A Family of Power-Factor-Correction Controllers

Zheren Lai and Keyue M. Smedley Department of Electrical and Computer Engineering

University of California at Irvine Irvine, California 92697

Ahslract - This paper presents a family of constant-switching- frequency pulse-width-modulated controllers for single-phase power-factorcorrection circuits that operate at continuous- conduction-mode. Both trailing-edge and leadingedge pulse- width modulation are used. These controllers do not require the muitiplier and the rectifid-linevoltage sensor, which are needed by traditional control methods, and can be implemented with a unified control circuit to achieve simplicity. Controller examples are analyzed and verified experimentally.

L Introduction

A single-phase diode bridge followed by a dc-dc converter can form a rectifier with active power-factor correction (PFC). The dc-dc converter controller forces the average input current i to have the same shape as the input voltage I'

The multiplier approach and voltage-follower approach are two traditional control strategies for the resistor emulator[ 1 1. Examples of the multiplier approach include the average-current control[ 21 and the peak-current control1 3 ) . etc. Examples of voltage-follower approach include buck-boost converters operating at dtscontinuous- conduction mode(DCM)[1]. Cuk converters at DCM (51, and boost converters at the boundary of DCM and continuous- conduction mode (CCM)[6], etc. Rectifiers under multiplier- approach control usually operate at CCM while rectifiers undcr \,oltage-follower-approach control generally operate at DCM or at the boundary of DCM and CCM. Converters operate at DCM are usually used for low-power-level applications. The CCM becomes necessary when power level goes higher. because the current stress on a switch and the current ripple in the inductor are too large for a single DCM converter to operate efficiently. The multiplier approach requires a multiplier in its current loop and to sense the rectified line voltage, hence, the control circuit is more complicated.

A number of papers have been dedicated to control methods for the CCM operation without a multiplier and the input-voltage sensor [7-11). Some methods can be implemented with a very simple control circuit under the penalty of higher current dlstortion. The non-linear-carrier control, proposed in [8, 91 for the boost converter and other topologies, has the simplicity of the voltage-follower approach and the performance of the multiplier approach.

g Such a rectlfier is called a "resister emulator." g .

However, the non-linear camer is very difficult to generate for some topologies. The carrier in [9] was approximated, with a fairly large minimum duty ratio &in (= 0.22) which leads to a limitation of load range at the light end. The Linear Peak Current Mode control proposed in [ IO] shows a simple and promising method for PFC application. A unified approach was proposed in [ 1 11 as an application example of a general pulse-width-modulation (PWM) method for a family of simple-and-high-performance CCM PFC controllers. The members of this family include the methods of [8] and [lo]. The objective of this paper is to further study these controllers.

In this paper, the general PWM modulator is briefed in section 11, followed by the derivation procedures of t h s controller f a d y . Derivation results of five commonly-used topologies are listed in a table. Section I11 takes the simplest controller among this family as example. Stability and current dtstortion are analyzed in detail and then the control method is verified experimentally. Section IV and V verify the validity of the controller family experimentally by applying them to the flvback and Cuk converters respectively. Finally conclusions are drawn in section VI.

Notation conventions are as follows. Capital letters are used for quantities associated with steady state unless indicated explicitly; lower case letters represent time-variant variables; a quantity in a pair of angular brackets is the local average of the quantity, i.e. the average in each switching cycle.

U. Unified Realization of a Family of PFC Controllers

Fig. 1 shows the block dlagram for PFC circuits with the proposed controller family. Notice that no rectified-line- voltage sensor and multiplier appear in the diagram. The proposed PFC controllers are similar to the current-mode controllers for dcdc converters.

The PWM modulator and the current sensor in Fig. 1 are distinct from those used in typical multiplier-approach controllers. The PWM modulator in Fig. 1 is the general PWM modulator which can perform leading-edge modulation as well as trailing-edge modulation. The sensed current can be valley inductor current or valley switch current besides peak and average inductor or switch current. One can see shortly that for trailing-edge modulation, the sensed current need to be the peak current while for leadtng- edge modulation, the sensed current is the valley current.

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Fig. 1 . Block diagram of PFC circuits with proposed controllers.

The peak or valley inductor current can be considered as the instant inductor current and the peak or valley switch current can be considered as the instant switch current. Thus the proposed control methods is summarized as instant- inductor-current control, average-inductor-current control, instant-switch-current control, and average-switch-current control respectively according to the way that the current is sensed. The instant inductor current and the instant switch current are not equal in converters with more than one inductor, such as the Cuk converter.

