Abstract:
A “three-point” measurement technique precisely determines dynamic electrical parameters of individual cells/batteries and/or interconnecting conductors embedded in a larger series, parallel, or series-parallel battery/electrical system. Three measuring points are defined. Two of these points comprise terminals of the subject cell/battery or interconnecting conductor. The third measuring point is an adjacent point that is separated from one of the other two measuring points by a single conducting path that may include one or more cells or batteries. By measuring dynamic parameters between alternate pairs of these three measuring points, three dynamic parameter measurements are acquired. A mathematical computation combines the three measurements and uniquely determines the desired dynamic parameters of one or two subject elements—thus effectively “de-embedding” the subject elements from the remainder of the system. A “four-point” extension of this technique permits measuring individual dynamic parameters of single cells/batteries disposed internally in a multiple-unit string of parallel-connected cells/batteries.

Description:
[0001]    The present application is a Continuation-In-Part of and claims priority of U.S. patent application Ser. No. 09/662,092, filed Sep. 14, 2000, the content of which is hereby incorporated by reference in its entirety. 
     
    
     
       BACKGROUND OF THE INVENTION  
         [0002]    The present invention relates to electronically testing electrochemical cells and batteries. More specifically, it relates to precisely determining an individual dynamic parameter of a particular cell, battery, or interconnecting conductor that is embedded in a larger system of series, parallel, or series-parallel connected cells/batteries, loads, and chargers.  
           [0003]    Electrochemical cells and batteries, such as primary cells/batteries, secondary (i.e., storage) cells/batteries, and fuel cells/batteries are important sources of electrical energy. Connecting such units together in series causes their voltages to add. Connecting them together in parallel causes their currents to add. Accordingly, series-connected cells/batteries, parallel-connected cells/batteries, and series/parallel combinations of cells/batteries are routinely found in many applications including automotive, traction, heavy equipment, standby, and telecommunication applications.  
           [0004]    Precisely determining a dynamic electrical parameter (i.e., complex impedance, complex admittance, real resistance, or real conductance) of an individual cell/battery or interconnecting conductor embedded in a larger system without disconnecting the subject element from the system has traditionally posed an important challenge. One approach to this challenge has been described by Burkum et al. in U.S. Pat. No. 4,697,134. This patent teaches passing a known ac current through the parallel combination of a string of series-connected cells, a battery charger, and a load. It then determines the “impedance” of an individual cell or inter-cell connector by measuring the ac voltage developed across the cell or connector and taking the ratio of this measured ac voltage to the known ac current. However, this procedure determines only the magnitude of the complex impedance. Furthermore, the disclosed technique is subject to significant errors due to current-shunting by the charger and the load. The disclosed method also ignores the influence of other series strings in parallel with the desired string in a multi-string installation—such as would commonly be found in a telephone central office.  
           [0005]    Methods and apparatus for measuring complex impedance and complex admittance of electrochemical cells and batteries as well as general electrical elements have recently been disclosed by Champlin in U.S. Pat. No. 6,002,238, U.S. Pat. No. 6,172,483, U.S. patent application Ser. No. 09/503,015 and U.S. patent application Ser. No. 09/710,031. However, the techniques disclosed therein apply generally to measuring isolated elements. If the subject element is embedded in a larger battery/electrical system, the loading imposed by the system could influence the results of the measurement.  
           [0006]    The inventions disclosed herein, however, remove this system influence by mathematically de-embedding a subject “element”, i.e., cell, battery or interconnecting conductor, from the remainder of the battery/electrical system. The disclosed method and apparatus permit precisely determining a dynamic parameter of an embedded element without actually disconnecting the element from the battery/electrical system.  
