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({circumflex over (H)}·ejωt) is generated in space 35 between the two current-carrying conductors of section 5. Ampere's Law states that phasor (complex) quantities {circumflex over (H)} and ÎAB are related by{circumflex over (H)}·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 {circumflex over (H)} 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 {circumflex over (H)}, the complex ac voltage {circumflex over (V)}CD induced into the voltage-sensing circuit is{circumflex over (V)}CD=jωμ0{circumflex over (H)}·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                          =                                                            V                ⋒                            CD                                      jω              ⁢                                                          ⁢                                                I                  ⋒                                AB                                              =                                                    μ                0                            ⁢                              ∯                                                                            H                      _                                        ^                                    ·                                                            ⅆ                      S                                        _                                                                                                      I                ⋒                            AB                                                          (        3        )            
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 beZMEAS=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.