the measured capacitances, which can be converted into indications of the spatial distribution of the substance with a postprocessing module. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing will be apparent from the following more particular description of example embodiments of the invention, as illustrated in the accompanying drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating embodiments of the present invention. 
         FIG. 1  is a schematic illustration of a conventional electronic capacitance tomography (ECT) system. 
         FIG. 2  is a schematic illustration of a conventional ECT system that uses grouping techniques to improve resolution without sacrificing signal-to-noise ratio. 
         FIG. 3  is a schematic illustration of a multiple excitation capacitance polling (MECaP) ECT system according to embodiments of the present invention. 
         FIGS. 4A and 4B  show schematic illustrations of a circuit model and a finite element model, respectively, for evaluating an MECaP ECT system. 
         FIG. 5  shows a circuit diagram for an equivalent circuit for measuring C 12 . 
         FIGS. 6A and 6B  are plots of output current and signal magnitude, respectively, for measurement of C 12 . 
         FIGS. 7A and 7B  are a schematic diagram and a circuit diagram, respectively, of an MECaP evaluation model for measuring C 13  and C 23 . 
         FIGS. 8A and 8B  are plots of output current and signal magnitude, respectively, for measurement of C 13  and C 23 . 
         FIG. 9  is a circuit diagram for an equivalent circuit for measuring an arbitrary number of capacitances. 
         FIG. 10  is a bar graph showing scanning speed with MECaP techniques for different numbers of simultaneous excitations. 
         FIG. 11  is a plot of calculation time (circles) and scanning period (triangles) using MECaP techniques as a function of the number of simultaneous excitations. 
     
    
    
     DETAILED DESCRIPTION 
     A description of example embodiments of the invention follows. 
     Traditional alternating current (AC) electronic capacitance tomography (ECT) involves applying an excitation signal at each time instance to one of the M electrodes of an ECT sensor and measuring the voltage at the other electrodes to determine the inter-electrode capacitance. Such a timely sequential, single-electrode excitation method is time consuming, especially when M is relatively large. As an alternative, embodiments of the present inventive methods apply multiple excitation signals to multiple electrodes at each time instance, thus enabling simultaneous measurement of more than one inter-electrode capacitance, thereby improving the efficiency of AC-based ECT measurement systems. The present inventive method, which is known as Multiple Excitation Capacitance Polling (MECaP), involves: 
     (1) simultaneously applying N AC excitation signals, each of which is at a different frequency, e.g., f 1 , f 2 , . . . f N , to N respective electrodes in an ECT sensor with M electrodes (N≦M−1); 
     (2) detecting the signals from each of the N excited electrodes with one of the remaining electrodes (M−N); 
     (3) filtering the received signals with a bank of N bandpass filters whose central frequencies are tuned to f 1 , f 2 , . . . f N , respectively; and 
     (4) calculating the individual inter-electrode capacitance from either the ratio of the excitation signal magnitude to that of the received signal or phase changes in the received signal. 
       FIG. 3  shows an MECaP ECT system  300  capable of making multiple simultaneous inter-electrode capacitance measurements according to the present inventive techniques. As shown in  FIG. 3 , the system  300  includes a sensor  310  with eight electrodes  311   a - 311   h  (of course, other sensors may include any suitable number and configuration of electrodes). Oscillators  340  apply excitation signals to a subset of the electrodes  311   a - 311   h ; here, each oscillator  340  applies a respective signal to each of electrodes  311   a - 311   e , where the respective signals are at different frequencies, i.e., f 1 , f 2 , . . . f N . Electrode  311   g  detects the signals from electrodes  311   a - 311   e  and couples them into bandpass filters  350 , which transmit signals near their respective center frequencies, f 1 , f 2 , . . . f N , to amplifiers  360 . The amplified, filtered signals can then be used to compute capacitances between the various pairwise combinations of electrode  311   g  and each of electrodes  311   a - 311   e.    
     For eight electrodes, as shown in  FIG. 3 , there are a total of M(M−1)/2=(8×7)/2=28 capacitance measurements would need to be performed within each scanning frame using the traditional AC techniques. In comparison, MECaP techniques involve only seven measurements as specified in Table 1. Because the delay caused by the additional bandpass filtering electronics will be on the order of nanoseconds, which is far less than the period of each measurement (typically 10 μs), the MECaP method enables a scanning speed increase of approximately three times relative to the traditional method. Other embodiments may enable even larger increases in scanning speed. 
     Table 1 shows that the excitation signals are applied to the electrodes in an increasing order instead of all at once. If all the excitation signals are applied at the same time, then the equations may become unsolvable. This progressive or successive excitation, which is more than just “parallel” vs. “serial” excitation, distinguishes MECaP from other ECT techniques. 
     