         SUMMARY OF THE INVENTION  
         [0007]    The present invention employs a “three-point” measurement technique to precisely determine dynamic electrical parameters (i.e., complex impedance, complex admittance, real resistance, or real conductance) of individual elements (i.e., cells/batteries or interconnecting conductors) embedded in a larger series, parallel, or series-parallel battery/electrical system. The system may include a load and/or a battery charger. Three measuring points are defined. Two of these points comprise terminals of the subject cell/battery or interconnecting conductor. The third measuring point is an adjacent point that is separated from one of the other two measuring points by a single conducting path that may include one or more cells or batteries. By measuring dynamic parameters between alternate pairs of these three measuring points, three dynamic parameter measurements are acquired. A mathematical computation combines the three measurements and uniquely determines the desired dynamic parameters of one or two subject elements—thus effectively “de-embedding” the subject elements from the remainder of the system. A “four-point” extension of this technique permits measuring individual dynamic parameters of single cells/batteries disposed internally in a multiple-unit string of parallel-connected cells/batteries. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0008]    [0008]FIG. 1 is a schematic representation depicting the impedance of a cell/battery being measured while the cell/battery is connected to a load.  
         [0009]    [0009]FIG. 2 is a simplified equivalent circuit representing the measurement arrangement of FIG. 1.  
         [0010]    [0010]FIG. 3 is a schematic representation of a first impedance measurement of the circuit of FIG. 1 in accordance with a particular aspect of the present invention.  
         [0011]    [0011]FIG. 4 is a schematic representation of a second impedance measurement of the circuit of FIG. 1 in accordance with a particular aspect of the present invention.  
         [0012]    [0012]FIG. 5 is a schematic representation of a third impedance measurement of the circuit of FIG. 1 in accordance with a particular aspect of the present invention.  
         [0013]    [0013]FIG. 6 is a simplified equivalent circuit of the circuit of FIG. 1 illustrating the three impedances measured in the measurements of FIGS. 3, 4, and  5 .  
         [0014]    [0014]FIG. 7 depicts measuring the impedance of a cell/battery embedded in a series string of such batteries with a plurality of series strings arrayed in parallel.  
         [0015]    [0015]FIG. 8 is a schematic representation of a first impedance measurement of the circuit of FIG. 7 in accordance with a particular aspect of the present invention.  
         [0016]    [0016]FIG. 9 is a schematic representation of a second impedance measurement of the circuit of FIG. 7 in accordance with a particular aspect of the present invention.  
         [0017]    [0017]FIG. 10 is a schematic representation of a third impedance measurement of the circuit of FIG. 7 in accordance with a particular aspect of the present invention.  
         [0018]    [0018]FIG. 11 depicts measuring the impedance of a cell/battery and/or interconnecting conductor embedded in the system of FIG. 7 using a special “three-point” impedance meter in accordance with a particular aspect of the present invention.  
         [0019]    [0019]FIG. 12 is a block diagram representation of the special “three-point” impedance meter depicted in FIG. 11.  
         [0020]    [0020]FIG. 13 is a block diagram representation of a special “n-point” impedance meter in accordance with a particular aspect of the present invention.  
         [0021]    [0021]FIG. 14 is a flow chart of a control algorithm for de-embedding M single elements using the apparatus of FIG. 13.  
         [0022]    [0022]FIG. 15 represents a part of a battery system and demonstrates general rules to be followed when choosing measuring points.  
         [0023]    [0023]FIG. 16 a  is a schematic representation of two cells/batteries connected in parallel and depicts the determination of a first cell/battery impedance and a first interconnecting conductor impedance in accordance with a particular aspect of the present invention.  
         [0024]    [0024]FIG. 16 b  is a schematic representation of two cells/batteries connected in parallel and depicts the determination of a second cell/battery impedance and a second interconnecting conductor impedance in accordance with a particular aspect of the present invention.  
         [0025]    [0025]FIG. 17 a  is a schematic representation of a parallel string of multiple cells/batteries and depicts the determination of the impedances of a cell/battery and interconnecting conductor on a first end of the string in accordance with a particular aspect of the present invention.  
