Transformer-Induced Low-frequency Oscillations in the Series-resonant Converter Ieee_klesser_1991

8/10/2019 Transformer-Induced Low-frequency Oscillations in the Series-resonant Converter Ieee_klesser_1991

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KLESSER AND KLAASSENS: TRANSFORMER-INDUCED OSCILLATIONS IN SERIES-RESONANT CONVERTERS

327

--

I

E S

DO1 DO3

U0t

4

DO4 DO2

Fig . 1 . Circuit d iagram of

fu l l

br idge conf iguration

of

ser ies- resonant dc-dc conver ter .

t

Fig .

2.

Simplif ied schematic of

f u l l

br idge conf iguration of ser ies- resonant conver ter .

network, in which the resonant circuit can be connected

to the dc-voltage sources

E ,

and

U,.

T L O w av efo rm s

of

the resonant curren t il are ge nerated in the configurations

depicted in Fig. 3(f) and (g). These configurations will

be arranged at the moment that the resonant current i l

equals the magnetizing c urrent

i,

under the condition that

the absolute value of the (primary) transforme r voltage

up

is smaller than the voltage

U ,

on the filter capacitor

C O ,

so

that the rectifier consequently becomes inactive and the

resonant circuit is decoupled from the voltage source

U,.

Any of the network configurations of F ig . 3can be de-

scribed mathematically by the set of relations

2),

pro-

vided that appropriate choices are made of the ternary

variables

j

and

k:

j E ,

=

ucl

+

uLI

+ u p

i l

=

C lduc l /d t

uLI= Lldil/dt

up = 1

k2)L,di,/dt

+ k U ,

j , k a [ l , 0, - 1 1 .

( 2 )

Clearly, the second order network configurations dis-

played in Fig. 3(a)-(d) will genera te sine waves with ra-

dial frequency

WO

=

=l/JL,C,

(3)

while the configurations displayed in Fig. 3(f) and (g)

generate sine waves

of

radial frequency

U,

= l / J ( L , + L,)CI

7)

generated i n discontinuou\ mode o f o p er a t io n

Doninin ol

z \ i \ t c n i ~ . 5

+ 0 . 3 7

X

[l

~ cI (x 2 ) ] / [1

+

X I > -4.

(A21)

Note that if this inequality is transformed into the equality

it determines the limit case in which mode I cha nges over

to mode 11. As can be seen from (2), the voltage [l

-

uCI(x2)] 1 +

X I

is equivalent to

This means that (A21) can be reformulated as

In the following the angle

a = a A ,

q )

will be approxi-

mated f rom (A 2 9 , af ter which , v ia (A3) and (A18) the

amplitude iI2

=

i I 2 ( h ,

q )

can be obtained in order to put

the inequality (A22) solely in

A

and

q,

which determines

the domain o f ex is tence o f mode

I.

For small values of X


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KLESSER AN D KLAASSENS: TRANSFORMER-INDUCED OSCILLATIO NS IN SERIES-RESONANT CONVERTERS

relation

(A20)

can be rewritten in second-order approxi-

mation as

4(1

q)a

= 4qX(a CY) ] q) + q2x7r*CY

or,

CY =

Xqn/[1

-

X{q27r2/4(1 4) q}].

(423)

Substitution of

(A23)

in

(A3)

after replacing sin CY by

CY

leads to

Zlill = 1 X q ~ / 4 ( 1

9).

(A241

Substitution of

(A24)

in

(A18)

results in

EZI:~: 1

+

Xq2a2/4(1 4 . (A25)

After substituting

(A25)

in the inequality

(A22)

and after

recalling that XE = (1 + A), one obtains

1 + Xqa2/4(1 4) < q (X +

1).

(A26)

With algebraic assistance (A27) can be approximated for

A


11/12

336

IEEE

TRANSACTIONS ON POWER

ELECTRONICS, VOL. 6 NO.

3, JULY

1991

Combining this last relation with

(A38)

and

(A41)

leads

to

In the following it is demonstrated that for increasing

values of

8

the length of the T LO waveform of mode I11

A combination of (A39) and (A42) will result in

assumptions, and employing

(A3), (A36),

and

(A46)

one

can write

a

=

O{

A},

y

= O{ 1 e } and

6

= O {

1

/ e 2 . This

means that the relations

(A35)

and

( k 44)

respectively can

be approximated by

(A45)

Z l & =

1

UCI X3).