       
         
               
             
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Measurement sequence for an eight-electrode sensor using the 
               
               
                 MECaP method. 
               
             
          
           
               
                 Measurement 
                 Simultaneously Measured 
               
               
                 Sequence (n) 
                 Capacitance 
               
               
                   
               
               
                 1 
                 C 12   
               
               
                 2 
                 C 13 , C 23   
               
               
                 3 
                 C 14 , C 24 , C 34 , 
               
               
                 4 
                 C 15 , C 25 , C 35 , C 45 , 
               
               
                 5 
                 C 16 , C 26 , C 36 , C 46 , C 56   
               
               
                 6 
                 C 17 , C 27 , C 37 , C 47 , C 57 , C 67   
               
               
                 7 
                 C 18 , C 28 , C 38 , C 48 , C 58 , C 68 , C 78   
               
               
                   
               
             
          
         
       
     
     Table 2 shows a comparison among traditional AC techniques, grouping techniques, and MECaP techniques with and without grouping. When grouping is not used, MECaP can reduce the total number of measurements to be performed in each scanning frame by (M−1)(M−2)/2. If the electrodes are grouped into K sections, MECaP techniques may reduce the number of measurements by (M/K)(K−1)(K−2)/2. For the traditional and grouping cases, the reduction in the number of measurements is proportional to the square of the parameter M or K. As a result, increasing the number of electrodes and/or sections in the sensor significantly increases the scanning speed. 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Numbers of Measurements for Different ECT Techniques 
               
             
          
           
               
                   
                 No. of Meas. Per 
                   
               
               
                 ECT Techniques 
                 Frame 
                 Reduction 
               
               
                   
               
               
                 Traditional method 
                 M(M − 1)/2 
                 (M − 1)(M − 2)/2 
               
               
                 (no grouping) 
               
               
                 MECaP (no grouping) 
                 (M − 1) 
               
               
                 Traditional method 
                 (M/K)K(K − 1)/2 
                 (M/K)(K − 1)(K − 2)/2 
               
               
                 (grouped into K sections) 
               
               
                 MECaP (grouped 
                 (M/K)(K − 1) 
               
               
                 into K sections) 
               
               
                   
               
             
          
         
       
     