         [0026]    [0026]FIG. 17 b  is a schematic representation of a parallel string of multiple cells/batteries and depicts the determination of the impedances of a cell/battery and interconnecting conductor on a second end of the string in accordance with a particular aspect of the present invention.  
         [0027]    [0027]FIG. 18 a  is a schematic representation of a parallel string of multiple cells/batteries and depicts conditions of a first set of three-point measurements used in the determination of the impedances of an interior cell/battery and interconnecting conductor in accordance with a particular aspect of the present invention.  
         [0028]    [0028]FIG. 18 b  is a schematic representation of a parallel string of multiple cells/batteries and depicts conditions of a second set of three-point measurements used in the determination of the impedances of an interior cell/battery and interconnecting conductor in accordance with a particular aspect of the present invention. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0029]    Consider FIG. 1. This figure illustrates measuring the impedance of cell/battery  10  embedded in a very simple battery system comprising cell/battery  10  connected to load  20  through interconnecting conductors  30  and  40 . Impedances Z 1 , Z L , Z C1 , and Z C2  represent the impedances of cell/battery  10 , load  20 , interconnecting conductor  30 , and interconnecting conductor  40 , respectively. Impedance meter  50 , which may be of the type disclosed by Champlin in the patents and patent applications referenced above, contacts the two terminals of cell/battery  10  with Kelvin contacts  60  and Kelvin contacts  70 . As is well known, Kelvin contacts comprise two separate electrical connections to each terminal—one for current and one for voltage—and negate the effects of contact and lead-wire resistance. Although Kelvin contacts are usually required to obtain accurate measurements with the very small impedance values encountered in most battery systems, the “three-point” measurement technique disclosed herein does not depend upon having Kelvin contacts. Single connections to each terminal will suffice if impedance values are sufficiently large.  
         [0030]    [0030]FIG. 2 shows an equivalent circuit representation of the simple battery system of FIG. 1. Because the series combination of Z L , Z C1 , and Z C2  parallels impedance Z 1 , the impedance Zm “seen” by impedance meter  50  is not actually Z 1  but is instead the composite impedance:  
               Z                 m     =       Z1   ·     (       Z   C1     +     Z   C2     +     Z   L       )         Z1   +     Z   C1     +     Z   C2     +     Z   L                 (   1   )                               
 
         [0031]    The influence of the impedances Z C1 , Z C2 , and Z L  upon the measured impedance Zm is clearly observed in equation (1).  
         [0032]    Now consider performing the three impedance measurements shown in FIGS. 3, 4, and  5 .  
         [0033]    First measure Zab (FIG. 3) with both sets of Kelvin contacts of impedance meter  50  directly contacting the two terminals, a and b, of the cell/battery being measured. Terminals a and b comprise the first two measuring points of a three-point measurement technique.  
         [0034]    Next measure Zbc (FIG. 4) with one set of Kelvin contacts contacting one of the cell/battery&#39;s terminals, terminal b, and the other set bridging across the adjacent interconnecting conductor impedance Z C2 , to contact the load at point c. Point c comprises the third measuring point of the three-point measurement technique.  
         [0035]    Finally, measure Zca (FIG. 5) with one set of Kelvin contacts contacting measuring point c and the other set contacting measuring point a.  
         [0036]    [0036]FIG. 6 shows an equivalent circuit relating the three measured impedances Zab, Zbc, and Zca to the three system impedances Z 1 , Z 2 , and Z 3 . Note that this particular choice of measuring points makes system impedance Z 2  equal the interconnecting conductance impedance Z C2  while system impedance Z 3 =Z L +Z C1  combines the load impedance with the other interconnecting conductor impedance. An alternative choice of measuring points would make Z 2 =Z C1  and Z 3 =Z L +Z C2 .  
