Abstract:
A “three-point” measurement technique effectively removes system effects to determine impedance, admittance, resistance, or conductance of an individual cell, battery, or interconnecting conductor embedded in a series or series-parallel electrochemical battery or fuel cell system. Three electrical contact points are defined. Two of these points bound the subject element. The third point is separated from the other two by a conducting path that may include one or more cells or batteries. By measuring dynamic parameters between alternate pairs of contact points, three dynamic parameter measurements are acquired. A mathematical computation combines the measurements and determines the dynamic parameter of a subject element as if it were alone—thus effectively “de-embedding” the subject element from the remainder of the system. A “four-point” extension of this technique permits measuring a dynamic parameter of a cell/battery disposed in a multiple-unit string of parallel-connected cells/batteries.

Description:
The present application is a Continuation-In-Part of U.S. patent application Ser. No. 09/862,783, filed May 21, 2001 now abandoned, which is a Continuation-In-Part of U.S. patent application Ser. No. 09/662,092, filed Sep. 14, 2000 now abandoned, the content of which is hereby incorporated by reference in its entirety. 

   BACKGROUND OF THE INVENTION 
   The present invention relates to electronically testing electrochemical cells and batteries including fuel cells and batteries. More specifically, it relates to accurately determining a dynamic parameter of a particular element, i.e., cell, battery, or interconnecting conductor, embedded in a cell/battery system as if it were isolated from the system. Such a determination is referred to as “de-embedding” the subject element. 
   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. 
   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. Burkum, et al., in U.S. Pat. No. 4,697,134, described one approach to this challenge. 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 method of Burkum, et al., also ignores the influence of series strings that are in parallel with the desired string in multi-string installations—an arrangement commonly found in telephone central offices. 
   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. Nos. 6,002,238, 6,172,483, 6,262,563, and 6,294,896. 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 remainder of the system may influence the results of the measurement. 
   The inventions disclosed herein 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 accurately determining a dynamic parameter, i.e., impedance, admittance, resistance, or conductance, of an embedded element without actually disconnecting that element from the system. 
   SUMMARY OF THE INVENTION 
   The present invention employs a “three-point” measurement technique to determine a dynamic parameter of an individual element (i.e., cell/battery or interconnecting conductor) embedded in a series, or series-parallel, electrochemical cell, battery, or fuel cell system. Three electrical contact points are defined. Two of these points bound the subject cell/battery or interconnecting conductor. The third point is separated from the other two by a conducting path that may include one or more cells or batteries. By measuring dynamic parameters between alternate pairs of contact points, three dynamic parameter measurements are acquired. A mathematical computation combines the measurements and determines the dynamic parameter of a subject element as if it were alone—thus effectively “de-embedding” the subject element from the remainder of the system. A “four-point” extension of this technique permits measuring a dynamic parameter of a cell, battery, or interconnecting conductor disposed in a multiple-unit string of parallel-connected cells or batteries. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a schematic representation depicting the impedance of a cell/battery being measured while the cell/battery is connected to a load. 
       FIG. 2  is a simplified equivalent circuit representing the measurement arrangement of FIG.  1 . 
       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. 
       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. 
       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. 
       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 . 
       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. 
       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. 
       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. 
       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. 
       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. 
       FIG. 12  is a block diagram representation of the special “three-point” impedance meter depicted in FIG.  11 . 
       FIG. 13  is a block diagram representation of a special “n-point” impedance meter in accordance with a particular aspect of the present invention. 
       FIG. 14  is a flow chart of a control algorithm for “de-embedding” M single elements using the apparatus of FIG.  13 . 
       FIG. 15  represents a part of a battery system and demonstrates general rules to be followed when choosing electrical contact points. 
       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. 
       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. 
       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. 
       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. 
       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. 
       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 
   Consider FIG.  1 . This figure illustrates measuring the impedance of a cell/battery  10  embedded in a very simple 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 four patents referenced above, couples to the two terminals of cell/battery  10  with Kelvin input probe  60  and Kelvin input probe  70 . As is well known, Kelvin probes comprise two separate electrical connections to each contact point—one for current and one for voltage. Their purpose is to negate the effects of contact and lead-wire resistance. Although Kelvin probes 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 rely upon Kelvin probes. Single-conductor connections to each contact point will also suffice if impedance values are sufficiently large. 
     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  is in parallel with impedance Z 1 , the impedance Zm “seen” by impedance meter  50  is not actually Z 1  but is instead the composite impedance: 
             Zm   =       Z1   ·     (       Z   C1     +     Z   C2     +     Z   L       )         Z1   +     Z   C1     +     Z   C2     +     Z   L                 (   1   )               
The influence of impedances Z C1 , Z C2 , and Z L  upon the measured impedance Zm is clearly observed in equation (1).
 