       FIG. 4A  shows a circuit model  400  used to evaluate and simulate the electrode excitation process with MECaP techniques. In the example shown in  FIG. 4A , a measurement object is modeled as a 100 mm diameter, 100 mm thick cylinder  401 . Four electrodes  411   a - 411   d  are attached to the outer surface of the cylinder  401 . A synthesizer  430 , which is electrically coupled between ground and a first resistor  432 , generates a sinusoidal excitation u 1  at a frequency of 1 MHz. Applying u 1  to the first electrode  411   a  and measuring the resulting current across a second resistor  434  coupled to the second electrode  411   b  makes it possible to measure the capacitance C 12  between the first and second electrodes  411   a ,  411   b.    
       FIG. 4B  shows a meshed finite-element model  450  used with the circuit model  400  of  FIG. 4A  to evaluate and simulate the electrode excitation process with MECaP techniques. Using finite-element analysis, the cylinder  401  of  FIG. 4A  is decomposed into numerous finite elements  451  arrayed on a cylindrical lattice. The capacitance between any given pair of elements  451  can be simulated by modeling a second electrical connection between the pair. In  FIG. 4B , for example, the second electrical connection is shown as a first resistor  482  in series with a voltage source  480  and a second resistor  484 . The behavior of the MECaP techniques can be simulated by evaluating the finite-element model  450  using standard numerical methods. 
       FIG. 5  shows an equivalent circuit  500  that can be used to measure C 12 , the capacitance between electrodes  411   a  and  411   b  shown in  FIG. 4B . The stray capacitance C s1  and C s2  are assumed to be zero for the purpose of simplification. In the simulation, the current, i, going through the second resistor  434  is measured and used with other parameters to calculate the capacitance values between pairs of electrodes. 
     For the purposes of non-limiting example, the CMOS switch resistance is taken to be R on =35-60Ω; the excitation signal frequencies are f 1 =1 MHz, f 2 =2 MHz, . . . , f n =N MHz; and the excitation signal amplitude is taken to be 15 V. Further details for measuring the relevant capacitance values are presented below. 
     Measurement of C 12    
     During the first measurement, shown in  FIG. 4 , the first electrode  411   a  is excited by a sinusoidal wave u 1  with an amplitude of 15 V and a frequency of f 1 =1 MHz. Based on the Kirchhoff&#39;s law, the following relationship can be established between the capacitance, C 12 , the resistors, and the excitation signal: 
                     u   1     =     i   ·     (       1       j   ·   2     ⁢           ⁢   π   ⁢           ⁢     f   1     ⁢     C   12         +     3   ⁢           ⁢     R   on         )               (   1   )               
Rearrangement yields:
 
                     C   12     =     1     2   ⁢           ⁢   π   ⁢           ⁢     f   1     ⁢         1     a   2       -       (     3   ⁢     R   on       )     2                     (   2   )               
where a=|A(i)/A(u 1 )| is the ratio of the current amplitude to the voltage amplitude and A(u 1 )=15 V.
 
       FIGS. 6A and 6B  are plots of the output current and signal magnitude, respectively, going through the second resistor  434 . As shown in  FIG. 6B , the peak amplitude of the output current, A(i), is calculated as 8.714 mA. In practice, the peak value of the current can be measured by using a demodulator circuit. With the value of the second resistor  434  set to R on =60Ω, the capacitance C 12  is 92 pF. 
     Measurement of C 13  and C 23    
       FIG. 7A  is a schematic diagram of simultaneous measurement of capacitance between two different pairs of electrodes according to MECaP techniques. A first synthesizer  430  applies a first sinusoidal excitation signal u 1  at a frequency of f 1 =1 MHz to the first electrode  411   a . Simultaneously, a second synthesizer  440  applies a second sinusoidal excitation signal u 2  at a frequency of f 2 =2 MHz to the second electrode  411   b  via a resistor  442 . The third electrode  411   c  is used to collect the received signal. 
     Referring to the equivalent circuit shown in  FIG. 5 , the current going through the third electrode  411   c  can be measured by using the demodulator circuit referred to above. Thus, to determine the capacitance C 13  and C 23 , relationships between the excitation signals, u 1  and u 2 , and the output signal, i 3 , need to be established. 
       FIG. 7B  is a diagram of a circuit  700  that can be used to establish a relationship among the excitation signals, u 1  and u 2 , and the output signal, i 3 . When multiple voltage sources are present in a linear circuit (as is the case in the system of  FIG. 7A ), each of them can be analyzed independently by assuming the others to be held at zero potential. Thus, to analyze the effect of the input voltage u i  on the circuit, excitation signal  112  is first considered to be zero. Derivation yields the following relationship between the port currents, i 1  and i 2 , and the input voltage, u 1 : 
                   {             i   1     ⁢   2   ⁢     R   on             -       i   2     ⁡     (       2   ⁢     R   on       +     Z   12       )                           =     u   1                   i   1     ⁢   2   ⁢     R   on           +         +       i   3     ⁡     (       R   on     +     Z   13       )               =     u   1                 i   1           +     i   2             -     i   3             =   0                   (   3   )               
where Z 12 =1/(jωC 12 ) and Z 13 =1/(jωC 13 ). Since the capacitances to be measured are in the picofarad to femtofarad range, the corresponding impedance of these capacitances may be much larger than that of the CMOS switch resistance R on . For instance, for a capacitance of 100 pF, the impedance at 1 MHz is calculated as |Z C |=1,592Ω&gt;&gt;R on =60Ω. The impedance |Z C | may increase further if the capacitance decreases to the femtofarad level. Thus, the current flowing through the capacitances C 12  and C 23  is neglected in Eq. (3). This simplification is applied to the following analysis.
 