         [0037]    One can easily show from FIG. 6 that the three measured impedances are given by:  
               Z                 a                 b     =       Z1   ·     (     Z2   +   Z3     )         Z1   +   Z2   +   Z3               (   2   )                   Z                 b                 c     =       Z2   ·     (     Z3   +   Z1     )         Z1   +   Z2   +   Z3              
        and           (   3   )                 Z                 c                 a     =       Z3   ·     (     Z1   +   Z2     )         Z1   +   Z2   +   Z3               (   4   )                               
 
         [0038]    These three equations can be inverted mathematically to yield explicit expressions for Z 1 , Z 2 , and Z 3 , in terms of the measured quantities Zab, Zbc, and Zca. The results are  
             Z1   =       (       Z                 a                   b   2       +     Z                 b                   c   2       +     Z                 c                   a   2       -       2   ·   Z                   b                   c   ·   Z                   c                 a     -       2   ·   Z                   c                   a   ·   Z                   a                 b     -       2   ·   Z                   a                   b   ·   Z                   b                 c       )       2   ·     (       Z                 a                 b     -     Z                 b                 c     -     Z                 c                 a       )                 (   5   )               Z2   =       (       Z                 a                   b   2       +     Z                 b                   c   2       +     Z                 c                   a   2       -       2   ·   Z                   b                   c   ·   Z                   c                 a     -       2   ·   Z                   c                   a   ·   Z                   a                 b     -       2   ·   Z                   a                   b   ·   Z                   b                 c       )       2   ·     (       Z                 b                 c     -     Z                 c                 a     -     Z                 a                 b       )                 (   6   )                               
 
         [0039]    and  
             Z3   =       (       Z                 a                   b   2       +     Z                 b                   c   2       +     Z                 c                   a   2       -       2   ·   Z                   b                   c   ·   Z                   c                 a     -       2   ·   Z                   c                   a   ·   Z                   a                 b     -       2   ·   Z                   a                   b   ·   Z                   b                 c       )       2   ·     (       Z                 c                 a     -     Z                 a                 b     -     Z                 b                 c       )                 (   7   )                               
 
         [0040]    Equation (5) effectively de-embeds the subject cell/battery since Z 1  would be its measured impedance if it were, in fact, disconnected from the system.  
         [0041]    The three-point measurement technique disclosed above can be readily extended to the very important case depicted in FIG. 7. FIG. 7 illustrates an attempt to measure the impedance Z 1  of cell/battery  10  embedded in a series string of cells/batteries, with a plurality of such strings arrayed in parallel. The parallel array may also include a load  80  and a rectifier  90  as shown. This arrangement is typical of battery/electrical systems routinely found in telephone central offices. Again, the loading of the system will interfere with the direct measurement of Z 1  by impedance meter  50 .  
         [0042]    However, consider performing the three impedance measurements shown in FIGS. 8, 9, and  10 .  
         [0043]    First measure Zab (FIG. 8) with both sets of Kelvin contacts of impedance meter  50  directly contacting the two terminals, a and b, of the subject cell/battery. These two terminals comprise the first two measuring points of the three-point measurement technique.  
         [0044]    Next measure Zbc (FIG. 9) with one set of Kelvin contacts contacting one of the cell/battery&#39;s terminals, terminal b, and the other set bridging across an adjacent connector and an adjacent cell/battery to contact point C. Point c comprises the third measuring point of the three-point measurement technique.  
         [0045]    Finally, measure Zca (FIG. 10) with one set of Kelvin contacts contacting measuring point c and the other set contacting measuring point a.  
         [0046]    The experimental arrangements depicted in FIGS. 8, 9, and  10  have again divided the system into three impedances, Z 1 , Z 2 , and Z 3 . These three system impedances are identified in FIGS. 8, 9, and  10 . System impedance Z 1  is again the desired impedance of the subject cell/battery. System impedance Z 2  is an arbitrarily-defined adjacent impedance which includes the impedance of both an adjacent cell/battery and an interconnecting conductor; and system impedance Z 3  is the impedance of all of the rest of the battery system—not including system impedances Z 1  and Z 2 .  
