Abstract:
An EIT system includes a plurality of voltage sources for supplying a corresponding plurality of voltages to a corresponding number of other structures, voltage source calibration means for calibrating each voltage source, and switching means for individually connecting the calibration means to each voltage source in succession during a period when each other structure is in an inactive condition. Calibrating respective voltages and currents for each voltage source compensates for shunt impedance of each voltage source. A method for calibrating the system includes individually connecting the calibration means to each voltage source in succession during a period when each other structure is in an inactive condition for calibrating all of said voltage sources in the same way.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This U.S. patent application claims priority on, and all benefits available from U.S. provisional patent application 60/641,508 filed Jan. 5, 2005, all of which is incorporated here by reference. 
    
    
     STATEMENT OF GOVERNMENT INTEREST 
     Development of the present invention was supported, in part, by CenSSIS, the Center for Subsurface Sensing and Imaging Systems, under the Engineering Research Center Program of the National Science Foundation (Award number EEC-9986821). 
    
    
     FIELD AND BACKGROUND OF THE INVENTION 
     The present invention relates generally to the field of EIT, and in particular to a new and useful electrical impedance imaging apparatus and a method for calibrating the electrical impedance imaging apparatus. 
     Electrical impedance tomography is an imaging modality which displays the spatial distribution of the complex conductivity distribution inside a body. An excitation is applied to electrodes on the body surface, resulting in an electromagnetic (EM) field appearing within the volume. If the excitation consists of one or more currents, the voltages that appear at some or all of the electrodes are measured. If the excitation consists of voltages, then the currents at the electrodes are measured. The inverse problem describing the current to voltage relationship is then solved to determine the complex conductivity distribution that must have been present to produce the measured data set. The ill-posedness of this inverse problem manifests itself through the small changes in surface current or voltage that sometimes result from large changes in the interior impedance distribution. High precision electronics are needed to apply the excitations and measure the data that correspond to these changes. 
     For an electrical impedance tomograph with finite measurement precision, distinguishability is defined as the ability to detect an inhomogeneity in a homogeneous background, and is maximized for all conductivities and geometries when currents are applied to the surface electrodes and the resulting voltages are measured. Furthermore, distinguishability is maximized when multiple, independent current sources are used to apply spatial patterns of currents to the electrodes. Applying current patterns whose eigenvalues match the natural response (modes) of the body being interrogated maximizes the signal to noise ratio (SNR) of the resulting data set and therefore minimizes the amount of regularization necessary for the subsequent reconstruction. 
     Taken together, these two observations suggest that multiple, high-precision current sources are required to maximize the distinguishability and SNR of an impedance tomography data set. While such current sources have been developed for this application, they tend to have limited bandwidth, apply only sinusoidal excitations, and require a large number of high precision components. (See [1] R. D. Cook, G. J. Saulnier, D. G. Gisser, J. C. Goble, J. C. Newell, and D. Isaacson. ACT3: A high-speed, high-precision electrical impedance tomograph.  IEEE Transactions on Biomedical Engineering , vol. 41 (8): 713-722, August 1994; Also see [2] A. S. Ross, G. J. Saulnier, J. C. Newell, and D. Isaacson. Current source design for electrical impedance tomography.  Physiological Measurement , vol. 24(2):509-516, May 2003.) The result is an electronics package with a large system footprint, and high component, power, and cooling costs. 
     In contrast, precision voltage sources are generally easier and less costly to implement. However, as mentioned above, applying voltages and measuring currents produces a less optimal EIT system. In an attempt to gain the hardware simplicity of a voltage source along with the optimality of applied currents, algorithms have been developed for utilizing multiple voltage sources to apply a desired current pattern (See [3] M. H. Choi, D. Isaacson, G. J. Saulnier, and J. C. Newell. An iterative approach for applying multiple currents to a body using voltage sources in electrical impedance tomography. In Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, volume 4, pages 3114-3117, September 2003.) This algorithm takes into account the interaction between the sources which results in the current at any given electrode being a function of the voltages on all the electrodes. 
