CC BY
15
Y^K 621.3.01
Investigation of energy processes in circuits of oscillatory charge of supercapacitors
Biletskyi O. O., Kotovskyi V. I.
1 National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
E-mail: bilclsky27&gmail.com
Introduction. At this stage of the continuous development of combined power supply systems there is a problem of improving the methods and electrical devices aimed at the accumulation of energy and its dynamic transmission to consumers. In modern modes of operation of electric vehicles (EV), energy storage devices must withstand millions of cycles of charge/discharge without degradation of electrical characteristics. Supercapacitors (SP) can process at least one million cycles and can be used in combined power supply-systems of the EV and various electrical and electromechanical objects that are stochastically in need of high pulse power.
Problem statement. In combined power supplies from the SC and accumulator battery (AB) combine high performance with the specific power of the SC with high energy specific AB. which can easily provide high power at the beginning of the movement of the EV or at a sharp change in the speed of movement., while providing the required energy storage with AB in long motion. Using combined systems with SC and AB can significantly increase the life of AB and work with low losses in circuits oscillatory charge SC. The purpose of this work is to develop the theory of energy processes in the circle of the oscillatory charge SC from AB. which is based on the consideration of the dependence of the capacity of the SC on the voltage on their terminals and the purposeful change of the initial voltages of their charge, which improves the energy efficiency of the combined power supply systems.
Results. In this work, a study of the energy characteristics in the circuits of the oscillatory charge of supercapacitors from a storage battery, which is considered as a real source of electromotive force (EMF), has been carried out. A comparison of the power characteristics of circuits of oscillatory charge SC with different values of the quality factor of the charging circuit is carried out. The approximated solution of the nonlinear nonuniform differential equation of the second order for an oscillatory process of charge SC from AB. in which the capacitance is a linear function of the voltage at its terminals is obtained, which makes it possible to determine the dependence of energy losses in charge circles on the parameters of their elements. Conclusions. The conditions for increasing the energy transfer coefficient from AB to SC in the circuits of the oscillatory charge are analyzed. The features of the influence of the initial voltages, capacities and Q-fact.or of the charging circuit on the energy transfer coefficient from AB to SC are determined. The regularities of increasing the energy transfer coefficient and reducing the power losses in the circuits of the oscillatory charge from the SC from AB with the increase of the effective Q-fact.or of the charge circuit, and the initial voltages on the terminals of such a SC are established.
Key words: energy processes: charge: supercapacit.or: internal resistance: battery: power losses DOI: 10.20535/RADAP.2019.76.5-14
Introduction
At the stage of the steady development of combined power supply systems there is a problem of improving the methods and electrical devices aimed at the accumulation of energy and its dynamic transmission to consumers fl 4]. In modern modes of operation of electric vehicles (EV). energy storage devices must withstand millions of cycles of charge/discharge without degradation of electrical characteristics. Supercapacitors (equivalent denomination ionistors, ultra-capacitors, non-linear capacitors, or double-layer electrochemical capacitors) can process at least one
million cycles and can be used in combined power supply systems of the EV and various electrical and electromechanical objects that are stochastically in need of high pulse power [2,4 10,15,16].
Supcrcapacitor (SC) can provide currents and power tens of times larger compared to new lithium ion batteries and withstand a thousand times more cycles of charge/discharge without destroying [1,2,8,11]. Specific power of industrial designs of SC is 9 • 104 W/kg, which is 22 times more than for lithiumion accumulator battery (AB) [2, 11]. The duration of the charge processes of the SC is from 1 to 30 seconds, which is almost 1000 times less than in AB. In
modern samples of SC. the specific energy is almost 7 times lower than in industrial lithium ion batteries [2]. According to these features, in combined power supply-systems AB is used for long-term power modes, and SC for providing pulsed modes with high capacities.
An analytical review of the work related to the study of power characteristics in electric circuits of combined power supplies from the SC and AB confirmed that the studies in most cases were conducted without an analysis of the energy transfer coefficient from AB to SC. There is also no analysis of electrical energy losses in the circuits of the oscillatory charge of the SC from AB in combined systems under different initial conditions under voltage on the terminals of the SC fl 9]. Such approaches for a long time have prevented the carrying out of studies on increasing the energy performance of combined power supplies from the SC and AB.
1 Formulation of the problem
In combined power supplies from the SC and AB combine high performance with the specific power of the SC with high energy specific AB. which can easily provide high power at the beginning of the movement of the EV or at a sharp change in the speed of movement, while providing the required energy storage with AB in long motion [3.9]. Using combined systems with SC and AB can significantly increase the life of AB and work with low losses in circuits oscillatory charge SC.