         [0047]    The equivalent circuit of FIG. 6 again describes the relationships between system impedances Z 1 , Z 2 , Z 3  and measured impedances Zab, Zbc, and Zca. Accordingly, equations (5)-(7) again explicitly yield Z 1 , Z 2 , and Z 3 . Impedance Z 1  represents the de-embedded subject cell/battery and is of particular interest. The value of Z 1  is again given by equation (5):  
             Z1   =       (       Z                 a                   b   2       +     Z                 b                   c   2       +     Z                 c                   a   2       -       2   ·   Z                   b                   c   ·   Z                   c                 a     -       2   ·   Z                   c                   a   ·   Z                   a                 b     -       2   ·   Z                   a                   b   ·   Z                   b                 c       )       2   ·     (       Z                 a                 b     -     Z                 b                 c     -     Z                 c                 a       )                 (   5   )                               
 
         [0048]    In the example depicted above, the particular choice of measuring point c places both a cell/battery and an interconnecting conductor into impedance Z 2 . The measured value of Z 2  is thus an arbitrary quantity that may be of little interest. However, one could just as well have chosen measuring point c so that impedance Z 2  contains only the impedance of the adjacent interconnecting conductor. In that case, the interconnecting conductor impedance could be of considerable interest. Its value would be explicitly given by equation (6):  
             Z2   =       (       Z                 a                   b   2       +     Z                 b                   c   2       +     Z                 c                   a   2       -       2   ·   Z                   b                   c   ·   Z                   c                 a     -       2   ·   Z                   c                   a   ·   Z                   a                 b     -       2   ·   Z                   a                   b   ·   Z                   b                 c       )       2   ·     (       Z                 b                 c     -     Z                 c                 a     -     Z                 a                 b       )                 (   6   )                               
 
         [0049]    Impedance Z 3  describes the impedance of all of the rest of the battery system—not including impedances Z 1  and Z 2 . Its value is explicitly given by equation (7):  
             Z3   =       (       Z                 a                   b   2       +     Z                 b                   c   2       +     Z                 c                   a   2       -       2   ·   Z                   b                   c   ·   Z                   c                 a     -       2   ·   Z                   c                   a   ·   Z                   a                 b     -       2   ·   Z                   a                   b   ·   Z                   b                 c       )       2   ·     (       Z                 c                 a     -     Z                 a                 b     -     Z                 b                 c       )                 (   7   )                               
 
         [0050]    In principle, these three measurements could be performed in sequence using conventional impedance measuring apparatus such as apparatus disclosed by Champlin in the U.S. patents and patent applications referred to above. Readings could be simply recorded after each measurement, and a hand calculator or computer subsequently employed to evaluate the appropriate equation or equations that de-embed the subject element or elements.  
         [0051]    Alternatively, one could use a special three-point impedance meter  100  connected as shown in FIG. 11. One sees in FIG. 11 that three-point impedance meter  100  possesses three sets of system-contacting Kelvin conductors  110 ,  120 , and  130  which simultaneously contact measuring points a, b, and c, respectively. FIG. 12 discloses further that three-point impedance meter  100  contains a conventional two-point impedance meter  50  adapted to measure the impedance of an isolated element connected between its Kelvin input conductors  60  and  70 . Switching circuitry  140  is interposed between input conductors  60 ,  70  and system-contacting conductors  110 ,  120 ,  130 , and is adapted to selectively connect a pair of system-contacting conductors (either  110  &amp;  120 ,  120  &amp;  130 , or  130  &amp;  110 ) to input conductors  60  &amp;  70 .  
         [0052]    Under programmed control of microprocessor/controller  150 , switching circuitry  140  alternately selects each particular pair of system-contacting conductors and commands impedance meter  50  to measure the impedance between its input conductors  60  and  70 . The resulting three measured impedances are temporarily stored in storage memory  160  and then processed by computation circuitry  170 —which may, in fact, also comprise microprocessor/controller  150 —to determine the subject embedded impedance or impedances using one or more of equations (5),(6), and (7). Three-point impedance meter  100  therefore de-embeds the subject impedances directly, without operator intervention.  