Average or instant-switch-current controls are more desirable than inductor-current control due to simpler current-sensing circuitry. These two sensors can be implemented as shown in Fig. 2 . The capacitor voltage in Fig. 2(a) is reset cycle-by-cycle to provide the average- mitch-current information in that. switchng cycle. The switch current can be the current in the active switch, such as a MOSFET, or the passive switch, i.e. a diode. The diode current is for the first time utilized in a PFC controller.

A 7he General PWModulator

The general PWM modulator proposed in 11 11 is redrawn in Fig. 3. It contains a constant-fiequency clock generator. a flip-flop FF. a comparator CMP, and a few stages of integrator-with-reset. The integrator performs conventional integration unless the control input R is at logical-high state, which as result resets the integrator output to zero. The time constant of the integrator is selected to equal the switching period. The general PWM modulator can realize control functions expressed by a general modulation equation

V I = ( vz d + v3 d2 + ~4 d3 + ...... ) (1)

where v l . v2. ... are control inputs to the modulator and are linear combinations of the sensed current and the modulation voltage vm in the PFC controller case, shown in Fig. 1. These control inputs are slow signals compared to the switching frequency. To find the d solution for (l), the general modulator can be applied with Q of the flip-flop as its trailing-edge modulation duty-ratio output. Equivalent leading-edge modulation for the same control function can

(a) average-switch-current sensing

- @)instant-switch-cumnt sensing

Fig. 2. Switch-current-sensing schemes.

be obtained by re-organizing (1) into a polynomial in d'. where d' = 1-d, with Q of the flip-flop as d' output, i.e. a is the duty-ratio output. Th~s can be achieved by applying VI'. v2'. . . . to v 1. v2, . . . of the modulator respectively 11 11. where VI'. 9') ... satisfy

- 1 -1 - 1

Sign(v1') is the sign of voltage v1' to guarantee V I ' to be positive. For some applications, leadmg-edge modulation may lead to simpler control circuitry, such as the boost PFC circuits shown in section 111.

B. Derivation of the PFC Controller Family

This derivation is to find control inputs to the general modulator so that the input current of the dc-dc converter is proportional to the input voltage, with the proportional coefficient controlled by the modulation voltage vm. Trailing-edge modulation equations are found first, then convert to the leadlng-edge modulation equations by ( 2 ) . The following three assumptions are made for simplicity.

1) The switching frequency and the linc frequency are well separated so that the line current and voltage can be considered slow signals and the power stages operate at quasi-steady state.

2) The current ripple in an inductor is negligible. This is a stronger condltion than CCM operation. Current distortion will be higher if this assumption is not satisfied, however, some amount of distortion is tolerable for PFC applications.

3) The energy conversion efficiency is assumed to be 100% so that the efficiency term is not carried in the derivation.

The dcdc converter should be able to step-up in order to be suitable as a PFC circuit, since the output voltage will be lugher than the input voltage when the line voltage is near zero [ 11. This condition excludes the buck and buck-derived converters, such as the forward converter, as PFC circuits.

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R R + I Inlegrator I integrator

w/ reset W i Imet

Fig. 3 , The general constant-frequency pulse-width modulator.

Dcdc converters are classlfed into two types according to the device right after the &ode bridge when the E M filter is ignored. ' h s device can be a switch or an inductor, as shown in Fig. 4. Fig. 4(a) represents converters that have a switch in series with the &ode bridge, such as the buck- boost converter. Fig. 4@) shows converters having an inductor in series with the bridge, such as the boost or the Cuk converters, etc.

The control goal for both types of converters is

<i g > = v g b . (3)

vg vom(d), (4)

where a line cycle, the duty ratio d needs to be controlled to satisfy

is the emulated resistance. At quasi-steady state in

where M(d) = P(d)/Q(d) is the conversion ratio of the dcdc converter. P(d) and Q(d) are polynomials in d. M(d) for five commonly-used converters are listed in Table 1. Thus the general PFC control function is

where R,<ig > = vnl/M(d). ( 5 )

vn, = %v& (6) and R, is the equivalent current-sensing resistance. For a dc- dc con\.erter at CCM steady state, the instant active-switch (MOSFET) current IT does not always equal the inductor current iL uhen the active switch is on. instead

iT = N(d) iL . (7)

iD = N(d) iL ( 8 )

Similarly the instant &ode current

when the diode is on. Function N(d) is the ratio of two polynomials in d. similar to M(d). N(d) for the five topologies are also listed in Table 1.

