Method and apparatus for testing cells and batteries embedded in series/parallel systems

[0001] 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.

[0002] 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.

[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. 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.

[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. 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.

[0006] 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.

[0007] 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.

[0029] 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 Z1, 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.

[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 is in parallel with impedance Z1, the impedance Zm “seen” by impedance meter 50 is not actually Z1 but is instead the composite impedance: $\begin{array}{cc}\mathrm{Zm}=\frac{\mathrm{Z1}·\left(Z_{\mathrm{C1}}+Z_{\mathrm{C2}}+Z_L\right)}{\mathrm{Z1}+Z_{\mathrm{C1}}+Z_{\mathrm{C2}}+Z_L}& \left(1\right)\end{array}$
The influence of impedances Z_C1, Z_C2, and Z_L upon the measured impedance Zm is clearly observed in equation (1).

[0031] Now consider performing three impedance measurements as shown in FIGS. 3, 4, and 5.

• • [0032] 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 Z1 and comprise the first two electrical contact points of a three-point measurement technique.
  • [0033] 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.
  • [0034] Finally, measure Zca (FIG. 5) with one Kelvin probe coupled to point c and the other coupled to point a.

[0035] FIG. 6 shows an equivalent circuit relating the three measured impedances, Zab, Zbc, and Zca, to the three system impedances Z1, Z2, and Z3. Note that this particular choice of electrical contact point c makes system impedance Z2 equal the interconnecting conductance impedance Z_C2 while system impedance Z3=Z_L+Z_C1 combines the load impedance with the other interconnecting conductor impedance. The alternative choice of contact point c would make Z2=Z_C1 and Z3=Z_L+Z_C2.

[0036] One can easily show from FIG. 6 that the three measured impedances are given by: $\begin{array}{cc}\mathrm{Zab}=\frac{\mathrm{Z1}·\left(\mathrm{Z2}+\mathrm{Z3}\right)}{\mathrm{Z1}+\mathrm{Z2}+\mathrm{Z3}}& \left(2\right)\\ \mathrm{Zbc}=\frac{\mathrm{Z2}·\left(\mathrm{Z3}+\mathrm{Z1}\right)}{\mathrm{Z1}+\mathrm{Z2}+\mathrm{Z3}}\mathrm{and}& \left(3\right)\\ \mathrm{Zca}=\frac{\mathrm{Z3}·\left(\mathrm{Z1}+\mathrm{Z2}\right)}{\mathrm{Z1}+\mathrm{Z2}+\mathrm{Z3}}& \left(4\right)\end{array}$
These three equations can be inverted mathematically to yield explicit expressions for Z1, Z2, and Z3, in terms of the measured quantities Zab, Zbc, and Zca. The results are $\begin{array}{cc}\mathrm{Z1}=\frac{\begin{array}{c}({\mathrm{Zab}}^2+{\mathrm{Zbc}}^2+{\mathrm{Zca}}^2-2·\mathrm{Zbc}·\mathrm{Zca}-\\ 2·\mathrm{Zca}·\mathrm{Zab}-2·\mathrm{Zab}·\mathrm{Zbc})\end{array}}{2·\left(\mathrm{Zab}-\mathrm{Zbc}-\mathrm{Zca}\right)}& \left(5\right)\\ \mathrm{Z2}=\frac{\begin{array}{c}({\mathrm{Zab}}^2+{\mathrm{Zbc}}^2+{\mathrm{Zca}}^2-2·\mathrm{Zbc}·\mathrm{Zca}-\\ 2·\mathrm{Zca}·\mathrm{Zab}-2·\mathrm{Zab}·\mathrm{Zbc})\end{array}}{2·\left(\mathrm{Zbc}-\mathrm{Zca}-\mathrm{Zab}\right)}\mathrm{and}& \left(6\right)\\ \mathrm{Z3}=\frac{\begin{array}{c}({\mathrm{Zab}}^2+{\mathrm{Zbc}}^2+{\mathrm{Zca}}^2-2·\mathrm{Zbc}·\mathrm{Zca}-\\ 2·\mathrm{Zca}·\mathrm{Zab}-2·\mathrm{Zab}·\mathrm{Zbc})\end{array}}{2·\left(\mathrm{Zca}-\mathrm{Zab}-\mathrm{Zbc}\right)}& \left(7\right)\end{array}$

[0037] Equation (5) effectively “de-embeds” cell/battery 10 since Z1 would be its measured impedance if it were, in fact, completely disconnected from the system.

