Page 153 - Lindens Handbook of Batteries

6.6 PRINCIPLES OF OPERATION

The constraint on the voltage across each branch yields the following equations:

V = V + 0 I R + I R (6.5)
3
ct
q
V = V + 0 I R + (6.6)
C D
Equations (6.3) through (6.6) above can be rearranged to obtain the relationship between the change
in the applied current, I, with time, dI/dt, and the resultant voltage drop, (V - V )
0
dI 1  R  dV 1
-
R +  1 +  I = + ( VV ) (6.7)
dt C  R  dt RC D 0
ct
ct
D
Equation (6.7) now contains the component models [Eqs. (6.3) through (6.6)] for each element of
the battery model. The mathematical solution for Eq. (6.7) can be found in standard references. For
the case of constant current, the solution takes the following form: 4
Q
/
V = 0 e - tR C + V + 0 I R + IR (1 - e - / tR C D ) (6.8)
ct
ct
ct
C D
This model equation relates the change in cell voltage, V, to the input current, I. The parameter Q
0
refers to the total capacity of the battery. The change in cell capacity during a charge or discharge is
calculated by integrating the current passed as follows:
t
=
QQ - 0 ∫ Idt (6.9)
0
The values for the circuit elements such as V , C , R, and R are adjusted to represent the experi-
ct
0
D
mental data of interest adequately.
A typical set of such values for the NiMH battery is shown in Table 6.1. These parameters were
obtained by comparing results from the charging and discharging of a 10-cell battery pack with 85 Ah
capacity, at 64 A current. Figure 6.4 shows the cell voltage versus the capacity of the cell during
charge and discharge at various rates. Shown also are some experimental data for comparison. The
results for a wide range of C-rates are obtained by utilizing the circuit parameters extracted using
data at one discharge (or charge) rate.
TABLE 6.1 Parameters for the Equivalent Circuit Representation of a NiMH Battery 4
Parameter Discharge Charge
E (kcal/mol) 6 6
a
r , mΩ 1.45 2.70
0
r , mΩ — -5.15
1
r , mΩ — 6.23
2
nj (See expression for R ) 0 2
ct
τ 24
C (F) R /τ
D
ct
R, mΩ 0.786
 nj  -  
j
R , mΩ  ∑ r SOC  exp  E a  - R
)
(
ct  j 
=
 j0   RT 