CC BY
5
The Analysis of Energy Transition Processes
in Boost Converter
Martynyuk V. V., Kosenkov V. D., Geydarova O. V., Fedula M. V.
Khmolnvtskvi National University E-mail: fcdula&kh-nu. km.ua
Introduction. This article presents the analysis of energy transition processes in boost converter under the conditions of load changing. The boost converter laboratory layout is investigated with forms of the transient process caused by load changing. The transient currents and voltages are measured in harmonic and anharmonic modes. The parameters of boost converter transient modes are estimated. The analysis of energy transition processes is performed using two boost converter models.
Problem statement. The boost converter circuits are used in the areas of electronics where the minimization of energy losses is strictly required. The boost converter energy losses depend on the form of transient processes caused by load changing. The purpose of energy losses minimization requires exact analysis of energy transition processes which appear under load changing. The presented work describes the responses of boost converter circuit to changing the load twice with four different transient modes. Results. The analysis of energy transition processes is performed with two models. The first model is based on the iterative mapping technique, where the currents and voltages of each period of gate driving signal are determined from the circuit parameters and initial conditions given by the previous period of driving signal. Such model allows to perform an exact analysis of current and voltage ripple transient processes. But the iterative mapping model is characterized by numerical approximation errors and requires more computation time. The second model describes the envelopes of boost converter current and voltage transient processes analytically. This model is based on the state-space averaging method which is widely used for modelling of switching circuits. Such model does not take into account the waveforms of current and voltage ripples, but it provides a more simple description of transient process envelopes which are useful for the circuit design purposes. The modeling results obtained from iterative mapping and state space averaging, match with the experimental data.
Conclusions. The performed analysis shows that the energy losses depend on the transient process mode significantly. During the load transient process time, the smallest energy losses can be obtained under the critical load transient process form which is situated between periodic and aperiodic modes. Such result is obtained experimentally and confirmed by the both iterative mapping and state-space averaging models. The analytical results are confirmed by optimization procedure in MATLAB environment.
Key words: boost converter: energy losses: transient process: critical mode: modelling
DOI: 10.20535/RADAP.2019.77.17-29
Introduction
The boost converters are used in many fields of modern electronics fl 3]. Mostly, boost converters function as sources and converters in power supplies of different electrical devices [3,4]. There are many different circuit realizations of boost converters [2,3,5 9]. But the main principles of DC-to-DC power conversion with output voltage increase correspond to the basic circuit [3,10,11] that is shown in Fig. 1, where the capacitor voltage is increased regarding to the inductor that functions as a current source.
The analysis of transient processes and minimization of energy losses in boost converter circuits are important problems of modern power systems research [3, 12]. Such problems appear, for an example, in renewable energy sources [4,13,14], smart grids [14,15],
and other devices with high requirements to energy consumption [2,16,17]. Also, one of most important boost converter application 3X6 clS IS the photovoltaic energetics [4, 18 20]. The work [21] shows that the photovoltaic battery exploitation efficiency can be improved significantly by using optimizer circuit based on a boost converter. The new directions of boost converter research are related to the investigation of its transient responses under different control forms [20, 22 28]. Various models of boost converters are investigated due to the practical application features [29 34]. So, in these works the fractional order models for DC-DC converters and other circuits are analyzed in accordance with parameters of real electrical parts. One of the most important fields of boost converter development is the analysis of transient responses
(a) (b)
Fig. 1. The boost converter circuit diagram (a) and laboratory realization (b)
under varying load [4,13,17,26]. Thus, it causes transient processes that influence the loss characteristics.
In this paper the analysis of boost converter losses and responses to load transients is performed using analytical techniques and optimization procedure based on iterative mapping models.
1 The Considered Boost Converter
The boost converter circuit diagram [3,11,28] and its realization are shown in Fig. 1.
In the presented work, the boost converter is realized on the base of 2mH inductor with magnetodi-electric core (sendust, = 60) and electrolytic capacitors with ESR < 70ra0 and parasitic inductance less than 10^H. The switch is designed on the base of power MOSFET transistor STW88N65M5 [35]. The rectifier VD is power Shottky diode 30CPQ150 [36]. The source voltage is E = 12.87V. A significant parasitic resistance is modelled by resistor R0 = 10. The converter is investigated with load resistances 250, 500, 1000. The driving signal is generated by UNI-T UTG1010A functional generator.