[0038] 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 Z1 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 Z1 by impedance meter 50.

[0039] However, consider performing the three impedance measurements shown in FIGS. 8, 9, and 10.

• • [0040] 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 Z1 and comprise the first two electrical contact points of the three-point measurement technique.
  • [0041] 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.
  • [0042] Finally, measure Zca (FIG. 10) with Kelvin probe 70 coupled to point c and Kelvin probe 60 coupled to point a.

[0043] The experimental arrangements depicted in FIGS. 8, 9, and 10 have again divided the system into three impedances, Z1, Z2, and Z3. These three system impedances are identified in FIGS. 8, 9, and 10. System impedance Z1 is again the desired impedance of cell/battery 10. System impedance Z2 is an arbitrarily-defined adjacent impedance which includes the impedance of both an adjacent cell/battery and an interconnecting conductor; and system impedance Z3 is the impedance of all of the rest of the battery system—not including system impedances Z1 and Z2.

[0044] The equivalent circuit of FIG. 6 again describes the relationships between system impedances Z1, Z2, Z3 and measured impedances Zab, Zbc, and Zca. Accordingly, equations (5)-(7) again explicitly yield Z1, Z2, and Z3. Impedance Z1 represents the “de-embedded” subject cell/battery and is of particular interest. The value of Z1 is again given by equation (5): $\begin{array}{cc}\mathrm{Z1}=\frac{\begin{array}{c}({\mathrm{Zab}}^2+{\mathrm{Zbc}}^2+{\mathrm{Zca}}^2-2·\mathrm{Zbc}·\mathrm{Zca}-\\ 2·\mathrm{Zca}·\mathrm{Zab}-2·\mathrm{Zab}·\mathrm{Zbc})\end{array}}{2·\left(\mathrm{Zab}-\mathrm{Zbc}-\mathrm{Zca}\right)}& \left(5\right)\end{array}$

[0045] In the example depicted above, the particular choice of contact point c places both a cell/battery and an interconnecting conductor into impedance Z2. The measured value of Z2 is thus an arbitrary quantity that may be of little interest. However, one could just as well have chosen point c so that impedance Z2 contains only the impedance of the adjacent interconnecting conductor. In that case, impedance Z2 could be of considerable interest. Its value would be explicitly given by equation (6): $\begin{array}{cc}\mathrm{Z2}=\frac{\begin{array}{c}({\mathrm{Zab}}^2+{\mathrm{Zbc}}^2+{\mathrm{Zca}}^2-2·\mathrm{Zbc}·\mathrm{Zca}-\\ 2·\mathrm{Zca}·\mathrm{Zab}-2·\mathrm{Zab}·\mathrm{Zbc})\end{array}}{2·\left(\mathrm{Zbc}-\mathrm{Zca}-\mathrm{Zab}\right)}& \left(6\right)\end{array}$

[0046] Impedance Z3 describes the impedance of all of the rest of the battery system—not including impedances Z1 and Z2. Its value is explicitly given by equation (7): $\begin{array}{cc}\mathrm{Z3}=\frac{\begin{array}{c}({\mathrm{Zab}}^2+{\mathrm{Zbc}}^2+{\mathrm{Zca}}^2-2·\mathrm{Zbc}·\mathrm{Zca}-\\ 2·\mathrm{Zca}·\mathrm{Zab}-2·\mathrm{Zab}·\mathrm{Zbc})\end{array}}{2·\left(\mathrm{Zca}-\mathrm{Zab}-\mathrm{Zbc}\right)}& \left(7\right)\end{array}$

[0047] 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.

[0048] 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 & 120, 120 & 130, or 130 & 110) to input probes 60 & 70.

[0049] 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.

[0050] 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.

[0051] 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.

[0052] 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, Z1. 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 Z2) must contain only that one adjacent element.

[0053] 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 Z1 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.

[0054] If only the value of Z1 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.

[0055] 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.

