3@1s with high pf

7/28/2019 3@1s with high pf

http://slidepdf.com/reader/full/31s-with-high-pf 1/8

Three-Phase Single-Switch Boost PFC Converter with High Input Power Factor

Kai Yao1, Xinbo Ruan

1,2, Senior Member, IEEE, Chi Zou

1, Zhihong Ye

3

1

Aero-Power Sci-tech Center, College of Automation Engineering, Nanjing University of Aeronautics andAstronautics, Nanjing 210016, China (e-mail: [email protected], [email protected])2 College of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074,

China (e-mail: [email protected])3 Lite-On Technology Power SBG ATD-NJ R&D Center, Nanjing 210019, China (e-mail: [email protected])

Abstract— Three-phase single-switch boost

power-factor-correction (PFC) converter features zero-current

turn-on for the switch, no reverse recovery in diode, constant

frequency operation, simple control and low cost, and it is

suitable for the low-to-medium power conversions. However,

the input power factor (PF) is relatively low when the duty cycle

is constant in the whole line cycle. This paper analyzes the

operation principle of the three-phase single-switch DCM boostPFC converter, and the expressions of the input current and PF

are derived, based on which, a variable duty cycle control is

proposed so as to improve the PF and reduce the input current

harmonics. A method of fitting the duty cycle is further

proposed for simplifying the circuit implementation. Besides a

higher PF and lower input current harmonics, the proposed

variable duty cycle control achieves a lower output voltage

ripple and a higher efficiency over the constant duty cycle

control. A 3kW prototype has been built and tested in the lab,

and the experimental results are presented to verify the

effectiveness of the proposed method.

Index Terms — Power-factor-correction, Three-phase,

Discontinuous-current-mode, Pulse width modulation, Variable

duty cycle control.

I I NTRODUCTION Power factor correction (PFC) converters have been

widely used in ac-dc power conversions to achieve high power factor (PF) and low harmonic distortion. Themethods of achieving PFC can be classified into activeand passive types. Compared with passive PFC converter,active one can achieve a high PF and a small size [1].

There are different topologies and control strategies for implementing active three phase PFC converters [2].Specifically, three- phase six-switch boost PFC converter can achieve unit PF and very low harmonic distortion of

the line current. The converter is mainly used in high andmedium power applications due to the many number of switches, the complexity of the control and high cost [2].

Three-phase single-switch boost PFC converter operates indiscontinuous current mode (DCM) to achieve PFC.Featuring zero-current turning on for the switch, noreverse recovery of diode, simple control and low cost,the converter is mainly used in medium and low power,cost-sensitive applications [3]. However, If the duty cycle

is constant in a line cycle, the average value of theinductor current in a switching cycle is not sinusoidal.The PF is low and the input current contains rich 5th and7th harmonics.

To date, a number of techniques for the input currentharmonics reduction of three-phase single-switch boostPFC converter have been proposed [4-9]. Adding the 5 th harmonic trap in the input side can reduce the 5 th harmonic in the input current and improve the PF.However, the efficiency decreases as high circulatingcurrent flows in the loop consisting of the trap filter andthe ac source [5]. Variable switching frequency controlcan reduce the input current harmonic distortion.However, the inductor and EMI filter design is a littlecomplicated [6-7]. Injecting 6th harmonic into the dutycycle is an effective technique [4, 8-9]. The proposed 6th harmonic injection circuit, which employs a band-passfilter, has a severe phase-shift problem and the effect isnot so ideal [4]. Ref. [8] offers another solution, wherethe harmonic injection is realized with the out voltage

ripple. The method requires complicated and expensiveadditional circuitry such as phase-detecting and phase-locking circui ts to properly synchronize theinjected signal with the input current. The solution proposed in [9] is that the duty cycle is modulated bysubtracting a fraction of the ac component of the rectifiedinput voltage from the error signal of the output voltagecontrol loop. However, the input current harmonicsdistortion can not be achieved at minimum value over theinput voltage range.

This paper analyzes the operation principle of thethree-phase single-switch boost PFC converter, based onwhich, a variable duty cycle control is proposed so as toimprove the PF. A method of fitting the duty cycle is further proposed for simplifying the control. Besides a higher PF andlower input current harmonics, the proposed variable dutycycle control achieves a lower output voltage ripple and ahigher efficiency over the constant duty cycle control.

II. OPERATION PRINCIPLE OF THREE-PHASE BOOST PFC

CONVERTER .

Fig. 1 shows the main circuit of a three-phase

978-1-4244-5287-3/10/$26.00 ©2010 IEEE 2921



single-switch boost PFC converter, where La= Lb= Lc= L. The input voltage is defined as

sina m

v V t ω = (1)

( )sin 2 3b m

v V t ω π = − (2)

( )sin 2 3c m

v V t ω π = + (3)

where V m and ω are the amplitude and angular frequencyof the input voltage.

