Work aimed at modeling the charge voltage and current characteristics of nickel-cadmium cells subject to taper charge is presented. Work reported at previous NASA Battery Workshops has shown that the voltage of cells subject to constant current charge and discharge can be modeled very accurately with the equation: voltage = A + (B/(C-X)) + De to the -Ex where A, B, D, and E are fit parameters and x is amp-hr of charge removed during discharge or returned during charge. In a constant current regime, x is also equivalent to time on charge or discharge

Mcdermott, P. P.

English

NASA Technical Reports Server

MODELING TAPER CHARGE WITH A NON-LINEAR EQUATION '- 
P a t r i c k  P. McDermott, Ph.D. 
B-K Dynamics, Inc. 
Rockv i l le ,  Maryland 
I n t  roduct  i on 
Th is  i s  a repor t  o f  work which has been i n  progress far s i x  months aimed a t  
model i ng t h e  charge vo l tage and c u r r e n t  c h a r a c t e r i s t i c s  o f  nickel-cadmium 
c e l l s  subject  t o  t ape r  charge. Work repor ted a t  previous NASA B a t t e r y  
Workshops has shown t h a t  t h e  vo l tage o f  c e l l s  sub jec t  t o  constant  cu r ren t  
charge and discharge can be modeled very accura te ly  w i t h  t h e  equation: 
B 
Voltage = A + - + ~ e - E x  
C - X  
where A, 0,  I>, and E are f i t  parameters and x i s  amp-hrs o f  charge removed 
du r ing  discharge o r  returned dur ing charge. I n  a constant cu r ren t  regime, 
x  i s  a l so  equ iva len t  t o  t ime on charge o r  discharge. 
The present study i s  aimed a t  d e r i v i n g  equat ions f o r  f i t t i n g  t h e  charge 
tape r  p o r t i o n  o f  t he  charge curve s ince t h e  p rev ious l y  der ived equations 
are  not appropr iate f o r  t h i s  app l ica t ion .  F igu re  1 shows a t y p i c a l  cur- 
r e n t  vs t ime  p l o t  f o r  a c e l l  i n  constant  c u r r e n t  discharge w i t h  a constant 
cur ren t  charge t o  some predetermined vol tage 1 i m i t  (VL). The cur ren t  i s  
he ld  a t  -16 amps f o r  30 minutes du r ing  t h e  discharge p o r t i o n  and then 
jumps t o  +16 amps f o r  approximately 20 minutes o f  charge. A t  t h i s  po in t ,  
t h e  vo l tage h i t s  t h e  l i m i t ,  and t h e  c u r r e n t  drops r a p i d l y  over t he  next 
10 minutes f l a t t e n i n g  out t o  around 2 amps by  t h e  t i m e  t h e  charge t e r -  
minates a t  approximately 90 minutes. As we s h a l l  see i n  t h i s  repor t ,  t h e  
shape o f  the  tape r  current  w i l l  vary w ide ly  depending on the  t e s t  condi- 
t i ons .  The s lope a t  which t h e  c u r r e n t  drops i n i t i a l l y  i s  q u i t e  v a r i a b l e  
as i s  t he  cur ren t  when the  charge curve f l a t t e n s  out a t  t h e  end o f  t he  
charge. 
Test M a t r i x  
Data f o r  t h i s  study i s  taken from a t e s t  o f  50 amp-hr c e l l s  being conducted 
a t  t h e  Goddard Space F l i g h t  Center. This t e s t  i s  unique i n  t h e  sense t h a t  
t he  same gr0u.p o f  c e l l s  i n  t he  c e l l  pack a r e  being tes ted  under var ious  
cond i t i ons  o f  charge, discharge, temperature and vo l tage  l i m i t .  The c e l l s  
a re  cyc led under one se t  o f  cond i t i ons  (8  cyc les)  t o  reach equ i l ib r ium,  
then re turned t o  basel ine c y c l i n g  regime before  being subjected t o  a new 
s e t  o f  environmental t e s t  cond i t ions .  The data f o r  t h e  study i s  taken from 
t h e  e igh th-cyc le  a t  each t e s t  condi t ion.  
