Modeling of the Boost Power Factor Correction Rectifier in Mixed Conduction Mode Using PWA Approximation F. Tahami, B. Gholami, H. M. Ahmadian Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran
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[email protected] Abstract-- PFC converters for higher power are commonly designed for continuous conduction mode (CCM). However, at light load, discontinuous conduction mode (DCM) will appear close to the crossover of the line voltage, causing the converter to switch between the two conduction modes. As a result, the converter dynamics change abruptly, producing input current distortion. In this article a piecewise affine approximation has been employed in modeling a boost power factor correction converter operated in the mixed conduction mode. This approach makes the new model very useful in large signal analysis of PFC rectifiers and design of controller. The results obtained from the proposed model are compared with those of linear approximation and the exact nonlinear model. Index Terms— Power Factor Correction rectifier, circuit averaging, Piecewise affine approximation.
I. INTRODUCTION The power-factor-correction (PFC) rectifiers are becoming an integral part of switching power supplies connected to the AC networks providing input-line harmonics in accordance with harmonic distortion standards. The simplest and less expensive approach to realize a near-ideal rectifier is to employ a full-wave rectifier followed by a boost converter as shown in Fig. 1. The input boost inductor in a PFC converter can be operated in either discontinuous conduction mode (DCM) or continuous-conduction-mode (CCM). Input current waveforms with very low harmonic distortion can be achieved, as long as a continuous inductor current is assured. However, when these converters are operating at light load, DCM will appear close to the crossover of the line voltage, causing the converter to switch between CCM and DCM. This operating mode is called mixed conduction mode (MCM) [1]–[4]. As a result of this mode of operation, the converter dynamics change abruptly, producing input current distortion. Moreover, when the load is further decreased, the converter will operate in DCM during the entire line period. Since the input current controller is designed for operation in CCM, and the corresponding system transfer functions in CCM and DCM differ, the input current tracking will not be satisfactory. To avoid these problems, a large input inductor L can be exploited. Nevertheless, a converter with a low value of the input inductor L is desirable to reduce the weight and allow an easier design of the EMIfilter.
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Each time voltage approaches zero, systems goes to DCM (twice in each period). If DCM analysis is not employed, the PFC rectifier can not perform power factor correction properly near zero. Therefore current waveform will not resemble sinusoidal waveform for amounts near zero. As a result to access high quality in power factor correction it is essential to produce and employ a MCM model of rectifier. Hence, the control algorithm needs to be able to deal with sudden changes between these two dynamics. A conventional PFC rectifier based on a PWM converter controlled by two interconnected feedback loops (Fig. 1), a wide bandwidth current loop and a slow voltage loop [5]. For design of this controller, the rectifier can be modeled using the loss-free resistor (LFR) concept. The assumption that the inner current loop operates ideally is usually sufficient to linearize the equations of the average voltage controller, but the nonlinear time-varying nature of the system does not allow this assumption for most of the converter topologies [6, 7]. There are few major shortcomings in these analyses. The averaged model of a multi-loop dc–dc boost converter may have a quadratic nonlinearity. Hence, this system may have more than one equilibrium solution. If two of these solutions are stable, then the system will have two operating points, one of which is the nominal solution. This possibility is completely ignored in linearized averaged models. Consequently, the small-signal model cannot predict the post-instability dynamics. Besides, controllers that are designed based on this model may be
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Fig. 1. Realizing of an ideal rectifier by a PWM converter [5]
conservative and may not achieve globally stable closed-loop systems. Therefore, it is essential to investigate the stability of PFC converters by using a non linear model. Over the last few years, several new methods have emerged for the analysis of piecewise-linear (PWL) and piecewise-affine (PWA) systems. These systems represent a powerful model class for nonlinear systems. They are based on linear dynamics in the presence of saturations or hybrid systems where the continuous dynamics within the different discrete states is linear or affine [8]. The authors have already investigated the effectiveness of employing PWA approximation in analysis of PFC rectifiers operating in CCM [9]. In this paper a PWA model for PFC rectifiers is introduced which is suitable to predict the system behavior under large-signal perturbation in mixed conduction mode. Simulation examinations show that the results obtained from this modeling are very similar to those of nonlinear analysis obtained from the PSPICE program. II. MODELING THE AC-DC BOOST RECTIFIER Ideal rectifiers can perform the function of low-harmonic rectification, without need for low-frequency reactive elements. In an ideal rectifier, it is desirable that the rectifier present a resistive load to the ac power system. v iac = ac (1) Re Where Re is the emulated resistance. This leads to unity power factor rectification and the ac line current having the same wave shape as voltage. The power apparently consumed by Re is transferred to the dc output port. Average rectifier power is controlled by variation of the emulated resistance. The control network varies the duty cycle, as necessary to cause the converter input current to be proportional to the applied input voltage. For a sinusoidal input voltage, the rectified voltage Re is: v g (t ) = VM sin (ωt )
(2)
It is desired that the output voltage of the near-ideal rectifier be a constant voltage V . Hence the converter conversion ratio should vary as: V (3) M (t ) = VM sin (ωt ) To avoid distortion near line voltage zero crossings, converter should be capable of producing M (t ) approaching infinity. DC/DC converters capable of increasing the voltage can be adapted to the ideal rectifier application. The boost converter is most often chosen. If the boost converter operates in continuous conduction mode, then (3) implies that the following function should held for duty ratio: VM sin (ωt ) d (t ) = 1− (4) V To come to the dynamic model of the rectifier we only need to model the dominant behavior of the system, i.e. to predict how slow variations in the control signal, load and input affect the rectifier output. For this purpose, we average signals over
the switching period to remove the switching harmonics. This provides us with a large signal time-invariant model. However, the model is nonlinear because of multiplication of the timevarying quantity d(t) with other time-varying quantities such as i(t) and v(t). This non-linearity reflects the difficulty of analysis and design [7].
