The Family of Commutation Modes of Converters

Copyright © I FAC Control in Power Electronics and Electrical Drives . Lausanne , Switzerland , 1983

THE FAMILY OF COMMUTATION MODES OF CONVERTERS F. Kehtari Department of DC Tractz"on Substatz"ons & Converters Engz"neerz"ng, BBC-Secheron Ltd, Geneva, Swz"tzerland

Abstract. The variety of converters and their connection to the AC network power supply, leads to different commutation process and behavories of the converter. In order to define the static characteristic of converters, depending essentially on their commutation process, it is interesting to define the Family of Commutation Modes comprehending all commutation process of converters and determine general methods that can be applied to find their static characteristic. These methods, based on some electrical considerations, are defined in a mathematical aspect which can be used easily without the necessity of sophisticated calculators. To demonstrate the appliacation of the theory, the static characteristic of a twelve pulse bridge rectifier, composed of two three phase bridge rectifiers in serial, is calculated for different magnetic coupling of the main power supply transformer. Keywords. Power supplies; power converters; commutation process; mathematical analysis; static characteristics. INTRODUCTION

1.

To determine the DC voltage at a certain load should be known:

In DC controlled systems, where the DC power supply is obtained by means of converters, it is of great advantage to know the static characteristic: u f(i ) of the converter. d d In this relationship u and id are respectid vely the direct voltage and current per-unit 1 values. This characteristic depends on the converter and the AC network including the transformer. Fig. 1 illustrates a typical connection of a converter from AC to the DC side.

AC

firing angle overlap angle inductive voltage drop, per-unit value DC current, per-unit value. In fact the overlap angle and so the inductive voltage drop depends on the commutation process of the converter. The commutation process of high pulse number (p ~ 6) converters becomes easily complicated considering the transformer and the AC network. Specially when such a converter is composed of several others in parallel or serial as the commutation process in each converter might be influenced by the one of another. This leads to different Modes of Commutation. A commutation mode is defined by the following relationships:

NETWORK

( 1)

TRANSFORMER

(2) CONVERTER

Of course it is possible to study each particular case and find the above relationships. Therefore to simplify the calculations it's of great advantage to apply some mathematical methods, so that by knowing a commutation mode, it can help to find out the relationships to another one. This is precisely the aim of this article which defines the "Family of Commutation Modes" and illustrates these mathematical

+

Fig. 1.

Typical connection of a converter.

lAppendix A defines symbols and p.u. values 159

F. Kehtari

160

methods, based on certain electrical considerations. In this study, the inductance on the DC side is supposed infinitely elevated and the c ommutation impedance purely reactive.

"\ ~., a

DEFINITION

2.

Let us define C as the family of commutation modes, in which each of its elements a, b, c, .•. is a particular mode of commutation of a converter and its transformer. In this family, the only element known is "a" which is the simple and traditionnal commuta tion mode defined by: d

x

i

d

= 1/2

cosa-i

(cos (a) - cos(a+ 11»

d

b

e

x

Fig. 3.

Application o .

3.

COMMUTATION MODES IN PLANE (id' lid)

(3)

(4)

During the commutations, to go from the ini) to its final tial point (id = 0, u = u dia d point (i , u ), the converter follows a df df trajectory composed of different elements of C (different modes of commutations), in other words it goes from one commutation mode to another. The beginning element is always "a" as mentionned above.

Figure 4 shows the plane (id' u ). In this d plane, regarding relationships 3 and 4 it can be noticed that in mode "a" of commutation: the lieu of points where a = constant is a straight line defined by: u

d

cosa- id

the lieu of points where 11 is an ellipse defined by:

=

constant

(5)

Figure 2 compares such trajectories for 3 converters "1", "2" and "3" in which converter "1" follows aefg, converter "2" aebcd and

Fig. 2.

Commutation trajectories.

converter "3" ahid. The three converters start from mode "a" where u = udiaand finish in d different modes where u = u • The passage d df from one mode to another can be possible or not. In the above example the passage of "a" to "e" or "a" to "h" is directly possible while the passage of "a" to "b" should go throught "e". So the aim is to know these other modes (b, c, d, e, ••• ) knowing "a" by means of an application 0 which corresponds an element of C to another. (i.e. Fig. 3) To limit the complexity of the matter, elements of C are chosen in which in the plane (id' u ) d straight line and ellipses have the same properties as in mode "a".