Fig. 2. Input voltage waveform.

0

ia

v gs

0

D y

icib iap1

t/T s

t/T s

D R1 1

icp2

ibp2

icp1

ibp1

D R2

Fig. 3. Inductor current waveform in a switching cycle during [0, π/6].

Obviously, the voltage and current meets the following:0a b c

v v v+ + = (4)

0a b ci i i+ + = (5)

Fig. 2 shows the input voltage waveform. A line cycleof 2π can be divided into 12 intervals. In each interval of π/6, the voltage direction and relative size of the three phase input voltage is the same. Fig. 3 shows the inductor current waveform in a switching cycle.

(1) Switching mode 1

When Qb turns on, D1、 D5、 D4 conducts. Fig. 4(a) showsthe equivalent circuit, from which it can be seen that

a a b b c cv L di dt v L di dt v L di dt − = − = − (6)

From (4) to (6), the rising rate of the inductor current is

a adi dt v L=

b bdi dt v L=

c cdi dt v L= (7)

When Qb turns off, the inductor current reaches its peak value, which is

1

a y sa

ap y s

v D T dii D T

dt L= = (8(a))

1

b y sbbp y s

v D T dii D T

dt L= = (8(b))

1

c y sccp y s

v D T dii D T

dt L= = (8(c))

where D y is the duty cycle and T s is the switching cycle.


When Qb turns off, Db conducts. Fig. 4(b) shows theequivalent circuit, from which it can be seen that

a b

a b o

di div v V L

dt dt

⎛ ⎞− − = −⎜ ⎟

⎝ ⎠ (9(a))

c bc b o

di div v V L

dt dt

⎛ ⎞− − = −⎜ ⎟

⎝ ⎠ (9(b))

where V o is the output voltage.

From (4), (5) and (9), the falling rate of the inductor current is

( )3a a odi dt v V L= − (10(a))

( )2 3b b odi dt v V L= + (10(b))

( )3c c odi dt v V L= − (10(c))

ia reaches zero first, the duty cycle corresponding tothis falling period is

1

1

31

3ap a

R y

a s o a

i v D D

di dt T V v= =

−(11)

When ia reaches zero, ib and ic is

( )2 1 1

2

3

y so b ab

bp bp R so a

D T V v vdi

i i D T dt V v L

+= + =

− (12(a))

( )2 2

2

3

y so b a

cp bp

o a

D T V v vi i

V v L

+= − = −

− (12(b))


When ia reaches zero, The equivalent circuit is shown inFig. 4(c), from which it can be seen that

Fig. 1. Main circuit of three-phase single-switch boost PFC converter .

o

Lava

vb

vc

ia

ib

ic

Lb

Lc

V o+ _

(a)

switching mode 1. ( b) switching mode 2. (c) switching mode 3.Fig. 4. Equivalent circuit of switching mode 3.

2922



c b

c b o

di div v V L

dt dt

⎛ ⎞− − = −⎜ ⎟

⎝ ⎠ (13(a))

b ci i= − (13(b))

From (13), the falling rate of the inductor current is

2b c o b c

di di V v v

dt dt L

+ −= − = (14)

From (12) and (14), the duty cycle corresponding tothis falling period of ib and ic is

( )

( )( )

2

2

2 21

3

bp o b a

R y

b s o b c a o

i V v v D D

di dt T V v v v V

+= =

+ − − (15)


During this period, the inductor currents of three phase areall zero, the output capacitor supply the energy to the output.

From Fig.3, during [0, π/6], the average inductor currentin a switching cycle can be derived as

( ) ( ) 1

_ 1 1 0

sin

2 3 3sin

ap

a av y R

i t i t D D I

t

ω

ω

= + =

−

(16(a))

( ) ( )( )

( )( )

_ 1 1 1 1 2 2 2

0

1

2

sin 2 sin 3

3 3sin cos

b av y bp R bp bp R bpi t D i D i i D i

t M t I

M t M t

ω ω π

ω ω

⎡ ⎤= + + +⎣ ⎦

− +=

− −

(16(b))

( ) ( )

( )

( )( )

_ 1 1 1 1 2 2 2

0

1

21

cos 6 sin 22

3 3sin cos

c av y cp R cp cp R cpi t D i D i i D i

M t t

I M t M t

ω π ω

ω ω

⎡ ⎤= + + +⎣ ⎦

+ −

=

− −

(16(c))

where ( )20 2

y o s I D V Lf = ， ( )3

o mV V = , f s = 1/T s is the

switching frequency.The operation principle of other interval is similar to that

of [0, π/6], the average inductor current in a switchingcycle can also be derived. During [0, π], the averageinductor current of phase a is

