Analysis, Design and Implementation of a Resonant Solid State Transformer
Transformer-Induced Low-frequency Oscillations in the Series-resonant Converter Ieee_klesser_1991
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Transcript of 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
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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 .