https://ntrs.nasa.gov/search.jsp?R=19850023077 2020-03-20T17:24:04+00:00Z

F igu re  2 shows t h e  t e s t  m a t r i x  f o r  t h i s  study. There a re  e s s e n t i a l l y  
four cond i t i ons  of d i  scharge ( .1C, .2C, .5C, and .8C), four  cond i t ions  
of charge ( .2C, .E, .6C and .8C), t h r e e  temperatures (O°C, 10°C, and 
20°C), and t h r e e  vo l tage  l i m i t s  (VL 3, VL 5, and VL 7) .  The box i n  t h e  
f i g u r e  shows t h e  vo l tage  a t  which the  c e l l  goes i n t o  taper  f o r  t h e  
t h r e e  vo l tage  l i m i t s  a t  t h e  t h r e e  d i f f e r e n t  t e s t  temperatures. Depth 
o f  d ischarge (noD) cou ld  a l so  be considered a t e s t  v a r i a b l e  which i s  
t o t a l l y  dependent on d ischarge cu r ren t  s ince  d i  scharge t i m e  i s  a1 ways 
30 minutes. Don ranges from 5% a t  t h e  .1C discharge r a t e  t o  40% f o r  
the  .8C discharge rate.  
The 4 x 4 m a t r i x  i n  F igu re  2 shows a1 1 o f  t h e  combinations o f  charge 
and d ischarge which were employed i n  t h e  tes t .  Note, however, t h a t  
t h r e e  o f  t h e  combinations were not tes ted (see boxes which are  crossed 
out )  because o f  extreme condi t ions.  The charge ra tes  were no t  s u f f i -  
c i e n t  t o  r e t u r n  t h e  amount o f  charge required a t  the  25% and 40% DoD 
leve ls .  
Each box i n  t h e  4 x 4 m a t r i x  represents an add i t i ona l  se t  o f  n ine  t e s t  
cond i t i ons  which are  t h e  va r ious  combinations of t h e  temperature ( th ree  
l e v e l s )  and vo l tage l i m i t  ( t h r e e  l e v e l s ) .  A l l  t o l d ,  t h e r e  a r e  117 pos- 
s i b l e  combinations o f  these var iab les  i n  t h e  t e s t i n g  scheme. 
Time t o  Voltage L i m i t  
The var ious  combinations o f  charge and discharge ra tes  had a s i g n i f i -  
cant impact on t h e  t ime  i n  which t h e  c e l l  remained i n  constant cu r ren t  
charge be fo re  h i t t i n g  t h e  vo l tage  1 i m i t .  F igu re  3 shows t h e  t ime  t o  
vo l tage  1 i m i t  f o r  t h e  var ious  combinations o f  charge and discharge 
ra te .  The ve ry  sho r t  t imes ( l e s s  than 2 minutes) are seen t o  occur a t  
t h e  h i g h  charge ra tes  f o r  those c e l l s  which were subjected t o  lower 
discharge ra te ,  and, there fore ,  lower depths o f  discharge (5% and 10%). 
The longer  t imes t o  vo l tage l i m i t  occur f o r  var ious  combinations o f  
charge r a t e  and discharge r a t e  a1 ong a diagonal from t h e  1 ower l e f t  
hand s ide  o f  each m a t r i x  t o  t h e  upper r i g h t  hand side. A maximum o f  
30 minutes, one ha1 f o f  t h e  t o t a l  t i m e  i n  discharge, i s  observed a t  
VL 7 f o r  c e l l s  a t  .8C discharge and .6C charge. 
The shape o f  t h e  cur rent  taper, as w i l l  be seen l a t e r  i n  t h e  report ,  i s  
roughly c o r r e l a t e d  t o  t h e  amount o f  t ime  t h a t  t h e  c e l l  stays i n  constant 
cur rent  charge before  h i t t i n g  t h e  taper  l i m i t .  The drop-o f f  i n  cur rent  
i s  much sharper f o r  t h e  c e l l s  which have a shor t  t ime  t o  vo l tage l i m i t ,  
e s p e c i a l l y  those a t  l e s s  than 2 minutes. The parameters o f  t h e  f i t  
equations f o r  t h i s  p o r t i o n  o f  t h e  charge curve are h i g h l y  dependent on 
t h i s  i n i t i a l  . ra te  o f  drop i n  the  cur rent  a f t e r  h i t t i n g  the  vo l tage l i m i t .  