[0056] First, consider a simple system of two cells/batteries connected in parallel. FIG. 16a depicts such a system and identifies a choice of electrical contact points that simultaneously “de-embeds” the cell/battery on the right of FIG. 16a and the interconnecting conductor on the bottom. With the experimental arrangement shown, impedances Z1 and Z2 are given by equations (5) and (6), respectively. FIG. 16b shows a choice of electrical contact points that simultaneously “de-embeds” the other two elements. With the experimental arrangement shown in FIG. 16b, impedances Z1′ and Z2′ 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.

[0057] 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.

[0058] First consider FIGS. 17a and 17b. 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 Z1 and Z1′ are given by equation (5) and interconnecting conductor impedances Z2 and Z2′ 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.

[0059] Now consider FIGS. 18a and 18b. 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.

[0060] Equation (6) unambiguously yields the interconnecting conductor impedances Z2 and Z2′ in the two experiments. However, because contact point b does not satisfy the “no-branch rule”, equation (5) does not yield Z1 directly in either experiment. Instead, equation (5) yields Z1 in parallel with Z4 in the first experiment and yields Z1 in parallel with Z4′ in the second experiment. However, Z4=Z2′+Z3′ is known from equations (6) and (7) of the second experiment, and Z4′=Z2+Z3 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 Z1 as either $\begin{array}{cc}\mathrm{Z1}=\frac{\mathrm{M1}·\left({\mathrm{Z2}}^{\prime }+{\mathrm{Z3}}^{\prime }\right)}{\left({\mathrm{Z2}}^{\prime }+{\mathrm{Z3}}^{\prime }\right)-\mathrm{M1}}\mathrm{or}& \left(8\right)\\ \mathrm{Z1}=\frac{{\mathrm{M1}}^{\prime }·\left(\mathrm{Z2}+\mathrm{Z3}\right)}{\left(\mathrm{Z2}+\mathrm{Z3}\right)-{\mathrm{M1}}^{\prime }}& \left(9\right)\end{array}$
where M1, Z2, and Z3 are the results of evaluating equations (5), (6), and (7), respectively, in the first experiment, and M1′, Z2′, and Z3′ are the results of evaluating equations (5), (6), and (7), respectively, in the second experiment.

[0061] 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.

[0062] 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 Y1, Y2, and Y3 in terms of measured admittances Yab, Ybc, and Yac can be written $\begin{array}{cc}\mathrm{Y1}=\frac{2\mathrm{YabYbcYca}\left(\mathrm{YbcYca}-\mathrm{YabYbc}-\mathrm{YcaYab}\right)}{\begin{array}{c}{\mathrm{Yab}}^2{\mathrm{Ybc}}^2+{\mathrm{Ybc}}^2{\mathrm{Yca}}^2+{\mathrm{Yca}}^2{\mathrm{Yab}}^2-\\ 2\left({\mathrm{Yab}}^2\mathrm{YbcYca}+{\mathrm{YabYbc}}^2\mathrm{Yca}+{\mathrm{YabYbcYca}}^2\right)\end{array}}& \left(10\right)\\ \mathrm{Y2}=\frac{2\mathrm{YabYbcYca}\left(\mathrm{YcaYab}-\mathrm{YbcYca}-\mathrm{YabYbc}\right)}{\begin{array}{c}{\mathrm{Yab}}^2{\mathrm{Ybc}}^2+{\mathrm{Ybc}}^2{\mathrm{Yca}}^2+{\mathrm{Yca}}^2{\mathrm{Yab}}^2-\\ 2\left({\mathrm{Yab}}^2\mathrm{YbcYca}+{\mathrm{YabYbc}}^2\mathrm{Yca}+{\mathrm{YabYbcYca}}^2\right)\end{array}}\mathrm{and}& \left(11\right)\\ \mathrm{Y3}=\frac{2\mathrm{YabYbcYca}\left(\mathrm{YabYbc}-\mathrm{YcaYab}-\mathrm{YbcYca}\right)}{\begin{array}{c}{\mathrm{Yab}}^2{\mathrm{Ybc}}^2+{\mathrm{Ybc}}^2{\mathrm{Yca}}^2+{\mathrm{Yca}}^2{\mathrm{Yab}}^2-\\ 2\left({\mathrm{Yab}}^2\mathrm{YbcYca}+{\mathrm{YabYbc}}^2\mathrm{Yca}+{\mathrm{YabYbcYca}}^2\right)\end{array}}& \left(12\right)\end{array}$