   Now consider performing three impedance measurements as shown in  FIGS. 3 ,  4 , and  5 .
         First measure Zab ( FIG. 3 ) with both Kelvin input probes of impedance meter  50  coupled to the two terminals, a and b, of the cell/battery being measured. Terminals a and b effectively “bound” cell/battery impedance Z 1  and comprise the first two electrical contact points of a three-point measurement technique.   Next measure Zbc ( FIG. 4 ) with Kelvin probe  60  coupled to terminal b, and Kelvin probe  70  bridging across interconnecting conductor  40  to couple to load  20  at contact point c. Point c comprises the third electrical contact point of the three-point measurement technique.   Finally, measure Zca ( FIG. 5 ) with one Kelvin probe coupled to point c and the other coupled to point a.       

     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 electrical contact point c 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. The alternative choice of contact point c would make Z 2 =Z C1  and Z 3 =Z L +Z C2 . 
   One can easily show from  FIG. 6  that the three measured impedances are given by: 
             Zab   =       Z1   ·     (     Z2   +   Z3     )         Z1   +   Z2   +   Z3               (   2   )                 Zbc   =       Z2   ·     (     Z3   +   Z1     )         Z1   +   Z2   +   Z3         ⁢     
     ⁢   and           (   3   )               Zca   =       Z3   ·     (     Z1   +   Z2     )         Z1   +   Z2   +   Z3               (   4   )             
 
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   =             (       Zab   2     +     Zbc   2     +     Zca   2     -     2   ·   Zbc   ·   Zca     -                     2   ·   Zca   ·   Zab     -     2   ·   Zab   ·   Zbc       )             2   ·     (     Zab   -   Zbc   -   Zca     )                 (   5   )                 Z2   =             (       Zab   2     +     Zbc   2     +     Zca   2     -     2   ·   Zbc   ·   Zca     -                     2   ·   Zca   ·   Zab     -     2   ·   Zab   ·   Zbc       )             2   ·     (     Zbc   -   Zca   -   Zab     )           ⁢     
     ⁢   and           (   6   )               Z3   =             (       Zab   2     +     Zbc   2     +     Zca   2     -     2   ·   Zbc   ·   Zca     -                     2   ·   Zca   ·   Zab     -     2   ·   Zab   ·   Zbc       )             2   ·     (     Zca   -   Zab   -   Zbc     )                 (   7   )             
 
   Equation (5) effectively “de-embeds” cell/battery  10  since Z 1  would be its measured impedance if it were, in fact, completely disconnected from the system. 
   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 systems routinely found in telephone central offices. Again, the loading associated with the remainder of the system can hinder accurate measurement of Z 1  by impedance meter  50 . 
   However, consider performing the three impedance measurements shown in  FIGS. 8 ,  9 , and  10 .
         First measure Zab ( FIG. 8 ) with both Kelvin probes of impedance meter  50  coupled to the two terminals, a and b, of the subject cell/battery. These two terminals effectively “bound” the desired impedance Z 1  and comprise the first two electrical contact points of the three-point measurement technique.   Next measure Zbc ( FIG. 9 ) with Kelvin probe  60  coupled to contact point b, and Kelvin probe  70  bridging across an adjacent connector and an adjacent cell/battery to couple to contact point c. Point c comprises the third contact point of the three-point measurement technique.   Finally, measure Zca ( FIG. 10 ) with Kelvin probe  70  coupled to point c and Kelvin probe  60  coupled to point a.       

   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 cell/battery  10 . 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 . 
   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   =             (       Zab   2     +     Zbc   2     +     Zca   2     -     2   ·   Zbc   ·   Zca     -                     2   ·   Zca   ·   Zab     -     2   ·   Zab   ·   Zbc       )             2   ·     (     Zab   -   Zbc   -   Zca     )                 (   5   )             
 
   In the example depicted above, the particular choice of contact 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 point c so that impedance Z 2  contains only the impedance of the adjacent interconnecting conductor. In that case, impedance Z 2  could be of considerable interest. Its value would be explicitly given by equation (6): 
             Z2   =             (       Zab   2     +     Zbc   2     +     Zca   2     -     2   ·   Zbc   ·   Zca     -                     2   ·   Zca   ·   Zab     -     2   ·   Zab   ·   Zbc       )             2   ·     (     Zbc   -   Zca   -   Zab     )                 (   6   )             
 
   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   =             (       Zab   2     +     Zbc   2     +     Zca   2     -     2   ·   Zbc   ·   Zca     -                     2   ·   Zca   ·   Zab     -     2   ·   Zab   ·   Zbc       )             2   ·     (     Zca   -   Zab   -   Zbc     )                 (   7   )             
 