The purpose of this work is to develop the theory of energy processes in the circle of the oscillatory charge SC from AB. which is based on the consideration of the dependence of the capacity of the SC on the voltage on their terminals and the purposeful change of the initial voltages of their charge, which improves the energy efficiency of the combined power supply systems.
2 Energy processes in the circuits of oscillatory charge of supercapacitors
Let us investigate the oscillatory charge of the SC from the lithium ion battery, respectively, we will consider only the charging circuit of the combined power supply represented by the equivalent scheme in Fig. 1.
According to the equivalent scheme of the combined power supply EM (Fig. 1). the SC is charged from the lithium ion battery due to the active resistance of the charging circuit Rs = RAB + Ri + Rw, inductance coil L and switch. In this scheme, as a constant voltage source is used a lithium-ion battery with a nominal voltage Un = 2,3 V and an internal resistance RAB = 0,012 ohm. SC is represented as the equivalent circuit with the parallel branches with different time constants t = RC f ], resistance of the wires
Rw =0,01 ohm. This scheme with three branches with sufficient accuracy reflects the energy processes in the SC with the duration transients up to 30 minutes. The first branch is represented by the capacity, the value of which depends on the voltage. This branch consists of elements Ci mid Ri, the values of which do not change and the element Cv (Ui), whose value depends on the applied voltage to the SC. The branch has such a small constant time that its capacities are recharged in a few seconds. The second branch with unchanged parameters C^d R2 is used to display transient processes that take minutes. The third branch has the biggest time constant and reflects transient processes lasting more than 10 minutes, and it is assumed that the parameters C3 mid R3 are unchanged, that is, they are not voltage dependent on the terminals of the SC. To take into account the self-discharge of the SC in the equivalent circuit, a resistor R4 is used f , ].
r-
Rae E
I 1 1
R1
Rnp
C,(U1)
R2
C2
R3
C3
R4
Fig. 1. Equivalent circuit for a combined power source with AB and SC
The dose of energy, that is selected from lithium ion battery ctt 3. charge of SC from zero initial conditions Use (t = 0) = 0,i (t = 0) = 0 for the final voltage Use (t = to) = UAB, i (t = to) = 0, can be found by-expression:
CO UAB
Wab = Juab ■ i(t) ■ dt = Uab j C(U) ■ dU, (1)
0 0
where UUAB the voltage of the battery.
The change of the dose of electric energy entering the SC during the charge from zero initial conditions Use (t = 0) = 0,i (t = 0) = 0 for the final voltage Use (t = to) = UAB, i (t = to) = 0(and, accordingly, change the value of charge Q on each of the plates from 0 ao Q fin) can be found from the expression:
» Qfin
A Wse = J U (t) ■ i(t) ■dt = J U ■dQ =
00 Uab
= J U ■ (Ci + 2kU) ■ dU. (2)
0
In the work [12], when analyzing the energy-processes of the charge of a linear capacitor, variants of increasing the energy characteristics were proposed, using non-zero initial conditions under stress on terminals of linear capacitors, but the energy processes in
L
U
_ 2
the circuits of the chargc SC. with non-zero stress conditions, were not considered.
To study the expedient operating modes of a combined power supply system with SC and AB, it is necessary to analyze the energy characteristics in the process of oscillatory charge of SC from lithium-ion AB at non-zero initial conditions under voltage at the terminals of the SC, in the range — Uab < Uosc < +Uab .
In the study, the initial and final conditions for the current in the charge circuit were identical: i(t = 0) = i (t = <x) = 0. The dose of energy entering the SC is analyzed: the dose of energy selected from AB: the energy of losses in the circuit of the charge SC and the coefficient of energy transfer from AB in the charge from the source of the constant voltage. The points in the range are studied — Un, -0,9 • Un,..., +Un. The total capacity of the SC was represented by the sum of the constant capacity C\ = const and the capacity Cv (U) = k • |U|, which is linearly dependent on the voltage value U [3 7,10,11,13]:
C (U) = Ci + k ■ \U|.
(3)
Qf
1
R
E
{c^ü)
> 0, 5.
(4)
The expression for the current in the circuit of charge SC can be written in the form [13]:
z(t) _ (Ci +2k\USC (i)\)
fdUsc (t)
V dt
.
(5)
Given the second Kirchoff law for the scheme of the electric circuit of the oscillatory charge of the SC can be written:
E _ URc (t) + UL (t) + Use (t),
(6)
where URTl (t) = URab (t) + URl (t) + URw (t) - voltage drop on the resistive elements of the charging circuit. The expressions for URS (t) mid Ul (t) can be given:
URs (t) = Rs • i(t) =
= Rs • ((C\ +2k lUsc m • (^f^)) , (7)
UL (t) _ L
di (t) dt .
