Analysis of an Ac to Dc Voltage Source Converter Using Pwm With Phase and Amplitude Control

8/9/2019 Analysis of an Ac to Dc Voltage Source Converter Using Pwm With Phase and Amplitude Control

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ANALYSIS

O F

AN AC

TO

DC VOLTAGE SOURCE CONVERTER

USING PWM WITH PHASE AND AMPLITUDE CONTROL

Rusong Wu, S.B.Dewan,

G.R.

Slemon

Depar tment of Electrical Engineering

University of Toronto, Canada , M5S 1A4

Abstract

This paper presents a comprehensive analysis of

a

pulse-

width modulated AC to DC voltage source convecter under

phase and ampl itude control. A general mathematica l model

of the converter, which is dicontinuous, time-variant, and non-

linear, is first established. To obt ain closed-form solutions, the

following three techniques are used: Fourier analysis, transfor-

mation of reference frame and small signal linearization. Th ree

models, namely,

a

steady-state DC model,

a

low frequency

small signal AC model and a high frequency model, are con-

sequently developed. Finally, three solution sets, namely, the

steady-state solution, various dynamic transfer functions and

the high frequency harmonic components, are obtained from

the th ree models. The theoretical results are verified experi-

mentally.

1.

Introduction

Increasingly, AC to DC converters a re required t o provide

good input power factor, low line current distortion and regen-

eration. The pulse-width modulated AC to DC voltage source

converter has such good features. Several control strategies

of this kind of converter have been proposed [1]-[9]. One of

them is th e phase and amplitude control (PAC)

[1)-[4],

hich

has

a

simple structure and provides a good switching pattern.

This paper will present

a

comprehensive analysis

of

the con-

verter under this control, including steady-state, dynamic and

harmonic aspects.

The procedure of analysis is shown diagrammatically in Fig.

1. A

general mathematical model

of

the AC to DC voltage

source converter has been derived in a previous paper[9]. It is

useful in computer simulation to get

a

detailed time response of

the converter. But this model is time-variant, non-linear and

includes switching functions. It is therefore difficult to get ana-

lytical closed-form solutions. To solved it, three techniques are

applied. The first one is the Fourier analysis to get rid of dis-

continuities. After that, the general model is divided into two

models, the low frequency model and the high frequency model.

Both a re continuous and the boundary separating them is the

switching frequency. The second technique applied is the tr ans-

formation to a rotating frame of reference, synchronized with

the utility frequency, making the system time-invariant. T he

third technique is small signal linearization to linearize within

a

small area around the DC operating point. The system is fur-

ther divided into two parts, t he steady-state DC model and the

small-signal AC model. From them, the steady-stat e opera ting

point and various dynamic responses can be solved seperately.

Finally, the input current harmonic and the output voltage

ripple can be calculated from the high frequency model.

2. General Model of the AC t o DC Voltage Source

Converter

Th e main circuit of an AC to DC voltage source converter

This circuit will be analyzed under the

(1) the utility is

a

three phase balanced, sinusoidal voltage

is shown in Fig.2.

following assumptions:

source;

G e n e r a l M o d e l

E s t a b l i s h m e n t

E q u a t i o n s w i t h

Low F r e q u e n c y S w i t c h i n g F u n c t i o n

High Frequency

N o n l i n e a r

T r a n s f o rm a t i o n I n p u t o u t p u t

C u r r e n t V o l t a g e

o a R o t a t i o n

Frame of Rip p le

armonic P@ '

A n a l y s i s A n a l y s i s

e f e r e n c e

T i m e -

1

n v a r i a n t

E q u a t i o n s ,

S w i t c h i n g

P a t t e r n

s m a l l - S ig n a l C r e a t e d b y

L i n e a r i z a t i o n PAC C o n t r o l

L i n e a r

E q u a t i o n s

S t e a d y - S t a t e S m a l l - S i g n a l

DC Model AC m o d e l

s o l u t i o n

Fig.1: Procedure of analysis

(2) the filter inductors L are linear; saturation is not con-

sidered;

(3)

the DC load is equivalent to

a

resistance ro in series

with a electromotive force eh.