     When using a voltage source in EIT, it is necessary to know both the applied voltage and the resulting current with high precision. If it is desired to have the applied voltage remain unchanged for a wide range of load impedances, then it is necessary to have a voltage source with low output impedance. In the case where voltages are being applied to produce specific currents, the applied voltage will be adjusted, generally in an iterative way, to compensate for changes in the load impedance. Consequently, low output impedance less important than having the ability to precisely measure the actual applied voltage and current. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide a system and method for using voltage sources to produce a desired current pattern in an EIT system. 
     It is a further object of the present invention to directly measure the actual applied voltage and resulting current in an EIT system with high precision. 
     It is yet another object of the present invention to measure the actual applied voltage and resulting current in an EIT system with high accuracy. 
     Accordingly, a voltage source circuit, calibration circuit and calibration procedure are provided for use in EIT. The source incorporates direct measurement of the applied voltage and measurement of the applied current. The calibration procedure results in a set of calibration factors and parameters that allow the suppression of errors due to stray shunt impedance, passive element values and non-ideal properties of active devices. Since all voltage sources in an EIT system will be calibrated using a single calibration circuit, the approach results in nearly identical performance of all voltage sources. 
     The EIT system of the present invention includes a plurality of voltage sources for supplying a corresponding plurality of voltages to a corresponding number of other structures, voltage source calibration means for calibrating each voltage source, and switching means for individually connecting the calibration means to each voltage source in succession during a period when each other structure is in an inactive condition. Each voltage source includes an operational amplifier having an output and a sense resistor connected to the output of the operational amplifier. Each voltage source also includes a buffer amplifier having an input connected to the sense resistor for outputting a measured voltage, and an instrumentation amplifier having one input connected to the operational amplifier output and another input connected to the sense resistor for outputting a measured current. The calibration circuit includes an operational amplifier, a buffer amplifier, and switching means for selectively connecting an output of each voltage source to only one of the operational and buffer amplifiers to compensate for shunt impedance of each voltage source. 
     The method for calibrating the system includes individually connecting the calibration means to each voltage source in succession during a period when each other structure is in an inactive condition for calibrating all of said voltage sources in the same way. The method includes the steps of determining a gain for the buffer amplifier, determining a gain for the instrumentation amplifier, determining a value for the sense resistor, determining an output shunt impedance for each voltage source, and determining an actual current delivered to the body by the voltage source. 
     The various features of novelty which characterize the invention are pointed out with particularity in the claims annexed to and forming a part of this disclosure. For a better understanding of the invention, its operating advantages and specific objects attained by its uses, reference is made to the accompanying drawings and descriptive matter in which a preferred embodiment of the invention is illustrated. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In the drawings: 
         FIG. 1  is a block diagram of an applied-voltage EIT system with L number of electrodes; 
         FIG. 2  is a schematic circuit diagram of a voltage source with a current measuring capability; 
         FIG. 3  is a schematic circuit diagram of a voltage source with a current measuring capability and a calibration circuit for allowing precision voltage and current measurements; 
         FIG. 4  is a schematic circuit diagram according to  FIG. 3  and further showing the switch setting for the first step in the calibration method; 
         FIG. 5  is a schematic circuit diagram according to  FIG. 3  and further showing the switch setting for the second step in the calibration method; 
         FIG. 6  is a schematic circuit diagram according to  FIG. 3  and further showing the switch setting for the third step in the calibration method; 
         FIG. 7  is a schematic circuit diagram according to  FIG. 3  and further showing the switch setting for the fourth step in the calibration method; 