Fig. 4.

Plane (id' u ) and Mode "a" of d Commutation.

If a straight line or an ellipse satisfies one of these relationships it can be considered as a part of the static characteristic in mode "a". The main questin is now the following: If a straight line or an ellipse does not satisfy these relationships and so it belongs to another mode of commutation, how can this mode be defined ? Let us remind that to define a mode is to find

The Family of Commutation Modes of Converters

161

out the general relationships (1) and (2).

5.

To get the answer to the above question the case of straight line is treated separately to that of the ellipse.

We consider ellipse e1 given by: (see Fig. 6)

STRAIGHT LINES

4.

2

(6)

2

(ud/v) + (id / w) for

We consider strai ght line d1 g iven by: (see Fig. 5)

for et

ELLIPSES

\.I

=

=

1

(10)

].la

defined in a mode "g" of commutation so that for the two points A and B the firing angle et is eta and et 1'

eta

defined in a mode "s" of commutation so that for the two points A and B the overlap an gle is \.10 and \.11 ' A (a=a,) MODE " 5 "

Fig. 6.

Ellipse e1 in Mode "g".

Assuming the same form s uch as relationship (7) for id = feet, \.I) we obtain Fig. 5.

y + y cos (et +

Straight line d 1 l.n Mode "s".

( 11 )

2

Again to determine the three paramaters Y1' To define this mode we should find out relationships (1) and (2). Relationship (1) is given by (6). For relationship (2) we can assume that along d 1 it is of form:

° +° 1

2

co S (\.1+

Y2 and

(7)

do

Y + Y cos( et +

point B: id1

( 12) (13)

To determine the 3 parameters 01' 02 and

(8)

do

6.

DETERMINATION OF
(9)

•
=

f(et o)

=
to be defined further on.

Knowing
d

=

f(].l) is defined in mode "s".

In fact the value of
F. Kehtari

162

By seeking the maximum value for id' we obtain:

o +

\.I

max

+ 1>

0

'TT -+

"/s.

where uo

Straight line d 1 , d 2 and ellipse eo are shown

tPo

(14)

in Fig. 7.

So ~o denotes the maximum value that the overlap angle \.I could reach. For bridge converters for example a p = 2 or p = 6, the angle ~o = u o' and in case of a three phase bridge rectifier where the commutation process is no more the mode "a", then
7.1.

STATIC CHARACTERISTIC OF A THREE PHASE BRIDGE RECTIFIER

id

The last part of the characteristic straight line give by:

_

~

__

.5

Modes "h" and "a".

This defines the first point as mentionned in § 4. To find the second point, we use point F which is the final operating point of the rectifier. At this point the rectifier is shortcircuited and the direct current is maximum. So relationship (7) should present a maximum at this point. That means:

a

~s

____

~

L-__________

B

Fig. 7.

It is known that the static characteristic of a three phase bridge rectifier is divided in three parts from the beginning:

to its end

B

~

Two examples have been chosen to demonstrate the application of the above theory .

~

APPLICATIONS

~

7.

.5

+ P

p

cos(\.I +

~)

o

(15)

We try to define this mode "h" of commutation. As d 1 corresponds to the third part of the static characteristic of the bridge rectifier, and the second part is an ellipse in mode "a", the passage from mode "a" to mode "h" should directly be possible. Then we search an ellipse of form of relationship (5), tangent at d 1 • This leads to the ellipse el

=

since:

~ = ~o

\.I

+

max

2u o

=

TT/3(see § 6)

2TT/3

1

and we obtain \.10 =TI/ 3 and the tangency point:

dl

TT-~

~max

This defines point F such as

such as:

T (i

-io-

=/3/4

u

'dl

Now the relationship (7) can be calculated as below:

=13/4).

The corresponding firing angle u at point T on the ellipse is given by the straight line d 2 (in mode "a")

point T:/3/4 point F:/3/4 +

and: cos u o

Pl - P2

P1 = -P2 =13/6

i

d

= 13/6(1

- COS(\1+TT/§»

( 16)

Figure 8 shows in mode "h" the characteristic id = f(\.I,uo) regarding (16), and gives also a

The Family of Commutation Modes of Converters comparison with mode "a".

163

We define the ratio k such as: k

= Xp /(X +X ) ps

( 17)

depending on the magnetic characteristic of the transformer and also the reactance of the AC network. The ratio k is comprised between o and 1. When k=O, meaning X =0, the two secondaries p

.S

a!!