From Fig.

7(a)

can be seen that

zli;2 exq (A53)

z,i;2 4 =

- t ~ l i ; 3

COS E ) .

(A541

e x q

=

COS (1

(A551

Combination of these last two relations results in:

+

xb

- x 2 - T ) ]

sin

(y)

or ,

Elimination of 13 f rom (A55) and (A47) leads to

T CY) in

(6) = a+

y

6)

sin (y)

(A46)

t g c

=

[[Zl;12/ q - O X q ) ]

sin

(6). (A56)

Inspecting

(A56)

one sees that for increasing values of 8

up to 1 / X the values of E will increase

to a / 2 ,

for

8 >

l / h the value of E will exceed

~ / 2 .

ubstitution of E =

T / 2

in

(A54)

and

(A471

and combining the results leads

zIfl3

q

sin (6)

or,

and

(A47)

I2 sin

(6)

= ll3in E).

Elimination of the terms cos

E )

and sin E) f rom (A44)

q2

2 q ~ ~ i ; ~

os

(6) +

Z;iy2 +

( Z ~ ? , / X >

in2 (6)

and

(A47),

results in to

tzliI3 t q sin

(6). (A57)

F o r

eh = 1

the relation

(A51)

can be combined with^

(A571

giving

the

t q

sin

(6) = 2 - 4q. (A581

=

t22;iy3.

(448)

F~~

6 = 1 ,

neglecting small

values of CY = o{

(where

O{ }

= order of) one finds successively from

(A36)

and

(A46)

the angular values:

X

= 1.26

rad

6 = 0 .296

rad.

Substitution of the valye found for 6 in

(A48)

and elimi-

nation of the term

Z l i 12

rom the result making use of

(A35)

for 0 =

1

and neglecting

a

yields

( t z l i ; 3 ) 2= q2[X2(7r2

+

1) 5.37X

+

11

=

q2(1

-

2.69X)2

or

[Z l l l3 (1 2,69h)q.

(A491

0450)

Substitution of

(A34)

with neglecting

a

n

(A14)

yields

Ucl o) 1 3q - ehq.

Because of 6

= O{ X2 }

and 5

=

0

can be written as

2 - 4 9

=

O{

X

},

relation

(A58)

3 L

} or,

= 1 / 2

o { x ~ / ~ } .

(A591

This last relation represents the domain-line for which

the length of the TLO waveform segment of mode I11

measures

t7r/2

normalized time units. Remark: addi-

tional analyses (not presented here), show that the order-

term in

(A59)

can be specified as

O{

X 3 I 2 }

=

( T / ~ ) X ~ / ~ .

It has already been mentioned that the existence of mode

IV requires the validity of the inequality

(A601

1 ucI(xb)l/[l + XI > 9.

In the limit case where m ode

I11

changes over to mode IV

and vice versa the inequality is transformed into the

equality:

From (A50), (A9), and (A45) is obtained:

tz,i;,(e) = 2 3 9 - ehq.

For

8

=

1

the last relation can be combined with

giving the result:

2 4q + 1.69Xq = 0

nr

[ 1

~cI(xB)I/[1

+

XI

= 4.

(A611

Substitution of

(A43)

with neglected 6 =

0 { 1 / 0 2 }

in

(A61)

results in

[Z,i;2 41 = (1

+

X ) q .

(A621

Combination of

(A62)

and

(A53)

leads to

- -

eXq

-

q

=

(1

+

X)q

or

e = (2

+

X ) / X (A63)

(A@)

q = 0.5

+

0.21X.

(A52)

Setting E

= T

in

(A54)

yields

This last form describes the left domain boundary of mode

I1 and the right domain boundary of mode

111.

t ~ l i ; 3= ~ l i ; 2 4 .


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KLESSER AND KLAASSENS: TRANSFORMER-INDUCED OSCILLATIONS IN SERIES-RESONANT CONVERTERS

337

Combining (A 9), (A14), and (A45) leads to

z & l 2 z&

q.

zli;l

= 2 z&.