         [0053]    One could also construct measuring apparatus  180  (FIG. 13) which is similar to three-point impedance meter  100 , but is extended to have an arbitrary number n of system-contacting conductors—where n is any integer between 3 and the number of interconnection points in the system. Under programmed control of microprocessor/controller  150 , switching circuitry  140  alternately selects appropriate system-contacting conductors in groups of three. By consecutively performing three-point measurements upon each selected group, and evaluating one or more of equations (5), (6) and (7) after each set of three impedance measurements, extended apparatus  180  could potentially de-embed every element in the entire system without operator intervention.  
         [0054]    [0054]FIG. 14 discloses a flowchart of a control algorithm for de-embedding M single elements using the apparatus of FIG. 13. The algorithm begins at step  200 . Step  210  initializes a measurement counter i, and step  220  initializes an element counter j. At step  230 , a particular pair of system-contacting conductors is selected by switching circuitry  50 . The corresponding impedance between these conductors is measured at step  240  and stored in memory  160  at step  250 . At step  260 , the measurement counter is tested. If it has not reached  3 , the measurement counter is incremented and the measurements are repeated with a different pair of system-contacting conductors. If the measurement counter has reached  3 , computation circuitry  170  calculates the impedance of one de-embedded element from the three measured impedances stored in memory  160  at step  270 . The element counter is then tested at step  280 . If it has not reached M, the element counter is incremented and the procedure is repeated to de-embed another element. However, if the element counter has reached M, all M elements have been de-embedded, and the procedure terminates at step  290 .  
         [0055]    One sees from the discussion regarding FIGS. 8, 9, and  10  that measuring point c can be chosen rather arbitrarily if one is only interested in the impedance of a single element, Z 1 . This single element could be either a cell/battery or an interconnecting conductor. If one desires to additionally measure the impedance of the nearest adjacent element (interconnecting conductor or cell/battery), the interval between b and c (i.e., impedance Z 2 ) must contain only that one adjacent element.  
         [0056]    The general rules to be followed in choosing measuring points can be understood with reference to FIG. 15. Measuring points a and b define the two terminals of a subject element whose impedance Z 1  is desired to be measured. Furthermore, at least one of those two terminals must have no more than one conducting path proceeding from it. That single-path terminal is chosen as measuring point b. Measuring point c can then be any point along this single conducting path that can be reached without encountering an intervening branching path. There can be additional paths branching from point c itself; as there can also be from measuring point a. These two possibilities are illustrated in FIG. 12. However no paths can branch from point b or from any intermediate junction point between b and c.  
         [0057]    If only the value of Z 1  is desired, the number of cells/batteries and conductors disposed between measuring point b and measuring point c is unrestricted. However, as a result of the “no-branch” rule, an element on the end of a series string in a multi-string parallel array must have its measuring point c on the side of the element that is farthest from the parallel connection. An interior element of a series string, however, can have its measuring point c on either side.  
         [0058]    An extension of this three-point measurement technique can be used to de-embed elements of parallel strings of batteries—such as are frequently employed in trucks and heavy equipment.  
         [0059]    First, consider a simple system of two cells/batteries connected in parallel. FIG. 16 a  depicts such a system and identifies a choice of measuring points that simultaneously de-embeds the cell/battery on the right of FIG. 16 a  and the interconnecting conductor on the bottom. With the experimental arrangement shown, impedances Z 1  and Z 2  are given by equations (5) and (6), respectively. FIG. 16 b  shows a choice of measuring points that simultaneously de-embeds the other two elements. With the experimental arrangement shown in FIG. 16 b , impedances Z 1 ′ and Z 2 ′ are given by equations (5) and (6), respectively. Thus, all four elements of this simple parallel system can be de-embedded with two sets of three-point measurements.  