( ) _ 0

11,2, 6

6 6a av n

n ni t k I t nπ ω π

−⎛ ⎞= ≤ ≤ =⎜ ⎟

⎝ ⎠" (17)

where

( ) ( )1 6

sin

3 3sin

t k t k t

M t

ω ω ω

ω

= =

−

(18(a))

( )2

1 2sin sin 2

2 3

23 3sin sin

3 6

M t t

k t

M t M t

π ω ω

ω

π π ω ω

⎛ ⎞+ −⎜ ⎟

⎝ ⎠=

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞− + − +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥

⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

(18(b))

( )3sin sin 2 3

23 3sin sin

3 6

M t t k t

M t M t

π ω ω

ω

π π ω ω

⎛ ⎞+ +

⎜ ⎟⎝ ⎠=⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞

+ + − +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

(18(c))

( )4

sin sin 23

53 3sin sin

3 6

M t t

k t

M t M t

π ω ω

ω

π π ω ω

⎛ ⎞− −⎜ ⎟

⎝ ⎠=

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞− + + +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥

⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

(18(d))

( )5

1 2sin sin 2

2 3

53 3sin sin

3 6

M t t

k t

M t M t

π ω ω

ω

π π ω ω

⎛ ⎞− +⎜ ⎟

⎝ ⎠=

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞+ + + +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥

⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

(18(e))

According to the above analysis, Fig. 5 shows the

instantaneous waveform, the peak value envelope and theaverage value of ia. It can be seen that the shape of the peak inductor current is sinusoidal, however the shape of the average inductor current is not sinusoidal and there isdistortion in it. The normalized average value of ia (withthe base of I 0) is plotted in Fig. 6, from which, it can beseen that the shape of the average inductor current is onlydependent on M , and the bigger M is, the closer tosinusoidal the current shape is.

From (1) and (17), the average input power of phase ais derived as

4 _ _ 0

0 6 3 21 2 30

6 3

1

4

2 sin sin sin

lineT

in a a a av

line

m

P v i dt T

I V k td t k td t k td t

π π π

π π ω ω ω ω ω ω

π

=

⎛ ⎞= + +⎜ ⎟

⎝ ⎠

∫

∫ ∫ ∫(19)

where T line is the line cycleSubstitution of (17) and (19) into the PF leads to

3 261 2 30

_ 6 3

2 2 23 26 _ 1 2 30

6 3

2sin sin sin

2

in a

ma rms

k td t k td t k td t P

PF V

I k d t k d t k d t

π π π

π π

π π π

π π

ω ω ω ω ω ω

π

ω ω ω

⎛ ⎞+ +⎜ ⎟⎜ ⎟

⎝ ⎠= =

+ +

∫ ∫ ∫

∫ ∫ ∫

(20)

Fig. 5. Inductor current waveform of phase a in a

half line cycle. Fig. 6 Normalized average input current waveform

of phase a in a half line cycle.

Fig. 7. Relationship between the input PF

and M.

2923



where I a_rms is the rms value of the current of phase a.According to (20), the input PF is plotted as shown in

Fig. 7. It can be seen that the larger M is, the higher thePF is. When the output voltage is 750 V, the PF is only0.971 at the input voltage of 264 VAC, the input currentcontains rich harmonics. New control method should be proposed for the PF improvement.

III. VARIABLE DUTY CYCLE CONTROL TO IMPROVE I NPUT

POWER FACTOR

A. Variable Duty Cycle for PF=1

By observing (17), in order to achieve unity PF, theduty cycle should be variable as

( ) 0

1sin / 1,2, 6

6 6 ya n

n n D t D t k t nω π ω π

−⎛ ⎞= ≤ ≤ =⎜ ⎟

⎝ ⎠" (21)

where Do is a coefficient, which will be explained later.Substitution of (21) into (17) leads to

( )20

_ sin2

o

a av

s

D V i t t

Lf ω = (22)

From (1) and (22), the average input power is

( ) ( )204

_ 0

313

4 4

lineT

m o

in a a av o

line s

V V D P v t i t dt P

T Lf = ⋅ = =∫ (23)

From (23), D0 can be obtained as

0 23

s o

m o

Lf P D

V V = (24)

The ideal duty cycle which makes ib and ic sinusoidalcan also be derived as

( ) '0

2 1sin / 1,2, 6

3 6 6 yb n

n n D t D t k t nω π π ω π

−⎛ ⎞ ⎛ ⎞= − ≤ ≤ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠" (25)

( )''

0

2 1

sin / 1, 2, 63 6 6 yc n

n n

D t D t k t nω π π ω π

−⎛ ⎞ ⎛ ⎞= + ≤ ≤ =

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠"

