IEEE
TRANSACTIONS
ON
POWER ELECTRONICS,
VOL.
6,
NO.
I,
JANUARY
1991
141
Constant-Frequency Control
of
Quasi-Resonant
Converten
Dragan MaksimoviC
Member, IEEE,
and Slobodan Cuk,
Member, IEEE
Abstract-An additional independent control needed to eliminate the
undesirable variable switching frequency of quasi-resonant converters
can be obtained if the output rectifier is replaced by an active switch.
The concept is applicable to all classes of quasi-resonant converters. It
is demonstrated that in addition
to
operation at constant switching fre-
quency, selected classes of constant-frequency quasi-resonant
(CF-QR)
topologies feature extended range of accessible conversion ratios and
load currents. A practical example
of
a constant-frequency multi-res-
onant
(CF-MR)
buck converter operating at
2
MHz
is described.
I.
INTRODUCTION
UASI-RESONANT (QR) converter topologies are gener-
ated by addition
of
resonant elements to a PWM (square-
wave) converter. Consider, for example, the buck PWM con-
verter in Fig. l(a). The buck converter, just as all other basic
PWM topologies, has two switches-an active (transistor)
switch
S
and a rectifier (diode) switch
S.
If a resonant inductor
and a resonant capacitor are added as shown in Fig. l(b), the
modified topology becomes a member of the zero-current quasi-
resonant (ZC) class
[
11. Note that the s-switch needs to be con-
verted into a two-quadrant switch.
A
current-bidirectional
switch is used in this particular example. Other positions
of
the
resonant elements, or addition of more than two resonant ele-
ments result in other possible QR topologies, such as zero-volt-
age (ZV)
[2],
quasi-square-wave (ZC-QSW, ZV-QSW)
[3],
[4],
or multi-resonant (ZC-MR, ZV-MR)
[5].
In contrast to PWM topologies, where the switch waveforms
are approximately rectangular, QR converters exhibit smooth,
quasi-sinusoidal waveforms and switching transitions at zero
voltage or at zero current. For example, in the ZC topology,
tFe S-switch is turned
ON and
OFF
at zero current, while the
S-switch is turned
ON
and
OFF
at zero voltage, so that the power
losses associated with switching transitions are greatly reduced.
As
a result
of
the reduction in the frequency-dependent part of
the total power loss, higher switching frequency can be utilized,
and correspondingly smaller values and sizes of energy-storage
components should lead to smaller and lighter power convert-
ers. This is the main motivation behind the introduction of var-
ious classes of QR converters.
In comparison to PWM topologies, several unfavorable fea-
tures
of
QR converters can be identified: voltage and current
Manuscript received September
20,
1989. This paper was presented at
the 1989 High Frequency Power Conversion Conference, Naples FL, May
14-18. This work was conducted under the Power Electronics Program
supported by grants from Boeing Electronics Company, GTE Communi-
cation Systems Corporation, Rockwell, Inc., and EG&G Almond Instru-
ments Inc.
D.
MaksimoviC is with the Faculty of Electrical Engineering, University
of Belgrade, Belgrade, Yugoslavia.
S.
Cuk is with the Power Electronics Group, California Institute of
Technology, Pasadena, CA 91
125.
IEEE Log Number 9040465.
Q
I
I
vo
-
T
zc
I
I
I
I
Zero-Current Quasi-Resonant buck
(b)
I
1
I
I
I
1
I
I
Constant-Frequency ZC buck
(C)
Fig.
1.
Modification of PWM buck (a) yields quasi-resonant buck (b). Ad-
dition of controllable rectifier results in constant-frequency quasi-resonant
buck topology (c). In each step, added elements are highlighted.
stresses on switching devices are higher; conduction losses are
increased; load range and/or range of attainable conversion ra-
tios are limited. Yet another disadvantage is that the switching
frequency has to be varied in order to control the output voltage
against variations in input voltage and load. The variable-fre-
quency control is undesirable because optimal utilization of
magnetic components is not possible. Furthermore, it is more
difficult to handle the generated noise (conducted and radiated).