         [0060]    Multi-cell/battery parallel strings present a special challenge. Both terminals of a cell/battery in the interior of a parallel string have more than one conducting path leading from them. Accordingly, neither terminal satisfies the “no-branch rule” that must be satisfied by a measuring point b. However, the standard three-point measurement technique can still be applied to the interconnecting conductors and to the two cell/batteries on the ends of the string; and an extended form of the technique, a four-point, five-measurement, technique, can be applied to the cells/batteries in the interior.  
         [0061]    First consider FIGS. 17 a  and  17   b . These figures identify measuring points used to de-embed the cells/batteries and interconnecting conductors on the ends of a multi-element parallel string. With the experimental arrangements shown, cell/battery impedances Z 1  and Z 1 ′ are given by equation (5) and interconnecting conductor impedances Z 2  and Z 2 ′ are given by equation (6). By simply re-arranging the measuring points, the impedances of the other two interconnecting conductors at the ends of this string can be similarly determined.  
         [0062]    Now consider FIGS. 18 a  and  18   b . These figures depict two experimental three-point measurement sets performed on a cell/battery and its interconnecting conductors disposed in the interior of a parallel string. Note that measuring point c shifts from one side of the subject cell/battery to the other in the two experiments. However, Zab, the impedance measured between points a and b is the same in the two experiments. Thus, only five measurements are required to perform the two experiments.  
         [0063]    Equation (6) yields the interconnecting conductor impedances Z 2  and Z 2 ′ in the two experiments. However, because measuring point b does not satisfy the “no-branch rule”, equation (5) does not yield Z 1  directly in either experiment. Instead, equation (5) yields Z 1  in parallel with Z 4  in the first experiment and yields Z 1  in parallel with Z 4 ′ in the second experiment. However, Z 4 =Z 2 ′+Z 3 ′ is known from equations (6) and (7) of the second experiment, and Z 4 ′=Z 2 +Z 3  is known from equations (6) and (7) of the first experiment. Accordingly, by combining results of the two experiments, one can write the subject unknown cell/battery impedance Z 1  as either  
               Z1   =       M1   ·     (       Z2   ′     +     Z3   ′       )           (       Z2   ′     +     Z3   ′       )     -   M1              
          o                 r             (   8   )               Z1   =         M1   ′     ·     (     Z2   +   Z3     )           (     Z2   +   Z3     )     -     M1   ′                 (   9   )                               
 
         [0064]    where M 1 , Z 2 ,and Z 3  are the results of evaluating equations (5), (6), and (7), respectively, in the first experiment, and M 1 ′, Z 2 ′, and Z 3 ′ are the results of evaluating equations (5), (6), and (7), respectively, in the second experiment.  
         [0065]    A special four-point impedance meter similar to three-point impedance meter  100  disclosed in FIG. 12, but having four sets of connections, could advantageously perform this four-point, five-measurement, procedure and de-embed the interior cell/battery without operator intervention.  