(26)where

( ) ( ) ( ) ( )' '1 3 2 4/ 3 / 3k t k t k t k t ω ω π ω ω π = + = +

(27)( ) ( ) ( ) ( )' '3 5 4 6/ 3 / 3k t k t k t k t ω ω π ω ω π = + = +

( ) ( ) ( ) ( )' '5 1 6 22 / 3 2 / 3k t k t k t k t ω ω π ω ω π = − = −

( ) ( ) ( ) ( )'' ''1 5 2 62 / 3 2 / 3k t k t k t k t ω ω π ω ω π = + = +

( ) ( ) ( ) ( )'' ''3 1 4 2/ 3 / 3k t k t k t k t ω ω π ω ω π = − = −

( ) ( ) ( ) ( )'' ''5 3 6 4/ 3 / 3k t k t k t k t ω ω π ω ω π = − = −

Fig. 8. Ideal duty cycle that ensures PF =1.

From (21), (25) and (26), D ya, D yb and D yc can be plotted,as shown in Fig. 8. As seen, the unified duty cycle whichensures the PF of three phases to be unity can not berealized. However, a certain duty cycle which enables thePF to be near unity can be expected. The coefficient j1, j2 and j3 are introduced here to generate the duty cycle:

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

1 _ 2 _ 3 _

3 _ 2 _ 1 _

2 _ 3 _ 1 _

2 _ 1 _ 3 _

3 _ 1 _ 2 _

1 _ 3 _

0 / 6

/ 6 / 3/ 3 / 2

/ 2 2 / 3

2 / 3 5 / 6

y a y b y c

y a y b y c

y a y b y c

y

y a y b y c

y a y b y c

y a y b

j D t j D t j D t t

j D t j D t j D t t j D t j D t j D t t

D t j D t j D t j D t t

j D t j D t j D t t

j D t j D t

ω π

π ω π

π ω π

π ω π

π ω π

+ + ≤ ≤

+ + ≤ ≤

+ + ≤ ≤=

+ + ≤ ≤

+ + ≤ ≤

+ ( ) ( )2 _ 5 / 6 y c

j D t t π ω π

⎧⎪

⎪⎪⎪⎨⎪⎪⎪⎪ + ≤ ≤⎩

(28)

When the output voltage is 750 V and the input voltageis 264 VAC, M =1.16. From (1), (17), (20), (21), (25), (26),(28) and M =1.16, we can plot the relationship between PFand j1, j2, as shown in Fig. 9. As seen, j1=0 enables theoptimal PF with a certain value of j2. When j1=0, therelationship between PF and j2 can be plotted, as shown inFig.10, from which it can be seen that the highest PF can beobtained when j2=0.7, j3=0.3.

B. The Fitting Duty Cycle

The duty cycle expressed in (28) is complicated toimplement because a few of multiplier, divider andsquare root extractor are needed, it is necessary to seek afunction that fits (28), which can be more easilyimplemented. Take [0, π/6] for example, defining y=cosω t , ω t ∈[0, π/6], the ideal duty cycle of phase b and c in thisinterval can be rewritten as

k1 0:=

0.87 0.90 0.93 0.96 0.99

0.985

0.986

0.987

0.988

0.989

0.990

0.966 y0

Fig. 9. Surface of the input PF as the function of j1

and j2 Fig. 10. Relationship between PF and j2 when

j1=0 Fig. 11. Relationship between PF and y0 when M =1.16.

2924



( )( )( )( )2 2

_ 1 0 2 2

1 3 3 3 1

4 1 1 3 y b

y M y M y D y D

y y M y My

− − − − − −

=

− − − −

(29(a))

( )( )( )( )2 2

_ 1 0 2 2

1 3 3 3 1

3 1 2 1 y c

y M y M y D y D

My M y y y

− − + − − −

=

− − − −

(29(b))

Based on Taylor’s series

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

'

0 0 0

2''0 0 0 0

1 1

2! !

nn

f x f x f x x x

f x x x f x x xn

= + ⋅ −

+ ⋅ ⋅ − + + ⋅ ⋅ − +" "

(30)

Reserving only the first derivative item, (28) isapproximated as

( )

( )

1_ _ 1_ _ 1_

' ' _ 1 0 0 _ 1 0 _ 1 0 0 _ 1 0

1 0

( ) 0.7 0.3 ( )

0.7 ( ) ( ) ( ) 0.3 ( ) ( ) ( )

= 1 ,

y fit y b fit y c fit

y b y b y c y c

D y D y D y

D y y y D y D y y y D y

D h y M y

= +

⎡ ⎤ ⎡ ⎤= + − + + −⎣ ⎦ ⎣ ⎦

⎡ ⎤−⎣ ⎦

(31)

where ( ) ( )' '1 _ 1 0 0 _ 1 0 _ 1 0 0 _ 1 00.7 ( ) ( ) 0.3 ( ) ( ) y b y b y c y c D D y y D y D y y D y= − + − ,

