# Modeling, analysis, and control of a rectifier with power factor correction in half-bridge configuration/Modelamiento, analisis y control de un rectificador con correccion de factor de potencia en configuracion de medio puente.

1. IntroductionThe interest in improving the quality of the current absorbed from the electric generator by electronic equipment increases every day. Most of these equipment use a supply source that consists of a full-wave rectifier followed by a condenser [1], [2], [3], [4], which produces a non-sinusoidal input current and decreased power factor that hinders extracting the maximum mean power that can be delivered by the generator [3] and complying with standards like IEC61000-2-3 and IEE519 [5], [6]. Additionally, the high harmonic distortion of the current waveform causes electromagnetic interference (EMI) problems and generates harmonic voltages that interfere with other equipment connected to the same electric network [1] [2] [3] [4].

Hence, rectifiers with power factor correction (RPFCU) are the best option to overcome these inconveniences [2], [3], [7]. Several topologies exist to implement the RFPCU; the most commonly used is the half-bridge boost (RPFCUHBB) because it only has a semiconductor in series, meaning better efficiency with respect to other topologies [8].

The main control techniques of this topology are: average current, hysteretic, peak current, and discontinuous mode [2], among these, the average current method was selected for this work because of its good performance and high immunity to noise; also, the RPFCU-HBB model has been studied by several authors [1], [7], [8], [9], [10],. In this paper the RPFCU-HBB modeling considers the losses and it is obtained by averaging the equations of state [11], [12]; in addition, a detailed analysis in stationary state is shown along with analytical results useful for its design.

In [8] and [9] the authors observed voltage unbalance of output condensers and analyzed its causes, also proposing a control scheme to eliminate it. This work presents and analyzes in detail a scheme similar to that proposed in [8], but using an integral proportional controller.

2. Average model

An RPFCU-HBB is an AC-DC converter composed of two switches (Q1 and Q2), two condensers (C1 and C2), an inductance (L), and a load resistance (R), as shown in Fig. 1. Its functions are: control inductance current (iL) waveform for it to follow the alternate voltage ([v.sub.g]) waveform, regulate the output voltage ([v.sub.8]) at a specific value, and eliminate the voltage unbalance of the condensers, that is, make the voltage difference (vd) equal to zero; besides, iL, [v.sub.8] and vd are the variables of state and the useful cycle (h) is the input variable of the RPFCU-HBB.

[FIGURE 1 OMITTED]

Q1 and Q2 are alternately commutated through SPWM modulation; this produces a linear circuit for each time subinterval, as illustrated in Fig. 2, from which equations of state are obtained with their input and state variables averaged as suggested in [11].

L[d<[i.sub.L]>/dt] = (-[r.sub.L] - [r.sub.ds] - [rcR/[R + 2[r.sub.c]]] - [[r.sup.2.sub.c]/[R + 2[r.sub.c]]])<[i.sub.L]> (1)

[(2h - 1)R/2(R + 2[r.sub.c])]<[v.sub.s]> + <[v.sub.g]> - [<[v.sub.d]>/2] C[d<[v.sub.s]>/dt] = R(2h - 1)/[R + 2[r.sub.c]]]<[i.sub.L]> - [2/[R + 2[r.sub.c]]]<[v.sub.s]> (2)

C[d<[v.sub.s]>/dt] = <[i.sub.L]> (3)

Ecuation (1) represents the voltages around the grid containing inductance L, voltaje sources (<[v.sub.g]> and <[v.sub.d]>/2) and resistances ([r.sub.L], [r.sub.ds], [[r.sup.2.sub.C]/(R + 2[r.sub.C])] y [R/2]) equation (2) describes the currents that flow in the node joined to the source of current ((2h - 1) <[i.sub.L]>) and the resistances ([r.sub.C] y R/2)), lastly, equation (3) describes the currents flowing in the node where the C condenser and the source of current (<[i.sub.L]>) are; therefore, upon relating the grid to the two nodes we obtain the model of the average circuit, as illustrated in Figure.

[FIGURE 2 OMITTED]

3. Analysis in stationary state

The purpose of this analysis was to obtain the design equations to select the components of the power circuit of the RPFCU-HBB [8], [9]; for this, the following basic assumptions were considered:

* Assume that ([v.sub.g]) is an undistorted sinusoidal expressed as [V.sub.p] sin ([omega]t), with [V.sub.p] the voltage peak and [omega] the line angular frequency.

* C1 and C2 are big, thus, the voltage in both condensers is constant and the voltage notch for the commutation and line frequencies can be depreciated.

* If (iL) follows ([v.sub.g]), then the result is a unity power factor and (iL) is an undistorted sinusoidal given by lp sin ([omega]t), where lp is the line peak current.