The more detailed specifications of designed boost converter elements are presented in Table 1.
Such elements are selected for achieving a wide range of experimental modes and more exact analysis of different boost converter modeling techniques.
Fig.
(b)
2. The equivalent circuit: (a) the switch is closed; (b) the switch is open
In the state 1 the inductor current increases up to the maximum value that is defined by parasitic resistance R0. The capacitor is discharged to the load resistance R. Such as the boost converter is separated into two first-order circuits, the transient processes have exponential form in the state 1. Accordingly to the state-space averaging method [3,11,31,37,38], the boost converter (Fig. 1) model can be represented in the form (1).
L
C
diL dt dvc dt
-Ro -(1 - D)
a - d ) - £ .
+
2 The Boost Converter Modelling
The boost converter circuit (Fig. 1) can be represented by two equivalent circuits [3] for open and closed MOSFET switch that is shown in Fig. 2.
X +
VC
1
X
0 0
(1)
where t is the time L is the inductance, C is the capacitance, E is the source voltage, R is the load resistance, is the inductor current, vc is the capacitor voltage, D is the duty cycle of switching (Fig. ). The model ( ) allows to obtain averaged voltage and current transient processes waveforms. For more exact modelling with
Table 1 The main parameters of the designed boost converter (Fig. 1) elements
Parts Parameter Value
inductance 2 niH
magnetic core material sendust
Magnetic core material parameter a:
initial magnetic permeability 60
Inductor eddy current loss coefficient 250 • 10-9 1/Hz
L hysteresis loss coefficient 5•10-3
additional loss coefficient 2•10-3
the operation frequency range 10 kHz
maximum loss angle tangent 0.01
Capa- capacitances 8mF, 40pF, 2.1 nil-". 1.94mF
citors maximum ESR at 10kHz 70 mü
C maximum equivalent inductance 10 pH
part number 30CPQ150
Diode type Shottky
VD repetitive peak reverse voltage VrrM 150 V
maximum average forward current Ip ( av) 30 A
part number STW88N65M5
Tran- type MOSFET
sistor maximum drain-to-source voltage Vpss 710 V
Q drain current 84 A
maximum drain-source on resistance RnS(on) 29 mü
accounting of curront and volt age ripple, tho boost convcrter behaviour cari be described by tho iterativo niapping (2) and (3) expressed in accordance with fl 3.39]. The inductor curront is deterniined by (2):
the inductance, R is the load resistance. The coefficients i^d Bn (n = 0,1,2, 3,4,5,6) are defined by the following expressions:
A _ -^( t-tk)
+
L(t-*fc )
state 1
iL(t) = <
(2)
Ro Rq
+ ÍL(tk )e
(AO + Aiepi((-tk) + A2eP2((-tk^ E+ (.A3ePl(t-1 ) + A4eP2( 1 -1 fc)) ^(¿k)+ (A5Gp 1(t-fc) + AeeP2(t-1^{h),
state 2.
The capacitor voltage is deterniined by (3):
vc(tk) • e-(t-*), state 1
(b0 - BxePl(t-1 fc) - B2eP2(*-tk^ E+
B3ePl((-tfc) + B4eP2((-t)) iL(tk)+
B5epi( (-t fc) + B6eP2( (-t )) vc (tk ), (3)
Ao = d c Bo = 1 - AqRQ
Ai = d 2 c -1 - b \ Q < + (QL Bi = (Ro + Lpi) Ai
A2 = d 2 c (-1 + b 1 Q) i QL B2 = (Ro + LP2) A2
= i ( 2 V 1 + 1 + Q< ) + R0 QL B3 = (Ro + Lpi) A3
A4 = i ( 2 V 1 - A 1 Q, - R0 QL B4 = (Ro + LP2) A4
A.5 = i QL B5 = (Rq + Lpi) A5
= - QL
B6 = (Rq + Lp2) Ae
where b and c are the state 2 characteristic equation ( ) coefficients, which are defined as follows:
b =
rrqc + L RLC '
R + Rq RLC
(4)
s 2 + b • s + c = 0.
