Method and apparatus for evaluating stored charge in an electrochemical cell or battery

A testing device applies time-varying electrical excitation to a cell or battery and senses the resulting time-varying electrical response. Computation circuitry within the device uses voltage and current signals derived from the excitation and response signals as inputs and computes values of elements of an equivalent circuit representation of the cell or battery. The relative charge SOC) of the cell or battery is calculated from the value of the conductance component G of a particular parallel G-C subcircuit of the equivalent circuit. The absolute charge (Ah) contained in the cell or battery is calculated from the value of the capacitance component C of a different parallel G-C subcircuit. Relative or absolute charge values are then either displayed to the user or are used to control an external process such as charging of the battery.

BACKGROUND OF THE INVENTION
 The present invention relates to testing of storage batteries. More
 specifically, the invention relates to evaluating stored charge in an
 electrochemical cell or battery.
 Stored charge is an important parameter in many applications of
 electrochemical cells and batteries. With traction batteries, stored
 charge represents an electric vehicle's fuel supply and thus determines
 how far the vehicle can travel before recharging. With stationary standby
 batteries, the level of stored charge determines how long a critical load
 can continue to function in the event of a power failure or disconnection
 from the ac mains. In automotive applications, stored charge determines
 the length of time that the lights and accessories can be operated when
 the engine is off, or when the charging system has malfunctioned.
 With lead-acid batteries, relative stored charge, or state-of-charge (SOC),
 has been traditionally evaluated by observing either the battery's
 open-circuit voltage, or the specific gravity of the battery's
 electrolyte. However, neither of these measurements yields an absolute
 determination of the amount of stored charge. Furthermore, specific
 gravity measurements are messy and altogether impossible to perform on
 sealed lead-acid cells; and open-circuit voltage is difficult to determine
 under load conditions, and is imprecisely related to SOC since it is
 greatly affected by both "surface charge" and temperature.
 Because of these problems, several techniques for correcting the voltage of
 lead-acid batteries to obtain SOC have been proposed. These include the
 techniques described by Christianson et al. in U.S. Pat. No. 3,946,299, by
 Reni et al. in U.S. Pat. No. 5,352,968, and by Hirzel in U.S. Pat. No.
 5,381,096. However, such voltage correction methods are not very accurate.
 Furthermore, with electrochemical systems other than lead-acid, they may
 be of little help since battery voltage often bears very little
 relationship to stored charge.
 Because of the problems with traditional methods for determining relative
 charge (SOC), many techniques based upon measuring ac impedance have been
 suggested. For example, U.S. Pat. No. 3,984,762 to Dowgiallo purports to
 determine SOC directly from the phase angle of the complex impedance at a
 single frequency. In U.S. Pat. No. 4,743,855, Randin et al. assert that
 SOC can be determined from the argument (i.e., phase angle) of the
 difference between complex impedances measured at two different
 frequencies. Bounaga, in U.S. Pat. No. 5,650,937, reportedly determines
 SOC from measurements of the imaginary part of the complex impedance at a
 single frequency. Finally, Basell et al., in U.S. Pat. No. 5,717,336
 purport to determine SOC from the rate of change of impedance with
 frequency at low frequency. However, the fact that none of these ac
 impedance methods has gained wide acceptance suggests that they may not be
 altogether satisfactory methods for determining SOC.
 The absolute stored charge or amp-hour capacity of batteries has been
 traditionally measured by timed-discharge tests. However, because of the
 expense and the time involved in performing such tests, ac techniques for
 determining amp-hour capacity have been proposed. Sharaf, in U. S. Pat.
 No. 3,808,522, teaches a method for determining the capacity of a
 lead-acid battery from measurements of its ac internal resistance. Yang,
 in U.S. Pat. No. 5,126,675, also uses measurements of internal resistance
 to determine battery capacity. Muramatsu reports, in U.S. Pat. No.
 4,678,998, that he can determine both the remaining amp-hour capacity and
 the remaining service life of a battery from measurements of the ac
 impedance magnitude at two different frequencies. Fang, in U.S. Pat. No.