This load model can represent

a

wide variety of loads. Th e

converter can work either in the rectifying mode

(

e t

< V d

)

or in the regenerating mode

(

e h

>

Vd ). The series ro,

e t

equivalent load can simulate current source loads by use of large

TO and

e t

values. I t can also represent a voltage source load,

if

ro

is set low, or represent

a

pure resistance load by setting

e L

zero. When t he equivalent load includes some inductive

or

capacitive elements, t he whole system will have a higher order.

Th e analysis will be more complicated. bu t t he procedures to

be shown are still applicable.

Fig.2:

Main circuit of the AC to DC voltage source

converter

89CH2792-0/89/0000-11S6$01.00 0 1989 IEEE


2/8

A' =

L O O 0

O L O O

O O L O

o o o c

Z =

(1

- d; T

(1 + d , ) r

0 -R 0

-(d:- ;Ed:)

Fig.3: Symmetrical double-edge modulated switching

3)

0 0

-R

-(d:-

E d ; )

function

d;

d; d; -1lro

1 0 0 0

0 1 0 0

0 0 1 0

B =

where

d,

is the average value (or duty rat io) of the switching

function

dt

within one switching period. Therefore

(4)

a.

=

LJ l+ds)r

1 d(w,t) = d,

277 l - d , ) n

a, = 0

b,

= -sin(nd;r)r

2

(9)

(10)

2

nr

m

(5)

df = di

+

c(-l) .sin(nd,a) cos nw,t

n =l

where

R

=

RL

+

Rs

is the total series resistance in one phase,

and df i = 1, 2 or 3) is the switching function of the switching

device Si. When the Si is on, d: = 1. Otherwise df = 0.

Because no specific restriction was imposed on the switch-

ing function dr during the derivation, this mathematical model

is a general one, and universally applicable to six-step, var-

ious forms of pulse-width modulation or to other switching

strategies. It provides an exact solution at any moment if the

switching function d: is defined, and it is especially useful in

a computer simulation to obtain a detailed waveform in the

time domain. The problem with this model is that it can give

only a piece-wise solution instead of a continuous closed-form

one owing to the existence of the switching function. I t is diffi-

cult to use this model to evaluate the steady-state or dynamic

performance of the system analytically.

3. Fourier Analysis Applied to the Converter Model

To obtain continuous equations to describe the converter,

The Fourier series of

a

periodical time function is

the Fourier analysis can be applied to the model.

m m

f ( w t ) =

a0 +

ansinnwt

+

b, cos nut (7)

n =l n = l

For a natural sampling sinusoidal pulse-width modulation,

the switching points within one switching period are not sym-

metrical. However, when the switching frequency is much

higher then the utility frequency, the modulating wave can be

regarded as

a

constant within each switching period. Therefore

the switching pattern is close to a symmetrical one, as shown

in Fig.3. The switching funtion df can be expressed as follows:

3 3

2

n r

C O

Ed:

= E d ; +

c[c(-l) .

sin(nd,r)]cosnw,t (11)

Substitution of eq.10 and 11 into the matrix

A*

(eq.3) yields:

r l i=l n = l i=l

A ' = A + A h (12)

where

-R 0 0

- ( d l E:=i)

and

0 -R 0

- (& -+E d ; )

0 0

-R

-(d3-:Edi)

n=l

,--,

2

O

A4k = [(-1) . ~sin(ndka)cosnwst] I =

1,

2, or3(16)

n l

The matrix A describes the low frequency property of the

converter and the matrix Ah gives the characteristics in the

range equal to and higher than the switching frequency. The

variable vector x also can be partitioned into two parts corre-

spondingly,

2 = 2

+ xh

(17)

and Therefore eq.1 becomes:

1157


3/8

Z(xr + h ) = ( A + Ah)(Xi+ X h ) B e

(18)

It can be regarded as

a

combination of two models: one is

the low frequency model in the range lower than the switching

frequency F,,

Z i l = Ax , +

B e (19)

The other is the high frequency model in the range of

F

1 ,,

Z X h

=

AhXl

+

AXh -kAhXh

(20

1

Eq.19 can also be derived using the state-space averag-

ing technique [lo] which ignores high frequency components.