     The relationship between the output current, i 3 , and the first input voltage, u 1 , can be expressed as: 
                     a   13     =              H   13     ⁡     (   jω   )            ⁢     
     ⁢           =                i   3     ⁡     (   jω   )           u   1     ⁡     (   jω   )              ⁢     
     ⁢           =          1       2   ⁢     R   on       +       (       R   on     +     Z   13       )     ⁢       (       4   ⁢     R   on       +     Z   12       )     /     (       2   ⁢     R   on       +     Z   12       )                              (   4   )               
Similarly, the relationship between the output current, i 3 , and the second input voltage, u 2 , can be expressed as:
 
                     a   23     =              H   23     ⁡     (   jω   )            ⁢     
     ⁢           =                i   3     ⁡     (   jω   )           u   2     ⁡     (   jω   )              ⁢     
     ⁢           =          1       2   ⁢     R   on       +       (       R   on     +     Z   23       )     ⁢       (       4   ⁢     R   on       +     Z   12       )     /     (       2   ⁢     R   on       +     Z   12       )                              (   5   )               
Since the value of C 12  is known from the first measurement, the values for capacitances C 13  and C 23  can be found by substituting the values of |H 13 | and |H 23 | into Eqs. (4) and (5), respectively.
 
       FIGS. 8A and 8B  are plots of the output current and signal magnitude, respectively, going through the third resistor  436 . The signal magnitude may be obtained by Fourier transforming the output current. As shown in  FIG. 8B , there are two peaks in the output current: a first peak of 3.973 mA at 1 MHz corresponding to C 13  and a second peak of 6.326 mA at 2 MHz corresponding to C 23 . These currents can be used to calculate the capacitance values as C 13 =92 pF and C 23 =75 pF. These are consistent with results obtained from measurement using a signal excitation. 
     Measurement of C 14 , C 24 , and C 34    
     The above described measurement may be continued by connecting electrodes  411   a ,  411   b , and  411   c  to three excitation sources (not shown) simultaneously. The ratio of the output current to the input voltage, for each input signal, is then determined as above. Since the capacitance C 12 , C 13 , and C 23  are all known in the first two measurements, the capacitance between each of the excited electrodes  411   a - 411   c  and electrode  411   d  are the only unknown parameters in the above equations. They can be found by establishing the relationship between the three excitation voltages and the three filtered currents. For example, by considering that three AC voltages u 1 , u 2 , and u 3  are used to excite electrodes  411   a ,  411   b , and  411   c , respectively, the relationship between u 1  and the current going through electrode  411   d  can be established as: 
                   {             i   1     ⁢   2   ⁢     R   on             -       i   2     ⁡     (       2   ⁢     R   on       +     Z   12       )                                       =     u   1                   i   1     ⁢   2   ⁢     R     on   ⁢                                   -       i   3     ⁡     (       2   ⁢     R   on       +     Z   13       )                           =     u   1                   i   1     ⁢   2   ⁢     R   on                                     +       i   4     ⁡     (       2   ⁢     R   on       +     Z   14       )               =     u   1                 i   1           +     i   2             +     i   3             -     i   4             =   0                   (   6   )               
where Z 14 =1/jωC 14  is the impedance of C 14 . The equations can be solved in a similar fashion as described with respect to Eqs. (4) and (5) to derive an expression for the capacitance C 14  as a function of the amplitude of i 4 . Consequently, by measuring the amplitude of i 4  at the frequency of u 1 , the capacitance C 14  can be determined. The same method can be applied to the determination of C 24  and C 34 . As a result, a total of six capacitance values (C 12 , C 13 , C 23 , C 14 , C 24  and C 34 ) are determined in three measurement steps within a scanning frame.
 