vc (t) = <
vc (tk )e rH
R 1 C (t ífc )
state 2 state 2,
iL(t) = 0 (tk is the time when iL(t) reaches value 0),
where t is the time, tk is the time of the last commutation to state 1, k is the number of the last commutation, E is the electromotive force of the voltage source, L is
Q is the square root of characteristic equation discriminant, i.e. Q = \Zb2 — 4c, where s is the complex frequency, and the coefficient d is defined as d = 1/(RLC). The characteristic equation ( ) roots arepi = (—b + Q)/2, and p2 = (—b — Q)/2. In the state 2, the dynamics of transient processes depend on the values of the roots p^d p2. If the roots are real numbers, then the inductor current ih(t) and capacitor voltage vc (t) change with time by exponential laws that correspond to ( ). If the roots p^d p2 are complex (4c > b2), then the transient processes are damping harmonic oscillations. The roots of characteristic equation are
Pi,2
= - 6±
3"f,
(5)
c =
where S = b/2, Uf = \fb2/4 — c. In such case, the inductor current and capacitor voltage include harmonic cornp orient s:
iL{t) = E/(Ro + R) + Ge-St ■ sm("ft + VO, (6)
vc (t) = E + Ge-St■
■ cos (uft — (R0 — 5L)/(Lwf)) ■
(bjf)2 + (SL — Ro)2, (7) where G is the amplitude of harmonic oscillations.
3 The Approximate Calculation of Boost Converter Parameters
If the ripple level is low and transient processes have the form of almost straight lines, then the parameters of boost converter can be obtained by approximate techniques related to the ones described in [3]. Some of such techniques are presented below.
t0 At! t, At2
1)
Fig. 3. Almost linear form of inductor current (a) and capacitor voltage (b) transients
If the forms of inductor current and capacitor voltage transients are almost linear (Fig. 3), then the initial conditions for the state 1 are given by the expressions:
ih{t0 ) = IL,m VC (to) = VC,
II ,avg — ^-J-L
- Ml Vc,avg + Mc ,
(8)
where IL,avg is the averaged inductor current, VCavg is the averaged capacitor voltage, AIL mid AVC are the ripple estimations, t0, ti, t2 are the commutation time moments, accordingly to Fig. 3.
Therefore, the initial conditions for the state 2 are defined by the following expression:
, , __E
%L\ti) = +L, max +
Ko
E
(t
+ I 1L, min Tt ]
V Ko)
_ ^ (ti to)
(9)
VC (ti) = Vc, min = vc (to) • e RC! .
The periods of the state 1 and state 2, Ati and At2 (Fig. 3) can be obtained from the conditions for the ripple of inductor current (AIL) and capacitor voltage (AVC). If we s et AIL = Ki • IL, avg, wher e Ki is a real
positive coefficient, then, for t current is determined by (10):
i-L(ti) = Il,avg + Ki • Il,avg =
ti, the inductance
= E/Ro + (Il, min - E/Ro) e-t0
At i
(10)
The above expressions allow to obtain the equation (11).
-4oAt1 = Il, avg + KiIl, avg — E/Ro Il, avg — KiIl, avg — E/Ro '
(ID
which provides the state 1 duration Ati given by the following formula:
Ati = —— ln
Ko
(1 + Ki)lL,avg — E/Ro
(1 — Ki)lL,avg — E/Ro'
(12)
The averaged capacitor voltage can be determined using [26,31,33] for Ro = 0:
Vc,avg = Kg • E,
(13)
where Kg = (Ati + At2)/At\. The averaged current lid, avg ^d power Pid,avg values are given by (14):
lid, avg ,avg /R , Pid,avg = Vc avg /R.