Percent o f  Charge Returned Dur ing t h e  Charge Cyc le  
f i g u r e  4 shows t h e  percentage o f  charge returned t o  t h e  c e l l  s dur ing  
t h e  charge p o r t i o n  o f  t h e  cycle. As w i t h  t h e  t ime  t o  vo l tage 1 i m i t ,  
t h e r e  i s  a s i g n i f i c a n t  v a r i a t i o n  i n  the  percent r e t u r n  based on the  
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d i f f e r e n c e s  i n  charge and d ischarge ra tes,  and on t h e  v a r i a t i o n  i n  v o l t a g e  
l i m i t .  For t h e  ma t r i ces  shown i n  F i g u r e  4 ,  t h e  temperatures a re  averaged 
t o g e t h e r  f o r  each s p e c i f i c  t e s t  c o n d i t i o n .  A t  VL 3, t h e  percent  re tu rned  
i s  l e s s  than  110% f o r  most o f  t h e  cond i t i ons .  A t  VL 7, on t h e  o t h e r  hand, 
most o f  t h e  t e s t  c o n d i t i o n s  show a  percent  r e t u r n  g r e a t e r  than  l l O X ,  w i t h  
over  ha1 f o f  t h e  t e s t  c o n d i t i o n s  showing g r e a t e r  than  125%, a t  t h e  lower  
DoDs. Again, t h e r e  i s  a  rough c o r r e l a t i o n  between t h e  t i m e  t h a t  t h e  c e l l s  
remain i n  charge be fo re  h i t t i n g  t h e  v o l t a g e  l i m i t  and t h e  percentage o f  
charge re tu rned  t o  t h e  c e l l .  The c e l l s  a t  t h e  h i g h  charge r a t e s  b u t  modest 
DoDs show v e r y  s h o r t  t imes  t o  v o l t a g e  l i m i t ,  bu t ,  a1 so, much h ighe r  percent -  
ages o f  r e t u r n .  
F i t t i n g  t h e  Paper P o r t i o n  o f  t h e  Charge 
It was no ted  b y  t r i a l  and e r r o r  t h a t  t h e  s lope  o f  t h e  c u r r e n t  i n  t a p e r  
p l o t t e d  aga ins t  t h e  cu r ren t  y i e l d e d  a  curve  which was approx imate ly  
1  inear .  I t was a1 so noted t h a t  t h e  1  i n e a r i t y  was improved i f  a  cons tan t  
t e rm  was sub t rac ted  f rom t h e  cu r ren t .  The constant  term i n  many cases 
was c l o s e  t o  t h e  c u r r e n t  a t  t h e  end o f  t a p e r  charge, suggest ing t h a t  t h e  
constant  was r e l a t e d  t o  t h e  va lue  o f  c u r r e n t  t o  which t h e  experiment 
c u r r e n t  was approaching as a  l i m i t .  Th i s  observa t ion  suggested t h a t  
t h e  phenomenon cou ld  be f i t  b y  a  f i r s t  o rde r  d i f f e r e n t i a l  equa t ion  o f  
t h e  form: 
where I i s  c u r r e n t  and k  and a  a r e  constants .  The s o l u t i o n  t o  t h i s  
equa t ion  was t hen  used i n  a  non - l i nea r  regress ion  f i t  i program t o  
c a l c u l a t e  f i t  parameters i n  t h e  equat ion:  I = (1)exp 12y8 + ( 3 )  where 
I i s  c u r r e n t  and ( I ) ,  (2 ) ,  and (3)  a r e  t h e  f i t  parameters. 
Th i s  equa t ion  worked q u i t e  we1 1  i n  f i t t i n g  da ta  such as t h a t  shown i n  
F igu re  5a, t h e  t ape r  charge o f  t h e  c e l l  a t  .5C d ischarge,  .5C charge, 
VL 7  and O°C ( F i t  5570 -- See F igu re  1 f o r  code). Parameter 3, as 
suggested above, i s  c l o s e  t o  t h e  p o i n t  where t h e  c u r r e n t  i s l e v e l i n g  
o f f ,  and Parameter 3  p l u s  Parameter 1 a r e  approx imate ly  equal t o  
25 amps, t h e  c u r r e n t  f o r  t h e  c e l l  as i t  h i t  t h e  v o l t a g e  l i m i t .  