However, through the use of Fourier analysis, one can get bo th

the low and the high frequency models.

For simplicity, the subscript

I

in all low frequency equa-

tions will be omitted henceforth, i.e.

Zx = Ax + B e

(21)

The corresponding variables, i l ,

2 2 ,

iJ,

vd

represent only the

respective low frequency components. Th e high frequency com-

ponents, i.e. the harmonics, will be expressed with a subscript

h.

In the high frequency model, eq.20, the harmonic compo-

nent

x h

is usually much smaller than the low frequency compo-

nent x , consequently, the second and third items on th e right

side of eq.20 can be neglected to get an approximation,

Z X h

N

Ah2 (22)

Two separate models obtained in eq.21 and 22 pave the

way for solving the system analytically both in the low and

high frequency ranges.

4. State-Space Equations

in

Rotation Frame of

Reference

To solve eq.21, the function of the duty ratio d; in matrix

A

should be known. For the PAC control, d; is controlled by the

phase shift

11

and modulation index

m.

From Fig.4 it is seen

th at , if the modulating wave of phase

1

is

m

cos(wt- ), and

the switching frequency is much higher than the modulating

frequency, the du ty ratio

di

can be expressed for phase i

as:

e =

2T 1

d;

=

-

os

w t -

1

-

2

-

1)-3

+

-

[

3 2

e ,

cos wt

e ,

cos(wt

-

9)

e , cos(wt

9)

el

e2

-

e3

eL eL

Because d ; is a function of time, matrix A of eq.13 is time-

variant. Fortunately, the duty ratio d; of PAC control is

a

cosine function of time synchronized with the utility frequency.

It is therefore possible to transform the system

to a

rotating

frame of reference, in which it appea rs time-invariant.

First, apply a transformation to the voltage vector e .

e

= T e ,

or

e, = T - e

(24)

where the subscript r represents the variable, vector or ma-

trix in

a

rotating frame

of

reference. The transformation ma-

trix

T

and its inverse matrix

T-

re:

1.0

0

-1.0

u

Fig.4: Switching function d; and duty ratio

dl

in the

phase and ampl itude control

l o 0

For a three phase balanced system,

After transformation, the voltage vector in the rotating frame

of reference

e,

becomes:

(27)

Now, the voltage vector does not change with time in the rotat-

ing frame of reference. Th e zero sequence component eo equals

zero owing to the balanced condition.

Both the forward and

backward components e and eb equal &e,/2. Th e dc side

electromotive force eL is not affected by the transformation.

The next step is applying the transformation to the state

variable vector,

x

=

T x ,

(28)

(29)

or

x , = T- x

i.e.

1158


4/8

L&

Lib

CCd

La f

39)

r

- -

- R 0 0 0 io

0

-

%A

( 3 3 )

0 0 - R - j w L i b

+=

- R + j w L

0 e e j * if

e,-,

eJ

1

V d

9

0

-

70

-

5. Small Signal Linearization and DC,

AC

Models

After rotating of the reference frame, the converter repre-

sented by

eq.34

becomes a time-invariant system. But it is

still a nonlinear one. Small signal linearization around its DC

operating point can be applied for solution. Let:

x = x .rr

(41)

O =

i.e.

- R + ] R L

0 -Fe, f

*

0

- R - ] R L

- F e - , *

Ib

+

*

E L

Ra - -

d

R a -

Fe-,*FeJ* _ -

Arr

=

1159

- ~ + j w ~ o + e ] +

0

- R - ~ ~ Li e - j + (38)

-1

+ z e - j + + , j + -

o -


5/8

In the lossless situation

( R

= 0 r

= 7r/2) ,

L i L 0 0 1

I

:

:L

:J

, =

0 0 0

(53)

6. Steady S t a t e So lut ions

It is important to obtain steady-state solutions not only

because they give knowledge abou t t he relationsh ip between

state variables and system parameters in a steady-state oper-

ation, but also because the dynamic response of this nonlinear

converter is related to the steady -state operating point.