(14)
Besides, we can use a more exact formula for averaged load power:
t, sec
Pi
id, avg
T
(vc (t ))2 R
dr,
(15)
but a simple analysis shows small error under Ati = At2 = T/2, where T is the period of commutation.
e
Slldl clS clt the state 1, we obtain Vc = vc(t0) — 4AV ■ t/T, then, during the time T/2, the averaged load power is
Pi
i d,avg
T/2 g
r (Vc,max — 4AV • t/T)2
T/2 J
R
dt =
= 1(Vê,avg + 1AV2). (16)
For instance, if the ripple AV is limited by condition AV = 0,1VC,avg (voltage oscillation level is significant), then the error is only 0.33%, due to ( ). Thus, for Ro = 0 obtain
PS = Pid or E • lL,avg = Vc,avg • Ii
i d,avg
Vca
Il.
a v g
E
Ii
i d, avg
Kg.
Vc,
Kg • E/(1 + KgRo/R),
'2.5 ■ 10 sec, then we obtain Ati = 5 ■ 10 sec from the formula (24). On the other hand, let's determine the parasitic resistance R0 value that makes boost converter losing advantages in comparison with direct load-to-source connection, i.e. Vc,avg > E ■ R/(R0 + R)
KsE/(1 + ■ RQ/R) > E ■ R/(Ro + R). (25)
After several transformations of expression (25), we obtain the condition:
Ro < R/Kg.
(26)
(17)
and the inductor current is IL, avg = Iid,avg •Vc,avg/E = lid, avg • Kg • E/E = Kg • Iid,avg-
So, the boost converter circuit is a step-up transformer with transformation coefficient Kg.
(18)
Now, let's analyze the influence of resistance Ro on the averaged values of capacitor voltage Vc,avg and inductor current IL, avg. If Rj = 0, then equivalent source voltage is
K = E — Il, avg • Ro (19)
and the load voltage is defined by (20):
Vc,avg = Kg • (E — IL,avg • Ro) =
= Kg (E — Kg • lid,avg • Ro) =
= Kg (E — Kg Vc, avg Ro/R). (20)
Hence, the averaged load voltage Vc,avg and averaged load current Iid,avg are determined by the following expressions:
(21)
If R0 = R/Kg, then the boost converter load voltage is equal to the voltage of direct load-to-source connection. For the state 2, the boost converter circuit must satisfy the condition At2 << 2n/wf to avoid high losses and load voltage distortions. The performed analysis and measurements show, that under Ati = At2 = 2-K/uf > 3 ■ RC, the capacitor discharges almost to zero, and we can consider VCm\n = 0. Then, Vc,avg = VCm&x/2. Thus, for the selection of capacitance C, we can set the condition At2 = At i = RC/(5...10). The value of Kg can be obtained from the following condition:
Sg = ik {K'gE/(1+K2Ro m=»■ <«)
hence, we obtain the transformation coefficient: Kg = vrR/R o
4 The Analysis of Boost Converter Transient Process Modes Under Varying Load
The boost converter models and laboratory layout are investigated with the parameters E = 12.87V, L = 2mH, R0 = m, Ati = At2 = 5 ■ 10-5 sec for load resistances 25il, 50^, 100^. The capacitance C impacts the boost converter transient process mode. It is obtained by four different techniques in subsections 4.1 4.4, respectively.
lid, avg = Vc,avg/R. (22)
Thus, the averaged inductor current is
Il, avg = Kg • Vc,avg/R =
= (Kg/R) • KgE/(1 + Kg2Ro/R). (23) Such as VCmin = VCmaxe-1/(RC), then we obtain
the following expression for the state 1 duration: Ati = —RC m ( ^,avg — Al ) .
i \VC, avg + AVJ
(24)
If we consider the voltage ripple A V = 0.1 ■ Vc,avg and state 1 transient process time constant RC =
4.1 The Critical Transient Process in the State 2 (Switch Off)
The aperiodic and periodic modes of capacitor charging are considered for fixed values of E,L,R0,R. If the characteristic equation (4) has a single solution pi,2 = —S, then the critical capacitor charging mode appears. The critical mode condition is defined by (29):
(L + RqRC)/(2RLC ) = (Rq + R)/(RLC). (29)
The solution of the equation (29) gives the following result:
L
C =
Rq2' R
(Ro + 2 R) + 2y/RoR + R2
(30)
Thus, the capacitance value (30) provides the critical form of state 2 transient process. Whereas, for periodic charging mode, the inequality (31) must be satisfied:
C <
L Ro r
(Ro + 2R) + 2^RqR + R2
ER
vc
+ Ge~6 1 sin (wf t +
G sin ^ = -ER/(R + R0), Gcos ^ = - (5/w0) ■ ER/(R + R0),
sin ^ = Uf / \J ^"j + S2 = Uf /u0, 2
^ = 5/+ 52 = 5/u0,
vc =
ER Ro + R
- ^e~6t sin(wf t + ^ .
