# Patent

## Title
Broad-band low-inductance cables for making Kelvin connections to electrochemical cells and batteries

## Abstract

A broad-band technique for reducing the distributed inductance of a four-conductor Kelvin cable is disclosed. A special inductance-canceling cable section is connected in tandem with the cable section contacting the cell/battery. Connections between the two cable sections are transposed such that conductors in each conductor pair of the canceling section connect to current-carrying and voltage-sensing conductors from different conductor pairs in the contacting section. The canceling section thereby exhibits a distributed negative mutual inductance between its current-carrying and voltage-sensing conductors that can effectively cancel the distributed positive mutual inductance introduced by the contacting section.
In one embodiment, conductor pairs comprise pairs of insulated wires which may be twisted together. In other disclosed embodiments, conductor pairs comprise shielded coaxial cables.

## Background
CROSS-REFERENCE TO RELATED APPLICATION
The present application is a Divisional of and claims priority of U.S. patent application Ser. No. 10/896,835, filed Jul. 22, 2004, the content of which is hereby incorporated by reference in its entirety.


BACKGROUND OF THE INVENTION
The present invention relates to measuring a dynamic parameter (e.g., impedance, admittance, resistance, reactance, conductance, susceptance) of an electrochemical cell or battery. More specifically, it relates to reducing the effects of cable inductance upon electrical measurements implemented with time-varying signals through Kelvin (i.e., four-point) connections.
Measuring automotive and standby cell/battery parameters with time-varying signals (i.e., measuring dynamic parameters) are now commonly accepted maintenance and diagnostic procedures. (See, e.g., U.S. Pat. Nos. 5,140,269, 6,262,563, 6,534,993, and 6,623,314). Because of the very small impedances of such cells/batteries, Kelvin connections are routinely employed to reduce the influence of contact and interconnecting cable impedance. Kelvin connections make contact with each cell/battery terminal at two separate contact points—one for current and one for voltage. Apparatus for measuring a two-terminal cell/battery by means of Kelvin connections therefore requires a four-wire interconnecting cable.
Kelvin connections very effectively remove the spurious effects of cable and contact resistances when measurements are made with static currents and voltages. However, when measuring with time-varying signals, distributed mutual inductance between current-carrying and voltage-sensing conductors in the interconnecting cable can introduce significant errors.
Consider FIG. 1. FIG. 1 depicts cell/battery 10 connected to measuring apparatus 20 by means of four-wire cable 30, Y-junction 40, and Kelvin conductors A, B, C, and D. Current-carrying conductors A and B couple to positive and negative cell/battery terminals at contact points 50 and 60, respectively. Voltage-sensing conductors C and D separately couple to positive and negative cell/battery terminals at contact points 70 and 80, respectively. During dynamic measurements, a time-varying current flows through current-carrying conductors A and B and also flows internally between the terminals along an internal current path 90.
FIG. 2 shows the arrangement of conductors employed in the apparatus of FIG. 1. This arrangement was first introduced by Champlin in U.S. Pat. No. 3,873,911 and has been commonly used in dynamic testing of lead-acid storage batteries since 1975. FIG. 2 discloses contacting cable section 5 comprising an A-C pair of insulated wires 120 coupling to the positive cell/battery terminal and a B-D pair of insulated wires 130 coupling to the negative cell/battery terminal. The two conductor pairs are necessarily spaced-apart at the cell/battery terminals but are brought into close proximity at Y-junction 40. These insulated wire pairs may, or may not, be twisted together in section 5. At Y-junction 40 the four wires are re-arranged for connection to zero-coupling cable section 15. Throughout section 15, the A-B current carrying conductors and the C-D voltage-sensing conductors are separately paired and twisted together, pair 140 and pair 150, respectively. The advantage of this pairing and twisting arrangement is that transverse magnetic fields —inherently present in space 35 of cable section 5—are virtually non-existent in space 75 of cable section 15 by virtue of the twisted current-carrying conductors A and B. In addition, the twisted voltage-sensing conductors C and D exhibit negligible coupling to whatever small magnetic fields do exist in space 75. Accordingly, over-all cable inductance is largely confined to contacting cable section 5 with virtually no contribution from zero-coupling cable section 15. Zero-coupling section 15 can therefore be of any length desired for convenience.
Because of the necessity for physically-separated current-carrying conductors and for physically-separated voltage-sensing conductors in cable section 5, inductance is unavoidable in that section. Let ω=2πf be the angular measurement frequency, j=√{square root over (−1)}, and let Re( ) stand for “the real part of”. For a time-varying current iAB(t)=Re(ÎAB·ejωt) flowing in conductors A and B, a time-varying transverse magnetic field H(t)=Re(Ĥ·ejωt) is generated in space 35 between the two current-carrying conductors of section 5. Ampere's Law states that phasor (complex) quantities Ĥ and ÎAB are related by

