The Analysis of Energy Transition Processes in Boost Converter

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.


Introduction
The boost converters are used in many fields of modern electronics [1][2][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][6][7][8][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 areas is the photovoltaic energetics [4,[18][19][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][23][24][25][26][27][28]. Various models of boost converters are investigated due to the practical application features [29][30][31][32][33][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 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 2 inductor with magnetodielectric core (sendust, 0 = 60) and electrolytic capacitors with ESR < 70 Ω and parasitic inductance less than 10 . 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 = 12.87 . A significant parasitic resistance is modelled by resistor 0 = 1Ω. The converter is investigated with load resistances 25Ω, 50Ω, 100Ω. 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.

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. In the state 1 the inductor current increases up to the maximum value that is defined by parasitic resistance 0 . The capacitor is discharged to the load resistance . 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).
where is the time is the inductance, is the capacitance, is the source voltage, is the load resistance, is the inductor current, is the capacitor voltage, is the duty cycle of switching (Fig. 1). The model (1) 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

mΩ
accounting of current and voltage ripple, the boost converter behaviour can be described by the iterative mapping (2) and (3) expressed in accordance with [1][2][3]39]. The inductor current is determined by (2): The capacitor voltage is determined by (3): ( ) = 0 ( is the time when where is the time, is the time of the last commutation to state 1, is the number of the last commutation, is the electromotive force of the voltage source, is the inductance, is the load resistance. The coefficients and ( = 0, 1, 2, 3, 4, 5, 6) are defined by the following expressions: where and are the state 2 characteristic equation (4) coefficients, which are defined as follows: is the square root of characteristic equation where is the complex frequency, and the coefficient is defined as = 1/( ). The characteristic equation (4) roots are 1 = (− + )/2, and 2 = (− − )/2. In the state 2, the dynamics of transient processes depend on the values of the roots 1 and 2 . If the roots are real numbers, then the inductor current ( ) and capacitor voltage ( ) change with time by exponential laws that correspond to (4). If the roots 1 and 2 are complex (4 > 2 ), then the transient processes are damping harmonic oscillations. The roots of characteristic equation are where = /2, = √︀ 2 /4 − . In such case, the inductor current and capacitor voltage include harmonic components: where is the amplitude of harmonic oscillations.

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.
where , is the averaged inductor current, , is the averaged capacitor voltage, ∆ and ∆ are the ripple estimations, 0 , 1 , 2 are the commutation time moments, accordingly to Fig. 3.
Therefore, the initial conditions for the state 2 are defined by the following expression: The periods of the state 1 and state 2, ∆ 1 and ∆ 2 (Fig. 3) can be obtained from the conditions for the ripple of inductor current (∆ ) and capacitor voltage (∆ ). If we set ∆ = 1 · , , where 1 is a real positive coefficient, then, for = 1 , the inductance current is determined by (10): The above expressions allow to obtain the equation (11).
which provides the state 1 duration ∆ 1 given by the following formula: The averaged capacitor voltage can be determined using [26,31,33] for 0 = 0: and power , values are given by (14): Besides, we can use a more exact formula for averaged load power: but a simple analysis shows small error under ∆ 1 = ∆ 2 = /2, where is the period of commutation.
Such as at the state 1, we obtain = ( 0 ) − 4∆ · / , then, during the time /2, the averaged load power is For instance, if the ripple ∆ is limited by condition ∆ = 0, 1 , (voltage oscillation level is significant), then the error is only 0.33%, due to (17).
Now, let's analyze the influence of resistance 0 on the averaged values of capacitor voltage , and inductor current , . If 0 ̸ = 0, then equivalent source voltage is and the load voltage is defined by (20): Hence, the averaged load voltage , and averaged load current , are determined by the following expressions: Thus, the averaged inductor current is Such as ,min = ,max − /( ) , then we obtain the following expression for the state 1 duration: If we consider the voltage ripple ∆ = 0.1 · , and state 1 transient process time constant = 2.5 · 10 −4 sec, then we obtain ∆ 1 = 5 · 10 −5 sec from the formula (24). On the other hand, let's determine the parasitic resistance 0 value that makes boost converter losing advantages in comparison with direct load-to-source connection, i.e. , > · /( + ) or After several transformations of expression (25), we obtain the condition: If 0 = / , 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 ∆ 2 << 2 / to avoid high losses and load voltage distortions. The performed analysis and measurements show, that under ∆ 1 = ∆ 2 = 2 / > 3 · , the capacitor discharges almost to zero, and we can consider ,min = 0. Then, , = ,max /2. Thus, for the selection of capacitance , we can set the condition ∆ 2 = ∆ 1 = /(5...10). The value of can be obtained from the following condition: hence, we obtain the transformation coefficient:

The Critical Transient Process in the State 2 (Switch Off)
The aperiodic and periodic modes of capacitor charging are considered for fixed values of , , 0 , . If the characteristic equation (4) has a single solution 1,2 = − , then the critical capacitor charging mode appears. The critical mode condition is defined by (29): The solution of the equation (29) gives the following result: 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: For the presented boost converter parameters, state 2 critical capacitance (30) is 8 mF.
In Fig. 4, the load transient processes are shown for the load resistance changing from initial value = 50Ω to = 25Ω (triangle markers) and to = 100Ω (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 = 25Ω and = 100Ω. Also, the ripple waveforms are plotted for the case, when the load transient is absent and the resistance remains still = = 50Ω (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.

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 under the initial conditions: . On the other hand, = (− /( 0 + )) / sin = = −( 0 / ) · · · /( 0 + ). (35) As a result, we obtain the expression for capacitor voltage: The maximum value of the voltage can be reached under = /2, i.e.
The capacitance, that provides the maximum voltage for = /2, can be defined by the differential equation The solution of equation (38) is The maximum voltage boost condition (39) gives the capacitance value C=40 F.

The Critical Averaged Envelope Transient Process Mode
The characteristic equation of state-space averaging model (1) is given by (40): Thus, for the critical envelope transient process, the condition (41) must be satisfied.

The Optimized Transient Process Mode
In boost converter circuit, the losses are estimated by the input energy and output energy . The boost converter efficiency estimation is the maximum energy loss between the cases when the load resistance does not change ( = 50 Ω), when it decreases twice ( = 25 Ω) and when it increases twice ( = 100 Ω). Such estimation allows obtaining the best parameters for boost converter under the varying the load resistance in the given range = 50 Ω ± 50%.
The input energy is the energy that passes from voltage source with electromotive force to the boost converter circuit from time 1 to time 2 : Respectively, the output energy is the energy transmitted to load resistance: (45) The energy loss coefficient is given in per cents: The boost converter efficiency is estimated by the expression where ,0 is the loss coefficient for the initial resistance ( ,0 = 50 Ω), ,1 is the loss coefficient for the resistance ,0 /2, and ,2 is the loss coefficient for the resistance = 2 ,0 .
Thus, the boost converter losses minimization problem can be represented as (48): under the condition 0 − ∆ < < 0 + ∆ , where The minimization procedure is performed in MATLAB environment using fmincon function. As the result, the capacitance C=1.94 mF is obtained.
The load transient processes and ripples are shown in Fig. 7.

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 where is the duration of the longest of the investigated transient processes (for the capacitance 8 mF). In Table 2, the experimental and 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 where exp is the experimental value, 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 Ω. 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 mF) shows the longest time during which the maximum loss coefficient is less than 10%. Very similar result is obtained by optimization procedure (C=1.94 mF). Thus, the boost converter efficiency can be increased by selection of output capacitance with accounting of estimated load transient processes duration.  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.