The t h r e e  parameter f i t  equat ion  d i d  no t  work so w e l l ,  however, f o r  
curves where t h e r e  was a  sharp drop i n  c u r r e n t  i m n e d i a t e l y  a f t e r  t h e  
c e l l  h i t  t h e  v o l t a g e  l i m i t .  T h i s  i s  r e a d i l y  seen i n  F i g u r e  6a which i s  
a  t h r e e  parameter f i t  o f  t h e  c e l l  a t  .1C d ischarge,  .5C charge, VL 3, 
and 20°C. As was mentioned e a r l i e r ,  many o f  t h e  t e s t  c o n d i t i o n s  where 
t h e  c e l l  h i t s  t h e  v o l t a g e  l i m i t  q u i c k l y  demonstrated t h i s  sharp i n i t i  a1 
d rop  i n  cu r ren t .  It was a1 so noted t h a t  a  p l o t  o f  d I / d t  vs I f o r  t e s t  
c o n d i t i o n s  l i ' k e  those  shown i n  F i g u r e  6  would y i e l d  n o t  one b u t  two 
l i n e a r  p o r t i o n s  w i t h  d i f f e r e n t  s lopes. Th i s  suggested t h a t  t h e r e  were 
two f i r s t  o rde r  e f f e c t s  be ing  demonstrated d u r i n g  t ape r  charge which 
sequ i  red  a  more complex equat ion  i n  o rder  t o  adequately f i t  t h e  data.  
Four and F i v e  Parameter F i t  Equations 
I n  order  t o  accommodate t h e  two e f f e c t s  mentioned above, a f o u r  parameter 
equat ion w i t h  two exponential  terms and a f i v e  parameter equat ion w i t h  
two exponential  terms and a constant  were used w i t h  t h e  non- l inear  
regress ion  program t o  f i t  the  data. These equations a re  shown below: 
Four Parameter Equation: I = ( l ) e ~ ~ ( ~ ) ~  + 
F i ve  Parameter Equation: I = ( l ) e x p ( 2 ) t  + (3 )exp (4 ) t  + (5)  
where ( I ) ,  (2) , ( 3 ) ,  (4) ,  and ( 5 )  a r e  t h e  f i t  parameters. 
These f o u r  and f i v e  parameter equations were able t o  f i t  bo th  the  se ts  o f  
data shown i n  Figures 5 and 6 more accura te ly  than t h e  t h r e e  parameter fit 
equation. Figures 5b and 6b a re  the  f i t s  obtained w i t h  the f o u r  parameter 
equat ion and Figures 5c and 6c are t h e  f i t s  obtained w i t h  t h e  f i v e  para- 
meter equation. The sum o f  squared res idua ls  decreases ( f rom 1.0 t o  0.15 
t o  0.026) f o r  f i t s  5a, 5b and 5c, suggest ing t h a t  t h e r e  i s  an i nc reas ing  
accuracy w i t h  the  f o u r  and f i v e  parameter equations. 
Tables 1 and 2 show parameters f o r  t h e  th ree  f i t  and t h e  f i v e  f i t  equat ions, 
f o r  many o f  t he  t e s t  cond i t i ons  i n  t h e  t e s t  ma t r i x .  There a re  some t rends  
ev ident  as one reads from l e f t  t o  r i g h t  ( i nc reas ing  VL) f o r  any p a r t i c u l a r  
chargeldischarge combination, o r  reads from t o p  t o  bottom ( i nc reas ing  tem- 
perature).  There i s ,  however, no s t rong pa t te rn  as y e t  which suggests 
s t rong c o r r e l  a t i o n  w i t h  the  environmental parameters. 
Other C h a r a c t e r i s t i c s  o f  t h e  Taper Charge 
F igu re  7 shows an i n t e r e s t i n g  c o r r e l a t i o n  between the  cur ren t  a t  the  
end o f  charge and t h e  percent o f  charge returned. When one p l o t s  one 
against  t h e  other ,  t h e  data p o i n t s  c l u s t e r  along s t r a i g h t  l i n e s ,  w i t h  
t h e  slope o f  t h e  l i n e s  increas ing  t h e  decreasing discharge r a t e  o r  DoD, 
The cu r ren ts  a t  t he  end o f  charge a r e  s i g n i f i c a n t l y  h igher  f o r  t he  a 
c e l l s  w i t h  40% DoD (4.0 t o  5.0 amps) than f o r  t h e  c e l l s  a t  5% DoD which 
range f rom 2.0 t o  3.0 amps. What i s  more s t r i k i n g ,  however, a re  t h e  
d i f f e rences  i n  percent re tu rn .  The 5% DoD c e l l  s range from 130% t o  
over 200% charge r e t u r n  whereas the  40% Do0 c e l l s  c l u s t e r  around 110% 
charge re tu rn .  It i s  i n t e r e s t i n g  t o  note t h a t  t h e  I n  slope o f  t h e  1 ines 
drawn through the  da ta  p o i n t s  when p l o t t e d  against  DoD ( I n  scale)  y i e l d s  
a s t r a i g h t  l i n e  as shown i n  t h e  i n s e r t  t o  F igure  7. 