 5,241,275, teaches a method for determining remaining capacity from
 complex impedance measured at two or three frequencies in the range from
 0.001 to 1.0 Hz. Finally, Champlin, in U.S. Pat. No. 5,140,269, has shown
 that percent capacity can be determined from the measured dynamic
 conductance at a single frequency if the dynamic conductance of a
 reference, fully-charged, identically constructed, new battery is known.
 This method, although quite accurate, requires that aprioi data be
 available.
 SUMMARY OF THE INVENTION
 A testing device applies time-varying electrical excitation to a cell or
 battery and senses the resulting time-varying electrical response.
 Computation circuitry within the device uses voltage and current signals
 derived from the excitation and response signals as inputs and computes
 values of elements of an equivalent circuit representation of the cell or
 battery. In one aspect, the relative charge (SOC) of the cell or battery
 is calculated from the value of the conductance component G of a
 particular parallel G-C subcircuit of the equivalent circuit. In another,
 the absolute charge (Ah) contained in the cell or battery is calculated
 from the value of the capacitance component C of a different parallel G-C
 subcircuit. In other aspects, relative or absolute charge values are then
 either displayed to the user or are used to control an external process
 such as charging of the battery.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
 A method and apparatus for quickly and accurately determining both relative
 charge (SOC) and absolute charge (Ah) that is unaffected by "surface
 charge" and does not require apriori data would be of great value. The
 present invention addresses this need. It is based upon teachings
 disclosed in U.S. Pat. No. 6,002,238 issued Dec. 14, 1999 and entitled
 "METHOD AND APATUS FOR MEASURING COMPLEX IMPEDANCE OF CELLS AND
 BATTERIES" and U.S. Pat. No. 6,037,777, issued Mar. 14, 2000, entitled
 "METHOD AND APATUS FOR DETERMINING BATTERY PROPERTIES FROM COMPLEX
 IMPEDANCE ADMITTANCE" which are incorporated herein by reference.
 FIG. 1 discloses a block diagram of apparatus for evaluating stored charge
 according to the present invention. Apparatus of this type is fully
 disclosed in U.S. Pat. No. 6,002,238 and U.S. Pat. No. 6,037,777.
 Measuring circuitry 10 electrically couples to cell/battery 20 by means of
 current-carrying contacts A and B and voltage-sensing contacts C and D.
 Measuring circuitry 10 passes a periodic time-varying current i(t) through
 contacts A and B and senses a periodic time-varying voltagev(t) across
 contacts C and D. By appropriately processing and combining i(t) and v(t),
 measuring circuitry 10 determines real and imaginary parts of a complex
 parameter, either impedance Z or admittance Y, at a measuring frequency
 f.sub.k ; where f.sub.k is a discrete frequency contained in the periodic
 waveforms of both i(t) and v(t).
 Control circuitry 30 couples to measuring circuitry 10 via command path 40
 and commands measuring circuitry 10 to determine the complex parameter of
 cell/battery 20 at each one of n discrete measuring frequencies, where n
 is an integer number. This action defines 3n experimental quantities: the
 values of the n measuring frequencies and the values of the n imaginary
 parts and n real parts of the complex parameter at the n measuring
 frequencies.
 Computation circuitry 50 couples to measuring circuitry 10 and to control
 circuitry 30 via data paths 60 and 70, respectively, and accepts the 2n
 experimental values from measuring circuitry 10 and the values of the n
 measuring frequencies from control circuitry 30. Upon a "Begin
 Computation" command from control circuitry 30 via command path 80,
 computation circuitry 50 uses algorithms disclosed in U.S. Pat. No.
 6,037,777 to combine these 3n quantities numerically to evaluate 2n
 elements of an equivalent circuit representation of the cell/battery.
 Computation circuitry 50 then calculates the relative charge (SOC) and/or
 the absolute charge (Ah) of the cell/battery from values of particular
 elements of this circuit representation. Finally, computation circuitry 50
 outputs the computed result to the user on display 90 and/or uses the
 result to control a process 100 such as a battery charger.
 In practice, a microprocessor or microcontroller running an appropriate
 software program can perform the functions of both control circuitry 30
 and computation circuitry 50.