(8)
For the oscillatory charge of the SC from AB, the parameters of the electric circuit are chosen such that the condition for the Q-factor is satisfied. Considering the expression (3), we have:
The derivative of time for expression (5) has the form:
^ = 1 ((ci+2 \ Usc m •( ^
= c + 2k\USC [t)\)/ ■ (Ï +
+ (Ci +2k \Usc (i)\)
fdUsc (t)
V dt
Taking into account the expressions (3)-(4), the Q-factor of the charging circuit is a function of the voltage on the terminals of the SC Qf (U). In the study, two different values of the inductance L\ = 1,697 H and L2 = 42,438 H (estimated values) and the total resistance of the charging circuit Rs = 0,0245 ohm were used. At the nominal voltage on the SC |U„| the Q-factor of the oscillating circle of charge (Fig. 1) is Qfi dUnD = 2 mid Qf2 dUnD = 10. The dependence of the Q-factor of the charge circuit from the voltage is taken into account in the study of combined power supply systems.
3 The solution of nonlinear nonhomogeneous differential equation
In fig. 2 is shown the voltage dependence on the terminals of the SC (a) and the current in the charge circuit (b) (at Q-factor Qf i () = 2) at a charge from zero initial conditions Use (t = 0) = 0, i(t = 0) =0 to the moment, when the switch is closed i(t) =0 and the voltage on the terminals will be equal Use (tf ) = Uf.
_2 JdUso (t)\2 f\USc (t)\\ + = 2k{ dt ) V USc (t)) +
+ (Ci +2k \Usc m (d2Usc (ty
(d2Usc (t)\ V dt2 ).
(9)
The expression on Kirchhoff's second law, taking into account the expressions (5)-(9), will have the form:
(dUsc (t)\2 (| tfsc m
Y f\Usc (t)\\ J V Use (t) J
2kl ■ I "rw I 1 I +
'd2Usc (t)
V dt J V Use (t) + L ■ (Ci + 2k\USC (t)\)
d 2
+
+ Rc ■ ((Ci + 2k\USC (t)\) •(^^
+
+ U ( ) _ E.
The following replacements should be introduced to simplify the appearance of the non-linear nonhomogeneous differential equation of the second order (10):
Use (t) _ U, \Usc (i)\_ \U\,
dUsc (t) _U/
(ID
d2USc (t) _ U// d,t2 U .
Fig. 2. The dependence of the voltage on the terminals of the SC (a) and the current in the charge circuit (b) at a charge from zero initial conditions to the moment of closing the switch
The nonlinear nonhomogeneous differential equation of the second order (10). after the use of substitution (11). will have the form:
L ■ (Ci + 2k |U|) ■ UH + 2kL ■ (UI)' ■^-U^l +
+ Rs ■ (Ci + 2k |U|) ■U1 + U = E. (12)
This nonlinear nonhomogeneous differential equation of second order can not be solved by direct methods. It is necessary to get the approximate value of the roots, using a numerical method in the package of applications MATLAB. Let's take the parameters of the equivalent circuit (Fig. 1) such that Q-factor is Qf i (Un) = 2. The oscillatory charge of SC occurs from zero initial conditions Use (t = 0) = 0, i(t = 0) = 0 to the moment when the switch is closed i(t) =0 and the voltage on the terminals will be equal Use (t f) = Uf.
4 Approximation of the voltage dependence on the supercapaci-tor terminals
The approximation of the voltage dependence on the terminals of the SC Use (t), in the process of oscillatory charge (Qf i (Uf) = 2), was implemented with the Curve Fitting Tool (MATLAB) application, which allows describing the approximated function by the given data vectors Use (t) mid t from the MATLAB workspace. An exponential function is selected as an approximated function in the Curve Fitting Tool application. The dependence of the voltage on the terminals of the SC in the process of oscillatory charge over a time interval from i = 0 to i = 180 s is depicted in Fig. 2 a. The voltage on the terminals of the SC. in the workspace of the application package MATLAB. is
given by the data vector (with the dimension of 1x4699 points) [14].
The solution of the nonlinear nonhomogeneous differential equation of the second order (12) will be written as the sum of two exponents:
Use (t) = a ■ ebt + c ■ edt, (13)
where a, b,c,d - the stable of integration, which are determined from the initial conditions.