Measurement of C 1N , C 2N , . . . , C (N−1)N  
 
       FIG. 9  shows a circuit model  900  that can be used to determine relationships between currents and capacitances for N simultaneously applied excitation signals. If (N−1)(N/2) capacitances have already been measured, then the capacitances, C 1N , C 2N , . . . , C (N−1)N , may be measured by applying N excitation signals {(u 1 ,f 1 ), (u 2 ,f 2 ), . . . , (u N , f N )} are applied to the electrodes  411   a - 411 N. The last electrode, e.g., electrode  411 (N+1), may be used as the receiving electrode. By assuming that u 2 =u 3 = . . . u N−1 =0, based on the Kirchhoff&#39;s law, the relationships between the excitation signal, u 1 , and the port currents, i 1 , i 2 , . . . , i N , are given by the following N+1 equations: 
                   {                 ⁢             i   1     ⁢   2   ⁢     R   on             -       i   2     (       2   ⁢     R   on       +                                                   =     u   1                               Z   12     )                                                                 i   1     ⁢   2   ⁢     R     on   ⁢                                   -       i   3     ⁡     (       2   ⁢     R   on       +     Z   13       )                                       =     u   1               …                               …                   …               i   1     ⁢   2   ⁢     R   on                                                 -       i     N   +   1       (       2   ⁢     R   on       +               =     u   1                                                                   Z     1   ⁢     (     N   +   1     )         )                           i   1           +     i   2             +     i   3                         -     i     N   +   1               =   0                     (   7   )               
where Z 1k =1/jωC 1k , (k,=2, . . . , N+1) is the equivalent impedance between the first electrode (e.g., electrode  411   a ) and the N+1 electrode (e.g., electrode  411   d ).
 
     Equation (7) can be rewritten in the matrix form:
 
Ψ 1   ·{right arrow over (i)}={right arrow over (u)}   1   (8)
 
where the matrices are given as follows:
 
     
       
         
           
             
               
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                 [ 
                 
                   
                     
                       
                         u 
                         1 
                       
                     
                   
                   
                     
                       
                         u 
                         1 
                       
                     
                   
                   
                     
                       … 
                     
                   
                   
                     
                       
                         u 
                         1 
                       
                     
                   
                   
                     
                       0 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Based on Cramer&#39;s rule, the ratio between the current, i N+1 , and the excitation signal voltage, u 1 , can be expressed as: 
                     a     1   ⁢     (     N   +   1     )         =              i     N   +   1         u   1            =            det   ⁡     [     Θ   1     ]         det   ⁡     [     Ψ   1     ]                        (   9   )               
where the det[ . . . ] is the matrix determinant and Θ 1  is an (N+1)×(N+1) matrix formed by replacing the N+1 column of Ψ 1  by the column vector {right arrow over (u)} 1 /u 1 :
 
     
       
         
           
             
               Θ 
               1 
             
             = 
             
               [ 
               
                 
                   
                     
                       2 
                       ⁢ 
                       
                         R 
                         on 
                       
                     
                   
                   
                     
                       - 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               R 
                               on 
                             
                           
                           + 
                           
                             Z 
                             12 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     1 
                   
                 
                 
                   
                     
                       2 
                       ⁢ 
                       
                         R 
                         on 
                       
                     
                   
                   
                     
                         
                     
                   
                   
                     
                       - 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               R 
                               on 
                             
                           
                           + 
                           
                             Z 
                             13 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     1 
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     … 
                   
                   
                     
                         
                     
                   
                   
                     1 
                   
                 
                 
                   
                     
                       2 
                       ⁢ 
                       
                         R 
                         on 
                       
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                       - 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               R 
                               on 
                             
                           
                           + 
                           
                             Z 
                             
                               1 
                               ⁢ 
                               N 
                             
                           
                         
                         ) 
                       
                     
                   
                   
                     1 
                   
                 
                 
                   
                     
                       2 
                       ⁢ 
                       
                         R 
                         on 
                       
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     1 
                   
                 
                 