 FIG. 2 discloses a six-element equivalent circuit representation of a
 typical automotive storage battery. This circuit representation was
 evaluated using apparatus of the type disclosed in FIG. 1 with n=3 by
 employing algorithms disclosed in U.S. Pat. No. 6,037,777. The three
 measurement frequencies were 5 Hz, 70 Hz, and 1000 Hz. One notes that the
 n=3 equivalent circuit comprises three subcircuits:
 A series G1-L1 subcircuit.
 A parallel G2-C2 subcircuit.
 A parallel G3-C3 subcircuit.
 One notes further that the three subcircuits are characterized by having
 very different time constants. The shortest time constant, .tau..sub.1
 =L1.multidot.G1=93.5 .mu.S, belongs to the series G1-L1 subcircuit. The
 next longest time constant, .tau..sub.2 =C2/G2=2.22 mS, belongs to the
 parallel G2-C2 subcircuit; and the longest time-constant, .SIGMA..sub.3
 =C3/G3=41.6 mS, belongs to the parallel G3-C3 subcircuit. Accordingly, the
 three subcircuits represent quite different physical processes and can be
 differentiated from one another by their time constants.
 FIG. 3 is a logarithmic plot of the three time constants defined above as
 functions of charge (ampere-hours) removed from the battery. One notes
 that the three time constants remain widely separated as charge is
 removed, and that the longest of the three, .tau..sub.3, is nearly
 independent of state-of-charge.
 FIG. 4 discloses the variation of conductance G3 with charge (amp-hours)
 removed from the battery. One sees that G3 approaches minimum near full
 charge and again near full discharge, while reaching maximum at about 50%
 state-of-charge. This variation is consistent with a theoretical model
 that associates the G3-C3 subcircuit with a linearized, small-signal,
 representation of the nonlinear electrochemical reaction occurring at the
 negative plates. For such a model, conductance G3 would be proportional to
 both the number of reaction sites available for the charge reaction (i.e.,
 PBSO.sub.4 sites) and also the number of sites available for the discharge
 reaction (i.e., Pb sites). Accordingly, G3 would be proportional to the
 product (SOC).multidot.(1-SOC). FIG. 4 shows a theoretical curve based
 upon this assumption. The agreement between the theoretical curve and the
 experimental points is seen to be excellent.
 The observed variation of G3 with SOC can be exploited to determine SOC
 from measurements of G3. Inverting the theoretical G3(SOC) curve leads to
 a quadratic equation for SOC(G3). This quadratic equation has two roots,
 identified as SOC.sup.+ and SOC.sup.- for G3=400 S in FIG. 4. The
 SOC.sup.+ root in FIG. 4 corresponds to approximately 85% SOC and the
 SOC.sup.- root corresponds to about 15% SOC. Thus, if one knows which root
 is the correct root, one can determine SOC from measurements of G3.
 Introducing a single piece of auxiliary information such as battery
 voltage, single-frequency conductance, or the value of one of the other
 elements of the equivalent circuit can readily identify the correct root.
 FIG. 5 discloses plots of measurements of conductance, i.e. the real part
 of the complex admittance, obtained at two different frequencies--8 Hz and
 22 Hz. These results were obtained by sequentially discharging a battery
 while measuring the conductance and the specific gravity after each
 discharge. The data are plotted as functions of the battery electrolyte's
 mean specific gravity. Although the effect is not as pronounced, one notes
 the same type of behavior at both frequencies as was noted above with
 reference to FIG. 4. That is, the measured conductance is smallest at both
 full charge and at full discharge, and reaches maximum near 50% SOC. This
 suggests that if accuracy is not too important, one may be able to
 determine an approximation to SOC from a simple analysis of measurements
 of conductance alone obtained at one or more appropriately chosen
 frequencies. Such an analysis would be much simpler than the rigorous
 mathematical analysis disclosed in U.S. Pat. No. 6,037,777 and would take
 advantage of the fact, disclosed above in FIG.3, that the time constants
 of the subcircuits are widely separated so that the subcircuits have
 little interaction.