For these initial conditions, the coefficients are defined with an accuracy of 95 %:
a = -2.21e + 04 (-1.648e + 11,1.648e + 11); b= -0.2949 (-336.9, 336.3); c = 2.21e + 04 (-1.648e + 11,1.648e + 11); d = -0.2949 (-336.9, 336.3).
The statistics of the data approximation in Curve Fitting Tool's application for the voltage dependence function on the SC terminals, depending on the time Use (t):
RAISE: 0.071 the standard error. A value close to 0 indicates that the approximation can be used successfully because the standard deviation of the sample mean value is within the normal range.
R-square: 0.997 the correlation area between the initial values and the approximate values. A value close to 1 indicates that the variance is negligible [14].
Adjusted R-square: 0.997 this is the number of degrees of freedom of the approximated correlation area. A value close to 1 indicates a good approximation.
The current in the charging circuit is determined in accordance with the expressions (5) and (13):
i(t) = (Ci + 2k (a ■ ebt + c ■ edt)) ■
■ (a ■&■ ebt + c ■ d ■ edt) . (14)
The approximate solution of the nonlinear nonhomogeneous differential equation of the second order (12) in the form (13) satisfies the requirements for approximation. The root of the mean-square error is close to 0: accordingly, the solution of this nonlinear differential equation in the form (13) can be successfully-used to analyze the energy processes in the circuits of the oscillatory charge SC from the source of a constant EMF.
The correlation area between the initial values and the approximate values and the number of degrees of freedom, of the approximated correlation area, which indicate a good approximation.
5 The analysis of the energy characteristics of the oscillatory charge of a supercapaci-tor
The dose of energy that is selected at the oscillatory-charge of the SC from the source of the electromotive force is determined by the expression
"t
WAB = JuAB • i(t) • dt,
(15)
where UAB - the voltage of the source of the electromotive force, in this case the battery.
After substituting in this expression the formula for the current in the charging circuit (14). we obtain:
WA B
f Uab (C + 2k (aebt + cedt))
u
• (a beb 1 + cdedt) dt. (16)
The dose of energy that enters the SC. with oscillatory charge from the initial voltage Ui to the final Uf
cJu] -Ui) 2k (uf -Ui) AWSC = -¿ + —^-¿ =
= ( Uf - Ui) • C (Uf + Ui) 2k (U/2 + UfU + Ui)
2
+
3
(17)
The energy transfer coefficient ■qsc is determined by the ratio of the energy received in the SC to the energy-
selected from AB for the entire time of the oscillatory-charge:
Vsc =
(Wsc (t f ) — Wsc (t i)) (Wab (ti) -Wab (tf ))
(Uf - Ui)
Ci (Uf + Uj) + 2k(u2 + UfUi + U?)
0 + Q
tf
(18)
J Uab • « (t) dt
respectively Wse(ti), Wse(tf) - the energies that were accumulated in the SC accordingly before switching and after the end of transient process of the oscillatory charge from the AB; A WAB = WAB(ti) -WAB (t f) - the energy given by AB during the oscillatory charge.
The energy of losses in the circuit of oscillatory-charge of the SC from AB can be determined from the expressions (15) (17). This is energy, which is the difference between the energy given by AB and the energy received SC during the oscillatory charge:
Wiosses = (Wab (t i) -Wab (t f)) -
- (Wse (t f) -Wse (U)) (19)
The dose of energy W^, that enters to the SC during the oscillatory charge from the initial voltage Ui to the final voltage Uf, is reduced to the value of W0AB, that is to the quantity of the dose of energy-selected from AB with an aperiodic charge of a fully-discharged SC (U0se (t = 0) = 0), is determined by the expression:
w<„ = (U'-U)
tf
J Uab • i(t) • dt
0
C (Uf + Ui) 2k (uf + UfUi + Ui)
2
3
.
The expression for the energy dose WIAB, which is taken from the AB during the charge from the initial voltage Ui to the final voltage Uf, has reduced to the value of W0AB, can be written as
W
A B
tf
UA B ( )
u ~tf
UA B ( )
0
tf
( )
u
~tf .
( )
0
(21)
s e
tli oscillatory charge of the SC from the initial voltage Ui to the voltage Uf, for the given values of energy-
doses WgC mid W'AB, has the form
r!
W
S C
S C
WAb
(22)
f
!
!