From the

D C

model of eq.44, the steady- stat e solutions can

be obtained:

where

E

= E , / 4

(57)

and

r =

tan-'

F)

Two other imp ortant variables to be observed are the power

factor angle 9 nd th e peak AC line current

I,.

They can be

derived from the forward and backward curren t components

I1

and Ib as

E s i n r - % si n( *+ +)

E c o s r - %cos(* + r)

Arg I1)=

tan-'

(59)

and the

RMS

value of th e stead y-s tate line current from eq.60

is

I

=

I m / 2 = d M

sin*

(for R =

0,9

= 0) (63)

2&RL

Th e function M 9 ) or keeping

9

= 0 is shown in Fig.5. It

is seen that the M Q ) unction is symmetrical with respect to

M axis assuming no losses. When th e resistance of the main

circuit is taken into account, th e modulation index M to keep

9

=

0

is nearly a const ant in the rectifying area, bu t

a

larger

variation is needed in the regenerating area. Therefore, in pa-

rameter selection, a sufficient design margin of the modulation

index has to be provided for regenerating operation.

M I

\ % = 5.0

1.0I

5.0

-20

-10 10 20

for keeping @

= 0

Fig.5: Modulation index M vcrsus

ijd

phase control Ik

7

Dynamic Response Analysis

The small signal AC model of the converter has been de-

rived in eq.46. Its Laplace transformation is:

?rr(S) =

Szrr

Ass)-I[Amxrr+(s)

+

A+Xrr$(s)+

AWX,SJ( )+B.,e^,(~)+A,oX,,io(

s)+B,oio

s)+B.L~L(

) ]

64)

(65)

and

and

(60)

2 = ( S L + R + j n L )

Z , = ( S L + R - j R L )

[ ( s L

+

R)'

+

R Z L z ]

+

~ ( s LM 2

+

R)(66)

[Ecosr- cos(* r)]'+ Esinr - sin(* + r)]

RZ

+

R2L2)/2

The converter is usually operated at unity power factor.

Th e necessary modulation index

A4

for keeping @ = 0 can be

obtained from Eq. 59,

In eq.64, the steady-state solution Xrrhas been solved in

the previous section. All the matrices are known. Therefore

any kind of transfer functions between the sta te variables

i f ,

~.

; b , t?d and the controls

l i z ,

4 oad disturbance o ,

?L

or input

disturbance

2 , 6,

can be obtained from this equation.

61)

for 0

= 0)

4 E in

I'

v d

sin(@+ I )

M =

I160


6/8

As examples, Fig.6, Fig.7 and-Fig.8 show the dynafnic re-

sponses between the variables

?&, q5

and the controls&, 1c in the

conditions of E = 63.5V,

R =

377, L = 6.43mH, C = 13.7mF,

v d N 197V, and cosq N 1.0. Th e calculated solutions from the

eq.64 are close to the experimental results, especially in the low

frequency area. Based on these dynamic responses obtained,

the regulators of a closed-loop control system can be designed

or adjusted properly.

-450

Fig.6: Frequency response of 6d/& at Ro

=

29.2ohm,

E L = 0, M

=

0.83,

rk

= 20.1 , R E 1.13ohm

- r - I - - l - - - t t t c - - - - c - - t c t ( -

I 0.1 1 10 100

-2701803

-360

EL

=

0,

M =

0.83,

9

= 13.2 ,

R

N

1.220hm

Calculated

Fig.7: Frequency response of / m at

Ro =

42.lohm,

Phase

Gl(s)= 239

(1 -

rn) 1

+A

-90

- 80

-270

. . Experimental

alculated

Fig.8: Frequency response of fd/G at Ro

=

42.lohm,

E L = 0, M = 0.83, \k = 12.6 ,R N 1.220hm

8. Inpu t Cur rent Harmonic Analysis

Previous sections were dedicated to the analysis in the fre-

quency range lower than the switching frequency. This section

and the following one will focus on the harmonic analysis of

the converter.