Fig. 2. The boost rectifier circuit
The first sequence of the state space averaging consists in identifying matrix coefficients corresponding to all the converter states. To illustrate this process, the large-signal model for the boost PFC rectifier (Fig. 2), in which the MOSFET and diode conduction losses, as well as inductor resistance are accounted for, is explained. A. Large Signal Model of the Boost PFC rectifier in CCM The averaged values of inductor current i g and energy storage capacitor voltage v over one switching period are chosen as state variables. Employing the averaging technique, the state space representation of the circuit can be described as follows [6]: x = Ax + b(t ) + g ( x )u (5) with: 1 RL − − L x1 i g L , x = = , A= 1 1 x2 v − R×C C v g − VD (6) b(t ) = L , u = d (t ) 0 VD + x2 − Ron × x1 L g (x ) = x1 − C Where Ron is the MOSFET ‘on’ resistance, RL is the inductor resistance and VD is the diode forward voltage drop. The above equations describe a system that is nonlinear in the input.
B. Large Signal Model of the Boost PFC rectifier in DCM The effective resistance of rectifier Re in DCM operation looking from the input, calculated as [5]: Re =
2L Ts × d 2
(7)
Here Ts is the period of switching. Considering Fig. 2, one can writes DCM equations as follow:
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ig dv v =− + + dt RC C di g dt
=
vg L
−
L
di g
− vg
dt CRe
(8)
2vi g
(9)
Ts vd 2 + 2 Li g
(5), (8) and (9) all include nonlinear terms. III. PIECEWISE AFFINE APPROXIMATION The systems considered in this paper fall under the framework of hybrid systems. These systems have multiple models, each model being valid in a given region of a certain space. These systems have a sate with hybrid nature that consists of both continuous-time physical variables and discrete-event logical variables. The class of nonlinear systems considered in this work is described by: z(t ) = f ( z ) + h(z )u (10) y (t ) = Cz (11) n Where z ∈ R contains the state variables of the nonlinear systems. The vector u ∈ R p is the control input and the vector y ∈ R q is the output. All the matrices in the equations are assumed to have the appropriate dimensions. It is further assumed that the mapping from z to z does contain an arbitrarily small neighborhood of the origin, so that the conditions of Brockett's theorem do not apply [10] and the linearization in a neighborhood of the origin is well defined. The method that will be studied is suited for controlling systems whose dynamics partition the state space R n into a finite number of closed polytopic regions. We will therefore assume that such a partition exists with polytopic χi , i ∈ I . Each cell is constructed as the intersection of a finite number pi of half spaces given by the following inequalities [11]:
{
χi = z hij T z < gij , j = 1,..., pi
}
For i = 1,..., M and j = 1,..., N i Where M is the cardinality of this set and Ni is the number of neighboring cells of cell i . Given a nonlinear system, there are two main steps in controller design: computing a piecewise-affine (PWA) approximation of the dynamics and designing a controller for this PWA model of converter. Finding the piecewise-affine approximation of a nonlinear function based on its values at the vertices of a simplicial grid is basically an estimation problem. Given f : Rn → Rn and n + 1 vertices belonging to a simplicial cell i , the objective is to find the matrix Ai and the vector bi such that the piecewiseaffine approximation of f in the polytopic cell i is described by: f (z) ≈ Ai z + bi (18) At each vertex of each simplex i , a linear equation of the form f (α )T = α T 1θ can be written, where f (α ) is a n + 1 vector, θ is a (n +1) × n matrix defined by: θ = [ATi | bTi ]T (19) where α represents the n ×1 vector with the coordinates of the vertex. If all the values f (α ) are stacked in a matrix F and all the rows [α T | 1] in a matrix X , the solution is then given f by θ = X −1 F . Since a simplicial partition of the domain of is being used, the matrix X is nonsingular [12]. Now we should define the Bi for each simplicial cell. We do this by introducing the Chebychev center wi (cheb) for each cell that is, the center of the Euclidean ball of maximum radius that can be fit inside the polytopic cell and then use the approximation [12].
[ ]
(
)
Bi = h wi (cheb ) (20) The principle of the proposed method is based on the procedure depicted in Fig. 3. The switching among plant models is governed by the state z (t ) and the input u (t ) .