The energy of losses w(osses in the circuit of the oscillatory charge SC from the initial voltage Ui to the final voltage Uf, has reduced to the value Woab, can be found from expressions (20)-(22)
W,
/,sses = (Wab(ti) -Wab(tf))/-- (Wsc(tj) - Wsc(ti))/ = Wab - W/sc. (23)
6 Processing and analysis of the results
Accordingly in tab. 1. the values of the energy characteristics in the circuit of the oscillatory charge of the SC from the initial voltage Ui to the final voltage Uf, are given at Q-factor of charging circuit Qf 1(Un) = 2(L = 1.697 H):
- dose of energy WSc, which enters to the SC during the oscillatory charge:
- dose of energy Wab, which is selected from the AB during the oscillatory charge:
energy of losses Wia llatory charge SC:
in the circuit of osci-
- coefficient of energy transfer from AB r]sc, under given conditions. In tabl. 2 shows similar energy characteristics, when the charge from the SC from AB in a oscillatory charging circuit with Qf2(Un) = 10 (L = 42.438H).
Numerical simulation in the MATLAB application package completely corresponds to mathematical dependencies (1). (23).
It is necessary to analyze the energy characteristics in the circuits of the oscillatory charge of the SC with variable initial voltages Uqsc- the dose of energy-entering the SC: the dose of energy selected from AB: energy of losses in the circuit: the coefficient of energy-transfer from AB.
From tabl. 1 it is evident that, when the Q-factor of the oscillatory charge circuit of SC is Qf 1 (Un) = 2, the coefficient of energy transfer from AB changes in the range from 27,13 %, when U0Sc = —Un, to values exceeding 90,0 %, when U0sc > 0,7 ■ Un.
At initial voltage on terminals of the SC U0sc = = 0,9 ■Un , the coefficient of energy transfer under the charge the SC (at Q-factor Qf 1(Un) = 2) is increased by 1,27 times, when compared with the coefficient of energy transfer at zero initial conditions of voltage: and when changing the initial voltage of the SC from Uosc = —0, 9 ■ U„ to Uosc = 0,9 ■ Un, the coefficient of energy transfer increases by 2,8 times.
The dose of energy Wsc, that entering to the SC at the initial voltage on the terminals U0sc =0, 5 ■ Un, is 2912,52 J, which is 1,4 times less than with the voltage U0sc = 0V, but the energy losses in the circuit of
oscillatory charge of the SC, under given conditions, will be smaller in 2,61 times. At initial voltage on the terminals of the SC U0Sc =0, 7 ■Un, the dose of energy Wsc, that entering to the SC, will be less than 2,032 times than the dose of energy, entering the SC with a oscillatory charge from zero initial conditions, but the energy of losses will be less in 6,35 times.
With Q-factor Qf 1(Un) = 2, the dose of energy-selected from AB, when charged from the initial voltage SC U0Sc = —0,5 ■ Un to the nominal voltage on the terminals of the SC, 1.33 times more than when charged from the voltage U0sc = 0 V. With a oscillatory charge of the SC from the voltage U0sc = 0,7 ■ Un to the voltage Un, the dose of energy selected from AB will be less than 2,43 times, in comparison with the energy dose at a charge from zero initial conditions.
With the Q-factor Qf2(Un) = 10 (L = 42,438H) of the oscillating charge circuit of the SC, the process of charge of the SC occurs at the coefficient of energy-transfer from AB rise > 57 % (Table ) and the higher the value of the initial voltage on the SC U0Sc, the higher the coefficient of energy transfer from AB.
At the charge of the SC from the initial voltage at the terminals U0sc = 0, 9 ■ U the coefficient of energy-transfer from AB increases by 5,56 %, compared with Un the coefficient of energy transfer from AB at zero initial conditions of voltage (Table 2). When changing the initial voltage on the terminals of the SC from U0sc = —0,8^Un to U0sc = 0, 8■Un, the coefficient of energy transfer changes 1,4 times and makes 98.23 %.
Accordingly, the dose of energy Wsc, accumulated during the charge in the SC from the initial voltage at the terminals U0sc = 0, 5 ■ Un to the nominal voltage Un, will be 1,55 times less than the dose of energy entering the SC during the oscillatory charge from zero initial voltage on the terminals of the SC: and the energy of losses Wiosses in the circuit of the oscillatory charge of the SC, while, will be less than 2,59 times. At the initial voltage on the terminals of the SC U0Sc =0, 9 ■Un, the dose of energy accumulated in the SC during the oscillatory charge will be 6,6 times less, than the dose of energy at the initial voltage of U0sc = 0 V; under these conditions, the energy of losses Wlosses, in the circuit of the oscillatory charge SC from the initial voltage on the terminals of the SC U0sc = 0,9 ■ Un, will be less than 49 times for the energy of losses at oscillatory charge from zero initial conditions.