                   
                     1 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     … 
                   
                   
                     1 
                   
                   
                     0 
                   
                 
               
               ] 
             
           
         
       
     
     The term det[Ψ 1 ] is a first-order linear function of impedance Z 1(N+1) =1/jωC 1(N+1) . Thus, Eq. (9) can be expressed as: 
                       α   ⁢           ⁢     C     1   ⁢     (     N   +   1     )       2       +     β   ⁢           ⁢     C     1   ⁢     (     N   +   1     )           +   γ     =            det   ⁡     [     Θ   1     ]              a     1   ⁢     (     N   +   1     )                   (   10   )               
where α, β, and γ are real constants calculated from the term, |det[Ψ 1 ]|. Given that the value of a 1(N+1)  can be measured experimentally, the capacitance C 1(N+1)  can be calculated by using the expression:
 
     
       
         
           
             
               
                 
                   
                     C 
                     
                       1 
                       ⁢ 
                       
                         ( 
                         
                           N 
                           + 
                           1 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             β 
                             2 
                           
                           + 
                           
                             4 
                             ⁢ 
                             
                               α 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                        
                                       
                                         det 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             Θ 
                                             1 
                                           
                                           ] 
                                         
                                       
                                        
                                     
                                     
                                       a 
                                       
                                         1 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             N 
                                             + 
                                             1 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   - 
                                   γ 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       - 
                       β 
                     
                     
                       2 
                       ⁢ 
                       α 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     As a result, for any M-electrode cylinder, by measurement N, there are N(N+1)/2 capacitance values that can be calculated by measuring the magnitude of output signal at each excitation frequency. As a result, for an M-electrode ECT sensor, the total number of capacitance values, M·(M−1)/2, can be measured in M−1 measurements with MECaP techniques. 
     By using the measured capacitance values, C ij , (i=1, . . . , N−1, j=2, . . . , N, i≠j), the two-dimensional distribution of material permittivity ∈(x, y) is given by: 
                   {               ∇     ·     ɛ   ⁡     (     x   ,   y     )           ⁢     ϕ   ⁡     (     x   ,   y     )         =   0                 -     ∇     ϕ   ⁡     (     x   ,   y     )           =   E                     Q   ij     /   Δ     ⁢           ⁢   V     =     C   ij                     (   12   )               
where Φ(x, y) is the potential distribution; E is the electric field; Q ij  is the charge across capacitance C ij  and can be calculated by using the Gauss integral law; and ΔV is the voltage of the excitation signal. Applying Eq. (12) makes it possible to calculate the spatial distribution of materials within a volume using ECT.
 
     Table 3 lists the maximum scanning speeds for different measurement techniques and numbers of electrodes. The maximum scanning speeds using MECaP and traditional AC techniques are calculated using optimal time per measurement values. Results from the two methods are compared in four different cases, where the number of the electrodes, M, is assumed to be 8, 12, 16, or 44 (grouped into 4 sections). 
     As shown in Table 3, when the number of simultaneously excited electrodes N approaches the upper limit N max =M−1, the maximum scanning speed achieved with MECaP techniques can be increased from about two to eight times that of traditional AC techniques (without grouping). For example, for a twelve-electrode sensor, the total number of measurements performed per frame is 12×11/2=66. In comparison, a reduction to eleven measurements is possible using MECaP techniques when the number of simultaneously excited electrodes N is eleven. For grouped electrodes, MECaP techniques may achieve 2,000 frames per second, doubling the optimal speed reported elsewhere. 
     