The maximum value of the voltage uc can be reached under t = Tf /2, i.e.
cTf/2 R + Ro
(31)
1 + exp I -
For the presented boost converter parameters, state 2 critical capacitance (30) is 8mF.
In Fig. 4, the load transient processes are shown for the load resistance changing from initial value Rinit = 50Q to R = 25Q (triangle markers) and to R = 100Q (square markers). The left plots show the ranges of experimental and modeled boost converter transient current and voltage. Right plots show the respective current and voltage ripple waveforms for the load resistances R = 25Q and R = 100Q. Also, the ripple waveforms are plotted for the case, when the load transient is absent and the resistance remains still R = Rinit = 50Q (circular markers). The final values of load resistance are denoted in the figure legend. The next, plots Fig. 5, Fig. 6, Fig. 7 are organized in the same way.
4.2 The Maximum Voltage Boost Mode
The following analysis is performed for obtaining the capacitance value, that provides the maximum speed of periodic capacitor charging. The boost converter circuit is connected to the voltage source E under the initial conditions: iL(0) = 0 vc (0) = 0. Under p1,2 = -S ± ju we obtain
L+R0RC 2 RLC
R0 + R RLC
( L+RpRC)2 4 R2L2C2
.
The capacitance, that provides the maximum voltage vCfor t = Tf /2, can be defined by the differential equation
dv ( y
C,lf/2 dt
0,
(38)
Ro + R
J,C = CGe~61 (-S (sin wt + + w cos (wi + .
' (32)
Next, the expressions ( ) are simplified for t = 0, as follows.
The solution of equation (38) is
C = L/(RoR) (39)
The maximum voltage boost condition (39) gives the capacitance value C=40^F.
Fig. 5 shows the respective transient processes where the load changes from 50 Q to R, similarly to Fig. 4.
4.3 The Critical Averaged Envelope Transient Process Mode
The characteristic equation of state-space averaging model (1) is given by (40):
'2 + ^l^' + R{Dr12 + * =0. <*»
Thus, for the critical envelope transient process, the condition (41) must be satisfied.
'RoRG + L\2 — 4R ■ (D - 1)2 + Ro = 0.
RLC
(R0RC + L\2 V RLC J
From equation (41), the critical capacitance is obtained as (42).
(33)
Ch2 = ■ [ro + 2fl(1 - D)2
±\ -Rl + (-Ro - 2R(1 - D)
where tg^ = Wf/5, ^ = arctg(wf/5). On the other hand.
(42)
(34)
Such as C2 reaches very tow values (near 10 7) under the given boost converter circuit parameters (R0 = 1 Q, R = 50 Q, L = 2mH, D = At1/(At1 + At2) = 0.5), then we select
G = (-ER/(R0 + R)) / sin ^ =
= -KM) ■ E ■ R ■ /(Ro + R). (35)
As a result, we obtain the expression for capacitor voltage:
C = C-_1 = ■ [Ro + 2fi(1 - D)2
±\ -R2 + \-Ro - 2R(1 - D)
(43)
(36)
The critical envelope mode capacitance C 2.1mF is obtained from (43).