The dose of energy Wab, which is selected from the AB during the oscillatory charge at Q-factor of the charge circuit Qf2(Un) = 10 (L = 42,438H), varies nonlinearly from 12178.00 J with the initial voltage at the terminals U0Sc = —0, 9 ■ Un up to 1064,82 J, at initial voltage at the terminals U0sc = 0,9 ■ Un- At the initial voltage at the terminals of the SC U0sc = 0,5Un, the dose of energy selected from AB will be 1,59 times less, and at initial voltage U0sc =0, 9 ■ Un- in
Table 1 Experimental data Qfi (Un) = 2
rise ^^losses,J Wab ,J Usemax,^ Wse ,J U0se ,V U0se/Uab
27.13 7051.18 9677.18 3.05 2626.00 -2, 30 -1,0
34.55 5963.24 9111.24 3.05 3148.00 -2, 07 -0, 9
41.91 4975.33 8565.33 3.05 3590.00 -1, 84 -0, 8
48.72 4127.39 8049.59 3.04 3922.2 -1, 61 -0, 7
54.91 3410.20 7563.10 3.02 4152.90 -1, 38 -0, 6
60.38 2814.39 7104.09 3.01 4289.70 -1,15 -0, 5
65.12 2329.00 6677.10 2.98 4348.10 -0, 92 -0,4
69.07 1943.40 6282.50 2.95 4339.10 -0, 69 -0, 3
72.19 1647.97 5926.07 2.92 4278.10 -0, 46 -0, 2
74.53 1429.92 5614.24 2.89 4184.32 -0, 23 -0,1
76.05 1282.41 5355.17 2.86 4072.76 0 0
77.45 1148.28 5091.78 2.83 3943.49 0, 23 0,1
79.12 994.27 4762.74 2.79 3768.47 0,46 0, 2
81.05 828.17 4370.43 2.75 3542.25 0, 69 0, 3
83.20 657.89 3916.85 2.70 3258.96 0, 92 0,4
85.56 491.37 3403.88 2.64 2912.52 1,15 0, 5
88.12 336.68 2833.01 2.58 2496.34 1, 38 0, 6
90.84 '201.95 2205.70 2.52 2003.73 1, 61 0, 7
93.74 95.42 1523.11 2.45 1427.69 1, 84 0, 8
96.78 25.31 786.36 2.37 761.05 2,07 0, 9
100.00 0 0 2.30 0 2, 30 1,0
Table 2 Experimental data Qf2 (Un) = 10
rise ^^losses,J Wab ,J Usemax,^ Wse ,J U0se ,V U0se/Uab
57.17 5462.00 12754.00 3.89 7292.000 -2, 30 -1,0
63.93 4393.00 12178.00 3.89 7785.00 -2, 07 -0, 9
70.24 3445.00 11575.00 3.87 8130.00 -1, 84 -0, 8
75.77 2659.80 10978.00 3.84 8318.00 -1, 61 -0, 7
80.51 202510 10389.00 3.81 8363.90 -1, 38 -0, 6
84.48 1523.29 9814.99 3.76 8291.700 -1,15 -0, 5
87.68 1140.87 9261.97 3.71 8121.10 -0, 92 -0,4
90.17 859.23 8737.33 3.65 7878.10 -0, 69 -0, 3
91.92 666.93 8253.03 3.59 7586.10 -0, 46 -0, 2
93.02 545.33 7818.65 3.53 7273.32 -0, 23 -0,1
93.51 483.19 7448.41 3.48 6965.22 0 0
93.87 433.08 7068.21 3.42 6635.13 0, 23 0,1
94.31 375.32 6596.47 3.35 6221.15 0,46 0, 2
94.81 313.13 6037.37 3.26 5724.24 0, 69 0, 3
95.39 248.93 5396.42 3.16 5147.49 0, 92 0,4
96.02 186.24 4675.76 3.05 4489.52 1,15 0, 5
96.70 127.84 3879.05 2.93 3751.21 1, 38 0, 6
97.44 79.91 3009.67 2.79 2932.76 1, 61 0, 7
98.23 36.56 2070.62 2.64 203A.06 1, 84 0, 8
99.07 9.86 1064.82 2.48 1054.96 2,07 0, 9
100.00 0 0 2.30 0 2, 30 1,0
6,99 times loss than the dose of energy, which is selected from AB. when charged from zero initial conditions by-voltage.
