Handbook of Energy Storage for Transmission or ... - W2agz.com

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The energy E, that is stored in an ideal capacitor at voltage V, is equal to:
E = 0.5 CV 2
The energy increases as the square of the applied voltage. When charged at a constant
current, the voltage of an ideal capacitor rises linearly with time. When charged at a
constant power, the stored energy rises linearly with time. In reality, the first order model
of a capacitor is a series combination of an inductor, a resistor, and a capacitor (see
Appendix for additional modeling details). The fundamental equations for all types of
capacitors are summarized in Table 1. Note that R s
, the series resistance, is also referred
to as the equivalent series resistance, ESR.
Table 1 Fundamental equations for all capacitors including electrochemical capacitors.
Stored Charge, Q Q = CV C = capacitance
Stored energy, E, ideal case E = ½ CV 2 V = applied voltage
ε = dielectric constant
Capacitance of parallel
C = εA/t A = area of the capacitor plate
plate capacitor, C
T = separation of the plates
Self-resonant frequency, f o ,
1
f
for RLC circuit
o
=
2 Π LC
L = inductance
Maximum power, P max P max = V 2 /4R s R s = series resistance (ESR)
Resistive charge or
R
= 100
discharge efficiency, η
RL
+ RS
Constant current charge or 100( Vr
− I
oRS
)
η =
discharge efficiency, η ( V + I R )
L
η R L = load resistance
r
o
Electrochemical Capacitors 9
S
Vr = rated voltage
I o = fixed current
For most practical applications in the utility industry, the inductance in the series-RLC
circuit can be ignored because operation is well below the self-resonant frequency. Thus,
a simple series-RC circuit is a good first-order model for the real capacitor.
It is important to understand the effect of the capacitor internal resistance (R s ) on the
efficiency of discharge. For example, modeling the capacitor as series-RC circuit being
discharged into a resistive load R L the efficiency of discharge in percent is equal to
100R L /(R S + R L ). Thus the efficiency is nearly 100% when the load resistance, R L , is
much greater than the internal resistance, R S . On the other hand, the efficiency is exactly
50% for the matched load, that is when R L = R S . That is, for a matched load half the
delivered energy is dissipated in the capacitor itself and not in the load. Similar
efficiency relationships can be calculated for constant current charge or discharge, as
listed in Table 1.
Electrochemical Capacitor Characteristics
What Is a Double-Layer Capacitor
Electrochemical capacitors consist of two electrodes, a separator, electrolyte, two current
collectors, and packaging. Within the electrochemical capacitor, charge is stored
electrostatically, not chemically as in a battery. It has, as a dielectric, an electrolyte
solvent, typically potassium hydroxide or sulfuric acid, and is actually two capacitors