         FIG. 8  is a schematic circuit diagram according to  FIG. 3  and further showing the configuration for applying the voltage source; 
         FIG. 9  is a schematic reed relay model where the contact resistance, R s , is 0.150Ω when the switch is “on” and 1GΩ when the switch is “off”; 
         FIG. 10   a  is a graph showing the error in the estimated load resistance when three sample voltage sources are operated without calibration according to a simulation; 
         FIG. 10   b  is a graph showing the error in the estimated load capacitance when three sample voltage sources are operated without calibration according to a simulation; 
         FIG. 11   a  is a graph showing the error in the estimated load resistance when three sample voltage sources are operated with full calibration according to a simulation; 
         FIG. 11   b  is a graph showing the error in the estimated load capacitance when three sample voltage sources are operated with full calibration according to a simulation; 
         FIG. 12   a  is a graph showing the error in the estimated load resistance when three sample voltage sources are operated with relative calibration according to a simulation; and 
         FIG. 12   b  is a graph showing the error in the estimated load capacitance when three sample voltage sources are operated with relative calibration according to a simulation. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Referring now to the drawings, in which like reference numerals are used to refer to the same or similar elements,  FIG. 1  is a block diagram of an applied-voltage EIT system  11  with L number of electrodes  15 . Each electrode  15  is connected to a circuit  21  that includes a voltage source  23  for generating the applied voltage as well as an ammeter  25  to measure the applied current and a voltmeter  27  to directly measure the applied voltage. A switching network enables a single calibration circuit to be connected to any of the voltage source/ammeter/voltmeter circuits  21  to allow the whole system  11  to be calibrated to a single reference. 
     Though not shown in  FIG. 1 , the voltage sources  23  (with ammeters  25  and voltmeters  27 ), switches, and calibration circuit  31  each interface to a digital controller which sets the system configuration and collects digital measurements of voltage and current. A series of calibration steps, to be described below, are performed to collect calibration data for each source  23 . Typically, the electrodes  15  are not in contact with a body to be imaged during these calibration steps. The digital controller utilizes the calibration data to determine actual applied currents and voltages from measurements that are made when collecting data for an EIT image. 
       FIG. 2  shows one embodiment for implementing a voltage source  923  with an integrated current measurement capability. The voltage source  923  provides a voltage V i  at an operational amplifier  941 , which is part of a measuring circuit that also includes a current-sensing resistor, R sense , included in the feedback loop. The operation amplifier  941  is a buffer amplifier, and specifically a unity gain buffer because the voltage that is transferred is unchanged. The measuring circuit also includes a shunt impedance, Z shunt , and a load resistance, R load . The signal, I actual , is the measure of the actual current that is going to the load while the output voltage, V o , is produced. 
     In one exemplary case, the shunt impedance, Z shunt , is infinite and the operational amplifier  941  is ideal, having infinite gain, infinite input resistance and zero input capacitance. Under these conditions, the voltage, V a , that is applied to the load resistance, R Load , equals the input voltage, V i . Also, the current delivered to the load, I actual , equals the current through R sense , denoted as I sense . This load current can be determined by measuring V sense  and evaluating I sense =V sense /R sense . 
     If the operational amplifier has finite gain, V o ≠V i  and it is necessary to directly measure V o  in a high-precision application. The introduction of Z shunt  creates a larger problem because it causes I sense  to be different from I actual . Z shunt  can include the capacitance introduced by wiring, printed-circuit board traces, and the input capacitance and resistance at the operational amplifier&#39;s non-inverting terminal. Also, the addition of other circuits, such as a voltmeter, connected to the output will insert additional capacitance and finite shunt resistance which can be grouped in Z shunt . In all cases, the presence of finite impedance to ground will result in some of I sense  flowing to ground through Z shunt  rather than into R Load . The error current is denoted by 
                     I   error     =         I   sense     -     I   actual       =         V   0       Z   shunt       .               (   1   )               
Rearranging, equation 2 is derived:
 