 FIG. 6 discloses a plot of measured values of capacitance C2 as a function
 of charge (amp-hours) removed from the battery. Note that this capacitance
 is observed to decrease monotonically with charge removed. I have found
 the value of capacitance C2 to be an excellent indicator of the absolute
 charge (Ah) contained in the battery. As such, its measurement could be
 used to implement a battery "fuel gauge". To illustrate this possibility,
 "EMPTY" and "FULL" indications have been added to the plot of FIG. 6.
 Although the relationship between C2 and remaining charge is not exactly a
 linear relationship, it appears to be fairly close to linear. One notes
 that, after all, many automobile fuel gauges are not exactly linear
 either.
 FIG. 7 discloses the total absolute charge in amp-hours as a function of
 measured capacitance C2 for six fully-charged batteries of varying sizes.
 The experimental amp-hour points plotted for each battery were obtained
 from actual timed discharge tests obtained with a constant discharge
 current of 25 amperes, to a final terminal voltage of 10.5 volts. Also
 plotted in FIG. 7 is an experimental linear equation of the form
 Ah=22+8.2.multidot.C2. One notes that the experimental points agree with
 the linear equation within approximately .+-.3 ampere-hours. This
 interesting result implies that the absolute charge capacity in
 ampere-hours can be quite accurately determined from measurements of
 C2--without the necessity of performing costly and time-consuming
 timed-discharge tests.
 FIGS. 8 and 9 illustrate the aspects of the invention. FIG. 8 illustrates
 the G3-C3 subcircuit and shows that the complex admittance of this
 parallel subcircuit is Y3=G3+j.omega.C3. Thus, my discussion above
 actually discloses a relationship existing between the real part of Y3 and
 the SOC of the battery. Similarly, FIG. 9 illustrates the G2-C2 subcircuit
 and shows that its complex admittance is Y2=G2+j.omega.C2. Thus, my
 discussion above actually discloses a relationship existing between
 absolute charge (Ah) and the imaginary part of Y2 at a particular
 frequency .omega.. Although it is true that complex Z and complex Y are
 reciprocals of one another, the same is not true of their real parts (R,G)
 or their imaginary parts (X,B). Accordingly, the results of any ac
 measurement must be expressed in complex admittance form--not complex
 impedance form--in order to observe the important relationships that I
 have disclosed herein.
 This fundamental result sets my work apart from the prior-art work of those
 sited above. Previous workers Dowgiallo (in U.S. Pat. No. 3,984,762),
 Randin et al. (in U.S. Pat. No. 4,743,855), Bounaga (in U.S. Pat. No.
 5,650,937), Basell et al. (in U.S. Pat. No. 5,717,336), Sharaf (in U.S.
 Pat. No. 3,808,522), Yang (in U.S. Pat. No. 5,126,675), Muramatsu (in U.S.
 Pat. No. 4,678,998), and Fang (in U.S. Pat. No. 5,241,275) have all
 attempted to derive either SOC or Ah information from the magnitude of the
 complex impedance, or from components of complex impedance measured at one
 or more frequency. However, none of these workers has suggested that the
 real or imaginary parts of a complex admittance were of any interest at
 all.
 Although my disclosure has relied upon particular apparatus and algorithms
 previously disclosed in U.S. patent applications Ser. No. 09/152,219 and
 Ser. No. 09/151,324, other methods will be apparent to one skilled in the
 arts. For example, one can employ bridges or other types of apparatus to
 measure complex admittance or impedance. Furthermore, if accuracy is not a
 strict requirement, one can take advantage of the fact that the various
 time constants are widely separated from one another and simply assume
 that the subcircuits are not coupled. Within this approximation, C2 and C3
 are treated as short circuits at frequencies near f.sub.01
 =1/2.pi..tau..sub.1, L1 and C3 are treated as short circuits at
 frequencies near f.sub.02 =1/2.pi..tau..sub.2, and at frequencies near
 f.sub.03 =1/2.pi..tau..sub.3, L1 are treated as a short circuit while C2
 is treated as an open circuit. Thus, with some batteries, it is possible
 to obtain satisfactory results with a very simple analysis of measurements
 at two or three frequencies. With certain batteries, it is even possible
 to obtain useful approximations to C2 or G3 from measurements of complex Y
 or Z=1/Y obtained at a single, appropriately chosen, frequency. 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 my invention.