( )( ) ( )

' ' _ 1 0 _ 1 0

0 ' '0 _ 1 0 _ 1 0 0 _ 1 0 _ 1 0

0.7 ( ) 0.3 ( ),

0.7 ( ) ( ) 0.3 ( ) ( )

y b y c

y b y b y c y c

D y D yh y M

y D y D y y D y D y

+=

− + −

The derivation of the fitting duty cycle of other interval is

similar to that of [0, π/6]. The expression of D1 and h( y0, M ) of other interval is the same with [0, π/6], as shown in (32)

( ) ( ) ( )

( ) ( ) ( )

1 0

_

1 0

1 , cos 0 / 6( )

1 , cos / 3 / 6 / 2 y fit

D h y M t t D t

D h y M t t

ω ω π

ω π π ω π

⎧ − ≤ ≤⎡ ⎤⎪ ⎣ ⎦= ⎨

− − ≤ ≤⎡ ⎤⎪ ⎣ ⎦⎩

(32)

Substitution of (32), (17) and (1) into (20) leads to

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

26

0 10

23

0 2

6

22

0 3

3

4261 00

4232 0

6

23

1 , cos sin

21 , cos /3 sin

1 , cos /3 sin

1 , cos

1 , cos /3

1

h y M t k td t

h y M t k td t

h y M t k td t

PF

k h y M t d t

k h y M t d t

k h y

π

π

π

π

π

π

π

π

ω ω ω

ω π ω ω

π

ω π ω ω

ω ω

ω π ω

⎧ ⎫⎪ ⎪⎡ − ⎤⎣ ⎦⎪ ⎪⎪ ⎪⎪ ⎪

+ ⎡ − − ⎤⎨ ⎬⎣ ⎦⎪ ⎪⎪ ⎪⎪ ⎪+ ⎡ − − ⎤⎣ ⎦⎪ ⎪⎩ ⎭

=

⎡ − ⎤⎣ ⎦

+ ⎡ − − ⎤⎣ ⎦

+ −

∫

∫

∫

∫

∫

( ) ( )4

20

3

, cos / 3 M t d t π

π ω π ω ⎡ − ⎤⎣ ⎦∫

(33)

(33) shows that PF depends both on M and y0. Consideringthe worst case of M =1.16, we substitute M =1.16 into (33) and

x y

z

v v

v

⋅

Fig. 13. Control circuit of the variable duty cycle control.

plot the curve of the relationship between PF and y0, asshown in Fig.11. From Fig.11 it can be seen that when y0=0.966, the maximum PF can be obtained.

Substituting y0=0.966 into h( y0, M ), the curve of therelationship between h( y0, M ) and M can be plotted, as shown

in Fig.12. As seen, when 1.16≤ M ≤1.74, ( )1 2 1 M − fits

well with h(0.966, M ). With ω t ∈[0, 2π], (32) is written as( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )

1

1

1

_ 1

1

1 cos / 2 1 0 / 6

1 cos / 3 / 2 1 / 6 / 2

1 cos 2 / 3 / 2 1 / 2 5 / 6

( ) 1 cos / 2 1 5 / 6 7 / 6

1 cos 2 / 3 / 2

y fit

D t M t

D t M t

D t M t

D t D t M t

D t

ω ω π

ω π π ω π

ω π π ω π

ω π π ω π

ω π

⎡ − − ⎤ ≤ ≤⎣ ⎦

⎡ − − − ⎤ ≤ ≤⎣ ⎦

⎡ − − − ⎤ ≤ ≤⎣ ⎦

= ⎡ − − − ⎤ ≤ ≤⎣ ⎦

− + ( ) ( )

( ) ( ) ( )

( ) ( )

1

1

1 1

1 7 / 6 3 / 2

1 cos / 3 / 2 1 3 / 2 11 / 6

1 cos / 2 1 11 / 6 2

11 1

2 1 3 2 3

g g

m o m

M t

D t M t

D t M t

v v D D

M V V V

π ω π

ω π π ω π

ω π ω π

⎧⎪⎪⎪⎪⎪⎪⎨⎪

⎡ − ⎤ ≤ ≤⎪ ⎣ ⎦⎪

⎡ − + − ⎤ ≤ ≤⎪ ⎣ ⎦⎪

⎡ − − ⎤ ≤ ≤⎪ ⎣ ⎦⎩

⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

(34)

where v g is the rectified voltage of the input.