Finally, it is not unusual that operation at constant frequency is
imposed by system requirements.
In the ZC buck converter of Fig. l(b), for example, the in-
stant at which the S-switch is turned
ON
can be determined by
an external control signal, just as in the original PWM parent
converter. However, the turn-off transition (at zero current) is
confined to an interval determined by zero-crossing of the res-
onant current waveform. As a result, one degree
of
freedom is
lost and only a variable-frequency control is applicable. The
same conclusion holds true for all other quasi-resonant topolo-
gies.
0885-8993/91/0100-0141$01.00
0
1991 IEEE
142
IEEE
TRANSACTIONS
ON
POWER
ELECTRONICS, VOL.
6,
NO.
I,
JANUARY
1991
Several methods for constant-frequency control of resonant
converters have been proposed. In all cases, an additional in-
dependent control is introduced in order to enable the operation
at constant switching frequency. Thus, in
[6],
an additional
switch is used to vary the apparent value of a resonant element.
Conceptually similar is the method where the nonlinear char-
acteristic of the magnetic material is used to control the value
of a resonant inductor. Another possibility is to use full-bridge
resonant topologies where a variety of control strategies can be
devised for the four active (controllable) switches. Examples of
constant-frequency
,
full-bridge resonant converters are dis-
cussed in
[7]
and
[8].
In order to introduce the approach presented in this paper,
suppose that the ZC buck converterAof Fig. I(b) is further mod-
ified as shown in Fig. I(c). The S-switch is converted into a
controllable rectifier
so
that its turn-off transition is no longer
subject to the circuit waveforms, i.e., the instant at which the
S-switch is turned
OFF
is determined by some external control.
Hence, in the constant-frequency zero-current (CF-ZC) topol-
ogy, two switching transitions are externally controllable-the
S-switch turn-on and the S-switch turn-off. The conversion ratio
can be controlled by varying the length of the interval between
the two controllable transitions, while the switching frequency
is kept constant.
It is interesting that in
[9], a controllable rectifier was used
to achieve the constant-frequency control of converters that can
be classified as members of the ZV-QSW class. In Section
11,
constant-frequency quasi-resonant converters are introduced in
more general terms-the controllable rectifier can be imple-
mented in any QR topology and obtained benefits are not re-
stricted to the possibility of the constant-frequency control. In
Sections
111
through V, operating modes, idealized waveforms,
steady-state characteristics and design considerations are dis-
cussed for the three most interesting CF-QR classes-CF-ZV,
CF-ZV-QSW, and CF-ZV-MR.
A
practical converter
(2
MHz
25
W CF-ZV-MR buck) is presented in Section VI together
with experimental results.
11.
QUASI-RESONANT CONVERTERS WITH
A
CONTROLLABLE RECTIFIER
Quasi-resonant converters are derived by adding two or more
resonant elements to
a
parent PWM converter. Basic PWM to-
pologies have two switches-an active (controllable) switch im-
plemented as a transistor (!he S-switch), and a rectifier
implemented as a diode (the S-switch). Depending on the po-
sition of resonant elements with respect to the switches, various
classes of QR converters can be identified.
A
complete account
of possible QR topologies is given in
[
131.
In conventional QR
topologies, the S-switch can be current bidirectional,
or
voltage bidirectional,
while the $-switch is a diode. It is well known that substantially
different operating modes (half-wave, full-wave) correspond to
the different realizations of the active switch. It is not surprising
that a number of novel operating modes can be found if both
switches in a QR topology are controllable. Most importantly,
the controllable rectifier provides the additional independent
control necessary for operation at constant switching frequency.
The term constant-frequency quasi-resonant (CF-QR) is used to
denote the family of QR topologies with a controllable rectifier.
In a CF-QR topology, the two controllable switches can be
current bidirectional
or
voltage bidirectional. In general, there
are three distinct switch realizations:
CC: current-bidirectional
&
current-bidirectional;
CV: current-bidirectional
&
voltage-bidirectional;
VV: voltage-bidirectional
&
voltage-bidirectional.