Ĥ·dl=ÎAB  (1)

where the integral extends over any closed contour surrounding a current-carrying conductor. For an ac current entering cell/battery 10 on conductor A and leaving on conductor B, the direction of the (complex) magnetic field vector Ĥ is as shown in FIG. 1. The transverse magnetic field therefore emerges from the plane of the conductors in space 35 of cable section 5 as is shown in FIG. 2.

Voltage-sensing conductors C and D in cable section 5 along with conducting path 90 through cell/battery 10 form a closed loop. According to Faraday's law of induction, any time-varying magnetic field linking this loop will induce a time-varying voltage into the voltage-sensing circuit. For a complex magnetic field vector Ĥ, the complex ac voltage {circumflex over (V)}CD induced into the voltage-sensing circuit is

{circumflex over (V)}CD=jωμ0Ĥ·ds  (2)

where μ0 is the magnetic permeability of free space, and dS is a differential area vector perpendicular to a surface bounded by the closed loop.

Thus, with time-varying signals, the time-varying magnetic field formed in space 35 of cable section 5 introduces distributed coupling between the current-carrying circuit and the voltage-sensing circuit. Such spurious coupling is fundamental to the geometry of FIGS. 1 and 2 and tends to defeat the effectiveness of the Kelvin connections.
One can define the mutual inductance between the current-carrying A-B circuit and the voltage-sensing C-D circuit as follows:




  
    
      
        
          M
          
            AB
            ,
            CD
          
        
        =
        
          
            
              
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                ^
              
              CD
            
            
              jω
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Mutual inductance MAB,CD is a distributed parameter—distributed over the entire length of contacting cable section 5. In any dynamic measurement, a magnetically-induced voltage

{circumflex over (V)}CD=jωMAB,CD·ÎAB  (4)

will be developed in the voltage-sensing circuit along with the normal ac voltage developed across cell/battery 10. Accordingly, as shown in FIG. 3, the complete cell/battery impedance measured with Kelvin connections appears externally to be

ZMEAS=ZBAT+jωMAB,CD  (5)