[0063] 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 R1, R2, and R3 in terms of measured dynamic resistances Rab, Rbc, and Rca are $\begin{array}{cc}\mathrm{R1}=\frac{\begin{array}{c}({\mathrm{Rab}}^2+{\mathrm{Rbc}}^2+{\mathrm{Rca}}^2-2·\mathrm{Rbc}·\mathrm{Rca}-\\ 2·\mathrm{Rca}·\mathrm{Rab}-2·\mathrm{Rab}·\mathrm{Rbc})\end{array}}{2·\left(\mathrm{Rab}-\mathrm{Rbc}-\mathrm{Rca}\right)}& \left(13\right)\\ \mathrm{R2}=\frac{\begin{array}{c}({\mathrm{Rab}}^2+{\mathrm{Rbc}}^2+{\mathrm{Rca}}^2-2·\mathrm{Rbc}·\mathrm{Rca}-\\ 2·\mathrm{Rca}·\mathrm{Rab}-2·\mathrm{Rab}·\mathrm{Rbc})\end{array}}{2·\left(\mathrm{Rbc}-\mathrm{Rca}-\mathrm{Rab}\right)}\mathrm{and}& \left(14\right)\\ \mathrm{R3}=\frac{\begin{array}{c}({\mathrm{Rab}}^2+{\mathrm{Rbc}}^2+{\mathrm{Rca}}^2-2·\mathrm{Rbc}·\mathrm{Rca}-\\ 2·\mathrm{Rca}·\mathrm{Rab}-2·\mathrm{Rab}·\mathrm{Rbc})\end{array}}{2·\left(\mathrm{Rca}-\mathrm{Rab}-\mathrm{Rbc}\right)}& \left(15\right)\end{array}$

[0064] Similarly, equations comparable to equations (5), (6), and (7) that give unknown dynamic conductances G1, G2, and G3 in terms of measured dynamic conductances, Gab, Gbc and Gca are $\begin{array}{cc}\mathrm{G1}=\frac{2\mathrm{GabGbcGca}\left(\mathrm{GbcGca}-\mathrm{GabGbc}-\mathrm{GcaGab}\right)}{\begin{array}{c}{\mathrm{Gab}}^2{\mathrm{Gbc}}^2+{\mathrm{Gbc}}^2{\mathrm{Gca}}^2+{\mathrm{Gca}}^2{\mathrm{Gab}}^2-\\ 2\left({\mathrm{Gab}}^2\mathrm{GbcGca}+{\mathrm{GabGbc}}^2\mathrm{Gca}+{\mathrm{GabGbcGca}}^2\right)\end{array}}& \left(16\right)\\ \mathrm{G2}=\frac{2\mathrm{GabGbcGca}\left(\mathrm{GcaGab}-\mathrm{GbcGca}-\mathrm{GabGbc}\right)}{\begin{array}{c}{\mathrm{Gab}}^2{\mathrm{Gbc}}^2+{\mathrm{Gbc}}^2{\mathrm{Gca}}^2+{\mathrm{Gca}}^2{\mathrm{Gab}}^2-\\ 2\left({\mathrm{Gab}}^2\mathrm{GbcGca}+{\mathrm{GabGbc}}^2\mathrm{Gca}+{\mathrm{GabGbcGca}}^2\right)\end{array}}\mathrm{and}& \left(17\right)\\ \mathrm{G3}=\frac{2\mathrm{GabGbcGca}\left(\mathrm{GabGbc}-\mathrm{GcaGab}-\mathrm{GbcGca}\right)}{\begin{array}{c}{\mathrm{Gab}}^2{\mathrm{Gbc}}^2+{\mathrm{Gbc}}^2{\mathrm{Gca}}^2+{\mathrm{Gca}}^2{\mathrm{Gab}}^2-\\ 2\left({\mathrm{Gab}}^2\mathrm{GbcGca}+{\mathrm{GabGbc}}^2\mathrm{Gca}+{\mathrm{GabGbcGca}}^2\right)\end{array}}& \left(18\right)\end{array}$
Since all four quantities, Z, Y, R, and G are measured with time-varying signals, they are referred to collectively as “dynamic parameters”.

[0065] 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.