�
�
�
�
�
28
Electrochemical Supercapacitors for Energy Storage and Delivery
V V
0 sin ω t
c =
d
(1.56)
The current passing through the capacitor can be expressed as
dV
�
c
I = C
(1.57)
c
dt
Combining Equations (1.56) and (1.57), the following current expression
is obtained:
�
� 0
� 0
I = ω CV cos ω t = ω CV sin( ω t + 90
0 )
(1.58)
c
d
d
d
d
A capacitive reactance X C can be defined and expressed as
1
X =
(1.59)
C
ω C
d
This capacitive reactance has the same unit (ohm) as that of a resistor.
Substituting Equation (1.59) into Equation (1.58) yields
� 0
V
0
I =
sin( ω t + 90°) = I sin( ω t + 90°)
(1.60)
c
d
d
X C
Thus, the AC current has a phase constant of 90° ahead of the driving potential. Similar to capacitive circuits, a phase constant of –90° for an inductive
circuit can be obtained, indicating the current lags the potential by one quarter cycle. An equivalent inductive reactance X L in a circuit only containing
an inductor can be defined as
X L = ω d L
(1.61)
1.6.6.1 Series Resistor–Inductor–Capacitor Circuit
With the use of brief derivations of current through AC circuit elements,
more explicit evaluation of the current amplitude and equivalent resistances
in an RLC circuit can be performed [2]. By applying symmetry of notation
and the Pythagorean theorem, the amplitude of these oscillating functions
shown by phasor notation in Figure 1.10 gives a relationship as
� o 2 � 2 �
�
( ) = V + ( − V
V
V
)
(1.62)
R
L
C