* No voltage unbalance of condensers exists, this means that (vd) is equal to zero.

[FIGURE 3 OMITTED]

a. Low-frequency voltage notch ([delta][v.sub.s]) Bearing in mind the prior assumptions and solving (1), the expression of the useful cycle in stationary state (H) is given by:

H = [[V.sub.p] + [r.sub.p][I.sub.p]/[rho][v.sub.s]]sin ([omega]t) - [[omega]L[I.sub.p]/[rho][V.sub.s]]cos([omega]t) + [1/2] (4)

Where [rho] and [r.sub.[rho]] are defined as:

[r.sub.[rho]] = -[r.sub.L] - [r.sub.ds] - [[r.sub.c]R/[2[r.sub.c] + R]] - [[r.sup.2.sub.c]/[2[r.sub.c] + R]] (5)

[rho] = R/[2[r.sub.c] + R] (6)

Replacing (4) in (2) we obtain the average current that crosses the condenser (C) which has double the line frequency and is given by:

<[i.sub.c]> = [([V.sub.p] + [r.sub.p][I.sub.p]/[V.sub.s]]cos(2[omega]t) - [[omega]L[I.sup.2.sub.p]/[V.sub.s]]sin(2[omega]t) - [2[V.sub.s]/[2[r.sub.c] + R]] + [([V.sub.p] + [r.sub.p][I.sub.p])[I.sub.p]/[V.sub.s]] (7)

The DC component of (7) must be equal to zero; consequently, equation (7) becomes:

<[i.sub.c]>[([V.sub.p] + [r.sub.p][I.sub.p])[I.sub.p]/[V.sub.s]]cos(2[omega]t) - [[omega]L[I.sup.2.sub.p]/[V.sub.s]]sin(2[omega]t) (8)

The condenser's low-frequency voltage notch ([delta][v.sub.s]) is equal to:

By multiplying the maximum value of (9) by 2, we obtain the peak to peak voltage of [delta][v.sub.s]:

[delta][v.sub.s] = [1/[omega]C][integral]<[i.sub.C]>d([omega]t) [([V.sub.p] + [r.sub.p][I.sub.p])[I.sub.p]/2[omega]C[V.sub.s]]sin(2[omega]t) + [L[I.sup.2.sub.p]/2C[V.sub.s]]cos(2[omega]t) (9)

By multiplying the maximum value of (9) by 2, we obtain the peak to peak voltage of ([delta][v.sub.s]):

[delta][v.sub.d, p [right arrow] p] = [1/C[V.sub.s]][square root of ([[([V.sub.p] + [r.sub.P][I.sub.p]).sup.2][I.sup.2.sub.P]/[[omega].sup.2]] + [L.sup.2][I.sup.4.sub.p])] (10)

b. Maximum current notch [delta][i.sub.L, p[right arrow]p]

During the time subinterval in which Q2 is on and Q1 off, we obtain the circuit from Fig. 2(b). The net change in inductance current ([delta]iL) is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Ignoring losses, on one side ([v.sub.L]) is equal to:

[v.sub.L] = [V.sub.P] sin([omega]t) + [[V.sub.s]/2] (12)

On another, and making the cosine coefficient equal to zero, (4) becomes:

H = [[V.sub.p]/[V.sub.S]]sin([omega]t) + [1/2] (13)

Substituting (12) and (13) in (11) and solving

[delta][i.sub.L] = [1/L[f.sub.S][V.sub.S] ([1/4][V.sup.2.sub.S] - [V.sup.2.sub.P][sin.sup.2]([omega]t) (14)

Deriving (14) with respect to sin ([omega]t) and finding the end points, we obtain:

[delta][i.sub.L, p[right arrow]p] = [[V.sub.S]/4L[f.sub.S] sin [omega]t = {0, [pi], 2[pi], 3[pi]} (15)

This means that in the crossings through zero ([delta][i.sub.L]) is maximum.

c. Power Balance

It was expressed in section 3.1 that the DC component of (7) must be equal to zero, hence:

[[V.sub.s.sup.2]/[2[r.sub.c] + R]]] = ([V.sub.p] + [r.sub.p][I.sub.p])[I.sub.p]/2 (16)

Equation (16) represents the input-output [8] [9] balance the terms on the left side is the power absorbed by the load and the term on the right is the power delivered by the line.

