76
Electrochemical Supercapacitors for Energy Storage and Delivery
V sc
0
− I cell R esr
t
T
fd = C dl
(
) .
I cell
Equations (2.58) and (2.58a) demonstrate that if the discharging time
is longer than t fd , negative cell voltage will result and this is impossible.
Therefore, both equations are valid only at t ≤ t fd .
2.5.4.6 Charging and Discharging Curves at Constant Current
Based on Equation (2.41) for constant charging and Equation (2.58) for constant
discharging, the charging and discharging curves can be drawn as shown in
Figure 2.17. Note that there are four variable parameters in Equations (2.41)
and (2.58): I R esr , R p and C
T
cell ,
dl . In each of the Figure 2.17 charts, one parameter is changed while the other three remain at fixed values.
In Figure  2.17a, R esr is changed and the voltage difference between the
charging curves corresponds to the value of I cell R esr , that is, ΔV cell = I cell R esr ,
from which R esr can be obtained. In Figure  2.17b, the charging and discharging curves at low R p are both drifted from linearity. This observation
suggests that when charging or discharging a supercapacitor at a constant
current, the faradic leakage current is large enough to make an impact if the
voltage–time curve is not linear. In Figure 2.17b, when R p is changed from
R p,1 to R p,2 , the voltage difference between the two charging curves at any
time can be expressed as:
⎡
⎛
⎞
(
⎛
⎞⎤
t
t
ΔV cell = I cell ⎢ R 1− exp ⎜ −
⎟ − R 1− exp ⎜ −
⎟⎥
(2.60)
⎜
⎟
⎜
⎟
⎢ ⎣
p,2
p 1
p,2
(
⎝ R C
T
dl ⎠
⎝ R C
T
p,1 dl ⎠ ⎥ ⎦
In Figure 2.17c, I cell is changed from I cell,1 to I cell,2 , and the voltage difference at
any time can be expressed as:
⎡
⎛
⎛
(
⎞ ⎞⎤
ΔV
⎢
cell
⎜
t
⎟⎥
cell = I ,2 − I cell,1 ) R R
⎢
esr + p 1− exp ⎜ −
⎟
(2.61)
⎜
⎜
⎟
⎝
⎝ R C
T T ⎟
⎣
p dl
⎥
⎠⎠ ⎦
In Figure 2.17d, C
T
dl is changed from C
T
T
dl,1 to C dl,2 , and the voltage difference
at any time can be expressed as:
⎡
⎛
⎞
⎛
⎞⎤
t
t
ΔV cell = I cell R p ⎢ exp ⎜ −
⎟
T
− exp ⎜ −
⎟⎥
⎜
⎟
⎜
T
(2.62)
⎟
⎢ ⎣
⎝ R C
p dl,1 ⎠
⎝ R p C C dl,2 ⎠ ⎥
⎦
, ,