A separate study of possible operating modes
is
necessary for
each of the three realizations. In this paper, we consider the
realization with two current-bidirectional switches, which is of
most practical interest for two reasons: voltage-bidirectional
switches are prone to higher conduction losses because of the
additional diode in series with the transistor; furthermore, the
zero-voltage switching (which is more favorable than the zero-
current switching) of a voltage-bidirectional switch is not pos-
sible in a practical converter because the diode in series with
the transistor prevents resonant discharge of the transistor’s par-
asitic capacitance.
The CF-QR classes that are particularly interesting for high-
frequency applications-zero voltage (CF-ZV), zero-voltage
quasi-square-wave (CF-ZV-QSW) and zero-voltage multi-res-
onant (CF-ZV-MR)-are studied in Sections
111 through V. The
terminology and notation used in the subsequent sections are
summarized as follows.
An operating state of a converter is associated with posi-
tion of the two switches. For a given QR converter topol-
ogy, there can be up to four operating states, denoted as
(0-0),
(1-O),
(0-l),
and
(1-1).
In each state, the position
of the S-switch and the position of the S-switch !re indi-
cated. For example, if the S-switch is on and the S-switch
is off, the converter is in the (I-0)-state, and
so
on.
An operating mode is a periodic sequence of operating
states. Of particular interest are the modes of CF-QR con-
verters in which all switching transitions are at zero volt-
age
or
at zero current,
so
that switching losses are
minimized, and in which two transitions are controllable,
so
that the constant-frequency control is applicable. The
interval between the two controllable transitions (relative
to the switching period) is defined as the control duty ratio
Dc
.
In the load-to-output plane, the dc conversion ratio of a
converter is shown against the load current. An operating
region is a part
of
the load-to-output plane where operation
in a given mode is possible. Take, as an example, the class
of conventional ZV converters. It
is
well known that the
zero-voltage switching is possible only if the converter’s
output current exceeds a certain minimum value
[2].
Thus,
the operating region of a ZV converter in the half-wave
mode
or
in the full-wave mode cannot include the zero-
load axis. Similar restrictions can be found for operating
regions of other QR topologies. For the design of QR con-
verters, boundaries of the operating region are of particu-
lar importance because one of the basic design
considerations is to fit the required range of load currents
and conversion ratios into the available operating region.
MAKSIMOVIC
AND
CUK:
CONSTANT
FREQUENCY
CONTROL
OF
QUASI-RESONANT
CONVERTERS
~
143
TABLE
I
NOTATION
FOR
NORMALIZED
QUANTITIES
USED
IN
THE
ANALYSIS
OF
CF-QR CONVERTERS
Normalized Quantity Notation
!-switch
voltage, current
U,,
i,
S-switch
voltage,
cuhent
U,/.
id
Load
current
6
Time
e
Switching
period
0,
Switching
frequency
f
The resonant inductor,
L,,
and the resonant capacitor,
C,,
form a resonant network with
R,
=
wcr>
(1)
fr
=
WJ27r
=
1/27rJL,c,. (2)
Voltages are normalized to the input voltage
V,,
currents
to
V8/Ro,
frequency to the resonant frequency
f,
and time
to
l/wr.
The notation for normalized quantities is sum-
marized in Table
I.
Switch voltage and current stresses are relevant for com-
parison of various topologies. In the discussions of Sec-
tions I11 through V, the switch voltage stress
VStreFS
is
normalized to the Voltage stress in the PWM parent. Sim-
ilarly, the switch current stress
Istress
is normalized to the
current stress in the PWM parent at the maximum output
current.
A.
A
Unibing Analysis
Method
Suppose that a QR converter has a dc conversion ratio
311.
(311.
=
V,,,/
V,),
and that
M(
D)
is the conversion ratio of the
PWM parent converter, where
D
is the duty ratio
of
the
S-switch. Recall that
M(D)
=
D
for buck,
I/(
1
-
D)
for
boost,
-D/(
1
-
D)
for buck-boost and Cuk converters, etc.
The equivalent duty ratio
rn
of the QR converter is numerically
equal to the duty ratio for which the PWM parent would have
the same conversion ratio
311.,
i.e.,
m
solves the equation
M(
in)
=
311..