I . IIc-LJc converters with Fig. 4(a) conjiguration

a) Trailing-edge modulation For the hpe of converters shown in Fig. 4(a). average currents

<i g >=<IT>. (9)

$<iT> = vm/M(d) = v,Q(d)/P(d).

Combining with ( 5 ) yields

t 10)

R I Mqwtur

W i reset

' r ' l T T -

@$T-Jqq+ V O

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@) Fig. 4. General power stage cmfigurations of smgle-phase PFC circuits.

(10) gives the trailingedge average-witch-current- control modulation equation. This equation is the basis of non-linear-carrier control. The left side of (10) is implemented n4h the average-current sensor shown in Fig. 2(b) while the right side is a non-linear carrier. The required non-linear camer vm/M(t/Ts) for 0 < t .c Ts is a rational function of (Os), therefore, is very drfficult to generate sometimes. However if (10) is re-arranged into the general format as (I) , it can be implemented with the general modulator. The re-arrangement is to move the denominator at one side of the equation to the other side. Mathematically. this re-arrangement will not change the solution value of d.

The local average of the switch current <iT> satisfies

(1 1)

(12)

clT> = 1T d.

R, iT d = v,/M(d) = vmQ(d)/P(d).

hence

Re-arranging ( 12) yields the trailing-edge instant-switch- current-control modulation equation. One can find out in Table 1 that the difference between instant-switch-current and average-switch-current control methods is to integrate the instant-switch current one more time than the average- switch current. These two methods are physically equivalent. They both are listed for convenience of application.

Substituting (7) into (12) results in

5 i~ d = vmW(a?N(d)l. (13) Re-arranging ( 1 3) yields the inductor-current-control modulation equation.

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b) Leadmg-edge modulation From (13) one can obtain a leading-edge inductor-current- control modulation equation by applying (2). During t E [0, d' T,], (8) holds. Substituting (8 ) into the inductor-current- control equation yields the instant-switch-current-control modulation equation. Finally, substituting

into the switch-current-control equation results in the average-switch-current-control modulation equation for leading-edge modulation.

2. Dc-dc converters with Fig. 4(b) conjguration

a) Trailing-edge modulation For the converter type shown in Fig. 4@),

Substituting (15) into ( 5 ) yields

This equation gives the inductor-current control

under the small current ripple assumption. Substituting (7) into (17) yields the instant-Awitch-current-control modulation equation

<iD> = d' iD (14)

<i > = <iL>. (15)

%<iL> = vm/M(d). (16)

R& = vmm(d) (17)

g

$ 1 ~ = [v"(d)l/M(d). (18)

Finally, switch-current-control modulation equation

substituting (11) into (18) yields the average-

&<iT> = [v, d N(d)]/M(d). (19)

b) Leading-edge modulation Applying (2) to (17) leads to the leading-edge inductor- current-control modulation equation. Substituting (8) into the inductor-current-control equation gives the instant- .witch-current control. Finally. one can use (14) to get the leading-edge average-switch-current-control modulation equation.

Derivation results for the five commonly-used converters are listed in Table 1. Same procedures can be used to obtain control circuits for other topologies. Notice that the trailing- edge a\.erage-switch-curn~ control for boost converter is the same control method as the non-linear-carrier control[8]. The instant-switch-current control for the trailing-edge boost converter. which turned out to be the same method as the Linear Peak Current Mode control proposed in [lo]. is an approximation of the peak-current non-linear-carrier control[ 81. Accurate instant-switch-current control can be obtained through the same derivation as (13) in [SI which can directly apply the general modulator.

For those controllers with input v4 = 0 in Table 1, two stages of integrators are required whle those with inputs v3 = VJ = 0 need only one stage of integrator. There are six possible controllers for each converter topology. Some of

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them arc obviously more complicated than others. Some controllers may be subject to stability problem. Complete analysis of the stability of all the controllers is beyond the scope of this paper. Modification of the controller is necessary when the instability happens.