(12)
Thus each cell can be characterized by the following vector inequality: H iT z − g i < 0 Where, H i = hi1hi 2 ...hipi
(13)
[ ] gi = [g i1 gi 2 ...gip ]
(14) (15)
i
Within each cell the dynamics is affine and defined through Gi by: z(t ) = Ai z (t ) + bi + Bi u (t ) Gi (16) y (t ) = Cz (t ) Each polytopic cell has a finite number of facets and vertices. Any two cells sharing a common facet will be called neighboring cells. A parametric description of the boundaries can then be obtained [11]:
{
χ i ∩ χ j ⊆ I ij + Fij q q ∈ R n −1
}
(17)
Fig. 3. The switching among PWA models
IV. MODEL DERIVATION The procedure in the previous section can be used to derive the dynamic model of each cell in CCM operation. Since in the PFC rectifier circuit the input current i g is proportional to the input voltage v g , no additional partitioning is required for v g .
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Hence each cell is defined by a rectangular as depicted in Fig. 4. Then the linear dynamic of the system in each cell can be described as [9]: x = Ai x + Bi u + bi (21) In which: 1 RL − − L i g L x = , Ai = (22) 1 1 v − R×C C V D + V cheb,i − R on I g ,cheb,i V g ,cheb,i − V D ,B = I g ,cheb,i bi = L i 0 C Thus the linear feedback design methods can be applied. For performing PWA approximation of DCM model, nonlinear equations of (8) and (9) are linearized in each cell about the values in the Chebychev centre, namely Vcheb,i ,
TABLE I The Parameters of the BOOST RECTIFIER Circuit Parameter Symbol Quantity Inductance (µH) L 368 Capacitance (µF)
C
Load resistance (Ω)
R
Inductor resistance (Ω)
RL
Switching period (µsec)
Ts
Maximum input voltage (V)
Vg-max
Angular speed
ω
Dc output voltage (V)
V
Dc input current (A)
I
470 260 1. 7 25
75 2 100π 200 0.75
The first step is to find the boundary between CCM and DCM regions. It is known that both load resistance and input voltage influence the operating mode of the rectifier. In this
I g , cheb,i and Vcheb,i in each cell. Applying this procedure to the DCM model results in the following coefficient matrices: 0 0 1 Ai = 1 C − CR 2 − 4 I g ,cheb,iV cheb ,i Ts Di 2 2 T sVcheb,i Di + 2 LI g ,cheb,i bi = 2 − 4I V LT D g ,cheb,i cheb,i s i 2 2 C T sVcheb,i Di + 2 LI g ,cheb,i V g ,cheb,i + V cheb,i Bi = L 0
(
)
(
(23) Fig. 4. The CCM/DCM boundary curve and the state space partitions. The horizontal band is the state trajectory in the equilibrium
)
In the above equations Di is defined by considering (4) in which the voltage drop on the diode ( V D ) and the transistor ‘on’ resistance ( Ron ) are neglected : Vg , cheb,i (24) Di = 1 − Vcheb,i V. SIMULATION RESULTS The effectiveness of the proposed method was investigated by conducting a series of simulations for the boost rectifier of Fig. 2. For better illustration of results, they are compared with a nonlinear analog circuit analysis and the small signal linearized model. The parameters of the rectifier circuit that are used for the simulation are tabulated in Table I.
simulation, load resistance is assumed to be constant. Once input voltage goes under a certain level, boost rectifier goes to DCM. This certain level is determined by intersection of CCM/DCM boundary and the load characteristics line [5]. The CCM/DCM boundary curve and partitioning pattern of state space are shown in Fig. 4. The controller varies the duty cycle d (t ) such that equation (1) is satisfied. The variation of duty cycle is shown in Fig. 5. The duty ratio of (4) is applied to the proposed model, as well as to the linear approximation and the full simulation model of PSPICE. The results are illustrated in figures 6 through 8. Fig. 6 shows the converter input current waveform. The PWA modeling better resembles to the PSPICE analog model. A magnified view of the state trajectory of the system is shown in Fig. 8. As the trajectory moves through the cells in DCM and CCM regions, different models are exploited in PWA approximation. It can be seen that the PWA model represents the analog model, much better than small signal linear approximation.
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Fig.8. State trajectory Fig.5. The variations of duty cycle d(t)
VI. CONCLUSIONS In this paper we investigated the piecewise affine approximation of large-signal modeling of PWM rectifiers operating in mixed conduction mode. This method is very useful in design of the current feedback loop for PWM rectifiers. Validation of the proposed approach carried out by conducting a series of simulations. The results compared with those obtained by the nonlinear analog model of PSPICE showing a good approximation. The results represent clear advantage of piecewise affine approximation over linear approximation in predicting the behavior of the system. The proposed modeling technique is an important tool for linear feedback design. REFERENCES
Fig. 6. Comparison of input current waveforms for nonlinear, PWA and linear models
[1]
[2]
[3]
[4] [5] [6]
Fig.7. Comparison of output voltage waveforms of nonlinear, PWA and linear models
[7] [8] [9]
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