   In principle, the three measurements could be performed in sequence using conventional impedance measuring circuitry such as the circuitry disclosed by Champlin in the U.S. patents referenced above. The complete measuring apparatus could comprise such circuitry along with a pencil and pad to record the readings and an external hand calculator or computer to evaluate the appropriate equation or equations that “de-embed” the subject element or elements. Alternatively, the apparatus could contain onboard memory circuitry to store the measurements and onboard computation circuitry to perform the “de-embedding” calculations after the required measurements have been made. 
   In addition, one could employ 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 probes  110 ,  120 , and  130  which simultaneously couple to electrical contact 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 coupled between its Kelvin input probes  60  and  70 . Switching or multiplexing circuitry  140  is interposed between input probes  60 ,  70  and system-contacting probes  110 ,  120 ,  130 , and is adapted to selectively couple a pair of system-contacting probes (either  110  &amp;  120 ,  120  &amp;  130 , or  130  &amp;  110 ) to input probes  60  &amp;  70 . 
   Under programmed control of microprocessor/controller  150 , switching/multiplexing circuitry  140  alternately couples each pair of system-contacting probes to input probes  60  and  70 , and impedance meter  50  measures the impedance between its input probes  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. 
   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 probes—where n is any integer between 3 and the number of interconnection points in the system. Under programmed control of microprocessor/controller  150 , switching/multiplexing circuitry  140  alternately selects appropriate system-contacting probes 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  can potentially “de-embed” every element in the entire system without operator intervention. 
     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 probes is selected by switching/multiplexing circuitry  140 . The corresponding impedance between these probes 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 probes. If the measurement counter has reached 3, computation circuitry  170  calculates the impedance of one 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 . 
   One sees from the discussion regarding  FIGS. 8 ,  9 , and  10  that electrical contact 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. 
   The general rules to be followed in choosing electrical contact points can be understood with reference to FIG.  15 . Contact points a and b must “bound” the subject element whose impedance Z 1  is desired to be measured. Furthermore, at least one of those two points must have no more than one conducting path proceeding from it. That single-path point is chosen as contact point b. Contact 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 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. 
   If only the value of Z 1  is desired, the number of cells/batteries and conductors disposed between contact point b and contact 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 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 contact point c on either side. 
   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. 
   First, consider a simple system of two cells/batteries connected in parallel.  FIG. 16   a  depicts such a system and identifies a choice of electrical contact 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 electrical contact 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. 
   Parallel strings of cells/batteries 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 contacting 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. 
   First consider  FIGS. 17   a  and  17   b . These figures identify three electrical contact 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 contact points, the impedances of the other two interconnecting conductors at the ends of this string can be similarly determined. 
   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 contact 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. 
   Equation (6) unambiguously yields the interconnecting conductor impedances Z 2  and Z 2 ′ in the two experiments. However, because contact 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         ⁢     
     ⁢   or           (   8   )               Z1   =         M1   ′     ·     (     Z2   +   Z3     )           (     Z2   +   Z3     )     -     M1   ′                 (   9   )             
 
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.
 
   A special four-point impedance meter similar to three-point impedance meter  100  disclosed in  FIG. 12 , but having four system-contacting probes, could advantageously perform this four-point, five-measurement, procedure and “de-embed” any interior cell/battery in a parallel string without operator intervention. 
   For purposes of clarity, the above discussions have only considered complex impedance Z. However, it will be apparent to workers skilled in the art that the disclosed 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   )             
 
   Furthermore, if reactive and susceptive effects can be ignored, the disclosed techniques can likewise be applied to measuring real dynamic resistance R and real dynamic conductance G. Equations comparable to equation (5), (6), and (7) that give 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   )             
 
   Similarly, equations comparable to equations (5), (6), and (7) that give 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   )             
 
Since all four quantities, Z, Y, R, and G are measured with time-varying signals, they are referred to collectively as “dynamic parameters”.
 
   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 probes rather than Kelvin probes could be employed if element impedances are sufficiently large. The required measurements could be simply performed using conventional measuring apparatus, results written down after each measurement and a hand calculator or computer subsequently employed to evaluate the appropriate equations. Alternatively, the measuring apparatus could itself contain onboard memory circuitry to store the measurements and onboard computation circuitry to perform the “de-embedding” calculations. Furthermore, the system-contacting probes could be multiplexed or switched, thus permitting the measuring apparatus to couple to the system at all of the desired contact points simultaneously. Although specific “de-embedding” examples employing three and five measurements have been disclosed, any number of measurements could actually be employed. The measurement steps could occur in any order or occur substantially simultaneously. “De-embedding” could be implemented with hand-held equipment carried to a site. It could also be implemented with permanently integrated measuring apparatus distributed throughout the entire system and adapted to automatically “de-embed” and monitor various elements of the system. These and other variations are believed to be well within the scope of the present invention and are intended to be covered by the appended claims.