                     I   actual     =       I   sense     -       V   0       Z   shunt                 (   2   )               
meaning that knowledge of Z shunt , V o  and I sense  enables the computation of the actual load current value.
 
     The voltage source and calibration circuit described below in conjunction with  FIG. 3  will be able to directly measure Z shunt  in order to enable I actual  to be determined with high precision.  FIG. 3  shows an improvement to the voltage source to enable calibration. In addition to a number of switches, a buffer amplifier (Buffer)  51  has been added to allow measurement of the output voltage and an instrumentation amplifier (IA)  53  has been added for determining the current flowing in R sense . A calibration circuit  31  has also been added that can measure either current or voltage. The calibration circuit  31  includes an operational amplifier  63  and a buffer amplifier  61 . The operational amplifier  63  is assumed to have a frequency independent transresistance of R f  and the voltage buffer is assumed to have a gain of unity. 
     In a complete system, there is only one calibration circuit  31  and individual voltage sources  23  can be connected to this calibration circuit  31 , one at a time. It is assumed that the calibration circuit  31  is itself calibrated to a standard, i.e. the current or voltage values that it reports are the true values. However, since all voltage sources are calibrated using a single calibration circuit, errors in the calibration circuit will only impact the overall accuracy of the system and not its precision. 
     The shunt impedance, Z shunt  includes any stray capacitance along with input capacitance and resistance for the instrumentation amplifier  53 , buffer amplifier  51  and the voltage source  23  itself. Also included is the switch capacitance to ground. The capacitance of solid-state switches tends to be large and vary significantly with switch position and applied voltage. Since switch positions will change between calibration settings and operation setting, low-capacitance reed relays are used in place of solid-state switches. Reed relays present capacitance that varies little with switch position. Additionally, it is assumed that the “on” resistance of the switches is negligible. 
     There are several steps for calibrating the voltage source  23 . During the calibration process, the gains of the buffer amplifier  51  and the instrumentation amplifier  53  are measured along with the precise values of R sense  and Z shunt . It is assumed that phase-sensitive voltmeters are able to measure the voltages V meas , I meas , V cal  and I cal . Consequently, all gains, voltages, currents and impedance values can be complex. The 4 steps in the calibration algorithm are explained below. 
       FIG. 4  shows the switch settings for step one of the calibration process. In this configuration, switches S 1 , S 2 , S 3  and S C2  are closed and all other switches are open. With S 1  closed, R sense  is shorted; with S 2  closed, the voltage source operating amplifier  41  is configured as a unity gain buffer. S 3  and S C2  connect the voltage source operating amplifier  41  output, which is also the input to the buffer amplifier  51  and both inputs to the instrumentation amplifier  53 , to the calibration circuit voltmeter. 
     An input voltage, V i , is applied and the voltages V cal , V meas , and I meas  are measured. From these measurements, the following gains can be determined: 
                       G   Buffer     =       V   meas       V   cal         ⁢     
     ⁢       G     IA   -   CM       =         I   meas       V   cal       .               (   3   )               
where G Buffer  is the gain of the Buffer amplifier  51  and G IA-CM  is the common-mode gain of the instrumentation amplifier  53 . In general, the output of an instrumentation amplifier  53  is determined by
 
                     I   meas     =         G     IA   -   DM       ⁡     (       V   +     -     V   -       )       +       G     IA   -   CM       ⁡     (         V   +     +     V   -       2     )                 (   4   )               
where G IA-OM  is the differential gain, V + −V −  is the differential input voltage and
 
                   V   ++     ⁢   V     -       2   ⁢                   
is the common-mode input voltage. Common-mode gain can be determined because the two inputs to the instrumentation amplifier  53  are at the same voltage (connected by S 1 ), meaning that the differential input voltage is zero.
 
     In the second calibration step, the differential gain of the instrumentation amplifier  53 , G IA-OM  is determined.  FIG. 5  shows the switch arrangement for this step. As can be observed, switches S 2 , S 3 , S 4  and S C1  are closed and all other switches are open. Opening S 1  and closing S 4  produces a voltage drop across R sense  when an input voltage, V i , is applied which appears as a differential input voltage to the instrumentation amplifier  53 . The input voltage at the non-inverting input is measured by the calibration circuit and equals
 
V + =V cal   (5)
 
while the voltage at the non-inverting input is
 
                     V   -     =         V   meas       G   Buffer       .             (   6   )               
The differential input voltage, then, is
 
                     V   DM     =         V   +     -     V   -       =       V   cal     -       V   meas       G   Buffer                   (   7   )               
and the common mode input voltage is
 
                     V   CM     =           V   +     +     V   -       2     =       1   2     ⁢     (       V   cal     +       V   meas       G   Buffer         )                 (   8   )               
Using equation 4, equation 9 can be solved for:
 
                     G     IA   -   DM       =         I   meas     -       G     IA   -   CM       ⁢     V   CM           V   DM               (   9   )               
and, using this result in combination with equations 7 and 8, G IA-DM  can be computed.
 
       FIG. 6  shows the configuration for the third step in the calibration process. Here, switches S 2 , S 5  and S C1  are closed and all other switches are open. Operational amplifier  63  in the calibration circuit is configured as a current-to-voltage converter (I-V converter) which, for an ideal operational amplifier, produces an output voltage that equals −Rf times the input current (current in S C1 ). Additionally, with an ideal operational amplifier and negative feedback the voltage at the inverting operational amplifier input terminal is forced to equal the voltage at the non-inverting input terminal which is at ground potential. This “virtual ground” in the arrangement of  FIG. 6  results in both ends of Z shunt  ideally being at ground potential. When implemented with a real operational amplifier, the circuit will result in a small voltage appearing across Z shunt . A calibration algorithm will account for this voltage in determining values for R sense  and Z shunt . 
     In this step, the voltages I cal , V meas  and I meas  are recorded. For clarity, these quantities will be denoted as I cal     —     3 , V meas     —     3  and I meas     —     3 . In addition, the differential voltage across R sense  is computed using 
                     V     DM_   ⁢   3       =           I     meas_   ⁢   3       -       G     IA   -   CM       ⁡     (       V     meas_   ⁢   3         G   Buffer       )           (       G     IA   -   DM       +       G     IA   -   CM       2       )       .             (   10   )               
Since the current through R sense  equals the current in Z shunt  plus the current in the I-V converter, equation 11 can be derived.
 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       
                         