For any given QR class and a specified operating mode,
m
(as a function of the normalized load
6
and control variables)
does not depend on the specific PWM parent topology
[
131. In
order to compute the
DC
conversion ratio for any particular QR
topology, one should only substitute
m
for duty ratio
D
in the
well-known function
M(
D)
of the PWM parent. This unifying
method, originally devised in
[lo]
for zero-current and zero-
voltage QR classes, and applied to all QR classes in [13], ren-
ders unnecessary and redundant repeated analyses of numerous
variations of QR topologies.
The results of Sections
111
through V are applicable to
$1
CF-QR converters constructed from various PWM parents. In
each section, a CF-QR buck tbpology (for which
311.
=
M(
m)
=
m)
is included only to provide a reference for the computed
waveforms and for the corresponding discussion.
111.
CONSTANT-FREQUENCY, ZERO-VOLTAGE
CONVERTERS
In a CF-ZV cpnverter, the S-switch is switching at zero volt-
age, while the S-switch
is
switching at zero-current,
just
as
in
conventional, frequency-controlled ZV converters. An example
of the CF-ZV buck converter is shown in Fig. 2(a). There are
two operating modes of interest:
1
...
-+
m-
pJ-+pJ+m+***,
II:
. . .
-+
11-11-
lo-1(-~-~-+*-.
The sequence of operating states in mode
I
is identical to the
one found for frequency-controlled ZV converters. If the cur-
rent-bidirectional switches are used, mode
I
is quite similar to
the well-known half-wave mode. The only difference is that the
transition from the
(0-0
)-state to the
(0-1
)-state occurs when
the $-switch is turned
ON,
before its voltage drops to zero,
whereas in the half-wave mode the S-switch (diode) starts to
conduct when its voltage reaches zero. Mode
I
is of less prac-
tical interest because the corresponding operating region is very
restricted
[
121.
The periodic sequence of states in mode
II
is not encountered
in the operating modes of frequency-controlled ZV converters.
The following qualitative description of the operation in mode
II
is with reference to the pertinent waveforms shown in Fig.
2(b).
Assume !hat initially the converter is in the (1-0)-state
and that the S-switch is turned
ON at the beginning of the
switching cycle. The turn-on is at zero current. In the subse-
quent
(
1-1 )-state, current
id
descends linearly until the s-switch
is turned
OFF
at
8
=
el,
at zero voltage, and the circuit enters
the (0-1 )-state in which
L,
and
C,
are allowed to resonate. The
(0-1 )-state terminates when the S-switch voltage reaches zero
again
so
that its antiparallel diode starts to conduct at
,e
=
81.
After the
(
1-1 )-state is re-entered, the transistor in the S-switch
needs to be turned
OFF
during the interval when current
id
is
positive. Thus, when
id
reaches zero at
8
=
03,
the S-switch
turns OFF, and the final (1-0)-state is entered. The transistor in
the S-switch has to be turned
ON before current i, becpmes pos-
itive. The switching cycle is completed when the S-switch is
turned
ON
again at
0
=
04.
Wavefo>rms
p,
and
pd
denote control signals for the transitors
in
S
and
S,
respectively. In the hatched areas, the state of the
corresponding transistor is irrelevant for the operation of the
circuit because the antiparallel diode is conducting.
Control duty ratio
D,
is defined a,s the interval between the
two controllable transitions-the S-switch turn-on and the
S-switch turn-off-relative to the switching period
e,,
D,
81/Op.
(3)
dc analysis of mode
I1
yields an expression for the equivalent
duty ratio
m
as a function of frequency
f,
load
6
and control
D,.
The operating region for mode
II
is shown in Fig. 2(c) with
normalized frequency
f
as a vaoing parameter. For lower
f,
the
operating region expands in both
6
and
m
directions. In contrast
to frequency-controlled ZV converters, operation at zero load
is possible. However, the range of attainable conversion ratios
is more restricted. At zero load, the equivalent duty ratio cannot
be greater than
1
-
f.