From (34), the fitting duty cycle can be plotted as in Fig.8.The control circuit can be implemented as shown inFig. 13. The input voltage of three phase is sensed through adifferential sampler, and vA=mv g , where m is the voltagesensor gain. R8, D7, C 1 and R9 are the circuit to obtain the peak value of the rectified input voltage, i.e., v B =

m 3 V m. The reasonable resistance selection of the feedforward circuit can enable the voltage of C and D point to

be m(2V o− 3 V m) and m(2V o− 3 V m−v g ) separately. vC, vD

2 2

Fig. 12. Curve fitting of h(0.966, M ). Fig. 14. Comparison of the input PF Fig. 15. Normalized amplitudes of the5 th, 7 th , 11th and 13th harmonics

2925



and the output voltage control loop error signal vEA aresent to the multiplier, and

( )( )

2 31

2 32 3

o m g g

P EA EA

o mo m

m V V v vv v v

V V m V V

− − ⎛ ⎞= = −⎜ ⎟⎜ ⎟

−− ⎝ ⎠ (35)

v P is sent to the PWM comparator and compared with thesaw-tooth carrier, and the desired duty cycle of (34)can be obtained.

IV. ADVANTAGES OF VARIABLE DUTY CYCLE CONTROL

A. Improvement of the Input PF

According to (20) and (33), the input PF with constantduty cycle control and variable duty cycle control aredepicted in Fig. 14. It can be seen that the input PF isgreatly improved with the variable duty cycle control.When the input voltage is 264 VAC, the PF is increasedfrom 0.971 to 0.989.

By Fourier analysis, the harmonics of the input currentcan be obtained as

( ) ( )0

1

( ) cos sin

2

in n n

n

ai t a n t b n t ω ω

∞

=

= + +⎡ ⎤⎣ ⎦∑ (36)

where

( ) ( ) ( )

( ) ( ) ( )

0

0

2cos 0,1,2,

2sin 1,2,3

line

line

T

n in

line

T

n in

line

a i t n t d t nT

b i t n t d t nT

ω ω

ω ω

= = …

= = …

∫

∫ (37)

Substituting (17) and (34) into (36), the harmonics of the input current with constant and variable duty cyclecontrol, respectively, can be figured out, in which thecosine part, the even and the triple sinusoidal part arezero, i.e.,

( )

( )2 3

0 0,1,2,

0 1,2,3

n

n n

a n

b n

= = …

= = ⋅⋅ ⋅、

(38)

The normalized amplitudes of the 5 th, 7th, 11th and 13th

harmonics to the base of the fundamental component areshown in Fig. 15. It should be noted that the negativeamplitude means that the corresponding harmonic has a phase of 180°. Fig. 15 illustrates that with the variableduty cycle control, the 5th and 11th harmonics of the inputcurrent is greatly reduced, the 7th and 13th harmonics is a

little increased, the total harmonics is decreased.

B. Reduct ion of the Inductor Current Ripple

Taking [0, π/6] for example, in order to ensure theconverter operates in DCM, the following conditionshould be met.

1 2 1 y R R

D D D+ + ≤ (39)

With constant duty cycle control, Substitution of (11)

and (15) into (39), yields

1( , ) 1 y D f M t ω ≤ (40)

where

( ) ( )1 , cos f M t M M t ω ω = − (41)

With variable duty cycle control, substitution of (34)into (40), leads to

( )1 2 , 1 D f M t ω ≤ (42)

where

( ) ( ) ( )2 , 1 cos / 2 1 cos f M t M t M M t ω ω ω = − − −⎡ ⎤⎣ ⎦ (43)

A family curves of f 1,2( M , ωt ) over ωt with different M can be plotted, as shown in Fig. 16. As seen, for any M with

1.16≤ M ≤1.74, the maximum value of f 1,2( M , ωt ) occurs atωt =0. Substitution of ωt =0 into (41) and (43), yields

( ) ( )1 ,0 / 1 f M M M = − (44)

( ) ( )2 ,0 2 / 2 1 f M M M = − (45)

Supposing the efficiency of the converter is 100%, i.e. P in _ a= P o/3, so (18), (19) and (34), we have

36 21 2 30

6 3

3 sin sin sin

o s y

o m

P Lf D

V V k td t k td t k td t π π π

π π

π

ω ω ω ω ω ω

=⎛ ⎞

+ +⎜ ⎟⎝ ⎠∫ ∫ ∫

(46)

( )

( )

1 2

610

2

32

6

2

23

3

cos1 sin

2 1

cos /33 1 sin

2 1

cos /31 sin

2 1

s o

o m

Lf P D

t k td t

M

t V V k td t

M

t k td t

M

π

π

π

π

π

π

ω ω ω

ω π

ω ω

ω π

ω ω

=⎧ ⎫⎡ ⎤

−⎪ ⎪⎢ ⎥−⎣ ⎦⎪ ⎪

⎪ ⎪⎡ − ⎤⎪ ⎪+ −⎨ ⎬⎢ ⎥

−⎣ ⎦⎪ ⎪⎪ ⎪

⎡ − ⎤⎪ ⎪+ −⎢ ⎥⎪ ⎪−⎣ ⎦⎩ ⎭

∫

∫

∫

(47)

From (40), (44) and (46), the critical boost inductor withconstant duty cycle control is obtained as

Fig. 16. Relationship between f 1( M , ωt ) and ωt. Fig. 17. Critical inductors over the input

voltage range.