Distributed mutual inductance MAB,CD is a positive quantity that appears in series with cell/battery impedance ZBAT. It is electrically indistinguishable from a lumped self-inductance LBAT internal to the battery. For sufficiently small ZBAT or sufficiently large ω, the part of Equation (5) associated with the Kelvin cables may dominate. This fact constitutes the fundamental problem with dynamic measurements performed through Kelvin connections.
PRIOR ART
A method for reducing the influence of inductance of Kelvin cables has been taught by Bertness in U.S. Pat. No. 6,172,505. This method is depicted in FIG. 4. Y-junction 40 in FIG. 4 contains transformer 90 having primary winding 95 placed in series with current-carrying Kelvin conductor A and secondary winding 100 placed in series with voltage-sensing Kelvin conductor C. Windings 95 and 100 are wound in opposite directions on an iron core 110 so that the voltage induced into coil 100 by current flowing in coil 95 opposes the voltage normally induced into the voltage-sensing C-D circuit by current flowing in the current-carrying A-B circuit. Accordingly, transformer 90 introduces a lumped negative mutual inductance that can be adjusted to cancel the distributed positive mutual inductance inherently present in cable section 5.
Although the inductance cancellation method of FIG. 4 can be quite effective, there are several problems with its use. First of all, for a given magnetic core, the only way to change the transformer's mutual inductance is to vary the number of turns on its windings. That makes the transformer's mutual inductance a difficult quantity to adjust. Secondly, the core's hysteresis and eddy current losses introduce resistive terms into the transformer's equivalent circuit that can cause significant measurement errors. Third, the magnetic permeability of a ferromagnetic core, and hence the transformer's mutual inductance, is frequency dependent. As a result, effective cancellation of cable inductance may only occur over a fairly narrow range of frequencies. These and other problems are solved by the invention embodiments disclosed below.
SUMMARY OF THE INVENTION
A broad-band technique for canceling the distributed inductance of a four-conductor Kelvin contacting cable section comprising two spaced-apart conductor pairs with each conductor pair comprising a current-carrying conductor and a voltage-sensing conductor. A special canceling cable section, also comprising spaced-apart conductor pairs, is connected in tandem with the contacting cable section. Connections between the two cable sections are transposed such that conductors in each conductor pair of the canceling section connect to current-carrying and voltage-sensing conductors from different conductor pairs in the contacting section. The canceling section thereby exhibits a distributed negative mutual inductance that can effectively cancel the distributed positive mutual inductance of the contacting section.
In one embodiment, conductor pairs comprise pairs of insulated wires which may be twisted together. In several other disclosed embodiments, conductor pairs comprise shielded coaxial cables.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a drawing illustrating dynamic parameter measuring apparatus connected to a cell/battery by means of Kelvin (four-point) connections.
FIG. 2 is a schematic representation showing the conductor arrangement most commonly employed with the apparatus of FIG. 1.
FIG. 3 is an equivalent circuit showing total ac impedance measured using the apparatus of FIGS. 1 and 2.
FIG. 4 is a schematic diagram depicting the method for canceling Kelvin cable inductance taught by Bertness in U.S. Pat. No. 6,172,505.
FIG. 5 is a schematic diagram depicting a method for implementing inductance cancellation in a Kelvin cable according to one embodiment of the present invention.
FIG. 6 is a schematic diagram depicting a method for implementing inductance cancellation in a Kelvin cable according to another embodiment of the present invention.
FIG. 7 is a schematic diagram depicting a method for implementing inductance cancellation in a Kelvin cable according to still another embodiment of the present invention.
FIG. 8 is a plan drawing showing the dimensions of shielded coaxial conductors comprising the conductor pairs in cable section 5 of the Kelvin cables depicted in FIGS. 6 and 7.
FIG. 9 is a calculated plot of distributed mutual inductance M1 as a function of conductor separation W1, at the cell/battery terminals for the shielded coaxial conductors of the cable section 5 illustrated in FIG. 8.
FIG. 10 is a plan drawing showing the dimensions of parallel shielded coaxial conductors comprising the conductor pairs in cable section 25 of the Kelvin cables depicted in FIGS. 6 and 7.
FIG. 11 is a calculated plot of normalized distributed mutual inductance (M2/L2) as a function of normalized conductor separation (W2/d0) for the parallel shielded coaxial conductors of the cable section 25 illustrated in FIG. 10.



