**Modeling, analysis, and control of a rectifier with power factor correction in half-bridge configuration**

**J.F. Bayona, J.A. Parra, J.E. Vera, J. Avendaño**

*Escuela Colombiana de Carreras Industriales ECCI*

Email: *jbayonan@ecci.edu.co*, *jparrap@ecci.edu.co*, *jvera@ecci.edu.co*, *jonathan@ecci.edu.co*

*Received: 22/05/2013 Accepted 28/10/2013 Published 30/12/2013*

**Abstract**

This paper presents the detailed analysis of a single-phase rectifier with high power factor correction in half-bridge boost configuration (RPFCU-HBB). The purpose of this work was to achieve a unity power factor and regulated output voltage. Modeling and linearization around the RPFCU-HBB point of operation are exposed in detail. The analysis and design considerations of the current controller and the output voltage using the average current method are given. The control scheme to eliminate the voltage unbalance of the two output condensers is discussed in detail. The theoretical results are checked through the simulation of the RPFCU-HBB switch model, as well as through experimental work. By using the following parameters in the experimental prototype: input voltage of 120 Vrms, output power of 80 W, and output voltage of 450 V, we obtain a power factor of 0.99 and a total harmonic distortion of 2.5%.

**Key words **:RPFCU-HBB, linearization, stationary state, THD, EMI.

**RESUMEN**

Este paper presenta el análisis en detalle de un rectificador monofasico en configuración de elevador en medio puente con alto factor de potencia (RPFU-HBB). El proposito de este trabajo es lograr un factor de potencia unitario y un voltaje de salida regulado. El modelamiento y linealizacion alrededor del punto de operacion del RPFU-HBB son expuestos en detalle. El analisis y consideraciones de diseño del controlador de corriente y de voltaje de salida utilizando el metodo de corriente promedio son entregados. El esquema de control para la eliminacion del desbalance del voltaje de los dos condensadores de salida se discute en detalle. Los resultados teoricos son comprobados por medio de la simulacion del modelo de interruptores del RPFU- HBB y tambien a traves del trabajo experimental. Utilizando en el prototipo experimental los siguientes parámetros: voltaje de entrada de 120Vrms, potencia de salida de 80W y voltaje de salida de 450V, se obtiene un factor de potencia de 0.99 y una distorsion armonica total del 2.5 %.

**Palabras clave **:RPFU-HBB, linealizacion, estado estacionario, THD, EMI.

**Form Of Summons: **F. Bayona, et. al. Modeling, analysis, and control of a rectifier with power factor correction in half-bridge configuration. TECCIENCIA, vol. núm.15, p. 51-58 Junio 2013

**1. INTRODUCTION**

The 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 (RPFCU-HBB) 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], [11]. 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 *(Q _{1} and Q_{2}), *two condensers

*(C*an inductance

_{1 }and C_{2}),*(L),*and a load resistance

*(R),*as shown in Fig. 1. Its functions are: control inductance current

*(i*waveform for it to follow the alternate voltage

_{L})*(v*waveform, regulate the output voltage

_{g})*(v*at a specific value, and eliminate the voltage unbalance of the condensers, that is, make the voltage difference

_{8})*(vd)*equal to zero; besides,

*i*and vd are the variables of state and the useful cycle

_{L}, v_{8}*(h)*is the input variable of the RP-FCU-HBB.

*Q _{1} and Q_{2}* 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].

Equation (1) represents the voltages around the grid containing inductance *L, *voltage sources and and resistances , equation (2) describes the currents that flow in the node joined to the source of current and the resistances lastly, equation (3) describes the currents flowing in the node where the *C *condenser and the source of current (iL) are; therefore, upon relating the grid to the two nodes we obtain the model of the average circuit, as illustrated in Fig. 3.

**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:

1. Assume that *(v _{g}) *is an undistorted sinusoidal expressed as

*Vp*sin (ωτ), with Vp the voltage peak and

*ω*the line angular frequency.

2. *C _{1}*and

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

_{2}3. If *(i _{L}) *follows

*(v*then the result is a unity power factor and

_{g}),*(i*is an undistorted sinusoidal given by Ip sin

_{L})*(ωτ),*where Ip is the line peak current.

4. No voltage unbalance of condensers exists, this means that *(v _{d} *is equal to zero.

**III-A. Low-frequency voltage notch ***(δv _{s})*

Bearing in mind the prior assumptions and solving (1), the expression of the useful cycle in stationary state (H) is given by:

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

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

The condenser’s low-frequency voltage notch (*δv _{s}*) eiss eigquuaall at:o:

By multiplying the maximum value of (9) by 2, we
obtain the peak to peak voltage of(*δv _{s}*)

**III-B. Maximum cur**re**nt notch ***δi _{L},_{p}__{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 *(δi _{L}) *is given by:

Ignoring losses, on one side *(v _{L}) *is equal to:

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

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

Deriving (14) with respect to sin ^{(ωτ}) and finding the end points, we obtain:

This means that in the crossings through zero ( *δi _{L}) *is maximum.

**III-C. Power balance**

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

Equation (16) represents the input-output balance [8], [9], i.e., the term 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:

Where x, u, p_{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:

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

**5. CONTROLLER DESIGN**

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

**V-A. Current controller**

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

The Ki constant is chosen so the stationary state error is below 1°/o, then this constant is substituted in (22) and using the MATLAB ® sisotool, the values of *a _{1}, a_{2}, T_{1} and T_{2}*are found bearing in mind the following conditions: phase margin above 45°, gain above 8 dB, and attenuation above 20 dB to the commutation frequency.

**V-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

Voltages *v _{1}*and

*v*are resupplied through the two gain blocks H

_{2}_{2}; signal vid is obtained through the summand, as shown in Fig. 4, it then enters the

*C*controller and it is, lastly, summed to the iLref signal to obtain:

_{2}Hence, íl *; *will now follow *i**L*^{ref }rather than *I**L*^{ref }^{;} thereafter, tetitutmg (233^{:} in (3) we obtain:

Solving (24), we obtain:

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; thereby, λ must be negative. The relationship between λ and gains *KP _{vd} y KI_{vd}* f the controller is shown ahead:

**V-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 _{L}*follows

*i*and from Fig. 4 we can extract:

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

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

It can be noted that (31) is of first order and represents the output voltage dynamics *(v _{s}) *with respect to the controller's output voltage (v

_{Cx}). An integral-proportional controller is proposed for the stationary state error in the sum voltage loop to be equal to zero; the proportional and integral constants are found by using the MATLAB ® sisotool.

**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.

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 TM-DX32028069USB.

**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 ® 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 *(vC _{1} and vC_{2}), *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 230_{VRMS} 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.1

Figure 9 shows the current waveforms ^{iL} and *i _{L}*

^{y}

*i*

*v*^{rej }of the simulation; note that

^{iLref }has vanations to 0.

^{p}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 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. 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.

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