MODE

11

MODE

IIhll

IBB

Fig. 8.

Characteristics

id=f(~,ao)

in Modes

"aft and "h"o

7.2.

TWELVE PULSE BRIDGE RECTIFIER

The connection of a twelve pulse rectifier composed of two three phase bridge rectifiers in serial, is presented in Fig. 9, and Fig. 10 shows a simplified connection of the AC power supply in which all the commutation reactances are summed up in X and X • P s

of the transformer are magnetically separated, and the commutation process in each rectifier are independent . In this case the static characteristic of the rectifier, in plane (id' u ), is identical to that of a d three phase bridge rectifier , Under the circumstances when k ~ 0 the commutation process in each rectifier influence the one of the other and the static characteristic changes, going tangent from a straight line to an ellipse and then to a straight line and so on, each part belonging to a particular commutation mode to be defined. The characteristic for a certain ratio k begins in mode "a" and a = 0: (straight line)

id

1/2 (1 -

cos ~ )

at ~ = n/6 it leaves this straight line and continues on the ellipse ~ = n/6: (still in mode "a")

id

1/2 (cosa - cos( a +n / 6»

until: a

Fig. 9.

Twelve pulse rectifier composed of two three phase bridge rectifiers in serial,

=

( 18)

ao

At this moment, the overlap angle ~ increases from n/6 to n/3 and the continuation of the characteristic is a straight line defined in a commutation mode depending on ao and so on k. When ~ = n/3 is reached, a increases to (a o + n/6) and the characteristic changes to a new ellipse. Attaining (ao + n / 6) the overlap angle ~ rises from n/3 to: ~max =

n - 2 (ao + n /6) which is the end-

ing point of the characteristic

~

u

= O. d Table 1 gives a summary of these different parts.

The last part of the characteristic is:

Fig. 10.

Commutation reactances.

part 3

~

st. line

~

if k

part 4

~

ellipse

~

if 2/3

~

k

~

part 5

~

st. line

~

if 0

~

k

~

2/3

=

1

F. Kehtari

164

TABLE 1

lpart!

I

Parts of the static characteristic

shape!

I

a

1in,l· . °

1" .

!

IfJ = TIll

1

31st. line/a = ao !elliPse lao ~a I la~a o +TI/6

5

1st. line a =a o +TI 16 TI ~ 3 ~fJ

I I

IO~ k ~

/TI/l~fJ~ TI/lIO~ k~ fJ = TI/3 lo~ k~

4

1

k

10,",,1, 10,

ellipse IO~a~ao

2

fJ

IO ~

IfJ- fJmax

k~

I

Figure 11 illustrates the static characteristic of such a rectifier for various k and Fig. 12 the characteristic a = f(id)'

Since the commutation impedance was considered purely inductive, the next step to develop the theory would be its application in the case of an inductive and resistive impedance of commutation. Then to complete the theory should be determined the passage of the commutation modes to voltages and currents waveform.

REFERENCES Buhler, H. (1978). In Georgi (Ed.), Electronique de puissance. Traite d'Electricite vol. xv, EPF, Lausanne. Wasserrab, Th. (1962). In Springer (Ed.), Schaltungslehre Stromrichtertechnik, Berlin. Uhlmann, E. (1975). In Springer (Ed.), Power transmission by direct current, Berlin.

APPENDIX A Symbols D

x

I

c

Id

inductive voltage drop. short-circuit current between two commutating phases, peak value. direct current. pulse number. direct voltage.

Fig. 11.

Static characteristic of a twelve pulse rectifier for various k.

6.

0 and a

direct voltage at Id

0 and a

O. ;t

O.

a

AC network equivalent reactance. primary side equivalent reactance •

••

.2 ~---k= o

2.

•

direct voltage at Id

~-L

•

____

.1

~~

.2

..

____

Fig. 12.

Characteristics a

8.

CONCLUSION

.,

~

secondary side equivalent reactance.

a

firing angle.

fJ

overlap angle •

___ ,•

The theory of the Family of Commutation Modes of converters as presented above, permits to define the static characteristic of converters and particularly high pulse number converters, by means of simple mathematical methods. The advantage of the theory is its general approach to the study of any converter using same methods and applying it in a universal

Per-unit values d

x