26
Electrochemical Supercapacitors for Energy Storage and Delivery
and Equation (1.45) can be simplified to the differential equation:
dE
d q
2
1
= L
+ q
2
= 0
(1.46)
dt
dt
C
The solution to this differential equation with a maximum charge Q (i.e.,
amplitude of oscillation) at time t = 0 can be written as:
q = Qcos(ωt + φ)
(1.47)
where ω is the angular frequency of electromagnetic oscillation and φ is the
phase angle constant. By taking the derivative of Equation (1.46) with respect
to t, a function describing the current through an LC circuit can be obtained:
I = −ωQsin(ωt + φ)
(1.48)
The amplitude ωQ is equivalent to the maximum current I passing through
the circuit. Solving for ω is done by substituting Equations (1.47) and (1.48)
into Equation (1.46):
1
ω =
(1.49)
LC
This ω is often called the natural angular frequency by which the LC circuit
will oscillate devoid of any external power source.
1.6.5 Resistor–Inductor–Capacitor Circuits
Circuits containing the elements of resistance, inductance, and capacitance
are called RLC circuits [6–7]. By integrating a resistor into an LC circuit, the
oscillating electromagnetic energy is no longer perpetual and will dissipate
as thermal energy. Thus, the differential equation relating the change in electromagnetic energy becomes
dE
d q
2
dq 1
= L
+ R + q = 0
(1.50)
dt
dt
2
dt C
Solving this second-order differential results in the following function:
q = Qe –Rt/2L cos(ωʹt + φ)
(1.51)
Because the resistor is present, the angular frequency is modified to
Electrochemical Supercapacitors for Energy Storage and Delivery
and Equation (1.45) can be simplified to the differential equation:
dE
d q
2
1
= L
+ q
2
= 0
(1.46)
dt
dt
C
The solution to this differential equation with a maximum charge Q (i.e.,
amplitude of oscillation) at time t = 0 can be written as:
q = Qcos(ωt + φ)
(1.47)
where ω is the angular frequency of electromagnetic oscillation and φ is the
phase angle constant. By taking the derivative of Equation (1.46) with respect
to t, a function describing the current through an LC circuit can be obtained:
I = −ωQsin(ωt + φ)
(1.48)
The amplitude ωQ is equivalent to the maximum current I passing through
the circuit. Solving for ω is done by substituting Equations (1.47) and (1.48)
into Equation (1.46):
1
ω =
(1.49)
LC
This ω is often called the natural angular frequency by which the LC circuit
will oscillate devoid of any external power source.
1.6.5 Resistor–Inductor–Capacitor Circuits
Circuits containing the elements of resistance, inductance, and capacitance
are called RLC circuits [6–7]. By integrating a resistor into an LC circuit, the
oscillating electromagnetic energy is no longer perpetual and will dissipate
as thermal energy. Thus, the differential equation relating the change in electromagnetic energy becomes
dE
d q
2
dq 1
= L
+ R + q = 0
(1.50)
dt
dt
2
dt C
Solving this second-order differential results in the following function:
q = Qe –Rt/2L cos(ωʹt + φ)
(1.51)
Because the resistor is present, the angular frequency is modified to