30
Electrochemical Supercapacitors for Energy Storage and Delivery
R
i
i
C
ε
B
i
L
FIGURE 1.11
Electric circuit schematic of resistor, inductor, and capacitor (RLC) connected in series.
According to the circuit loop rule, the sum of the potentials through each
element must be equivalent to the driving potential applied:
V � o − V � − − V � − V �
R
L
C = 0
(1.63)
If the R, C, and L are in series in the circuit, as depicted in Figure 1.11, and a
current I � passes through it, Equation (1.63) can be rewritten as
( )
� o 2 = ( )
�
2
V
IR + ( I � X �
L − IX C ) = 0
(1.64)
By rearranging Equation (1.64), a function for the current is then obtained:
V � o
�
I =
(1.65)
2
+ (
2
R
X L − X C )
The current is equal to the driving potential divided by a denominator term,
hereafter known as impedance Z R–L–C of an AC RLC circuit:
1
Z R L
2
− −C = R + (X
2
2
L − X C ) = R + (ω
2
d L −
)
(1.66)
ω d C
Equation (1.67) represents equivalent resistance or AC impedance of the
whole series RLC circuit. It indicates that when a circuit is operating at a resonant frequency (i.e., X L = X C ), the overall impedance is equal to a real resistance R. If the frequency is increased to exceed the resonance, the capacitive
reactance will be reduced and the inductive reactance will be increased. The
situation is likewise reversed at frequencies below resonance.
Electrochemical Supercapacitors for Energy Storage and Delivery
R
i
i
C
ε
B
i
L
FIGURE 1.11
Electric circuit schematic of resistor, inductor, and capacitor (RLC) connected in series.
According to the circuit loop rule, the sum of the potentials through each
element must be equivalent to the driving potential applied:
V � o − V � − − V � − V �
R
L
C = 0
(1.63)
If the R, C, and L are in series in the circuit, as depicted in Figure 1.11, and a
current I � passes through it, Equation (1.63) can be rewritten as
( )
� o 2 = ( )
�
2
V
IR + ( I � X �
L − IX C ) = 0
(1.64)
By rearranging Equation (1.64), a function for the current is then obtained:
V � o
�
I =
(1.65)
2
+ (
2
R
X L − X C )
The current is equal to the driving potential divided by a denominator term,
hereafter known as impedance Z R–L–C of an AC RLC circuit:
1
Z R L
2
− −C = R + (X
2
2
L − X C ) = R + (ω
2
d L −
)
(1.66)
ω d C
Equation (1.67) represents equivalent resistance or AC impedance of the
whole series RLC circuit. It indicates that when a circuit is operating at a resonant frequency (i.e., X L = X C ), the overall impedance is equal to a real resistance R. If the frequency is increased to exceed the resonance, the capacitive
reactance will be reduced and the inductive reactance will be increased. The
situation is likewise reversed at frequencies below resonance.