Fig. 6 shows corresponding transient processes and ripples, where load changes from 25 Q to 100 Q value.
v
X
cos
(b)
Fig. 4. The transient processes under state 2 critical capacitance C 8mF: (a) inductor currents: (b) output
capacitor voltages
(b)
Fig. 5. The transient processes under maximum voltage boost capacitance C 40 pF: (a) inductor currents: (b)
output capacitor voltages
(b)
Fig. C. The transient processes under critical envelope capacitance C 2.1rnF: (a) inductor currents: (b) output
capacitor voltages
4.4 The Optimized Transient Process Mode
In boost converter circuit, the losses are estimated by the input energy Win and output energy W0Ut. The boost converter efficiency estimation is the maximum energy loss between the cases when the load resistance does not change (R = 50 Q), when it decreases twice (R = 25 Q) and when it increases twice (R = 100 Q). Such estimation allows obtaining the best parameters for boost converter under the varying the load resistance in the given range R = 50 Q ± 50%.
The input energy Win is the energy that passes from voltage source with electromotive force E to the boost converter circuit from time ti to time 12: t2
Win =je ■ il(t)<1t. (44)
ti
Respectively, the output energy Wld is the energy-transmitted to load resistance:
(45)
The energy loss coefficient is given in per cents:
Kioss = 100 • (Win — Wid)/Win. (46)
The boost converter efficiency is estimated by the expression
Kioss,max = max(Kioss,0, Kioss, i, Kiossg), (47)
where Kioss,0 is the loss coefficient for the initial resistance (Rid,0 = 50 Q), Kioss,i is the loss coefficient for the resistance Rid,0/2, and Kioss,g is the loss coefficient for the resistance R = 2Rid,0.
Thus, the boost converter losses minimization problem can be represented as (48):
min (max(Kioss,0, Kioss, i, Kiossg)), (48) c
under the condition V0 — AV < vc <V0 + AV, where A V = 0.1 •V0,V0 = 24 V.
The minimization procedure is performed in MâTLâB environment using fmincon function. As the result, the capacitance C 1.94 niF is obtained.
The load transient processes and ripples are shown in Fig. 7.
5 The Comparison of Boost Converter Efficiency with Different Load Transient Process Modes
The boost converter losses (46), (47) are analyzed for the capacitances obtained from expressions (30), (39), (43) and (48) for the observation time t G [—Ttr, Ttr ] where Ttr is the duration of the longest of the investigated transient processes (for the capacitance 8mF). In Table 2, the experimental
arid modeled losses are presented with corresponding relative errors of modelling (the averaged envelope (1) is compared to the averaged experimental data). All modelling errors are calculated in per cents by the formula
e(X) = 100 •
l^exp — X,
model I
max(Xexp) - min ( ^exp )
(49)
where Vexp is the experimental value, Xmodei is a modeled value.
The loss coefficient time dependencies are presented in Fig. 8 for the four obtained capacitances under the source equivalent resistance 1 Q. The graphics show
that boost converter losses significantly depend on output capacitor value during the load transient. After the ending of the transient process, the maximum boost converter loss coefficients do not depend on the output capacitance significantly. The case of critical envelope mode (C 2.1 niF) shows the longest time during which the maximum loss coefficient is less than 10%. Very similar result is obtained by optimization procedure (C 1.94niF). Thus, the boost converter efficiency can be increased by selection of output capacitance with accounting of estimated load transient processes duration.
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capacitor voltages
Table 2 Experimental and modeled losses and modelling relative errors for different, boost, converter transient modes
Experiment Iterative model Averaged model
Transient, ^iode C losses (47) losses current. voltage losses current. voltage
(47) error (49) error (49) (47) error (49) error (49)
Critical for state 2 (30) 8,1 niF 14.26% 13.57% 1.10% 1.96% 13.39% 2.28% 1.42%
Maximum voltage boost. (39) 40 pF 11.42% 11.28% 2.31% 1.83% 11.04% 2.86% 2.52%
Critical envelope (43) 2.1 niF 9.08% 9.23% 1.58% 1.31% 9.12% 1.84% 1.78%
Optimized (48) 1.94 niF 9.31% 9.12% 1.76% 1.25% 9.17% 1.65% 2.03%
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Conclusions
The boost converter efficiency (energy losses) and transient processes are analyzed for four different modes: critical mode of switch-off state, maxinmm voltage boost mode, critical averaged envelope mode and optimized mode. These modes are characterized by different energy losses during transient processes caused by load varying.