A generalized analysis of functional dependencies (Table 1.2) with oscillatory charge of the SC from AB (with the Q-factors of the charging circuit Qf i(Un) = 2 and Qf2(Un) = 10), confirms that when changing the initial voltage on the SC within -Un < U0se < Un, the dose of energy WAB, which is selected from AB during the oscillatory charge, and the energy of losses Wlosses in the circuit of the oscillatory charge of the SC, are nonlinearly reduced from the maximum values at the initial voltage at the terminals U0se = -Un to
U0 s e Un
Under these conditions, the dose of energy Wse, which enters to the SC during the oscillatory charge with the Q-factor of the charging circuit Qfi(Un) = 2, will be the maximum Wse = 4348,10 J at the initial voltage at the terminals of the SC U0se = -0,4 ■ Un, and at Q-factor Qf2(Un) = 10 the maximum dose of energy Wse = 8363,90 J, at initial voltage U0se = -0,6■Un, after reaching these values, the dose of energy Wse entering the SC will be nonlinearly reduced with values U0se ^ Un.
The coefficient of energy transfer from AB with
s e
charge circuit Qfi(Un) = 2 and at Qf2(Un) = 10, increases nonlinearly from the minimum values at U0se = -Un to maximum at U0se ^ Un.
Moreover, the larger the Q-factor of the oscillatory-charging circuit Qf (Un), the higher is the dose of the energy WAB, which is selected from AB during the charge of the SC, and the dose of the energy Wse that enters the SC with a higher the coefficient of oriorgy s e
oscillatory charge of the SC Wlosses of a greater value of the Q-factor of the circuit will be less. Thus, with the oscillatory charge of the SC from a source of constant EMF (in this case, lithium ion AB), with a high value of Q-factor of the charge circuit Qf (Un), negative values of the initial voltage on the terminals of the SC can be used and this will be advantageous from the energy-point of view.
For example, with Q-factor of charging circuit Qf2(Un) = 10, you can accumulate large amounts of energy in the SC, with the coefficient of energy-transfer r]se > 57 % and the larger the value of the U0 s e
greater will be -qse. To increase the coefficient of energy-transfer ■qse, at small values of the Q-factors of the charge circuit Qf (Un), it is necessary to increase the U0 s e
SC.
When choosing positive initial voltages on the terminals of the SC, it should be kept in mind that this reduces the dose of energy W'se entering to the
U0 s e
Un
Conclusions
1. With the oscillatory charge of the SC from AB, the coefficient of energy transfer from AB varies in a nonlinear range from the minimum values, at
U0 s e = - Un U0 s e
Un
higher the Q-factor of the charge circuit of the SC Qf (Un), the higher will be the value of the
s e
2. The dose of energy W'se entering into the SC during the time of one oscillatory charge at the Q-factor of the charging circuit Qfi(Un) = 2 will be the maximum W'semax = 1,163, at initial voltage SC U0se = -0,4 ■ Un; and for Q-factor Qf2(Un) = 10, the maximum value of the energy dose is W'semax = 2, 236, with initial voltage U0se = -0,6 ■ Un; after reaching these values, the dose of energy W'se entering the SC will decrease nonlinearly at U0se ^ Un values. According to the analysis, with the oscillatory-charge of the SC from AB, with a high value of Q-factor of the charge circuit Qf (Un), negative values of the initial voltage on the terminals of the SC can be used and this will be advantageous from the energy point of view.
3. The analysis of the functional dependences in the oscillatory charge of the SC from AB, confirms that when changing the initial voltage on
- Un < U0 s e < Un the energy of losses in the circuit of the oscillatory charge SC is nonlinearly reduced from the maximum values at the initial voltage at the terminals U0se = -Un to the minimum values, at a U0 s e Un
the energy of losses W'losses is reduced to the value of WAB, with an oscillatory charge of the SC from zero initial voltage conditions, with a higher value of Q-factor of the charging circuit Qf2(Un) = 10, it will be less by 62,39 % (for W'losses at Qf i(Un) = 2). In the transition to the region of positive values of the initial voltages on U0 s e > 0
losses W'losses significantly decreases and at Q-factor Qf2(Un) = 10 (U0se = 0,7 ■ Un) energy-losses make up 2.6 % of the energy, which gives AB.
4. The energy doses that are selected from AB W'AB are given to the value W0AB, with a Q-factor of charging circuit Qf2(Un) = 10 and the initial voltage of the SC U0se = -0, 7^Un will be
U0 s e = 0
in 1,39 times more, than the value of the dose of energy W'AB at Q-factor Qfi(Un) = 2.
References
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Дослщження енергетичних процес1в в колах коливального заряду суперкон-денсатор1в
Бглецький О. О., Котовський В. И.