Fig. 18. Normalized instantaneous input power with constant duty cycle control and

variable duty cycle control.

2926



2

36 21 2 30

6 3

3 1sin sin sino m

o s

V V M L k td t k td t k td t

P f M

π π π

π π ω ω ω ω ω ω

π

⎛ ⎞−⎛ ⎞≤ + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∫ ∫ ∫ (48)

From (42), (45) and (47), the critical boost inductor withvariable duty cycle control is obtained as

( )2

610

2

322

6

2

23

3

2 1 cos sin

32 1 cos sin

4 3

2 1 cos sin3

o m

o s

M t k td t

V V L M t k td t

P f M

M t k td t

π

π

π

π

π

ω ω ω

π ω ω ω

π

π ω ω ω

⎧ ⎫⎪ ⎪⎡ − − ⎤⎣ ⎦⎪ ⎪⎪ ⎪⎡ ⎤⎪ ⎪⎛ ⎞

≤ + − − −⎨ ⎬⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎪ ⎪

⎪ ⎪⎡ ⎤⎛ ⎞⎪ ⎪+ − − −⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

∫

∫

∫

(49)

According to the specifications of the converter, whichwill be given in Section V, the critical boos t inductor over the input voltage range with constant duty cycle controland variable duty cycle control are depicted in Fig. 17,from which we choose Lb1 = 125 μH and Lb2 = 155 μH.The increment of the inductor is helpful for the reductionof the peak and rms current of the inductor, switch anddiode. Thus, the conduction loss of the converter can be

reduced, leading to a higher efficiency.

C. Reduct ion of the Output Voltage Ripple

With the base of P o, the normalized instantaneous input power is der ived as

( ) ( )* _ _ _ in a a av b b av c c av ot v i v i v i P = + + (50)

Same with (17), the expression of ib _ av(t ) and ic _ av(t ) canalso be derived. Substitution of ia _ av(t ), ib _ av(t ) and ic _ av(t ) into(50) leads to p*

in _1. Similarly, when variable duty cyclecontrol is employed, from (17), (34) and (52), thenormalized instantaneous input power is derived as p*

in _2 .With the cycle of π/3, the curves of p*

in _1 and p*in _2 during

[0, π/3] are in depicted in Fig. 18 .

When ( )*in t >1, the storage capacitor C o is charged,

and when ( )*in

t <1, C o is discharged. The energy

discharging C o (which equals the charged energy) in thecycle of π/3 with constant duty cycle control and variableduty cycle control are

( ){ } ( )1

_1

* *1 0

2 1 6in

t

line E p t dt T ⎡ ⎤Δ = − ⋅⎣ ⎦∫ (51(a))

( ){ } ( )2

_ 2

* *2 0

2 1 6in

t

line E p t dt T ⎡ ⎤Δ = − ⋅⎣ ⎦∫ (51(b))

respectively, where t 1 and t 2 are the time instants when

( )*in

p t crosses 1 with constant duty cycle control and

variable duty cycle control, respectively. *1 E Δ and *

2 E Δ can also be expressed as2 2

_1 _1

_1*1

1 1

62 2 2 2

6

o o

o o o o

o o o

o line o line

V V C V C V

C V V E

P T PT

Δ Δ⎛ ⎞ ⎛ ⎞+ − −⎜ ⎟ ⎜ ⎟

Δ⎝ ⎠ ⎝ ⎠Δ ≈ =

(52(a))

2 2

_2 _2

_2*2

1 1

62 2 2 2

6

o o

o o o o

o o o

o line o line

V V C V C V

C V V E

P T PT

Δ Δ⎛ ⎞ ⎛ ⎞+ − −⎜ ⎟ ⎜ ⎟

Δ⎝ ⎠ ⎝ ⎠Δ ≈ =

(52(b))

where ΔV o1 and ΔV o2 are the output voltage ripple withconstant duty cycle control and variable duty cyclecontrol, respectively.

2

Fig. 19. Output voltage ripple with constant duty cycle control andvariable duty cycle control.

From (51) and (52), the expressions of ΔV o1 and ΔV o2

are derived as

( ){ }1

_1

*

1 02 1in

t

o o o oV P p t dt C V ⎡ ⎤

Δ = − ⋅⎣ ⎦∫ (53(a))

( ){ }2

_ 2

*2 0

2 1in

t

o o o oV P p t dt C V ⎡ ⎤Δ = − ⋅⎣ ⎦∫ (53(b))

According to the specifications of the converter, whichwill be given in Section V, ΔV o1 and ΔV o2 can be figuredout as shown in Fig. 19. It can be seen that with variableduty cycle control, the output voltage ripple is reducedcompared to constant duty cycle control.