The first mode is given by the condition of critical transient process (between periodic and aperiodic modes) for the state 2 (switch-off). This transient mode appears under a great value of the circuit time constant, thus, it requires a great value of output capacitance. Such mode allows minimization of the losses during state 2. But the main disadvantage is that the state 1 transient processes are not taken into account. Thus, during the load transient process, energy loss is 14.26% under the state 2 critical mode.
The second mode is determined by the condition of maximum output voltage boost. In this case, the boost converter efficiency is higher. The losses are 11.42% during load transient process. The disadvantage of such mode is that the time-domain waveforms of transient processes are not taken into account. But this technique is useful, if the short-time load transient losses constraints are strict.
The third mode is obtained from the condition of critical transient process (between periodic and aperiodic modes) for averaged envelopes obtained by the state-space averaging method. Such transient process mode shows only 9.08% energy loss.
The fourth mode is obtained by using optimization procedure in MATLAB environment. The procedure is realized by MATLAB functions. The obtained result is closed to the critical transient process for the averaged envelope. The energy losses are 9.31%. The disadvantage of this technique is the presence of numerical calculation errors that appear in the optimization procedure.
The presented results show that different transient modes can be required for obtaining the best efficiency of energy transition under different load variation frequencies and durations. For example, if the load resistance changes for a short time, then maximum voltage boost mode is convenient with a small output capacitance which provides a short response time, as shown in Fig. 8(b). But. if the load resistance changes for a longer time, then critical envelope mode (Fig. 8(c)) is more efficient and provides a longer protection from high energy losses.
It should be noted that optimization procedure leads to parameters close to critical envelope mode. Probably, the minimum energy losses can be obtained in critical envelope mode. In such case, the difference between critical envelope mode parameters and optimization result can be caused by the calculation errors of boost converter models and optimization procedure.
Therefore, consideration of load transient process modes can provide a significant improvement of boost converter efficiency under varying load.
All modelling errors are lower than 3%. but the iterative model given by (2) and (3) does not match exactly with current and voltage ripples, because of the parasitic parameters that are not taken into account. Thus, for more exact boost converter ripples modelling, the more complex models should be used to fit the characteristics of semiconductor and passive elements. Especially, the investigation of fractional order capacitor models [40. 41] is interesting in accordance with greater capabilities of the ripple curves fitting without a significant increase of model parameters number.
References
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Анал1з процес1в передач! енергй" у шд-вищуючому 1мпульсному перетворю-вач!
Мартинюк В. В., Косенков В. Д., Гейдарова О. В., Федула М. В.
Вступ. У статт! описуеться апатз процеов передач! eneprii' у шдвищуючому перетворювач! за умов змши паваптажешш. Достджепо лабораторпий макет перетворювача при р1зпих формах перех1дпого процесу. викликапого змшою паваптажешш. Перех1дш струми i папруги вим1ряпо в гармошчпих i пегармошчпих режимах. Проведено оцшку параметр!в перех1дпих режим!в. Апал1з процеов переходу eneprii здшспепо за допомогото двох моделей шдвищуючого перетворювача.
Постановка проблеме. Кола шдвшцуючих иере-творювач!в використовуються в областях електрошки, де м!шм1зац1я втрат енергп е критично важливою. Втра-ти енергп у шдвшцуючому иеретворювач! залежать в!д форми перех!дних ироцеав, викликаних змшою на-вантаження. М1шм1зац1я втрат енергп вимагае точного анал!зу процеав передач! енергп, яш виникають при змь ш навантаження. Запропонована робота описуе реакцп схеми перетворювача на зм!ни навантаження вдв!ч! з чотирма р!зними режимами перех!дного процесу.