         [0066]    For purposes of clarity, the above discussions have only considered measuring complex impedance Z. However, it will be apparent to workers skilled in the art that the disclosed measurement techniques apply equally well to measuring the reciprocal of complex impedance, complex admittance Y. Equations comparable to equation (5), (6), and (7) that give the unknown admittances Y 1 , Y 2 , and Y 3  in terms of measured admittances Yab, Ybc, and Yac can be written  
             Y1   =       2        YabYbcYca        (     YbcYca   -   YabYbc   -   YcaYab     )               Yab   2          Ybc   2       +       Ybc   2          Yca   2       +       Yca   2          Yab   2       -     2        (         Yab   2        YbcYca     +       YabYbc   2        Yca     +     YabYbcYca   2       )                   (   10   )                 Y2   =       2        YabYbcYca        (     YcaYab   -   YbcYca   -   YabYbc     )               Yab   2          Ybc   2       +       Ybc   2          Yca   2       +       Yca   2          Yab   2       -     2        (         Yab   2        YbcYca     +       YabYbc   2        Yca     +     YabYbcYca   2       )                  
        and           (   11   )               Y3   =       2        YabYbcYca        (     YabYbc   -   YcaYab   -   YbcYca     )               Yab   2          Ybc   2       +       Ybc   2          Yca   2       +       Yca   2          Yab   2       -     2        (         Yab   2        YbcYca     +       YabYbc   2        Yca     +     YabYbcYca   2       )                   (   12   )                               
 
         [0067]    Furthermore, if reactive and susceptive effects can be ignored, the disclosed measuring techniques likewise apply to measuring real dynamic resistance R and real dynamic conductance G. Equations comparable to equation (5), (6), and (7) that give the unknown dynamic resistances R 1 , R 2 , and R 3  in terms of measured dynamic resistances Rab, Rbc, and Rca are  
             R1   =       (       Rab   2     +     Rbc   2     +     Rca   2     -     2   ·   Rbc   ·   Rca     -     2   ·   Rca   ·   Rab     -     2   ·   Rab   ·   Rbc       )       2   ·     (     Rab   -   Rbc   -   Rca     )                 (   13   )                 R2   =       (       Rab   2     +     Rbc   2     +     Rca   2     -     2   ·   Rbc   ·   Rca     -     2   ·   Rca   ·   Rab     -     2   ·   Rab   ·   Rbc       )       2   ·     (     Rbc   -   Rca   -   Rab     )                
        and           (   14   )               R3   =       (       Rab   2     +     Rbc   2     +     Rca   2     -     2   ·   Rbc   ·   Rca     -     2   ·   Rca   ·   Rab     -     2   ·   Rab   ·   Rbc       )       2   ·     (     Rca   -   Rab   -   Rbc     )                 (   15   )                               
 
         [0068]    Similarly, equations comparable to equations (5), (6), and (7) that give the unknown dynamic conductances G 1 , G 2 , and G 3  in terms of measured dynamic conductances, Gab, Gbc and Gca are  
             G1   =       2        GabGbcGca        (     GbcGca   -   GabGbc   -   GcaGab     )                     Gab   2          Gbc   2       +       Gbc   2          Gca   2       +       Gca   2          Gab   2       -               2        (         Gab   2        GbcGca     +       GabGbc   2        Gca     +     GabGbcGca   2       )                       (   16   )               G2   =         2        GabGbcGca        (     GcaGab   -   GbcGca   -   GabGbc     )                     Gab   2          Gbc   2       +       Gbc   2          Gca   2       +       Gca   2          Gab   2       -               2        (         Gab   2        GbcGca     +       GabGbc   2        Gca     +     GabGbcGca   2       )                             and             (   17   )               G3   =       2        GabGbcGca        (     GabGbc   -   GcaGab   -   GbcGca     )                     Gab   2          Gbc   2       +       Gbc   2          Gca   2       +       Gca   2          Gab   2       -               2   -     (         Gab   2        GbcGca     +       GabGbc   2        Gca     +     GabGbcGca   2       )                       (   18   )                               
 
         [0069]    Since all four quantities, Z, Y, R, and G are measured with time-varying signals, they are referred to collectively as “dynamic parameters”.  
         [0070]    Although the present invention has been described with reference to preferred embodiments, workers skilled in the art will recognize that changes can be made in form and detail without departing from the true spirit and scope of the invention. For example, single conductor contacts rather than Kelvin contacts could be employed under appropriate circumstances. Three- and four-point testing could be performed using analog circuitry, digital circuitry, or hybrid combinations of analog and digital circuitry. The necessary calculations could be performed with a hand calculator, a computer, or an on-board processor. Measurements could be simply implemented with hand-held test equipment carried to a site. They could also be implemented with integrated measuring apparatus distributed throughout an entire battery system and configured to automatically de-embed and monitor various elements of the system. These and other variations of embodiments are believed to be well within the scope of the present invention and are intended to be covered by the appended claims.