4. Linear model

A linear model of RPFCU-HBB should be obtained to design a current controller [13], [14] consequently, the expansion of the right side of (1), (2) and (3) in Taylor series until the first derivate around the stationary point state is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Where x, u, [P.sub.s], h y f are the states of vectors inputs, states and inputs in stationary state, outputs and functions, respectively these vectors are given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

The transformation of the RPFCU-HBB linear model representation in space of states in function of transference is:

[i.sub.L](s)/h(s) = C[[sI - A].sup.-1]B (21)

5. Controller design

Design of controllers is achieved via the response on frequency method because it determines the relative and absolute stability [13], [15].

a. Current controller

The control technique through average current is illustrated in figure 4. The current controller is composed of an integrator and two advance networks.

[c.sub.1](s) = [[K.sub.i]/s][[1 + [a.sub.1][T.sub.1]s]/[1 + [T.sub.1]s]][[1 + [a.sub.2][T.sub.2]s]/[1 + [T.sub.2]s]] (22)

The [K.sub.i] constant is chosen so the stationary state error is below 1% then this constant is substituted in (22) and using the MATLAB sisotool, the values of [a.sub.1], [a.sub.2], [T.sub.1], [T.sub.2] are found bearing in mind the following conditions: phase margin above 45[degrees], gain above 8 dB and attenuation above 20 dB to the commutation frequency.

b. Voltage difference controller

In [8] and [9]the existence of condenser voltage unbalance was indicated, explaining that it is caused by the current controller offset. A method to eliminate the unbalance using a proportional controller was suggested by [8] this work presents a similar method but with an integral- proportional controller.

[FIGURE 4 OMITTED]

Voltages v1 and v2 are resupplied through the two gain blocks H2; signal [v.sup.i]d is obtained through the summand, as shown in Fig. 4, it then enters the [C.sub.2] controller and it is, lastly, summed to the iLref signal to obtain:

[i.sub.L]ref' = [i.sub.L]ref + [C.sub.2][H.sub.2][v.sub.d] (23)

Hence [i.sub.L] will now follow [i.sub.L]ref' rather than [i.sub.L]ref'; thereafter, substituting (23) in (3) we obtain

C[d<[v.sub.d]>/dt] = [I.sub.p]sin([omega]t) + K[P.sub.vd][H.sub.2]<[v.sub.d]> + K[I.sub.vd][H.sub.2][integral]<[v.sub.d]>dt (24)

Solving (24) we obtain

<[v.sub.d]> = [[kappa].sub.1][e.sup.[lambda]t] + [[kappa].sub.2]t[e.sup.[lambda]t] - [[I.sub.p][omega]cos([omega]t)/C([[lambda].sup.2] + [[omega].sup.2])] (25)

The unbalance in condensers is originated by the initial conditions of the condensers [8]; consequently the exponential terms of (25) represent the unbalance and decay to zero asymptotically in stationary state; there by [lambda] must be negative. The relationship between [lambda] and gains K[P.sub.vd] and K[I.sub.vd] of the controller is shown ahead:

K[I.sub.vd] = -[K[P.sup.2.sub.vd][H.sub.2]/4C] (26)

[lambda] = K[P.sub.vd][H.sub.2]/2C (27)

c. Voltage sum controller design

[8] and [9] expressed that the dynamics of the current loop is rapid, due to this only the output voltage dynamics needs to be considered to obtain the transference function that describes its behavior hence [i.sub.L] follows [i.sub.L] ref' and from figure 4. We can extract:

[I.sub.p] = [H.sub.a][V.sub.P][v.sub.ex]/[H.sub.1] (28)

By averaging equations (2) and (28) over a line period, we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

C[d[<[v.sub.s]>.sub.2[pi]]/dt] = [([V.sub.p] + [r.sub.[rho]][I.sub.p])[<[i.sub.P]>.sub.2[pi]]/[V.sub.S]] - [2[<[v.sub.S]>.sub.2[pi]]/[R + 2[r.sub.C]] (30)

Inserting (29) in (30) and linearizing around the operation point, we obtain the transference function given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

It can be noted that (31) is of first order and represents the output voltage dynamics ([v.sub.S]) with respect to the controller's output voltage ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). An integral- proportional controller is proposed for the stationary state error in the sum voltage loop to the equal to zero; the proportional and integral constants are found by using the MATLAB sisotool.

[FIGURE 5 OMITTED]

6. Implementation of the RPFCU-HBB circuit

The values of the RPFCU-HBB circuit components are presented in Table I. The inductance (L) was constructed with a ferrite material 77 nucleus. The transistors used in the RPFCU-HBB were MOSFETs IRF840 mounted on heat sinks.