(A651

Substitution of (A64) in (A65) yields

(A66)

Neglecting CY in (A34) and combining the result with

(A66) and (A53 ) leads to

1

q

=

ehq.

(467)

Substitution of (A63) in (A67) finally results in (3 +

X ) q

=

1

or,

(A68)

= 1 / 3 h / 9 .

This last form describes the left domain boundary of mode

I11 and the right domain boundary of mode IV.

A C K N O W L E D G M E N T

The authors wish to exp ress their grat itude to the m em-

bers of the Power Electronics Laboratory of the Depart-

ment of Electrical Engineering, who have contributed to

this work. The effort of M r. Ke es Weyerm ans throughout

this work is especially appreciated. The authors wish to

thank Mrs. Annett Bosch for the manuscript preparation.

REFERENCES

F . C. Sch w ar z , A method of resonant current pulse modulation for

p o w er co n v er te r s ,

IEEE Trans.

Ind.

Electron. Conrr . Insrrurn. ,

vol.

1 7 , N o .

3,

M ay 1 9 7 0 , p p . 2 0 9 - 2 2 1 .

- A n improved method of resonant current pulse modulation for

p o w er co n v er te r s ,

IEEE Tran sacti ons Iridusrrial E/rcrrori. Corirr. I r i -

s r r u m . ,

v o l . 2 3 , n o . 2 , p p .

133-141,

1 9 7 6 .

F . C . Sch w ar z an d

J .

B .

K laassen s , A co n t ro l lab le s eco n d ar y niu l-

t ik ilowatt dc current source with constant maximum pow er factor in its

141

151

three phase supply line,

IEEE Trans.

Ind.

Elrcrrori . Coritr . Ins trum ..

vol. 23 , no . 2 . pp . 142-150, May 1976.

R .

J .

King and T. A. Stuar t. Transformer induced instabili ty of the

ser ies resonant conver ter ,

IEEE Trans. Aerosp. Electron.

S y s t . , vol.

AES-19, n o . 3 , p p. 4 7 4 - 4 8 2 . M ay 1 9 8 3 .

F.

C. Sch w ar z , E n g in eer in g in f or mat io n o n an an a lo g s ig n a l to d i s -

c r e te time in te r va l co n v er te r , N A SA CR - 1 3 4 5 4 4 ,

1 9 7 3 .

Hans W . Klesser

was born in Yogyakar ta, Indo-

nesia in 1934. He received the M . S . degree in

physics in 1962 f rom the Delf t U niversity of

Tech n o lo g y in th e N e th er lan d s .

After var ied industr ial exper ience he jo ined the

Delf t University of Techno logy in 1970 where he

presently is a lecturer in Powe r Electronic s. His

reseach in terests include: switching power sys-

t ems , s to ch as t ics and co n t r o l .

J .

Ben Klaassens

was born in Assen, the Neth-

er lands in 1942.

H e

received the

B . S . ,

M . S . a n d

Ph .D . d eg r ees in e lec t r ica l en g in eer in g f r o m th e

Delf t University of Technology in The Nether-

lands.

He is currently an Associate Prof essor at the

D el f t U n iv er s i ty o f Tech n o lo g y teach in g g r ad u a te

co u r ses in th e p o w er e lec t r o n ics a r ea . H is w o r k

has been concer ned with inver ter circuits , pulse-

width modulation and the contro l of electr ical ma-

chinery . His research work and professional pub-

lications are in the area of conver ter systems with h igh in ternal pulse

f r eq uen c ies f o r su b - meg aw at t p o w er lev e ls emp lo y in g th y r i s to rs an d p o w er

transistors .

Dr . Klaa ssens has published a var iety of papers on ser ies- res onant con-

ver ters for low and h igh power applications. He has designed and built

pro to types of the ear ly dc-dc to the recent ac-ac ser ies- res onant conve r ters

for a wide var iety

of

applic ations such as electr ic motors and generators ,

communication power supplie s , radar s ignal generators , arc welders and

sp ace ap p l ica t io n s .

Documents

Transformer-Induced Low-frequency Oscillations in the Series-resonant Converter Ieee_klesser_1991