       
         
               
             
               
               
             
               
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Maximum Scanning Speed Achieved with Different ECT Techniques 
               
             
          
           
               
                   
                 M (K) 
               
             
          
           
               
                   
                   
                 8 (0) 
                 12 (0) 
                 16 (0) 
                 44 (4) 
               
               
                   
                   
               
             
          
           
               
                 H 
                   28 
                   66 
                   120 
                   66 
               
             
          
           
               
                 Traditional 
                 # of Measurements/Frame 
                   28 
                   66 
                   120 
                   66 
               
               
                 Method 
                 Max Speed (Frame/sec) 
                 2,357 †   
                 1,000 †   
                   550 †   
                 1,000 †   
               
               
                 MECaP 
                 # of Measurements/Frame 
                   15 
                   34 
                   61 
                   44 
               
               
                 N max  = 2 
                 Max Speed (Frame/sec) 
                 4,400 
                 1,941 
                 1,082 
                  1500 
               
               
                   
                 Increase % 
                 187% 
                 194% 
                 196% 
                 150% 
               
               
                 MECaP 
                 # of Measurements/Frame 
                   11 
                   23 
                   41 
                   33 
               
               
                 N max  = 3 
                 Max Speed (Frame/sec) 
                 6,000 
                 2,870 
                 1,610 
                 2,000 
               
               
                   
                 Increase % 
                 254% 
                 286% 
                 293% 
                 200% 
               
               
                 MECaP 
                 # of Measurements/Frame 
                    9 
                   18 
                   32 
               
               
                 N max  = 4 
                 Max Speed (Frame/sec) 
                 7,333 
                 3,667 
                 2,063 
               
               
                   
                 Increase % 
                 311% 
                 366% 
                 374% 
               
               
                 MECaP 
                 # of Measurements/Frame 
                    8 
                   16 
                   26 
               
               
                 N max  = 5 
                 Max Speed (Frame/sec) 
                 8,250 
                 4,125 
                 2,530 
               
               
                   
                 Increase % 
                 350% 
                 413% 
                 460% 
               
               
                 MECaP 
                 # of Measurements/Frame 
                    7 
                   11 
                   15 
                   33 
               
               
                 N max  = M − 1* 
                 Max Speed (Frame/sec) 
                 9,429 
                 6,000 
                 4,400 
                 2,000 
               
               
                   
                 Increase % 
                 400% 
                 600% 
                 800% 
                 200% 
               
               
                   
               
               
                 Note: 
               
               
                 *For grouped electrodes, N max  = K − 1 
               
               
                   † The maximum scanning speeds for the traditional AC-method were calculated by referring to reported optimal time per measurement values. 
               
             
          
         
       
     
     In cases where N max &lt;M−1, when the number of activated electrodes reaches the upper limit, N max , the remaining measurements continue with simultaneous excitation of N max  electrodes until the end of the scanning frame. The expression of H in Table 2 is then modified as: 
                   H   =       N   max     +       [           M   ⁡     (     M   -   1     )       /     N   max       -     (       N   max     +   1     )       2     ]     INT               (   13   )               
where [X] INT  represents rounding the number X up to the nearest integer. When N max  reaches the upper bound, N max =M−1, the right-hand side of Eq. (13) reduces to M−1, which is consistent with the expression listed in Table 2. As an example, for N max =2, the number of simultaneous excitations remains at two after completion of the second measurement, and the remaining capacitance values are measured using two simultaneous excitations. Therefore, for M=12, the total number of measurements per frame is H=2+(66−3)/2=34. The corresponding scanning speed is 1,941 frames/sec.
 
       FIG. 10  is a plot of scanning speed versus the number of simultaneous excitations. The frequency bandwidth may limit the number of excitations that can be applied simultaneously to the electrodes. To illustrate the trend of scanning speed increase as a function of the number of simultaneous excitations, these two parameters are plotted in  FIG. 10 , for the four types of sensor configurations (M=8, 12, 16, and 44, with K=4).  FIG. 10  shows that significant increase in capacitance scanning speed can be achieved for the same sensor configurations when MECaP techniques are applied compared to traditional AC techniques. 
     Computation Time for MECaP 
       FIG. 11  is a plot of estimated scanning period (triangles) and estimated calculation time (circles) using MECaP techniques for different numbers of excitations N. Compared with the traditional ECT techniques, extra calculation steps may be needed to perform the calculations described above, which, in turn, may affect the application of MECaP techniques to monitoring the object in real time. 
     The calculations below are based on the assumption that each complex number calculation (accumulation/multiplication) can be finished in one computer cycle and that each cycle takes 0.5 ns (i.e., the processor speed is 2 GHz or more). In cases where N electrodes are excited using AC signals, the calculation time may include three sub-calculations:
         Calculation of matrix determinants in Eqs. (8) and (9). Both Θ 1  and Ψ 1  are (N+1)×(N+1) matrices whose first columns include N+1 elements, whose N th  columns include N elements, and whose other columns include two elements. Calculating each determinant requires one computer cycle per element, or (N+1)N·2 N−1  cycles, resulting a total number of cycles CYC det =(N+1)N·2 N      Calculation of absolute values in Eq. (10). One accumulation and one multiplication are needed for |det[Θ 1 ]|. For the term |det[Ψ 1 ]|, the value is tripled for the three terms α, β, and γ, yielding five calculations total, or CYC abs =5.   Calculation of second-order equation roots in Eq. (11). This may require nine computer cycles, or CYC abs =9.       