V. EXPERIMENTAL VERIFICATION

In order to verify the validity of the proposed variableduty cycle control, a prototype has been built and tested inthe lab. The specifications of the pro totype are as follows:·input voltage: vin = 220V±20% /50Hz;·output voltage: V o = 750 VDC;·output power: P o = 3 kW;·switching frequency: f s = 30 kHz. Figs. 20 and 21 show the experimental waveforms of

the input voltage, input current, boost inductor currentand output voltage with constant duty cycle control andvariable duty cycle control at 176 VAC, 220 VAC and264 VAC input respectively. It can be seen that comparedto constant duty cycle control, the input current is closeto sinusoidal shape with variable duty cycle control.

Figs. 22 to 24 show the measured input PF, outputvoltage ripple and efficiency curve with different inputvoltage, from which it can be seen that the PF isimproved, the output voltage ripple is reduced and theefficiency is increased with the variable duty cyclecontrol.

VI. CONCLUSIONS

The input PF of three-phase single-switch boost PFCconverter is relatively low when the duty cycle is constant in

2927



the whole line cycle, especially at high input voltage. Aimingat this, a variable duty cycle control is proposed so as toimprove the PF and reduce the input current harmonics. Amethod of fitting the duty cycle is further proposed for simplifying the circuit implementation. Besides a higher PFand lower input current harmonics, the proposed variableduty cycle control achieves a lower output voltage ripple anda higher efficiency over the constant duty cycle control.

R EFERENCES [1] O. Garcia, J. A. Cobos, R. Prieto, P. Alou, and J. Uceda, “Power factor

correction: A survey,” in Proc. IEEE PESC 2001, pp. 8-13.[2] F. C. Lee, D. Boroyevich, “Power factor correction circuits topologies

and control,” in Proc. IEEE APEC 1993, pp. 335-341.[3] A. R. Prasad, P. D. Ziogas, S. Manias, “An active power factor

correction technique for three-phase diode rectifiers,” IEEE Trans. Power Electron., vol. 6, no. 1, pp. 83-92, Jan. 1991.

[4] Q. Huang, F. C. Lee, “Harmonic reduction in a single-switch,three-phase boost rectifier with high order harmonic injected PWM,”

Proc. IEEE PESC 1996, pp. 1266-1271. [5] Y. Jang, M. M. Jovanović, “A comparative study of single-switch

three-phase high-power-factor rectifiers,” IEEE Trans. Ind. Appl .,vol. 34, no. 6, pp. 1327-1334, Nov. 1998.

[6] L. Simonetti, J. Sebastian, J. Uceda, “Single-switch three-phase power factor under variable switching frequency and discontinuous inputcurrent,” Proc. IEEE PESC 1993, pp. 657-662.

[7] J. W. Kolar, H. Ertl, F. C. Zach, “Space vector-based analytical

analysis of the input current distortion of a three-phasediscontinuous-mode boost rectifier system,” Proc. IEEE PESC 1993, pp. 696-703.

[8] J. Sun, N. FrÖhleke, H.Grotstollen, “Harmonic reduction techniquesfor single-switch three-phase boost rectifiers,” Proc. IEEE Industry

Applications Society Annu. Meeingt 1996, pp. 1225-1232.[9] Y. Jang, M. M. Jovanović, “A novel robust harmonic injection method

for single-switch three-phase discontinuous- conduction-mode boostrectifiers,” IEEE Trans. Power Electron., vol. 13, no. 5, pp. 824-834,Sep. 1998.

(a) (b) (c)

Fig. 20. Experimental waveforms of input voltage, input current, inductor current and output voltage with constant duty cycle control: (a) @ 176VAC input; (b) @ 220 VAC input; (c) @ 264 VAC input.

va: [200 V/div]

i La: [20 A/div]

ia: [20 A/div]

vo: [10 V/div]Time: [2 ms/div]

va: [200 V/div]

i La: [20 A/div]

ia: [10 A/div]

vo: [10 V/div]Time: [2 ms/div]

(a) (b) (c)

Fig. 21. Experimental waveforms of input voltage, input current, inductor current and output voltage with variable duty cycle control: (a) @ 176VAC input; (b) @ 220 VAC input; (c) @ 264 VAC input.

Fig. 22 Measured PF over the input voltagerange.

Fig. 23. Measured output voltage rippleover the input voltage range.

Fig. 24 Measured efficiency over the inputvoltage range.

2928

3@1s with high pf

Documents

Transcript of 3@1s with high pf