Input Current Quality Parameters Analysis of Modular AC-DC SEPIC Charger Based on Double Fourier Series

Review of AC chargers topologies with active power factor correction are considered in the paper. Single and double-stage charger topologies are highlighted and the feasibility of using the first group of converters is substantiated because of its smaller size and simpler structure. Among single-stage converters, the advantages of modular converters based on SEPIC topologies are emphasized because of minimal dynamic losses, a simple algorithm, and circuit implementation of power factor correction at the boundary condition mode (BCM) of input current and small dimensions. The operation of modular converters is analyzed on the example of a two-cell converter and the principle of improving the shape of the input current under the condition of shifting control pulses of each cell is shown. As a result, the functional dependence of the electricity quality parameters (power factor PF and the total harmonic distortion THD) on the pulse ratio of each converter cell is established. Since it is difficult to comply BCM in real converters, and in continuous mode the shape of the input current is distorted, such converters operate in the discontinuous conduction mode (DCM) of input current. Based on the analysis of equivalent circuits in three intervals of operation of the SEPIC converter interval of increase of input current, interval of decrease of input current and interval of zero pause its mathematical model was created. It is shown that the increase of relative zero current interval duration significantly worsens the value of the power factor PF and the total harmonic distortion THD. As a result, the PF and THD parameters are functions of duty cycle and the relative zero current interval duration and have a complex relationship. Therefore, in order to estimate the number of converter cells that provide the required PF and THD values, it is advisable to derive these dependencies in analytical form, based on the use of a Double Fourier series. The obtained dependencies simplify the selection of the number of cells of the converter for a given range of duty cycle and the relative zero current interval duration. PF and THD dependencies for two, three and four cells are constructed on the basis of the derived formulas.


INTRODUCTION
Batteries of electric vehicles usually charged from AC grid with two-stage ac-dc converters [1]. The first stage is power factor pre-regulator (PFP), another one is an isolated dc-dc converter. Nowadays for power applications a variety of one-stage chargers with improved efficiency are proposed [2][3][4]. Interleaved SEPIC converters operating in discontinuous conduction mode (DCM) have a number of advantages as chargers with built in power factor (PF) correction [5][6][7][8]: • it operates as a voltage follower, meaning that the input current naturally follows the input voltage; • a current control loop can be omitted; • galvanic isolation can be easily implemented and several isolated outputs provided; • the input current ripple continuous and defined during the design stage; • can operate as step-up or step-down converter • ZCS of the transistor and ZVS of the output diode is provided.
According to such advantages interleaved SEPIC topology were used as a charger for power-assist wheelchair [9]. A topology and simple analyze of the charger's operation modes are given in [10]. For clear chosen of the charger cell number it is necessary to research dependency input current PF and THD depend on cell number N and the converter duty cycle range γ.
An analysis of PF and THD based Fourier variable is inefficient because of necessity to recalculate current spectrum after each current parameter changing therefore it is proposed to use a Double Fourier series [11] that allows obtain PF and THD values for different N and γ.

INPUT CURRENT
Let assume an abstract two cell converter with an ideal shape input current in border current mode (BCM) that provide maximum PF, shown in Fig. 1. The input current iin is a total current of two cells icell1 and icell2 shifted on half period T/2. The advantage of BCM that sine of Електронні системи та сигнали Copyright (c) 2019 Вербицький Є. В. DOI: 10.20535/2523DOI: 10.20535/ -4455.2019 input current is provided with constant pulse width γT [12]. Let analyze the input current shape of SEPIC converter, Fig. 2. For the analysis, it is assumed that: • capacitors Ci and Co are large enough to neglect the voltage ripple across them; • PWM period T significantly less than input voltage period Θ, T << Θ, therefore input voltage uin during PWM period k, Tk has approximately constant value uin(k) = Uin = const; • all SEPIC elements are ideal.
The DCM SEPIC has three equivalent circuits as shown in Fig. 3.
When transistor VTi is switched on the cell input current icell rises from zero value to maximum value Icell_max as follows: where Li is input inductance of the cell, Uin is input voltage value.
On the second interval the cell input current declines to zero: where uCi is voltage on capacitor Ci, Uo is output voltage, uCi = Uin, t0 is duration of third interval.
On third interval the cell input current is neglected, icell = 0. Third interval icell = 0 worsens input current shape, so it should be minimized. However, continuous current mode also unfavorable for PF correction because significant current distortion [10]. So, PF and THD of input current should be analyzed.