     Since each excitation measurements measures N capacitors, the total number of cycles for each frame is: 
                   CYC   =       ∑     n   =   1     N     ⁢     n   ⁡     [         n   ⁡     (     n   +   1     )       ⁢     2   n       +   5   +   8     ]                 (   13   )               
Referring again to  FIG. 11 , when N≦8, the calculation time is shorter than the scanning period, which allows for data processing in real time. This corresponds to an achievable scanning speed of 9,400 frames/sec.
 
     Further embodiments of the present inventive MECaP technique may use different or optimal numbers of simultaneous excitations depending on trade-offs between the maximum scanning speed and the minimum differentiable frequency bandwidth. Embodiments may also use coding methods for sensing channel differentiation. For example, embodiments may use signal processing methods for channel coding that complement the frequency band allocation to ensure reliable channel separation and differentiation among multiple electrodes. Coding can enhance the immunity of the circuitry to noise interference and consequently, improve the robustness of the measurement circuitry. 
     In addition, different embodiments may employ different spatial distributions of electrodes. When used to sense capacitance of cylindrical bodies, such as engines, the electrodes may be distributed along the radial and axial directions to maximize the signal-to-noise ratio in sensing. Further embodiments may include embedded electronics for simultaneous electrode scanning. For instance, microcontroller-based parallel excitation and measurement circuitry with embedded firmware may control the switching of sensing channels and interfacing with a computer or other processor. 
     Embodiments of the invention can take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment containing both hardware and software elements. In a preferred embodiment, the invention is implemented in software, which includes but is not limited to firmware, resident software, microcode, etc. 
     Furthermore, embodiments of the invention can take the form of a computer program product accessible from a computer-usable or non-transitory computer-readable medium providing program code for use by or in connection with a computer or any instruction execution system. For the purposes of this description, a computer-usable or computer readable medium can be any apparatus that can contain, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device. 
     The medium can be an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system (or apparatus or device) or a propagation medium. Examples of a computer-readable medium include a semiconductor or solid state memory, magnetic tape, a removable computer diskette, a random access memory (RAM), a read-only memory (ROM), a rigid magnetic disk and an optical disk. Current examples of optical disks include compact disk-read only memory (CD-ROM), compact disk-read/write (CD-R/W) and DVD. 
     A data processing system suitable for storing and/or executing program code will include at least one processor coupled directly or indirectly to memory elements through a system bus. The memory elements can include local memory employed during actual execution of the program code, bulk storage, and cache memories which provide temporary storage of at least some program code in order to reduce the number of times code must be retrieved from bulk storage during execution. 
     Input/output or I/O devices (including but not limited to keyboards, displays, pointing devices, etc.) can be coupled to the system either directly or through intervening I/O controllers. 
     Network adapters may also be coupled to the system to enable the data processing system to become coupled to other data processing systems or remote printers or storage devices through intervening private or public networks. Modems, cable modem and Ethernet cards are just a few of the currently available types of network adapters. 
     While this invention has been particularly shown and described with references to example embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the invention encompassed by the appended claims. For example, the multiple excitation polling techniques disclosed herein may be applied to other forms of tomography, such as tomography performed with acoustic, optical, radio-frequency, or other suitable signals.