DETAILED DESCRIPTION OF THE ILLUSTRATIVE EMBODIMENTS
Consider the invention embodiment disclosed in FIG. 5. This figure depicts Kelvin connections made to battery 10 by means of a conductor arrangement similar to that of FIG. 2, but with an important modification. In FIG. 5, an additional cable section, canceling cable section 25, has been inserted in tandem between cable sections 5 and 15. Canceling cable section 25 comprises two spaced-apart conductor pairs, 160 and 170, each comprising two insulated wires which may, or may not, be twisted together. By virtue of transposed connections made in Y-junction 40 and in cable junction 55, the roles of conductor pairs in sections 5 and 15 are exactly the same as those in sections 5 and 15 of FIG. 2. Conductor pairs in section 25, however, comprise a current-carrying conductor B paired with a voltage-sensing conductor C (pair 160), and a current-carrying conductor A paired with a voltage-sensing conductor D (pair 170).
Because current-carrying conductors A and B in section 25 are transposed from those of current-carrying conductors of section 5, the direction of the magnetic field in space 45 is opposite to that of the magnetic field in space 35. Voltage-sensing conductors are, however, positioned the same in both sections. Thus, voltages induced into the voltage-sensing circuit by currents in the current-carrying circuit have opposite signs in the two sections. Section 25 therefore exhibits a negative mutual inductance that can be utilized to cancel the positive mutual inductance inherent to section 5. However, in contrast with the lumped negative mutual inductance introduced by the transformer in the prior art method of FIG. 4, the mutual inductance of section 25 is a distributed quantity.
This same distributed negative mutual inductance would occur if connections to the voltage-sensing conductors, rather than to the current-carrying conductors, were transposed in going from section 5 to section 25. A distributed negative mutual inductance in section 25 requires only that connections to two of the conductors be transposed between sections 5 and 25 such that a voltage-sensing conductor from one conductor pair of section 5 is paired with a current-carrying conductor from the other conductor pair of section 5, and vice-versa.
The distributed negative mutual inductance introduced by this method is a very broad-band property that is essentially loss-free over a wide range of frequencies. These are very desirable advantages over the prior-art method of producing lumped negative mutual inductance depicted in FIG. 4. The distributed negative mutual inductance of the FIG. 5 embodiment can be varied by changing either the length of cable section 25 or the spacing between conductor pairs 160 and 170. Therefore, the mutual inductance of a Kelvin cable of the type depicted in FIG. 5 is fairly easy to “tune” experimentally. One disadvantage of the embodiment depicted in FIG. 5, however, is that its insulated-wire geometry is not very amenable to exact mathematical analysis.
FIGS. 6 and 7 disclose invention embodiments that are amenable to exact mathematical analyses and therefore do not require experimental “tuning”. With both of these embodiments, the insulated-wire conductor pairs of cable sections 5 and 25 of FIG. 5 have been replaced by shielded coaxial cables.
In cable section 5 of FIGS. 6 and 7, a coaxial cable 180 replaces conductor pair 120 and couples to the positive terminal of cell/battery 10 at contact points 50 and 70, while a coaxial cable 190 replaces conductor pair 130 and couples to the negative terminal of cell/battery 10 at contact points 60 and 80. With the coaxial cables depicted, cable shields comprise the current-carrying conductors, A and B, and center conductors comprise the voltage-sensing conductors, C and D. However, the reciprocity theorem reveals that the distributed mutual inductance of section 5 will be unchanged by interchanging the roles of coaxial cable shields and center conductors. Therefore, the selection shown in FIGS. 6 and 7 is simply a matter of choice.
In cable section 25 of FIGS. 6 and 7, coaxial cable 200 replaces conductor pair 160 and coaxial cable 210 replaces conductor pair 170. Again, with the coaxial cables depicted, cable shields comprise current-carrying conductors A and B and center conductors comprise voltage-sensing conductors C and D. Again this selection is simply a matter of choice since the reciprocity theorem likewise reveals that the distributed mutual inductance of section 25 is unaffected by interchanging roles of the coaxial shields and center conductors.
In the embodiment of FIG. 7, the conductor pairs in zero-coupling cable section 15 have also been replaced by shielded coaxial cables. However, in the FIG. 6 embodiment, they remain twisted pairs as in the embodiment of FIG. 5. Either of these choices provides negligible coupling between current-carrying conductors and voltage-sensing conductors in cable section 15. Therefore, the selection of one of these two embodiments over the other is simply a matter of choice.
FIG. 8 is a plan drawing defining dimensions of the shielded coaxial cables that comprise conductors in contacting cable section 5 of the invention embodiments depicted in FIGS. 6 and 7. One sees that section 5 comprises two coplanar coaxial cables having radii di of their inner conductors, radii do of their outer conductors, lengths L1, and separated at the cell/battery terminals by a distance W1. The distributed mutual inductance of this geometry can be determined exactly from Maxwell's equations. The result is