III. INPUT CURRENT ANALYSIS BASED ON A DOUBLE FOURIER SERIES
A Double Fourier series [10,11] allows to separate impact of a modulated function and a carrier function into spectrum of a pulse modulated signal due to their independent representation. The carrier function angle variable x1 = ω1·t, and the modulated function angle variable x2 = ω2·t may have any relation, where ω and Ω are angular frequencies of the carrier and modulated functions, t is time variable. Thus a modulation multiplicity parameter P = x1 / x2 may have any positive value (integer or fractional). Therefore, Double Fourier Series provides a simple analytical model that allows: • define PF and THD for any modulation multiplicity parameter P in analytical form; • detect interharmonics caused the fractional value of parameter P; • minimize calculations.
Coefficients C(m1)(m2) of the Double Fourier Series are the spectral components of the signal with multiplicity m1 relatively to the carrier frequency and the multiplicity m2 relatively to the modulation frequency: Copyright (c) 2019 Вербицький Є. В. DOI: 10.20535/2523DOI: 10.20535/ -4455.2019 where f(x1, x2) is a function of two variables x1 and x2 whose spectrum is calculated, φ1 and φ2 are integration limits that define pulses position relatively variable x1.
The Double Fourier Series applying express's the spectral characteristics of the modulated signal in an analytical closed form for any multiplicity parameter P. The signal harmonic with number k, Ck is calculated as follows [13]: The cell ideal input current spectrum is defined as current sum on first interval x1 ∈ [0, 2π·γ) where the cell current growths and second interval x1 ∈ [2π·γ, 2π) where the current falls down to zero.
On the first interval, for defined angle x2 of modulated function the cell input current starts from zero value, icell(0) = 0, and achieves value Icell_max = Asin(x2), where angle x2 of modulation function corresponds angle 2π·γ of the carrier function. Constant A is the current amplitude for x2 = π/2, formula (1): where Uin_max is amplitude value of the input sinusoidal voltage.
On the end of second interval the cell ideal current starts from Icell_max and at x1 = 2π it always tends to zero, icell(2π) = 0. Therefore, the cell input current on i interval of carrier period i defined as follows: According to formula (3) the Double Fourier Series spectrum components C(m1)(m2) calculated as follows: In the second clause we will change the limits of integration to [0, π).
From the formula (8) we can conclude that the integral is zero for even m2, and for odd m2 it is simplified to: Spectral coefficient (9) has only two types of nonzero values: (1 ) ; .

( 1)
For a converter with N cells the total spectral coefficients are defined as follow:
When converter operates in DCM with relative zero pause duration γz and the cell current icell shape is defined as follows: the Double Fourier series expression for spectral component C(m1)(m2) is: After series of transformation the spectral components in DCM defined as follow: Copyright (c) 2019 Вербицький Є. В.
According to formulas (4) and (10) nonzero harmonic are calculated as follows: The current shape of the two-cell converter for γ = 0.4 and γz = 0.4 is shown in Fig. 5.
After developing formulas of the cell current for BCM and DCM modes it is possible to calculate the PF and THD for different cell number.

IV. ESTIMATION CURRENT QUALITY PARAMETERS
Let consider ideal input current PF and THD values for predefined number of cells N in BCM. In comparison with one cell in N modular converter high harmonics mainly eliminated except harmonics CmPN+1, PF and THD of the modular converter calculated based on formula (17): (1 ) .
The same formulas for RMS current value Iin_RMS(N)p is obtained for current with relative zero current interval γz.
60( ) 30( ) Functional behavior of PF and THD depend on duty cycle γ and cell number N for BCM is shown in Figure 6 a) and 6 b) respectively.
As we can see increasing of cells N significantly improves PF and THD and extends range of duty cycle γ where they have good enough value, γ Є [1/N; 1-1/N].
PF and THD behavior for DCM is more complex because of additional parameter of relative zero current pause γz. The functional dependences of PF are shown in Fig. 7 a), b), c) for N = 2, N = 3, N = 4 respectively.
The same functional dependence may be obtained for any N with formulas (24) and (25), therefore developed formulas may be used for estimation number of cells N for predefined PF or THD.

CONCLUSIONS
In the paper the formulas for THD and PF parameters based on Double Fourier series for modular SEPIC converter valid for different cell number N are obtained. The obtained functions allow to simplify to determine cell number to defined THD and PF values for duty cycle γ range and relative zero current interval duration γz.