IIL Leading-edge PFC Controller for Boost Converters

The simplest controller found in Table 1 is for the boost converter with leadtng-edge instant-switchcurrent control. Accordmg to Table 1, v1 = R&, v2 = vm, where R& is the sensed current and Vm is the error ampllfier output, so only one stage of integrator is required. The entire circuit is shown in Fig. 5(a) with the general modulator blocked in dotted lines. Notice that the sensed current is the &ode current for this case and 3 is the duty-ratio output because of the leadmg-edge modulation.

A. Circuit Operation

The operation waveforms are shown in Fig. 5@). When a clock pulse arrives, FF is set to logical hgh, hence, 0 is logical low. As result, the MOSFET is turned off and the inductor current flows through the &ode D. When the integrator output vint meets the sensed current, CMP outputs a signal to reset FF, therefore, 3 becomes logical high to turn on the MOSFET and reset the integrator. This operation repeats at the next clock pulse.

B. S t a dy-Sta te and Stability Analysis

Mapping theory [12] is use for t h s analysis. Assume the peak inductor current for nth cycle is Ipn and the inductor current changes linearly with time. The moment that FF is reset satisfies,

where Ti is the integrator time constant. Without losing generality, Ti can be selected to equal switching period Ts . If Ti # Ts. the only parameter that will be afFccted is vm which is adjusted automatically in a closed-loop system. Replacing t with dn'Ts results in

(2 1)

where f, = l/Ts is the switching frequency. Thus r; - I g

dn' = I ; ( & + ___ ) - I , (22) Rs Lfs

On the other hand

Fig 5 . Leading-edge modulated PFC redifier. (a) Circuit diagram. @)Operation waveforms.

The steady state can be found out by letting (23) equals 0, therefore,

v ,I)'= x, (26)

Yo

Combining (26) with (22) yields the steady-state peak inductor current

Valley inductor current I, is found to be

For constant V,, R, and Vo, the valley current is proportional to the input voltage. The average inductor current is then found to be

hence,

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16

14

12

i? 10

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6

4

2

0 0 20 40 60 80 100

powsr Level f%WI Loslb

Fig. 6. Calculated line current at various power levels. Fig. 7. Theoretical THD vs. power levels.

I I

Fig. 8 . Experimental waveforms at different load power levels. Top: duty ratio ( 0 3 d i v ) ; Middle: line voltage (110 Vrms); Bottom: l i e current, SA/div, ?,A/div, and O.SA/div respectively for (a), (b), and (c ) . (a) at full load; (b) at 40% of full load; (c)at 10% of full load.

Ks determines the stability of this control method [ 121, In order for the control method to be stable,

IKSI 1. (30) For a given value of Vm. Vg. and Vo, the inductor current w i l l reach a stead\ state. Any perturbation will drsappear gradually. If the inductor current is higher than the steadq- state Ialue, it takes longer for vint to reach R, iD for a given valuc of Vm. thus the inductor has longer time to dlscharge. If the inductor current is too small. vint reaches 9s iD sooner. resulting a larger duty ratio, hence. the inductor current increases. Sub-switching-frequency-harmonic oscillation occurs when (30) is not satisfied.

Combining (6). (25). (30). and (26) gives another form the stability condltion at steady state., i.e.

(3 1) indcates that the stabilih is load dependent

C. Application to PFC Control If the operation is stable. the PFC circuit will be approaching to quasi-steady state. Since the valley inductor current of the boost conyerter is proportional to the input yoltage vg. under small ripple assumption, the average inductor current is approximately proportional to vg, When

the ripple is not negligible, (29) can be used to find the actual average line current. An example is given with the following circuit parameters.

For vg = 110 Vrms line voltage, L = 520 pJ3, fs = 100 MIZ, and Vo =220 Vdc. The normalized line current is calculated and shown in Fig. 6 for R, = 30, 70, and 300 Ohms respectively. Take = 30 Ohms as the full load, the line current THD vs. power level is plotted in Fig. 7. The distortion improvement over the peak-current control with multiplier approach [3] is obvious.

When the voltage conversion ratio is h g h enough ( 2 2). the duty ratio is higher than 0.5, thus the current loop is always stable, as (31) indlcates. This condltion may not be satisfied at light load and low voltage conversion ratio. With certain minimal load condltion, the unstable situation can be excluded.

D. Eqxrirxnta 1 Verification

An experimental circuit has been built accordmg to Fig. 5(a) and tested. The line voltage is 110 Vrms and the output voltage is 220 Vdc. The switching frequency fs is 100 kHz. The inductance of the boost converter is 520 pH. An LC EM1 filter was inserted between the diode bridge and the boost converter with an inductance of 44 pH and a capacitance of 0.68 pF. Experimental waveforms for a full load of 350 W. 40% of full load, and 10% of the full load are shown in Fig. 8 (a), (b), and (c) respectively. The waveforms in each figure, from top to bottom, are the duty ratio

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Fig. 9. W s t d e n v e d PFC topologies employed in references[ 14 - 161

I I

Fig. 10. Flyback PFC converter with trailing- edge average-witch-current control.

I I M 1 0 W O N CH3 l N D V lime 5 M V

F i g . 1 1 E r g m e n t a l waveforms of the flyhach conxerter at 100 W power output Top dut\ ratio (0 61'div). middle h e xoltage ( I 10 \ mis) bottom lms current ( 1 , I d n )

Fig. 12 Cuk converter wlth insiant-mdudor- current mtrol.

Fig 13. Experimental waveforms of the Cuk converter with instant-indudor-current control at 150 W power output. Top: duty ratio (0.5idiv); middle: line voltage (1 10 Vrms); bottom: line current (2 Aldiv)

measured with Tektronis time-to-voltage converter TVCSO 1. the line koltage. and the line current with its scale marked in the figure respectively Notice that in (a) and (b) the current dstortion is not significant while in (c) the lstortion is noticeable The current shape in (c) is similar to the calculated na.ceform shown in Fig 6 It is &fficult to compare the measured THD wth the theoretical THD since the line \ oltage itself has some dlstortion. however, one can still see the trend of lstortion as load decreases

E Advantage\ of Lading-Edgr Mvdula hon

Compared to trailing-edge modulation. Icadlng-edge modulation has threc a& antages

1 ) The controller for boost converter is the simplest resister emulator controller ever

2) The current-sensing circuit for some boost-demed topologies u i th multiple switches is simplified Fig 9 shows some of these topologies emplo?ed in literature[lJ - 161 For trailingedge modulation. the switch current is the current in the actiLe snitches. i e in the MOSFETs in these eumples Since these topologies ha\e more than one active switch. at least two current transformers (CT) are required For leadmg-edge modulation there is a common path for the

310 un 3 1 0 U , , J U T

I / I 1 I - I I

Fig 14 CuL converter wrth average-switch- current control

Fig 15. Experimental waveforms of the Cuk converta with average-witch-currentt control at 150 W power output. Top: duty ratio (0.5idiv): middle: line voltage (1 10 Vrms): bottom: line current (2 Ndiv).

two or more diode currents, thus they can be sensed with one CT, as shown in each figure.

3) The switching ripple current in the output filtering capacitor is reduced. The boost converter is usually used as a pre-regulator and the post-regulator is normally trailing- edge modulated. This results in less switching ripple current in the capacitor C as indicated in [ 131.

IV. Flyback Converter

From Table 1 one can also find the control circuits for flyback PFC converters. Fig. 10 shows the flyback PFC converter of trailing-edge average-switch-current control (the error amplifier is not shown). Under small ripple assumption, the line current chstortion for the PFC circuit is negligible. Similar analysis as the boost converter can be carried out for the current distortion if the current ripple is not negligible. As indlcated in [lo], this control method is uncondltionally stable.

An experimental circuit was built and tested. The major circuit parameters are as follows: fs = 63 kHz, the primary inductance is 340 FH, flyback transformer turns ratio is 1 : 1, Lf = 100 pH, Cf = 0.68 pF. line voltage is 110 Vrms, the output voltage is regulated at 50 Vdc. The experimental

7 2

result is demonstrated in Fig. 11. Obwously the linecurrent dstortion is insignificant.

V. Cuk Converter

Cuk, Sepic, and Zeta converters were classified as the same category as the flyback converter in [9] and [IO]. Neither reference clearly addressed the control of the these converters. The derivation results in t h s paper indmtes that the control circuits for these converters are not always equivalent.

To verify the control methods for the Cuk converter, instant-inductor-current and average-switch-current controlled PFC circuits were built and examined. The circuits are illustrated in Fig. 12 and Fig. 14 respectively with their error ampldiers not shown. The same power stage were used for both control methods. The switchng frequency is 100 kHz and other major parameters are shown in the figures. The line voltage was 110 Vrms and both outputs were regulated at 50 Vdc. Fig. 13 and Fig. 15 are the experimental waveforms at 150-W power output. From top to bottom the waveforms are the duty ratio, line voltage, and line current respectively. Notice that both line currents are closely following the line voltage in shape.

It was observed that, for the average-switch-current control, the line current had some ringing at around 2 kHz when the rectified line voltage changed slope rate abruptly, i.e. when the line voltage crosses zero or is at its peak for the dstorted line voltage. The ringing is noticeable in Fig. 15. It is believed that thls ringing is caused by the high system order of Cuk converter and the average-switch-current control does not provide enough damping. With instant- inductor-current control. the ringing i s significantly damped for the same power stage. In other words, Merent control methods may provide different dynamic performance for the same power stage.

VI. Conclusions

A famil! of PFC controllers are presented in this paper based on a general PWM modulator. The derivation procedures are addressed in detail with derivation results for five commonly-used converters listed in table. Both trailing- edge and leading-edge modulation can be realized at constant switching frequenq . Leading-edge modulation can sometimes lead to simpler control circuitry as demonstrated in the boost converter example.

Both leadng and trailing-edge modulation have three basic control circuits for one converter accordmg to the way that the current is measured. Physically the average-smltch- current control is equivalent to the instant-switch-current control in the sense that the former is the integration or the latter.

The PFC circuits are ideal resister emulators when the switching-frequency inductor-current ripple is zero. In practice, the linecurrent dlstortion &e to the current ripple can be anal-wed for each specfic control method.

For this family of controllers, the rectified-line-voltage sensor, the error amplifier in the current loop. and the multiplier in the voltage feedback loop that exist in a traditional CCM PFC circuit are eliminated hence, the control circuitq is simplified. The performance of these PFC circuits is comparable to or improved over those traditional CCM converters with the multiplier-approach control. The most important advantage of these controllers is that their implementation is unified. One can see from the three converters and four control methods experimentally verified in t h s paper that they share an identical PWM modulator. Therefore, these controllers are well-suited for integrated- circuit implementation.

References I. Sebastian, M. Jaureguizar, J. Uceda, "An overview of power factor correction in single-phase off-line power supply systems," IECON'94. L. Dixon, "Average Current Mode Control of Switching Power Supplies," Unitrode Power Supply Design Seminar, 1990. R. Red1 and B. P. Erisman, "Reducing Distortion in Peak-Current- Controlled Boost Power-Factor Correctors," APEC'94. R. Erickson, M. Madigan, and S . Singer, "Design of a Simple High- Power-Factor Rectifier Based on the Flyback Converter," APEC'PO. M. Brokovic, and S. Cuk, "Input Current Shaper Using Cuk Converter." INTELEC'92, Washington D. C.. Oct.. 1992, pp. 532- 539. J . Lai and D. Chen. "Design Consideration for Power Factor Correction Boost Converter Operating a1 the Boundary of Continuous Conduction Mode and Discontinuous Conduction Mode," APEC'93. D. Maksimonvic, "Design of the Clamped-Current High-Power- Factor Boost Rectifier." APEC'94. D. Maksimonvic, Y. Jang, and R. Erickson, "Nonlinear-carrier control for high power factor boost rectifiers," APEC'95. R. Zane and D. Maksimovic, "Nonlinear-carrier control for high- power-factor rectifiers based on flyback, Cuk or SEPIC converters," APEC' 96.

1101 I . P. Gegner and C. Q . Lee. "Linear Peak Current Mode Control: A Simple Active Power Factor Correction Control Technique For Continuous Conduction Mode," PESC'96, June 28, 1996.

111 Z. hi and K. M. Smedley, "A General Constant Frequency Pulse- Width Modulator and Its Applications," HFPC'96, Las Vegas, Sept. 5. 1996.

12) F. C. Moon, Chaotic b7brations - An introduction for &plied Scientists and Engineers. John Wiley & Sons, Inc. ,1987.

131 R. A. Mammano, "New Developments in High Power Factor Circuit Topologies," HFPC'96, Las Vegas, Sept. 5, 1996.

[14] P. N. Enjeti and R. Martinez, "A High Performance Single Phase Ac To Dc Rectifier With Input Power Factor Correction," APEC'93.

I151 A. Pietiewicz and D. Tollik, "New High Power Single-Phase Power Factor Corrector With Sofi-Switching," Intelec'96.

1161 E. X. Yang. Y. Jiang, G. Hua, and F. C. Lee, "Isolated Boost Circuit for Power Factor Correction," APEC'93.

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