  
    
      
        
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Note that M1 is unaffected by the coax cables' center-conductor diameter di.

FIG. 9 shows a plot of distributed mutual inductance M1 calculated from Equation (6) by assuming that section 5 comprises two 12-inch (30.48 cm) lengths of conventional RG-6/U coaxial cable (do=0.25′′) (6.35 mm). One notes that M1 varies quite gradually with W1 over most of the range of W1. However, as W1 approaches its maximum value of 24 inches (60.96 cm), mutual inductance M1 drops precipitously to zero.
FIG. 10 is a plan drawing defining the dimensions of the shielded coaxial conductors comprising conductors in canceling cable section 25 of the Kelvin cables depicted in FIGS. 6 and 7. One observes two parallel coaxial cables having radii di of their inner conductors, radii do of their outer conductors, lengths L2, and uniformly separated by distance W2. The distributed mutual inductance of this geometry can likewise be determined exactly from Maxwell's equations. The result is




  
    
      
        
          M
          2
        
        =
        
          
            (
            
              
                
                  -
                  
                    μ
                    0
                  
                
                ·
                
                  L
                  2
                
              
              π
            
            )
          
          ·
          
            ln
            ⁡
            
              (
              
                1
                +
                
                  
                    2
                    ·
                    
                      W
                      2
                    
                  
                  
                    d
                    0
                  
                
              
              )
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




Note that M2 is likewise unaffected by the coax cables' center-conductor diameter di. 

FIG. 11 is a plot of normalized distributed mutual inductance (M2/L2) as a function of normalized conductor separation (W2/do) calculated from Equation (7). One sees that (M2/L2) decreases logarithmically as (W2/do) increases linearly throughout most of the range plotted.
One can utilize equations (6) and (7) to design broadband low-inductance Kelvin cables of the type disclosed in FIGS. 6 and 7. As a design example, consider test leads in section 5 comprising a pair of two 12′′ (30.48 cm) lengths of RG-6/U coaxial cable. Equation (6) predicts that the distributed mutual inductance of this section will be M1=0.39±0.05 μH over the range 5′′≦W1≦23.5′′. According to equation (7), one can realize a negative mutual inductance M2=−0.39 μH with a section 25 length L2=17.46′′ (44.35 cm) of parallel RG-6/U cables separated by distance W2=1′′ (2.54 cm). Thus, the tandem combination of these two sections would result in a broadband low-inductance Kelvin cable having total mutual inductance |M1+M2|≦0.05 μH over the entire range 5′′≦W1≦23.5′′. This value represents a minimum decrease in mutual inductance of nearly an order of magnitude.
The invention embodiments illustrated in FIGS. 5, 6, and 7 all depict a canceling cable section 25, inserted between a contacting cable section 5, and a zero-coupling cable section 15. It should be made clear, however, that the order of this tandem arrangement is somewhat arbitrary. The canceling cable section 25 could just as well be placed after the zero-coupling cable section in the cascade rather than before it. Furthermore, zero-coupling section 15 is only employed to make the total cable length convenient. In many cases, a zero-coupling section 15 may not be necessary at all.
Moreover, the invention is not limited to the particular geometries and examples disclosed herein. For instance, pairs of shielded coaxial cables and of twisted insulated wires have both been disclosed in illustrative examples of canceling cable sections. However, other passive transmission-line geometries such as elliptical lines, open-wire lines, strip-lines, microstrip lines, etc, can just as well be employed. Furthermore, distributed negative mutual inductance can also be obtained from a spatial distribution of either passive or active elements. Other means for implementing a distributed negative mutual inductance section will be apparent to those skilled in the art, and our invention includes any such section obtained by any means whatsoever inserted in tandem with a Kelvin connecting cable. Workers skilled in the art will recognize that these and other variations may be made in form and detail without departing from the true spirit and scope of our invention.