                           DM 
                           ⁢ 
                           _ 
                         
                         ⁢ 
                         3 
                       
                     
                     
                       R 
                       sense 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           I 
                           
                             
                               ca 
                               ⁢ 
                               l 
                               ⁢ 
                               _ 
                             
                             ⁢ 
                             3 
                           
                         
                         
                           R 
                           f 
                         
                       
                     
                     + 
                     
                       
                         
                           V 
                           
                             
                               meas 
                               ⁢ 
                               _ 
                             
                             ⁢ 
                             3 
                           
                         
                         
                           
                             G 
                             Buffer 
                           
                           ⁢ 
                           
                             Z 
                             shunt 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
       FIG. 7  shows the switch configuration for the last step in the calibration. This configuration is also employed when using the voltage source after calibration. Here, only S 1  is closed; all other switches are open. A voltage V i  is applied and both I meas  and V meas  are measured and denoted as I meas     —     4  and V meas     —     4 . Likewise, the differential voltage across R sense  is computed using equation 12: 
                     V       DM   ⁢   _     ⁢   4       =           I       meas   ⁢   _     ⁢   4       -       G     IA   -   CM       ⁡     (       V       meas   ⁢   _     ⁢   4         G   Buffer       )           (       G     IA   -   DM       +       G     IA   -   CM       2       )       .             (   12   )               
The current through R sense  flows in Z shunt , resulting in equation 13:
 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       
                         
                           DM 
                           ⁢ 
                           _ 
                         
                         ⁢ 
                         4 
                       
                     
                     
                       R 
                       sense 
                     
                   
                   = 
                   
                     
                       
                         V 
                         
                           
                             meas 
                             ⁢ 
                             _ 
                           
                           ⁢ 
                           4 
                         
                       
                       
                         
                           G 
                           Buffer 
                         
                         ⁢ 
                         
                           Z 
                           shunt 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     In a high precision application, the deviation of the value of R sense  from its nominal value and its variation with aging and temperature will significantly degrade performance. Using the results from steps 3 and 4, along with the calibration constants from steps 1 and 2, the values of both R sense  and Z shunt  can be estimated. 
     Substituting (13) into (11) gives equation 14: 
                       V       DM   ⁢   _     ⁢   3         R   sense       =       -       I       cal   ⁢   _     ⁢   3         R   f         +         V       meas   ⁢   _     ⁢   3         V       meas   ⁢   _     ⁢   4         ⁢       V       DM   ⁢   _     ⁢   4         R   sense                   (   14   )               
which can be solved for R sense  producing equation 15:
 
                     R   sense     =       (       R   f       I       cal   ⁢   _     ⁢   3         )     ⁢       (           V       meas   ⁢   _     ⁢   3         V       meas   ⁢   _     ⁢   4         ⁢     V       DM   ⁢   _     ⁢   4         -     V       DM   ⁢   _     ⁢   3         )     .               (   15   )               
Finally, Z shunt  can be computed using equation 13 rearranged as equation 16:
 
     
       
         
           
             
               
                 
                   
                     Z 
                     ahunt 
                   
                   = 
                   
                     
                       
                         
                           V 
                           
                             
                               meas 
                               ⁢ 
                               _ 
                             
                             ⁢ 
                             4 
                           
                         
                         ⁢ 
                         
                           R 
                           sense 
                         
                       
                       
                         
                           G 
                           Buffer 
                         
                         ⁢ 
                         
                           V 
                           
                             
                               DM 
                               ⁢ 
                               _ 
                             
                             ⁢ 
                             4 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     These values of R sense  and Z shunt  can be stored and used to determine the current being delivered the load in an actual voltage source application. 
     In operation, the goal is have precise measurements of the voltage applied to a load and the current delivered to this load.  FIG. 8  shows the configuration, with the delivered current denoted as I actual  and the applied voltage denoted as V o . The voltage at the load, V o , is determined by
 
