Self Oscillating resonant converter

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Design of self-oscillating resonant converters based on a variable structure
systems approach
ArticleinIET Power Electronics · August 2015
DOI: 10.1049/iet-pel.2014.0964
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Ricardo Bonache-Samaniego
Universitat Rovira i Virgili
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Carlos Olalla
Universitat Rovira i Virgili
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Hugo Valderrama-Blavi
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Design of self-oscillating resonant converters based on a variable structure
systems approach
Ricardo Bonache-Samaniego*, Carlos Olalla, Luís Martínez-Salamero,
Hugo Valderrama-Blavi
Department of Electrical, Electronic and Control Engineering, Rovira i Virgili University,
Campus Sescelades, 43007 Tarragona, Spain.
*[email protected]
Abstract: A mathematical model is derived for the parallel resonant converter in which a simple
comparator circuit applied to the inductor current is used to establish stable oscillations at the
resonant frequency in cases of high Q. Second order differential equations are solved to construct a
piecewise phase-plane trajectory explaining the generation of a stable limit cycle and predicting its
amplitude and period. The self-oscillating mechanism is explored in other resonant converters and
verified by simulation. In all cases, switching between the two circuit configurations of the converter
is produced by the change of the input inductor current sign. Measurements in a 100 W parallel
resonant converter prototype oscillating around 500 kHz are in good agreement with the theoretical
predictions.
Index Terms—Resonant converters, self-oscillation, variable structure systems, limit cycles.
1. Introduction
Self-oscillating control of resonant converters is becoming increasingly popular among designers
because it allows a much closer operation to the resonant frequency than conventional forced-frequency
control while preserving zero-voltage switching. However, in spite of the simplicity of the control
mechanism, the mathematical description of the self-oscillating mode is relatively complex as it is proved
by a number of works reported in the last years. A describing function-based analysis is reported in [1] to
predict the steady-state behaviour of the converter by separating the system in two parts, i.e. the resonant
stage and a non-linear block made up of the input bridge and the non-linear controller. Later, a closed-loop
frequency-domain analysis with a relay and the use of the Hamel locus is presented in [2] to determine the
mode of oscillation for a given value of the relay hysteresis width in a LCLC converter. The notion of
band-pass signal [3] is used in [4] under the form of a time-varying phasor describing a sinusoid whose
frequency and phase change with time. A large-signal phasor-transform model is then derived in a LCC
converter and later linearized assuming self-oscillation for power factor control. The information provided
by small and large-signal models is employed by a FPGA to appropriately determine the changes of
polarity of the series-inductor current. It is also shown that the power factor control in self-oscillating
mode can be combined with an additional control loop to obtain output voltage regulation. The start-up of
a series resonant converter using an ON-OFF control based on the changes of polarity of either the

2
inductor current or a linear combination of inductor current and capacitor voltage is justified in [5] by the
increase of the stored energy during the transient state and verified by means of simulation. The previous
idea is the basis for the two-loop control of the series resonant converter reported in [6]. In that work an
internal loop implements an ON-OFF control based on the changes of polarity of a linear combination of
the inductor current and its reference, which, in turn, is provided by an outer loop made up of a PI
compensator processing the output voltage error. A similar approach is found in [7], in which an external
loop establishes the constants of the mentioned linear combination which constitutes the inner loop.
The basic problem underlying the self-oscillating behaviour inherent in all the mentioned works is to
find the parametric conditions for the existence of the resulting limit cycle together with an accurate
prediction of its amplitude and period.
In this paper, we explain the generation of the limit cycle under the optics of a variable structure
system with a switching surface where no sliding regime can exists because the trajectories in either side
of the boundary move away from it [8]. The phase trajectories of the two substructures involved in the
parallel resonant converter are combined to generate a growing spiral starting from the origin of the
phase-plane and eventually becoming an ellipse when reaching the attracting perimeter of the limit cycle.
Similarly, it will be shown by simulation that starting from any initial condition outside the limit cycle
results in a trajectory eventually attracted by the ellipse. The ON-OFF control based on the change of
polarity of the inductor current is also applied to the series resonant converter and to the input inductor
current of higher order resonant converters like LCC and LCLC resulting in all cases in a stable limit cycle.
2. Self-Oscillating Parallel Resonant Converter
The power stage of a parallel resonant converter (PRC) and its equivalent circuit are shown in Figs. 1a
and 1b respectively.
a)
b)
Fig. 1. (a) Power stage of a parallel resonant converter. (b) Equivalent circuit based on first harmonic approach

3
A model based on the first harmonic approximation is used in the equivalent circuit to represent the
effect of the rectifier and load by
2
8
L
eq R
R
[9]. The differential equation relating input and output
voltages is given by
2
2()
1
C C C i
eq
d v dv v v t
dt CR dt LC LC
, (1)
where
()
Sg
v t V
during
ON
T
and
()
Sg
v t V
during
OFF
T
Assuming a damping factor such that
01
, the solution of (1) in
ON
T
interval for zero initial
conditions [10]-[11] can be expressed as follows
2
( ) cos sin
2
eq
t
RC g
C g d d g
eq d
V
v t e V t t V
RC
, (2)
where
2
1
do
,
o
and being respectively the natural oscillation frequency and the damping
factor given by
1
oLC
and
2eq
LC
CR
.
Current
()
L
it
in the same interval is given by
() CC
Ldv v
i t C dt R
, (3)
From (2) and (3), we derive
1
2
2
1
( ) cos sin
2
eq
t
RC g g g g
L d d
eq d eq eq
V V V V
i t e t t
R L R C R
, (4)
Similar expressions for
()
C
vt
and
()
L
it
can be obtained for
OFF
T
interval by simply changing
g
V
by its
opposite -
g
V
.
Assuming
()
Sg
v t V
t
, this would result in a steady-state regime with coordinates
Cg
vV
,
g
Leq
V
iR
,
where superscript (*) denotes steady-state. Reciprocally,
()
Sg
v t V
t
would lead to a symmetrical
steady-state point
Cg
vV
,
g
Leq
V
iR
. This fact is visualized in Fig. 2 illustrating the trajectory in the
phase plane
CL
vi
from zero initial conditions up to the equilibrium point in both situations. The system
is stable and the trajectories are converging spirals. It is worth noting that current
()
L
it
will be negative

4
during certain subintervals of
ON
T
provided that its steady-state coordinate is relatively near to zero ,
which implies a high value of the load resistance, or equivalently, a high quality factor
Q
since
0eq
Q CR
. The same conditions apply to ensure positive values of
()
L
it
during certain subintervals of
OFF
T
. This property will be used in the next subsection to describe a simple switching mechanism leading
to a stable oscillation. This mechanism consists in provoking the change of polarity in the input voltage
()
S
vt
in agreement with the change of sign of current
()
L
it
, so that when
()
L
it
is positive ,
()
Sg
v t V
, and
when
()
L
it
is negative
()
Sg
v t V
.
2.1. Limit cycle generation
Variable structure systems (VSS) are systems whose physical structure is changed in accordance with the
actual state of the system. A VSS may be synthesized in the PRC by composing the portions of the
trajectories of
ON
T
and
OFF
T
intervals as illustrated in Fig. 3.
The structure-control law may be stated as
Network topology
ON
T
holds for
( ) 0
L
it
Network topology
OFF
T
holds for
( ) 0
L
it
Fig. 2. MATLAB simulation of the PRC phase-plane trajectory assuming constant input voltage.
10.5C nF
,
8LH
,
R 400
eq 
20
g
VV
(left),
20
g
VV
(right).