Another i n t e r e s t i n g  e f f e c t  i s  shown i n  F igure  8 where percent recharge i s  
shown vs. temperature f o r  each combination o f  charge and discharge. For t h e  
h igher  Dons t h e r e  i s  l i t t l e  o r  no temperature e f f e c t  (40% and 25% DoD). The 
e f f e c t  o f  temperature i s  more dramatic f o r  t h e  5% and 10% Do0 t e s t  cond i t ions  
w i t h  increas ing  impact from l e f t  t o  r i g h t  w i t h  increas ing  charge ra te .  The 
g rea tes t  d i f f e rence ,  f o r  example, i s  shown f o r  t h e  . l C  discharge rate/.8C 
charge r a t e  combinat i o n  where percent recharge ranges from 140 t o  200%. 
TIPIE 
Figure 5a. Threc Parameter Fit for .5C Discharge, .5C Charge, V L  7 and 0' C 
PFiRHPlETt2 1= 16.27 
PHRHMETER 2= -. 16 
PARHMETER 3= 8 - 7 1  
F'HPAPtETEP 4= - 
T I I I E  
Figrire 5b. Four Parameter Fit for .5C Discharge, 5 C  Charge, VL 7 and OD C 
PHR~PIETER 1= 9 I . 0 5  
PARAMETER 2= - . & I  -, -
PHRHMETER 3 =  15. -3 j 
PHRHMETER 4=  -. 8:3 
PHPHPIETEP S=  -. .? 6 9 
TIME 
Figure 5c. Five Parameter Fit for .5C Discharge, .5C Charge, VL 7 and 0°C 
23.89 
PHEHNETER c- -. 2 4  
PAPHMETER 3= 1.15 
T IHE , 
Figure 6a. Three Parameter Fit for . lC  Discharge, .SC Charge, VL 3 and 20' C 
PHRHPlETER 3= 20.15 
FHPHMETEF: -. 36 
TIME 
Figure 6b. Four Parameter Fit for . I  C Discharge, .5C Charge, VL 3 and 20' C 
PHRAMETER 1" 
PARAMETER 2' 
PARAMETER 3' 
PARAMETER 4' 
TIME 
Figure 6c. Five Parameter Fit for .I C Discharge, .5C Charge, VL 3 and 20' C 
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Summary and Concl usions 
This study has shown that taper charge current profiles of a1 1 the environ- 
mental t e s t  conditions used in th is  t e s t  can be f i t  very accurately by a 
five parameter f i t  equation (Figures 5c and 6c). The study did not pro- 
gress to the point where these parameters could then be correlated t o  the 
environmental t e s t  conditions. However, there were certain trends evident 
which could be pursued in future modeling efforts. 
1. Percent charge returned and time to voltage limit show a distinct 
character which can be related to chargeldischarge rate combinations 
and the level of voltage 1 imit (Figures 3 and 4) .  
2. Cells with very modest 000s (5% and 10%) and higher rates of  charge 
generally showed a very short time to  voltage limit,  higher percent- 
age of charge returned, a very steep drop i n  current after  hitting 
the voltage limit and more modest currents toward the end of charge. 
3. Cells with higher DoDs and charge rates showed, as might be antici- 
pated, longer time t o  the voltage limit,  lower percent of charge 
returned, a more modest rate of drop in current after V L ,  and 
greater currents a t  the end of charge. 
4. Percent charge return, the relationship between Do0 and current 
a t  the end of charge, can be easily modeled (Figure 7 ) .  
5. The effect of temperature i s  observed more a t  the 1 ower dis- 
charge rates (and  DoDs) and increases with increasing charge 
rate (Figure 8 )  
Future work should concent rate on a more comprehensive mu1 t ip le  correl ation 
of tes t  conditions (discharge rate (Don), charge rate, vol tage 1 imit and 
temperature) with the effects of these conditions on battery performance 
(percent charge returned, charge efficiency, end of charge current, and 
current taper profile).  


Modeling taper charge with a non-linear equation

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