Вступ. В дапий час при сталому розвитку комб!-повапих джерел живлеппя випикае проблема полшше-ппя електричпих пристрош призпачепих для збер!га-ппя eneprii' i 11 дипам1чпо1 передач! споживачам. При сучаспих режимах експлуатацп електромоб!л1в (ЕМ), пристро! для пакопичеппя eneprii' повипш працювати мгльйопн цикл!в заряд/розряд без попршеппя епергети-чпих характеристик. Суперкопдепсатори (в л!тератур! в!дом1, як ioiiicTDpn, ультракопдепсатори, пелипйш коп-депсатори або двошаров! електрсшм!чш копдепсатори) можуть працювати по мепше мгльйопа цикл!в i можуть ycniniiio використовуватись в комбшовапих джерелах живлеппя ЕМ або в р1зпомаштпих електромехашчпих об'ектах, як! стохастичпо потребують велику 1мпульспу потужшеть.
Постановка задач!. В комбшовапих системах еле-ктроживлеппя з суперкопдепсаторами (СК) та акуму-ляторпими батареями (АВ) поедпуються висока питома потужшеть СК з високою питомою eneprieio АВ. Та-кий шднд дозволяв легко забезепечити пеобх1дпу потужшеть па початку руху електротрапепорту або при р!зкому зб1льше1ш! швидкост руху, при цьому забезпе-чуючи пеобх1дпий запас eneprii' 3 АВ, при тривалому pyci. Застосувашш комбшовапих систем з СК та АВ може зпачпо шдвищити термш служби АВ та дозволяв працювати з пизькими втратами eneprii в кол! коливального заряду СК. Метою ц1е"1 роботи е вдоскопалегшя теори епергетичпих процес!в в електричгшх колах коливального заряду СК в!д джерел nocTifuioi ЕРС (АВ), яка базуеться па врахувашп залежпост! емпост! СК в!д прикладепо! до його клем папруги та врахувагш! змшпих початкових умов по папруз! при заряд! СК, що полшшуе епергоефектившеть комб!пова1шх систем електрожгшлешш.
Результати. В дашй робот! досл!джепо епергетичп! характеристики в колах коливального заряду СК в!д реального джерела пост!йпо1 ЕРС (АВ). Проведено по-р!впяппя епергетичпих характеристик ил коливального заряду СК при р!з1шх добротпостях зарядного контуру. Зпайдепо иаближепе р!шеппя для пел!шйного пеоднор!-дного днфере1щ!алыюго р!впя1шя другого порядку при коливалыюму заряд! СК в!д АВ, при врахувагш!, що емшеть е л!шйпою фупкц!ею в!д папруги па клемах СК. Дапе р!шеппя дае можлив!сть визпачати залежшеть eneprii' втрат в колах коливального заряду пелшшпого конденсатора в!д електротехшчпих параметр!в елемеп-т!в.
Висновки. Проапал1зоваш умови, за яких зростае коефкцепт передач! еперги в!д АВ до СК, в колах коли-вального заряду СК. Визначено вплив початкових умов по напруз! на клемах СК, добротноста зарядного контуру, емноста СК на коефщ!ент передач! енерги в!д АВ до СК в процеа заряду. Доонджено, що при шдвпщенш добротноста зарядного контуру та зб1лыненш початкових умов по напруз! на клемах СК, можна збшынити коефкцент передач! енергп в!д АВ до СК та зменшити енерпю втрат при коливальпому заряд! СК в!д АВ.
Ключовг слова: енергетичш процесп; заряд; суперконденсатор; внутршнш ошр; акумуляторна батарея; втратп електроенерги
Исследование энергетических процессов в цепях колебательного заряда суперконденсаторов
Белецкий О. А., Котовский В. И.
В работе проведено исследование энергетических характеристик в цепях колебательного заряда суперкон-
денсаторов от аккумуляторной батареи, которая рассматривается как реальный источник ЭДС. Проведено сравнение энергетических характеристик цепей колебательного заряда суперконденсатора (СК) при различных значениях добротности зарядного контура. Получено аппроксимированые решения нелинейного неоднородного дифференциального уравнения второго порядка для колебательного процесса заряда СК от аккумуляторной батареи (АВ), у которого емкость является линейной функцией от напряжения на его клеммах, что дает возможность определять зависимости энергетических потерь в цепях заряда от параметров их элементов. Проанализированы условия увеличения коэффициента передачи энергии от АВ в СК в цепях колебательного заряда. Определены особенности влияния начальных напряжений, емкостей и добротностей зарядного контура на коэффициент передачи энергии от АВ в СК.
Ключевые слова: энергетические процессы; заряд; суперконденсатор; внутреннее сопротивление; аккумуляторная батарея; потери электроэнергии