An approximate high frequency model has been established

in eq.22. For the steady-state harmonics

zxh N AhX (67)

x h = [ I l h r IZh, 13/19

Vdh]

(68)

where

I i h

i = 1, 2 or 3) he steady-state harmonic of the

h e urrent

I;,

and Vdh he steady state ripple of the DC

output voltage

vd.

In a three phase balanced system, each phase current has

the same steady-state harmonic content. Hence, only one phase,

namely phase 1, need be discussed. The first row of the eq.67

is

This equation means that there is a high frequency voltage

A14Vd applied to the inductance L which creates the harmonic

I 1 h . Therefore, the steady-state harmonic current is

(70)

Ai4vd

Impedance of the inductor

l h =

1 3

= n = l

{(-l)n& [sin(ndin) - Csin(ndin)

i=l

Because the amplitude of the harmonic is inversely propor-

tional to the square of the order

n*,

t is reasonable to approx-

imate the harmonic current by the first order component with

a modifying coefficient

kh,

i.e.

For phase and amplitude control, the duty ratio di has the

expression shown in eq.23. Therefore eq.71 becomes

I l h

I l h m cos(wst)

(72)

where

The real waveform of the current harmonic is close to a tri-

angular wave instead of a sine wave, so that the coefficient k h

is used to modify the difference of the peak

values

between a

triangular wave and

a

sine wave with equal RMS value,

k h = Z 21 1.23

8

The maximum harmonic value can be calculated by taking the

derivative of the I l h m with respect to

Rt.

At ( R t

-

Q = f n / 2 ,

I l h m

has its maximum value

1161


7/8

Eq.72 to 74 provide mathematica l expressions of th e line cur-

rent harmonic. It is seen from eq.72 tha t the line current har-

monic is predominantly related t o switching frequency F,. The

envelope of th e harmonic amplitude is given in eq.73 and the

maximum amplitude is expressed in eq.74.

At no load operation, the line current equals the harmonic

component. As an example, consider the no load current un-

der the following conditions: vd

=

202b', F,

=

3100Hz, L =

6.43mH, A4 = 0.89, @ = 0 . From eq.74 the maximum peak

harmonic can be obtained

as

I I i h m l m a x = 0 .5444

The envelope of its amplitude is calculated from eq.73 and

plotted in Fig.9. The experimental current waveform is shown

in Fig.10 under the same conditions. Both waveforms are in

good agreement with each other, except that the real no load

current still has a small amount of fundamental component

owing to the losses of t he circuit.

When a load is applied to the converter, the phase con-

trol

Q

increases correspondingly, but the modulation index M

changes very little (see Fig.5). The maximum harmonic in

eq.74 is only related to t he value of M , not the value

of

a.

Therefore the harmonic component superimposed on the fun-

damental component has an amplitude similar to that in the

no load condition. The difference is tha t t he harmonic compo-

nent is phase sh ifted owing to the increase of the phase control

(eq.73). This shift can be seen from the current waveform

shown in Fig.11.

/ -

Envelope of the no load

/

current harmonic //

\

/.

\,-/,/

Line-to-neutral supply

voltage

as

a reference

Fig.9: Envelope of the no load input line current

Fig.11: Waveforms of the input line current an d t he

line-to-neutral utility voltage (t:2ms/div,

e:100

V/div, z:2.5

A/div)

An important specification of the converter is the relative

value of the current harmonic rather th an the absolute value.

This relative harmonic can be defined as the ra tio of maximum

peak harmonic

I I l h m l m a x

to th e peak value of th e nominal cur-

rent I,,,

Substitution of eq.63 and 74 in the above equation yields:

4khR [I

os

(F)]

I;

=

3$F,M sin

a,,

(75)

where

an

s the nominal phase control angle.

It is seen that the ratio of the switching frequency to the

utility frequency

(F,/R)

is the main factor in influencing the

relative harmonic value. The higher the ratio, the lower the

current harmonic. It is also interesting to note that the rela-

tive current harmonic is not directly related to the inductance

L.

The explanation is that, when the value

L

increases, the

absolute current harmonic reduces. But the converter can pro-

vide less nominal line current if the nominal phase control @

is kept constant. Therefore their rati o does not change with

the variation of inductance L.