Результати. Анализ ироцеав передач! енергп здш-снюеться з використанням двох моделей. Перша модель базуеться на методшц ¡теративного воображения, де струмп \ напруги кожного перюду сигналу керуван-ня затвором транзистора визначаються за параметрами схеми \ початковими умовами, заданими поиередшм перюдом керуючого сигналу. Така модель дозволяв проводи™ точний анализ перех1дпих процеав \ пульсацш струму та напруги. Проте, модель ¡теративного воображения характеризуеться похибками числово! апрокси-мацп, \ вимагае б!льше часу для виконання обчислень. Друга модель ашинтичио описуе огпнаюч! перех!дних процеав струму \ напруги перетворювача. Така модель базуеться на метод! усереднення у простор! сташв, що широко використовуеться для моделювання ¡мпульсних схем. Така модель не враховуе форми пульсацш струм!в \ напруги, але забезпечуе бшьш простий опис перех!дних процеав, яш корпсш для цшей схемотехшкп. Результати моделювання, отримаш з ¡теративного воображения та усереднення у простор! сташв, сшвпадають з експери-меитальпими дапими, ¡з в!дносною похибкою, яка не перевпщуе 3%.
Висновки. Проведений ашинз показуе, що втра-ти енергп у шдвшцуючому перетворювач! залежать в!д перех!дного процесу. Протягом перех!дного процесу, викликаного змшою навантаження, наймешш втра-ти енергп можуть бути отримаш за умови критично! форми иерех!дного процесу, що розташована м!ж гар-мошчним та негармошчним режимами. Такий результат отримано експериментально \ шдтверджено як ¡терацш-нпм воображениям, так \ моделями усереднення стану простору. Ашинтичш результати шдтверджуються процедурою оитим1зацп в середовищ! МАТЬАВ.
Ключовг слова: шдвищуючий ¡мпульснпй перетво-рювач; втрати енергп; перех!дний процес; критичний режим; моделювання
Анализ процессов передачи энергии в повышающем импульсном преборазо-вателе
Мартынюк В. В., Косенков В. Д., Гейдарова Е. В., Федула М. В.
Введение. В статье описывается анализ процессов передачи энергии в повышающем преобразователе при изменении нагрузки. Исследован лабораторный макет
преобразователя при различных формах переходного процесса, вызванного изменением нагрузки. Переходные токи и напряжения измерены в гармонических и негармонических режимах. Проведена оценка параметров переходных режимов. Анализ процессов перехода энергии осуществлен с помощью двух моделей повышающего преобразователя.
Постановка проблемы. Цепи повышающих преобразователей используются в областях электроники, где минимизация потерь энергии является критически важной. Потери энергии в повышающем преобразователе зависят от формы переходных процессов, вызванных изменением нагрузки. Минимизация потерь энергии требует точного анализа процессов передачи энергии, которые возникают при изменении нагрузки. Предложенная работа описывает реакции схемы преобразователя на изменения нагрузки вдвое с четырьмя различными режимами переходного процесса.
Результаты. Анализ процессов передачи энергии осуществляется с использованием двух моделей. Первая модель основана на методике итеративного отображения, где токи и напряжения каждого периода сигнала управления затвором транзистора определяются по параметрам схемы и начальными условиями, заданными предыдущим периодом управляющего сигнала. Такая модель позволяет проводить точный анализ переходных процессов и пульсаций тока и напряжения. Однако, модель итеративного отображения характеризуется погрешностями числовой аппроксимации, и требует больше времени для выполнения вычислений. Вторая модель аналитически описывает огибающие переходных процессов тока и напряжения преобразователя. Такая модель базируется на методе усреднения в пространстве состояний, который широко используется для моделирования импульсных схем. Такая модель не учитывает формы пульсаций токов и напряжений, но обеспечивает более простое описание переходных процессов, полезное при разработке схемотехники. Результаты моделирования, полученные из итеративного отображения и усреднения в пространстве состояний, совпадают с экспериментальными данными, с относительной погрешностью, которая не превышает 3%.
Выводы. Проведенный анализ показывает, что потери энергии в повышающем преобразователе зависят от переходного процесса. В течение переходного процесса, вызванного изменением нагрузки, наименьшие потери энергии могут быть получены при критической форме переходного процесса, расположенной между гармоничным и негармоничным режимами. Такой результат получен экспериментально и подтвержден как итеративным отображением, так и моделями усреднения состояния пространства. Аналитические результаты подтверждаются процедурой оптимизации в среде МАТЬАВ.
Ключевые слова: повышающий преобразователь; потери энергии; переходный процесс; критический режим; моделирование