Suppose that the required equivalent duty
ratio
m*
is determined from design specifications. Then, param-
eter
f
has
to
satisfy
f
<
1
-
m*.
The selection off affects the trade-off between the S-switch
voltage stress and the switch current stress. Thus, iff
+
1
-
144
1.0
0.5
IEEE TRANSACTIONS ON POWER ELECTRONICS,
VOL.
6,
NO.
I,
JANUARY
1991
9.0
I
m
--
P.
1
0.2
0.5
f
-
0.6
h\-;
0.
-
0.0
1.0
2.0
3.0
f
-
0.6
(b)
(e)
Fig.
2.
(a)
CF-ZV
buck converter; (b) typical waveforms in mode
[I;
(c) operating region for mode
21,
with normalized
fre-
quency
f
as
varying parameter; (d) load-to-output dc characteristics; (e) control-to-output dc characteristics.
m*,
the S-switch voltage stress tends to its minimum value
(V,,,,,
+
2), but the current stress becomes infinitely large. On
the other hand, iff
+
0,
the current stress tends to
1
(to the
same value as in the PWM parent), but the voltage stress blows
up. Within these limits, parameter
f
can be selected to adjust
the values of voltage and current stresses. Regardless of the
choice forf, the S-switch voltage stress is ideally equal to the
switch voltage stress in the PWM parent.
As an example, for
m*
=
0.6,
parameter
f
is computed
so
that the S-switch stress-product,
V,,,,,
x
I,,,,,,
is minimized.
The result is
f
=
0.32,
(6)
Vstres,
=
5.2,
(7)
I,,,,,
=
2.2.
(8)
A conventional, frequency-controlled ZV converter would have
the current stress equal to one. However, the CF-ZV converter
has an unlimited load range (no load to full load), while the
load range of the conventional ZV converter
is
numerically
equal to the voltage stress minus one.
DC characteristics in the normalized load-to-output and con-
trol-to-output planes are shown in Fig. 2(d), (e) for
f
=
0.6.
IV. CONSTANT-FREQUENCY, ZV-QUASI-SQUARE-WAVE
CONVERTERS
Basic features of CF-ZV-QSW converters are camed over
from their freqdency-controlled counter-parts-both devices are
switching at zero voltage and the switch voltage stress is equal
to
1
(equal to the stress in the PWM parent). An example of the
CF-ZV-QSW buck topology is shown in Fig. 3(a). An operat-
MAKSIMOVIC
AND
CUK:
CONSTANT
FREQUENCY
CONTROL
OF
QUASI-RESONANT
CONVERTERS
1.0
m
0.5
I.
..
!!
I,
-
i’
’
-0.8
\
0.7
..
\
0.
-
0.6
0.5
--
0.4
0.3
f
-
0.5
-
0.2
4
f
=
0.5
4
-
0.4
d
=
0.5
m
=
0.37
1.0
T
d
-
0.0
/.3
t
f
-
0.5
Fig.
3.
(a)
CF-ZV-QSW
buck converter; (b) typical waveforms in mode
I;
(c) operating region for mode
I,
with normalized
frequency
f
as varying parameter; (d) load-to-output dc characteristics; (e) control-to-output dc characteristics.
ing mode with all zero-voltage transitions (two of which are
controllable) is defined by the sequence
I:...+B+
m+pJ+m+
...)
For a qualitative description of operation in mode
I
refer to Fig.
3(b). Assume that the converter
is initially in the (0-1)-state.
When the $-switch is turned
OFF,
the (0-0)-state is entered. At
the beginning of the (0-0)-state, current
io,
is negative and the
$-switch voltage increases in a quasi-sinusoidal manner. Simul-
taneously, the S-switch voltage decreases until it reaches zero
and its antiparallel diode turns
ON
at
8
=
8,.
The succeeding
(1-0)-state lasts until the S-switch is turned
OFF
at
8
=
e,,
ini-
tiating the second (0-0)-state. Now, the S-switch voltage de-
scends in a quasi-sinusoidal manner until it drops to zero and
its antiparallel diode starts to conduct at
8
=
03.
The converter
remains in the
(0-1
)-state until the S-switch is turned
OFF
again