Eq.76 can be used to determine the necessary switching

frequency Fa, if a certain amount of relative harmonic value

I;

is required,

9. Output Voltage Ripple Analysis

The ou tpu t voltage ripple can be calculated in a similar

way from the high frequency model using the fourth row of the

eq.67,

V d h =

Vd hm

c0sw.t (78)

where

l / d hm = *

cos [+cos [.t -

k

- i - )-]]7f

+F,C

3

x cos R t - i - 1)-

79)

[ 2x }

Fig.10: Waveforms of the no load input line current

and t he lin e-to-neutral utility voltage (t:2 msfdiv, e:100

Vfdiv, i:2.5 Afdiv)

At no load operation and

@

= 0, the maximum voltage ripple

occurs at Rt = fk:, where I is an integer, an d t he maximum

outp ut voltage ripple

is

given by

1162


8/8

Ivdhmlmaz

= t cos ( ) - os (

(80)

Eq.78 to 80 give the expressions of the voltage ripple, including

its envelope Vdhm and the maximum ripple Ivdhmlmaz. The

necessary capacitance of the DC filter can be determined by

the requirement for DC voltage ripple Vdr, which is the double

of

Ivdhmlmar,

Conclusions

From the analysis of the phase and ampli tude controlled

PWM AC to DC converter the following conclusions can be

drown:

1) The general model of an AC to DC voltage source con-

verter gives detailed time response if the switching func-

tion is known and is therefore useful in computer simu-

lations. Analytical solution is however difficult because

of

the discontinuities, time variation and nonlinearity of

the model.

2) Closed form solution can be obtained using Fourier anal-

ysis, transformation of reference frame and small signal

linearization converting the general model into

a

steady-

state DC model, a small-signal AC model and

a

high

frequency model.

3) The steady-state DC model gives steady-state solutions.

The small-signal model, which is suitable n the frequency

range lower than the switching frequency, provides vari-

ous transfer functions between state variables, controls,

input

or

load disturbances.

The high-frequency model

provides information about input current harmonics and

output voltage ripple. From the solutions, the circuit pa-

rameters and the regulators of a closed-loop control can

be properly designed.

4) When the circuit resistance is taken into account, the re-

quired modulation index

M

to obtain unity power factor

is relatively independent-of load in the rectifying area.

However, a larger change in

M

is required in the regen-

erating area. Therefore enough margin should be left for

a

four quadrant operation.

The relative current harmonic is not directly related to

the value of the filter inductance L .

Both input current harmonic content and output volt-

age ripple are predominantly affected by the r atio of the

switching frequency F, to the utility frequency R. The

higher the ratio, the lower the relative current harmonic

and the DC voltage ripple.

The theoretical results were verified experimentally.

References

[I] Eugenio Wernekinck, Atsuo Kawamura and R.Hoft,

A

High Frequency AC/DC Converter with Unity Power

Factor and Minimum Harmonic Distortion, Conf. Record,

IEEE-PESC, 1987, pp. 264-270

[2] S.Manias, A.R.Prasad, P.D.Ziogas, A 3-Phase Induc-

tor Fed SMR Converter with High Frequency Isolation,

High Power Density and Improved Power Factor, Conf.

Record, IEEE-PESC, 1987, pp. 253-263

[3] S.B.Dewan, Rusong Wu, A Microprocessor-Based Dual

PWM Converter Fed Four Quadrant AC Drive System,

Conf. Record, IEEE-IAS, 1987, pp.755-759

[4] J.W.Dixon, B.T.Ooi, Indirect Current Control

of

a Unity

Power Factor Sinusoidal Current Boost Type Three-phase

Rectifier, IEEE Trans. on Indust rical Electronics, Vo1.35,

No.4, pp.508-515

[5] B.T.Ooi, J.C.Salmon, J.W.Dixon, A.B.Kulkarri, A 3-

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Power Factor, Conf. Record IEEE-IAS, 1985, pp. 1008-

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Analysis of an Ac to Dc Voltage Source Converter Using Pwm With Phase and Amplitude Control

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