 V   o   =V   meas   /G   Buffer   (17)
 
The current being delivered to the load, factual is found using equation 2 with V o , given by equation 17,
 
                           I   actual     =       ⁢       I   sense     -       V   0       Z   shunt                     =       ⁢           I   meas     -         G     IA   -   CM       ⁡     (       V   meas       G   Buffer       )       .           (       G     IA   -   DM       +       G     IA   -   CM       2       )     ⁢     R   sense         -         V   meas         G   Buffer     ⁢     Z   shunt         .                                             (   18   )                                                           (   19   )                                           
Circuit Simulation
 
     The circuit topology shown in  FIG. 8  was implemented in PSpice to provide data to demonstrate the behavior of the calibration algorithm. A model for an Analog Devices AD8610 operational amplifier was used to implement the voltage source, the I-V converter, and the voltage buffers. The AD8610 was configured as a voltage follower to implement the voltage buffers. A model for a Burr-Brown (Texas Instruments) PGA207 high-speed programmable gain instrumentation amplifier was used for the instrumentation amplifier. This instrumentation amplifier was configured for unity differential gain and the model provides a common-mode rejection ratio (CMRR) of approximately 100 dB at DC with a single pole roll-off with a pole at approximately 4 kHz. The I-V converter used a feedback resistance of 1 kΩ. A 50 pF capacitor was added in parallel with R f  to improve stability. The switches, assumed to be reed relays, were modeled as shown in  FIG. 9 . The model is based on the specifications for the Coto Technology 9401 surface mount relay. These relays present 1.1 pF of capacitance between each open contact to the coil and 0.2 pF across open contacts. Contact resistance has a maximum value of 0.15Ω. 
     The performance of the calibration algorithm was evaluated by collecting values for V meas , I meas , I cal , and V cal  as needed for each of the 4 calibration steps defined above (with R Load =∞). Additionally, values for V meas  and I meas  were collected with R Load  present. From this data, an estimate of R Load  was produced and compared to the actual R Load . All measured voltages are complex and the estimate of R Load  is a complex impedance which can be viewed as the parallel combination of a resistance and capacitance, where the capacitance can be positive or negative and ideally equals zero. Separate estimates for R Load  were generated for various instances of the voltage source having different values of Z shunt  and/or R sense . The calibration circuit was not changed at any time in order to accurately model the case where multiple voltage sources, each having somewhat different properties due to component variation and differences in physical layout, are calibrated using a single calibration circuit. 
     The parameters for the 3 test cases are shown in Table 1. The shunt impedance, Z shunt  is modeled as the parallel combination of R shunt  and C shunt . Case 1 is a reference case, having a R sense  with the nominal value of 1 kΩ and a significant capacitive component to Z shunt . Case 2 is used to illustrate the ability of the calibration algorithm to compensate for the variation in the true value of R sense  from the nominal value. Case 3 has a substantially different value for Z shunt  and is used to illustrate the ability to compensate for differences in shunt impedance. 
     
       
         
               
             
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Parameters for the 3 test cases 
               
             
          
           
               
                   
                 R sense   
                 R shunt   
                 C shunt   
               
               
                   
                   
               
             
          
           
               
                   
                 Case 1 
                 1 kΩ 
                 10 MΩ 
                 150 pF 
               
               
                   
                 Case 2 
                 1.1 kΩ   
                 10 MΩ 
                 150 pF 
               
               
                   
                 Case 3 
                 1 kΩ 
                  1 MΩ 
                 300 pF 
               
               
                   
                   
               
             
          
         
       
