70
Electrochemical Supercapacitors for Energy Storage and Delivery
which is the maximum charging current. When t = ∞ (the moment when the
supercapacitor is fully charged),
E
i cell =
.
R esr + R p
This suggests that even if the supercapacitor is fully charged, extra current density is still needed to overcome the self discharging caused by R p .
However, if the leakage reaction does not occur, R p → ∞, i cell can be simplified
to:
E
⎛
t ⎞
i cell =
exp⎜−
T
⎟ (Charging proc cess)
(2.40)
R esr
⎝ R C
esr dl ⎠
In this case, the charging current will approach zero when t → ∞, that is,
i cell → 0, suggesting that no extra current density is needed to overcome
self discharging. In practice, R p is normally much larger than R esr , that is,
R p >> R esr , so Equation (2.38) can be simplified to Equation (2.40).
According to Equation (2.38), when charging an ideal supercapacitor (R esr =
0, and R p = ∞) using a constant voltage, the current will be infinite as long as
voltage is applied to the supercapacitor. Even for the case where R esr → 0 and
R p ≠ ∞, the supercapacitor voltage will go to E immediately after the constant
voltage E is applied, as seen using Equation (2.39).
2.5.4.2 Charging at Constant Cell Current
Figure 2.15b shows supercapacitor charging at a constant current density.
The constant charging current density is I cell , and the cell voltage is V cell .
Assume that before the charging starts, the supercapacitor is at zero charge
state. When the switcher is closed at t = 0, the charging process is started.
When t ≥ 0, the supercapacitor charging voltage can be expressed as:
⎛
⎛
⎞⎞
t
V cell = I cell R esr + I cell R p ⎜ 1 e
− xp ⎜ −
⎟⎟ (Charging
T
process)
(2.41)
⎜
⎜
⎟ ⎟
⎝
⎝ R C ⎟
p dl ⎠⎠
Equation (2.41) shows that when t = 0, V cell = I cell R esr ; when t = ∞ (the moment
when the supercapacitor is fully charged), V cell = I cell R esr + I cell R p . In this case,
the voltage across the supercapacitor (V sc ) reaches its maximum value,
defined as V
o
sc , which is equal to I cell R p . If there is no leakage current, R p → ∞
and Equation (2.41) can be simplified to:
Electrochemical Supercapacitors for Energy Storage and Delivery
which is the maximum charging current. When t = ∞ (the moment when the
supercapacitor is fully charged),
E
i cell =
.
R esr + R p
This suggests that even if the supercapacitor is fully charged, extra current density is still needed to overcome the self discharging caused by R p .
However, if the leakage reaction does not occur, R p → ∞, i cell can be simplified
to:
E
⎛
t ⎞
i cell =
exp⎜−
T
⎟ (Charging proc cess)
(2.40)
R esr
⎝ R C
esr dl ⎠
In this case, the charging current will approach zero when t → ∞, that is,
i cell → 0, suggesting that no extra current density is needed to overcome
self discharging. In practice, R p is normally much larger than R esr , that is,
R p >> R esr , so Equation (2.38) can be simplified to Equation (2.40).
According to Equation (2.38), when charging an ideal supercapacitor (R esr =
0, and R p = ∞) using a constant voltage, the current will be infinite as long as
voltage is applied to the supercapacitor. Even for the case where R esr → 0 and
R p ≠ ∞, the supercapacitor voltage will go to E immediately after the constant
voltage E is applied, as seen using Equation (2.39).
2.5.4.2 Charging at Constant Cell Current
Figure 2.15b shows supercapacitor charging at a constant current density.
The constant charging current density is I cell , and the cell voltage is V cell .
Assume that before the charging starts, the supercapacitor is at zero charge
state. When the switcher is closed at t = 0, the charging process is started.
When t ≥ 0, the supercapacitor charging voltage can be expressed as:
⎛
⎛
⎞⎞
t
V cell = I cell R esr + I cell R p ⎜ 1 e
− xp ⎜ −
⎟⎟ (Charging
T
process)
(2.41)
⎜
⎜
⎟ ⎟
⎝
⎝ R C ⎟
p dl ⎠⎠
Equation (2.41) shows that when t = 0, V cell = I cell R esr ; when t = ∞ (the moment
when the supercapacitor is fully charged), V cell = I cell R esr + I cell R p . In this case,
the voltage across the supercapacitor (V sc ) reaches its maximum value,
defined as V
o
sc , which is equal to I cell R p . If there is no leakage current, R p → ∞
and Equation (2.41) can be simplified to: