sm pfc boost tan(11)

8/13/2019 Sm Pfc Boost Tan(11)

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General Control for Boost PFC Converter from aSliding Mode Viewpoint

Grace Chu, Siew-Chong Tan, Chi K. Tse and Siu Chung Wong

Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kongemail: [email protected]

Abstract In this paper, we propose a current controlmethodology for the boost PFC converter based on sliding modecontrol theory. The resulting control is very general in thesense that it provides input current tracking under all practicalconditions by systematically including all necessary controlfunctions that compensate for the possible nonlinearity suchas the zero-crossing distortion in high line frequency condition.By choosing suitable state control variables and a proper slidingsurface, the derivation arrives at a feedforward control which issimilar to a previously proposed method. The implementation

takes the form of a typical pulse-width modulation (PWM)control. It can be designed either as an individual currentcontroller or as feedforward control applied to a standard PFCcontroller.

I. I NTRODUCTION

In this paper, a general current control rule is derivedfor the boost PFC converter. The aim is to provide inputcurrent tracking under all practical conditions. Speci cally,the proposed general control systematically and automaticallyincludes the necessary compensation functions to eliminatethe phenomena due to the inherent nonlinearity, which maymanifest as zero-crossing distortion under certain line fre-quency condition [1]. The derivation is based on generalsliding mode (SM) control theory. A xed-frequency versionof sliding mode control [2] is employed and the resultingcontrol takes the form of a typical PWM control, which canbe readily implemented with analog circuits. Interestingly,this SM-based derivation is simpler and more straight forwardthan the conventional large signal approach, as complicatedcurrent loop modeling is not required. The resulting gen-eral control scheme turns out to be consistent with somevoltage feedforward schemes proposed previously, with allthe required feedforward signal paths included systemati-cally [4], [5]. SM operation is ensured by the existence

condition and stability of the current loop can be assessedby frequency response analysis. Experimental measurementsare presented to verify the analytical results.

II . T HEORETICAL D ERIVATION

In this section, a general current loop control rule for theboost PFC converter is derived. Conventional linear controlis adopted for the outer voltage loop to regulate the outputvoltage and to generate the half-sinusoidal reference current.Continuous conduction mode (CCM) is considered here as itis commonly used in medium to high power applications. Asimpli ed schematic is shown in Fig. 1

A. Deciding the Control State VariablesThe proposed current controller employs the current error,

x1 , the integral of the current error, x 2 , and the double

C u r r e n t l o o p

c o n t r o l l e r

L D

C

S W

P W M

s i g n a li

L

v

o

R v

i

i

r e f

i

L

v

g s

Fig. 1. A boost PFC converter with a general current controller for theinner current loop.

integral of the current error, x 3 , as the controlled statevariables. The double-integral term is included to furtheralleviate the steady-state error caused by the nite switchingfrequency [3]. The state variables are described as

x1 = i ref iL ; x2 = x1 dt ; x3 = x1 dtdt, (1)where i L denotes the instantaneous inductor current and i ref represent the reference input current, as shown in Fig. 1.

The switching function is de ned as:

u = 12

(1 + sign( S )) (2)

where u represents the logic state of the power switch.

B. Deciding the Sliding Surface

The state variables trajectory is de ned as a linear combi-nation of the state variables, i.e.,

S = 1 x 1 + 2 x2 + 3 x 3 (3)

where 1 , 2 and 3 represent the sliding mode coef cients.The sliding surface is de ned by setting S = 0 , i.e.,

1 x 1 + 2 x 2 + 3 x 3 = 0 . (4)

Considering that the boost converter is operating in CCM,the time differentiation of (1) gives the dynamic model of the proposed system as

x 1 = ddt

i ref 1L

(vi uv o ); x2 = x 1 ; x 3 = x 2 , (5)

where u = 1 u represents the inverse logic of u, and videnotes the recti ed input voltage. Under the invariance of SM control, S = 0 and S = 0 . Thus,

S = 1 x 1 + 2 x 2 + 3 x 3 = 0 . (6)

978-1-4244-1668-4/08/$25.00 2008 IEEE


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L D

C

S W

P W M

s i g n a l

ir e f

i L

v

o

R v

i

P W M

v

i

S M - d e r i v e d n o n - l i n e a r c u r r e n t c o n t r o l l e r

vc

vo

vr a m p

P I

L

i L

vg s

dt

1

3

1

2

dt

d

Fig. 2. A boost PFC converter with the current loop employing the proposedgeneral control rule.

C. Synthesizing the Control Signal by Equivalent Control

Under the equivalent control, the logic signal u is substi-tuted by the continuous signal u eq , such that

0 < u eq < 1. (7)

Substituting (5) and (6) into (7) gives

1di ref

dt

1L

(vi u eq vo ) + 2 (i ref iL )

+ 3 (i ref iL )dt = 0 . (8)Solving (8) for u eq gives

u eq = 1vo Ldi ref

dt + vi LK 1 x 1 LK 2 x 2 (9)

where

K 1 = 2 1

; K 2 = 3 1

.

Equation (9) is a general control rule for the boost PFCconverter and is illustrated in Fig. 2.

III . E XISTENCE C ONDITION

For sliding mode operation to occur, the existence condi-tion has to be satis ed, i.e., lim S 0 S S < 0.For S > 0, i.e., u = 1 , u = 0 and S < 0, we have

1di ref

dt

1L

vi + 2 x 1 + 3 x2 < 0. (10)

For S < 0, i.e., u = 0 , u = 1 and S > 0, we have

1di ref

dt

1L

(vi vo ) + 2 x 1 + 3 x 2 > 0. (11)

The derivative of the reference current can be obtained as

di ref dt

= 2V 2ref

RV id sin(t )

dt =

2V 2ref RV i

cos(t ), (12)

where is the line frequency in rads 1 . Here we assumevo = V ref and the converter has a very high ef ciency with 1.

Substituting (12) into (10) and (11), and re-arranging, theexistence condition becomes

K 1 x1 + K 2 x 2 < 2V 2ref RV i

cos(t ) + V iL

sin(t )

K 1 x1 + K 2 x 2 > 2V 2ref RV i

cos(t ) + V iL

sin(t ) V ref

L .

(13)

In the right hand side of (13), x 1 is the current error which isbasically the inverted function of the inductor current rippleand x2 is the integral of the current error. The right handsides of (13) de ne the upper and lower boundaries for theampli ed current error and its integral. Thus, satisfactionof the existence condition can be veri ed numerically byplotting the existence condition boundaries along with thecurrent error and its integral for a recti ed line cycle.

IV. F REQUENCY R ESPONSE A NALYSIS

Under the assumption of ideal sliding mode operation,the current loop behavior is con ned to the sliding surface

S = 0 de

ned in (4) and the dynamics of the system can bedescribed by S = 0 in (6). Re-arranging (6) givesdi Ldt

= di ref

dt + K 1 (i ref iL ) K 2 (i ref iL )dt. (14)

Due to the linear nature, (14) can be readily expressed ins -domain as

si L = si ref + ( i ref iL ) K 1 + K 21s

. (15)

The block diagram of (15) is depicted in Fig. 3 (a), which canbe manipulated into a general form as shown in Fig. 3 (b).Upon inspection of Fig. 3 (b), the round-trip (open-loop) loopgain is obtained as

G (s ) = 1s

K 1 + K 21s

=K 2 1 + K 1K 2 s

s 2 , (16)

which is a second-order system with dc gain of K 2 and azero at K 2K 1 rads

1 . In practice, the round-trip loop gain is

G (s) = G (s ) H (s), (17)

i

r e f

i

L

i

L

s

1

sK K 1

21 s

(a)

i

r e f i

L

i

L

sK K s

sK K

1

1

21

21

ssK K s 11

21

(b)

Fig. 3. (a) Block diagram of the current loop under the general controlrule; (b) current loop block diagram in general form.


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P

W

M

v

i

V m

V r a m p

G 1

s

i

m o

R

l

G 2

v

c

v

c a

v

g s

(a)

P

W

M

v

i

V

m

V

r a m p

G

1

s

im o

R l

G

2

V

m

v

c

v

c a

v

g s

(b)

Fig. 4. Implementation of feedforward control for (a) a LEM controller;(b) a TEM controller.

where H (s ) accounts for the non-ideal properties of thepower converter and the practical op-amps, such as the non-ideal integral and the limited dc gain, which will be discussedin detail in the following section.

V. I MPLEMENTATION

The resulting general control rule is readily implementedby discrete components as an individual current controller.Alternatively, it can be incorporated in standard PFC con-trollers, such as UC3854 and UCC3817, as an extra feed-forward control. To formulate the feedforward control, thecurrent variables i L i ref in (9) is replaced by the equivalentvoltages across the sensing resistance R s iL R l imo , whereimo is the current generated by the practical multiplier.The instantaneous output voltage vo is approximated by theconstant reference voltage V ref .

In doing so, all the integrators in (16) are non-ideal andthus the rst two poles are non-zero. Here we denote thefrequencies of the rst two poles by f p1 and f p2 . Besides,the dc gain is limited by the open-loop gain A o of practicalop-amps. For feasible implementation, a low-pass lter withpole frequency f p3 is inserted for ltering noise of frequencyhigher than half the converter switching frequency. Therefore,the practical loop gain transfer function becomes

G (s) =Ao 1 + K 1K 2 s

1 + s2f p 1 1 + s

2f p 2 1 + s

2f p 3

, (18)

where f p1 and A o can be obtained analytically based on theop-amps open-loop characteristics provided in the datasheet;f p2 is approximately calculated by r l + R sL rads

1 ; and r l andR s are the damping resistance and current sensing resistanceof L respectively. Experimental frequency responses mea-sured from our circuit (Figs. 5, 6 and Table I) give f p1 10 Hz, f p2 59 Hz, f p3 64 kHz and A o 90 dB. Basedon (18), Bode plots can be obtained by drawing asymptotesmanually or using softwares such as Matlab. Therefore, theSM coef cients can be selected by the desired crossoverfrequency and phase margin, as in the conventional Bodeplots design approach used in linear controllers [6].

1 m H

I R F L 4 6 0

L

o

a

d

0 . 0 1 F

5 0 0 H z

1 1 0 V r m s

0 . 2 5

2 2 0 F

S T W 2 6 N M 6 0

L

R

s

C

v

o

v

i

P F C C o n t r o l

C i r c u i t r y

D

S W

R

v

g s

Fig. 6. Schematic of the boost PFC converter.

The equivalent control signal for leading edge modulation(LEM) is denoted by u eq and is given by

u eq = 1

V ref R s LR l di mo

dt + R s vi LK 1 (R l imo R s iL ) LK 2 (R l imo R s iL )dt . (19)

The control voltage vc is obtained by multiplying (19) with

the amplitude of the ramp function V m , giving

vc = V m

V ref vi

V m LR lV ref R s

di modt

LK 1V m

V ref R s(R l imo R s iL )

LK 2V m

V ref R s (R l imo R s iL )dt. (20)On the other hand, the current ampli er output of a LEMcontroller, e.g. UCC3817, can be expressed as

vca = ( R s iL R l imo )H ca , (21)

where H ca is the transfer function of a type 2 ampli er, whichcan be expressed in the s-domain as

H ca = Rf

R l+ 1

R l C z s, (22)

where R f , R l and C z are the component values shown inFig. 5. Here the high-frequency pole for noise ltering isignored. If we design the component values such that

R f R l

= V m LRsV ref

K 1 ; 1R l C z

= V m LRsV ref

K 2 , (23)

the general control rule can be implemented as a kind of feedforward control scheme for a LEM controller. Simplify-ing (20) with consideration of (21)(23) gives

vc = G 1 vi

sG 2 R l imo + vca , (24)where G1 = V mV ref and G2 =

V m LV ref R s . The resulting feedfor-

ward control is illustrated in Fig. 4(a) and it is consistentwith that proposed in [5].

Alternatively, the general control rule is readily appliedto a trailing edge modulation (TEM) controller. The currentampli er output of a TEM controller can be represented by

vca = ( R l imo R s iL )(1 + H ca ) (R l imo R s iL )H ca .(25)

Following a similar procedure as that for LEM, the feed-forward control scheme for a TEM controller can be obtained

as vc = V m G 1 vi + sG 2 R l imo + vca , (26)

which is illustrated in Fig. 4 (b).


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U C C 3 8 1 7

2 1 . 5 k

1 2 0 p F

6 2 0 k

1 0 0 F

2 . 2 F

1 . 8 k

1 0 k

4 . 7 F

1 5 0 n F

0 . 0 1 F5 6 0 p F

5 6 0 p F

4 5 3 2 9

1 6

7

11 21 31 4

6

8

3 . 9 k

V c c

1 8 V

v

i

vO

V

r e f

R l

R m o

5 1 0 k

1 1

1 N 5 8 1 9

R

a c

R v i

R s

0 . 2 5

R v d

5 0 k

1 1 . 3 k

1 N 5 8 1 91 . 8 n F

1 N 5 8 1 9

6 2 0 k

vo

1 0 k

2 . 2 F

1 0 0 k

V

r a m p

1 0

vc a o

3 . 9 k

1 5

V r e f

+ 1 5 V

- 1 5 V

L M 3 1 1

M C 3 4 1 5 2

d r i v e r

- 1 5 V

L M 3 1 8

1 0 k

1 0 k

1 0 k

1 0 k

+ 1 5 V

- 1 5 V

L M 3 1 8

1 0 0 k

1 k

+ 1 5 V

- 1 5 V

L M 3 1 8

1 0 k

1 0 k

1 k

+ 1 5 V

- 1 5 V

L M 3 1 8

1 0 k

1 0 k

1 k

+ 1 5 V

- 1 5 V

L M 3 1 8

1 0 k

1 0 k

1 0 k

+ 1 5 V

1 0 0 n F

1 n F

1 k

1 n F

5 1 0 k

1 0 k3 0

k

vc a o

v

c

V

r a m p

1 0 0

1 0 k

1 0 k

1 0 0

v

g s

R f

C z

C p

v

c

Fig. 5. Schematic of the PFC control circuitry.

VI. E XPERIMENTAL V ERIFICATIONS

Two boost PFC converters are constructed to verify theproposed current control rule. One uses the general controlrule, which is implemented as feedforward control usinga standard LEM controller UCC3817. Another adopts theconventional ACM control using the same controller IC. Thecomplete schematics of the feedforward control are shownin Figs. 5 and 6. The key circuit parameters are shown inTable I.

Comparing Figs. 7 (a) with (b), under line frequency f of 800 Hz, severe zero-crossing distortion is observed underconventional ACM control. Under the general control rule,

v

i

i

L

(a)

v i

i

L

(b)

Fig. 7. Inductor current waveforms (500mA/div) resulting from (a) theconventional ACM control and (b) the proposed general control rule. f =800 Hz.

TABLE I

C IRCUIT PARAMETERS USED IN THE EXPERIMENTS

Parameters of PFC stage ValuesRecti ed line voltage V i , rms 110 VNominal line frequency f 500 HzReference output voltage V ref 270 VSwitching frequency f sw 100 kHzInductance L 1 mHOutput capacitance C 220 FNominal resistive load R 1.2 k

low harmonic distortion is observed in the input current.Fig. 8 (a) shows the measured power factor PF of thetwo converters as f varies from 50 Hz to 800 Hz. UnderACM control, PF drops to around 0.98 at f = 800 Hz,while PF under the proposed control is maintained above0.994. Likewise, as the input voltage V i , rms is increased from70 Vrms to 140 Vrms, under ACM control, PF drops by 0.03to around 0.97, while PF of the converter using the proposedcontrol only reduces for less than 0.01, as shown in Fig. 8

(b). This veri es the robustness of the general control againstthe variation of line frequency and line voltage.