     
     No Calibration:  FIG. 10  shows the error in the estimated load impedance when the three voltage sources, i.e. the three cases in Table 1, are operated without calibration. R sense  is assumed to be equal to its nominal value of 1 kΩ, the instrumentation amplifier is assumed to have unity differential gain and zero common-mode gain, the voltage buffer is assumed to have unity gain, and Z shunt  is assumed to be infinite. The load impedance is a pure resistance of value 1 kΩ. 
       FIG. 10(   a ) shows error in the estimated load resistance. At low frequencies, the errors for Cases 1 and 3 are small (approximately −0.08Ω and −1Ω, respectively) due to the fact that R sense  is equal to the assumed value of 1 kΩ and the primary source of error is Rshunt (10 MΩ and 1 MΩ, respectively) being in parallel with the load. The error for Case 2 is approximately −90 Ω 15 since R sense  is actually 1.1 kΩ. At high frequencies, the error increases for all cases due to gain roll-off in the instrumentation amplifier and voltage buffer as well as the presence of C shunt . None of the sources is able to provide good estimates of the load resistance at 1 MHz. 
       FIG. 10(   b ) shows the error in estimated load capacitance. Since the actual load capacitance is zero, the error reflects the actual measured load capacitance. The errors are large and vary with both R sense  and C shunt . Note that the capacitance that is observed is smaller than C shunt  as a result of the limited bandwidth of the instrumentation amplifier and voltage buffer. 
     Full Calibration: In this simulation, performance with full calibration is also considered, meaning that the calibration circuit itself has been calibrated. Pspice is used to find the complex transimpedance and complex gain of the I-V converter and voltage buffer, respectively, in the calibration circuit. These values were then used to correct the measured values of I cal  and V cal  to give accurate representations of the voltage and current seen by the calibration circuit. 
       FIG. 11  shows the estimated load resistance and capacitance with full calibration. Now, there is only a small error in the estimate of the load resistance at low frequencies (&lt;&lt;0.5 mΩ) and a few ohms of error at high frequencies. It is important to note that the error at high frequencies is nearly the same for all three cases. The errors in the estimate of the load capacitance are generally less than 0.2 pF, though they are somewhat higher at low frequencies. 
     The performance of the algorithm is limited somewhat by the imperfect virtual ground presented by the I-V converter. The analysis leading to equation 15 takes into account the current through Z shunt  that results from having a non-zero voltage across it, it does not account for current that flows through the input and stray capacitance between the input of the I-V converter and ground. Consequently, the I-V converter will provide a low value for the current. 
     Relative Calibration: In practice, the calibration circuit may not be perfectly calibrated. A feature of the calibration algorithm is that all voltage sources are calibrated using the same calibration circuit, so while errors in calibration circuit calibration will impact the accuracy of the voltage sources, it should not impact their precision. To illustrate this point, the voltage sources were calibrated using a completely uncalibrated calibration circuit. In the calibration algorithm, the I-V converter is assumed to have a frequency independent transresistance of R f  and the voltage buffer is assumed to have a gain of unity. 
       FIG. 12  shows the result with an uncalibrated calibration circuit. The absolute error in the estimate of the load resistance is small at low frequencies since R f  actually equals the assumed value of 1 kΩ. At high frequencies the absolute error is much larger due to the uncalibrated frequency response of the calibration circuit. The spread in the error in the estimate of the load resistance, however, is very small at low frequencies (≈0.5 mΩ) and somewhat larger (≈0.5Ω) at high frequencies. The estimate of the load capacitance shows a bias of approximately −50 pF. This bias is due to the fact that the I-V converter uses a 50 pF capacitance in parallel with R f =1 kΩ. Since the I-V converter was not calibrated, the calibration algorithm attributes this RC time constant (with a sign inversion) to the load resistance. Varying the load resistance results in a change in the estimated load capacitance which maintains this same RC time constant. The spread of the estimated capacitance values is less than 0.1 pF across the frequency range shown. 
     Generally, the calibration circuit would be calibrated, though imperfectly, resulting in smaller errors in accuracy than those observed in  FIG. 12 . However as  FIG. 12(   b ) shows, even with large errors in accuracy, the relative precision of the voltage sources which are all calibrated using the single calibration circuit remains high. 
     The results obtained using data from PSpice demonstrate the ability of the calibration system to determine the calibration parameters needed for high-precision voltage source performance. If the calibration circuit is properly calculated, the calibrated voltage sources will also be highly accurate. If the calibration circuit is imperfectly calibrated, the accuracy of the calibrated voltage sources is also imperfect but all sources maintain a high relative precision, since all sources are inaccurate in the same way. In an EIT system, this relative precision of the sources is much more important than the accuracy of the sources. If, for instance, the measured currents at each voltage source are scaled by a single common scaling factor, i.e. the system has high relative precision but does not have high accuracy, the impedance values in the resulting EIT image would also be scaled but the image itself would not be distorted. If, on the other hand, the measured currents at each voltage source are scaled by different scaling factors, representing the case of low accuracy and low relative precision, the image itself would be distorted. 
     While a specific embodiment of the invention has been shown and described in detail to illustrate the application of the principles of the invention, it will be understood that the invention may be embodied otherwise without departing from such principles.