[FIGURE 6 OMITTED]

The Texas Instruments TMDX32028069USB development card was used to control the inductance current, the sum voltage and the voltage difference in the RPFCU-HBB. The Texas Instruments UC2705 circuit was used to manage all the MOSFETs transistors gate. The operational amplifiers from Texas Instruments OPA2350 and OPA4350 were used to condition the voltage signals from each of the condensers, line voltage, and inductance current, to be sampled by the digital analog converter of the development card from Texas Instruments TMDX32028069USB.

7. Experimentation and results

The simulation of the model of switches and the RPFCU-HBB experiment were conducted with the parameters from Table I, the MATLAB R simulink tool was used for the simulation, the simulation and experimental waveforms are presented.

Figures 5 and 6 illustrate the inductance current waveforms (iL) and the conductance voltage waveforms (vC1 and vC2), besides comparing the simulation waveforms to the experimental waveforms, noting that they are quite similar. The PF and the values of the input current harmonic components are measured by using the FLUKE 43B line analyzer. As noted in Fig. 7, the value of PF and THD measured was 0.99 and 2.5%, respectively; in addition, Fig. 8 compares the values of the harmonics measured with the limits of the IEC1000-3-2 class C standard. Because the limits are given for input 230VRMS, they are then multiplied by a factor of 1.91 to obtain the harmonic levels for input 120VRMS. It can be observed that the values of the harmonic components are much below the limits of the standard.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Figure 9 shows the current waveforms iL and iLref of the simulation; note that iLref has variations to 0.7 A, and however iL follows it, evidencing good performance of the control.

8. Conclusions

This work presented the modeling, analysis, and control of a rectifier with power factor correction in half-bridge boost configuration. The average current control technique was used for the input current to follow the line voltage. Useful equations (4, 10, 15 and 16) were developed to define the stationary state. A model considering the losses was obtained (1, 2, and (3) and linearized around the stationary state point (21); additionally, an integral-proportional controller was proposed and analyzed in detail (25) in the scheme suggested in [8] to eliminate the voltage unbalance of the output condensers.

[FIGURE 9 OMITTED]

The experimental results revealed that the RPFCU-HBB obtained a high power factor of 0.99 and a THD of 2.5%, fulfilling standards IEC1000-3-2, EN61000, and IEEE 519--as evidenced in Fig. 7.

DOI: http:/dx.doi.org/10.18180/tecciencia.2013.15.4

References

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[5] IEEE Industry Applications Society, <<IEEE recommended practices and requirements for harmonic control in electrical power systems,>> IEEE, 1992.

[6] K. N. Sakthivel, S. Das y K. Kini, <<Importance of quality AC power distribution and understanding of EMC standards IEC 61000-3-2, IEC 61000-3-3 and IEC 61000-3-11,>> de Electromagnetic Interference and Compatibility. 8th International Conference, 2003.

[7] R. B. Ridley, <<Average small signal analysis of the boost power factor correction circuit,>> de VPEC Seminar proceedings, 1989.

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Received: 22 May 2013

Accepted: 28 Oct 2013

Published: 30 Dec 2013

J.F. Bayona (1), J.A. Parra (2), J.E. Vera (3), J. Avendano (4)

(1) Escuela Colombiana de Carreras Industriales ECCI, Bogota, Colombia, ibavonan@ecci. edu. co,

(2) Escuela Colombiana de Carreras Industriales ECCI, Bogota, Colombia, iparrap@ecci. edu.co

(3) Escuela Colombiana de Carreras Industriales ECCI, Bogota, Colombia, Jvera@ecci.edu.co

(4) Escuela Colombiana de Carreras Industriales ECCI, Bogota, Colombia, ionathan@ecci. edu. co

Table 1. Values of circuit parameter components Circuit Symbol Value parameters Copper loss of rL 0.452 the inductance Ignition rds 0.3452 resistance of MOSFETs Equivalent rC 1.08452 resistance in series of Cl and Cl Inductance L 5 mH Output C = Cl = C2 100 [micro]F condensers Line voltage [v.sub.g] [V.sub.p] sin (rat) V Inductance [I.sub.p] [I.sub.p] sin (rat) A current Line [omega] 2[pi][60.sup.rad.sub.s] frequency Commutation [f.sub.s] 50 kHz frequency Line voltage [V.sub.p] 120 [square root of 2V] peak Line current [I.sub.p] 0.9927 A peak Output sum [V.sub.s] 450 V voltage Output voltage [V.sub.d] 0 V difference Maximum current [delta]iL,p-p 0.45A notch Maximum sum [delta]vs,p-p 10 V voltage notch

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Author: | Bayona, J.F.; Parra, J.A.; Vera, J.E.; Avendano, J. |
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Publication: | Tecciencia |

Article Type: | Report |

Date: | Dec 1, 2013 |

Words: | 3327 |

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