Under the general current control, frequency responsesof the current loop are measured and plotted along withthe theoretical Bode plots in Fig. 9 (a). Here we intend todemonstrate an unstable current loop and the SM coef cientsare chosen such that the phase margin is less than 10 . Themeasured frequency responses generally agree with the theo-retical Bode plots and an oscillation at a frequency coherentto the crossover frequency is observed in the inductor currentwaveform shown in Fig. 9 (b). The small discrepancies can beattributed to the tolerance of the compensation componentsand the presence of parasitic components in the powerconverter, which have not been taken into account in ourderivation.


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0.975

0.98

0.985

0.99

0.995

1

0 200 400 600 800 1000

P o w e r

f a c t o r , P

F

Line frequency, f [Hz]

SM-based general controlACM control

(a)

0.97

0.975

0.98

0.985

0.99

0.995

1

60 80 100 120 140 160

P o w e r

f a c t o r , P

F

Input voltage V i, rms [V]

SM-base general controlACM control

(b)

Fig. 8. PF measured over the variations of (a) line frequency and (b) inputvoltage.

VII. C ONCLUSION

In this paper, we have presented the derivation, design andexperimental evaluation of a general current control rule forthe boost PFC converter. The resulting control rule achieveslow input current distortion against the variation of linecondition. It takes a common form of PWM control andcan be either implemented as an individual controller, orfeedforward control applied to a standard PFC controller. SMoperation can be ensured by the compliance of the existencecondition. The current loop gain is deduced from the slidingsurface equation. Stability and performance of the systemcan be assessed by Bode plots, and based on that the SM

-60

-40

-20

0

20

40

60

100 1000 10000 100000-360

-315

-270

-225

-180

-135

-90

-45

0

M a g n i

t u d e [ d B ]

P h a s e

[ d e g r e e ]

Frequency [Hz]

K1 = 1.9 104

K2 = 4.0 109

Gain

Phase

Experimental resultsTheoretical results

(a)

(b)

Fig. 9. (a) Bode plots of the current loop with an unstable set of SMcoef cients; (b) the corresponding inductor current waveform (500 mA/div).K 1 = 1 .9 10 4 , K 2 = 4 10 9 .

coef cients can be designed in conjunction with the existencecondition. Experimental measurements generally agree withthe analytical results. The alleviation of the zero-crossingdistortion and the robustness of the proposed control havebeen veri ed.

R EFERENCES

[1] J. Sun, On the zero-crossing distortion in single-phase PFC converter, IEEE Trans. Power. Electron. vol. 19, no. 3, pp. 685692, May 2004.

[2] S. C. Tan, Y. M. Lai, C. K. Tse, and K. H. Cheung , A xed-frequencypulsewith modulation based quasi-sliding-mode controller for buckcontrollers, IEEE Trans. Power. Electron., vol. 20, no. 6, pp. 13791392, Nov 2005.

[3] S. C. Tan, Y. M. Lai and C. K. Tse, Indirect sliding mode controlof power converters via double integral sliding surface, IEEE Trans.Power. Electron., vol. 23, no. 2, pp.600 611, Mar. 2008.

[4] S. Wall and R. Jackson, Fast controller design for single-phase power-factor-correction systems, IEEE Trans. Ind. Electron., vol. 44, no. 5,

pp. 654660, 1997.[5] M. Chen and J. Sun, Feedforward current control of boost single-phase PFC converters, IEEE Trans. Power. Electron., vol. 21, no. 2,pp. 338345, Mar 2006.

[6] L. H. Dixon, High power factor switching pre-regulator design opti-mization, Unitrode Power Supply Design Seminar Manual SEM700,1990.

sm pfc boost tan(11)

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