## Claims

1. Apparatus employing time-varying signals to test an electrochemical cell or battery comprising:
measuring circuitry adapted to couple to said cell or battery with separate current-carrying and voltage-sensing conductors,
a contacting cable section comprising first and second conductor pairs, said first conductor pair contacting a first terminal of said cell or battery with a first current-carrying conductor and a first voltage-sensing conductor, said second conductor pair contacting a second terminal of said cell or battery with a second current-carrying conductor and a second voltage-sensing conductor, said contacting cable section characterized by positive mutual inductance coupling between its current-carrying and voltage-sensing conductors; and,
a canceling cable section comprising third and fourth conductor pairs interposed between said contacting cable section and said measuring circuitry, said third conductor pair comprising a third current-carrying conductor and a third voltage-sensing conductor and said fourth conductor pair comprising a fourth current-carrying conductor and a fourth voltage-sensing conductor, said third and fourth current-carrying conductors adapted to couple current-carrying conductors of said contacting cable section to current carrying conductors of said measuring circuitry and said third and fourth voltage-sensing conductors adapted to couple voltage sensing conductors of said contacting cable section to voltage-sensing conductors of said measuring circuitry, said canceling cable section characterized by distributed negative mutual inductance coupling between its current-carrying and voltage-sensing conductors.

2. The apparatus of claim 1 wherein said first conductor pair comprises conductors of a first shielded coaxial cable, and said second conductor pair comprises conductors of a second shielded coaxial cable.

3. The apparatus of claim 1 wherein said first conductor pair comprises conductors of a first insulated wire pair, and said second conductor pair comprises conductors of a second insulated wire pair.

4. The apparatus of claim 3 wherein said conductors of said first insulated wire pair are twisted together and said conductors of said second insulated wire pair are twisted together.

5. The apparatus of claim 1 wherein said third conductor pair comprises conductors of a third shielded coaxial cable, and said fourth conductor pair comprises conductors of a forth shielded coaxial cable.

6. The apparatus of claim 1 wherein said third conductor pair comprises conductors of a third insulated wire pair, and said fourth conductor pair comprises conductors of a fourth insulated wire pair.

7. The apparatus of claim 6 wherein said conductors of said third insulated wire pair are twisted together and said conductors of said fourth insulated wire pair are twisted together.

8. The apparatus of claim 1 including a Y-junction interconnecting conductors of said contacting cable section and said canceling cable section, said Y-junction directly connecting first and second current-carrying conductors to third and fourth current-carrying conductors, said Y-junction transposing connections between first and second voltage-sensing conductors and third and fourth voltage-sensing conductors.

9. The apparatus of claim 1 including a Y-junction interconnecting conductors of said contacting cable section and said canceling cable section, said Y-junction directly connecting first and second voltage-sensing conductors to third and fourth voltage-sensing conductors, said Y-junction transposing connections between first and second current-carrying conductors and third and fourth current-carrying conductors.

10. Apparatus of claim 1 including a zero-coupling cable section interposed between said contacting cable section and said measuring circuitry, said zero-coupling cable section comprising current-carrying and voltage-sensing conductors and characterized by negligible mutual inductance coupling between its said current-carrying conductors and its said voltage-sensing conductors.