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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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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
1
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
2
A model based on the first harmonic approximation is used in the equivalent circuit to represent the
effect of the rectifier and load by Req 
RL 2
[9]. The differential equation relating input and output
8
voltages is given by
d 2vC
1 dvC vC vi (t )



,
2
dt
CR eq dt LC LC
(1)
where vS (t )  Vg during TON and vS (t )  Vg during TOFF
Assuming a damping factor  such that 0    1 , the solution of (1) in TON interval for zero initial
conditions [10]-[11] can be expressed as follows
vC (t )  e

t
2 Req C


Vg


sin d t   Vg ,
Vg cos d t 
2 Req Cd




(2)
where d  o 1   2 , o and  being respectively the natural oscillation frequency and the damping
1
LC
and  
.
2CReq
LC
factor given by o 
Current iL (t ) in the same interval is given by
iL (t )  C
dvC vC

dt
R
,
(3)
From (2) and (3), we derive
iL (t )  e

t1
2 R eq C

1
 Vg
cos d t 

d

 Req

Vg
Vg 
 Vg
,
  2  sin d t  
R
 L 2 ReqC 
eq


(4)
Similar expressions for vC (t ) and iL (t ) can be obtained for TOFF interval by simply changing Vg by its
opposite - Vg .
Assuming vS (t )  Vg t , this would result in a steady-state regime with coordinates vC  Vg , iL 
Vg
Req
,
where superscript (*) denotes steady-state. Reciprocally, vS (t )  Vg t would lead to a symmetrical
steady-state point vC  Vg , iL  
Vg
Req
. This fact is visualized in Fig. 2 illustrating the trajectory in the
phase plane vC  iL 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 iL (t ) will be negative
3
during certain subintervals of TON 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
Q  0CReq . The same conditions apply to ensure positive values of iL (t ) during certain subintervals of
TOFF . 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
vS (t ) in agreement with the change of sign of current iL (t ) , so that when iL (t ) is positive , vS (t )  Vg , and
when iL (t ) is negative vS (t )  Vg .
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 TON and TOFF intervals as illustrated in Fig. 3.
The structure-control law may be stated as
Network topology TON holds for iL (t )  0
Network topology TOFF holds for iL (t )  0
Fig. 2. MATLAB simulation of the PRC phase-plane trajectory assuming constant input voltage. C  10.5 nF , L  8  H ,
R eq  400  Vg  20V (left), Vg  20V (right).
4
Fig. 3. VSS made up of the two substructures of the PRC ( C  10.5 nF , L  8  H , Req  300  , Vg  20V ) .
No sliding regime exists since the trajectories on either side of the switching boundary iL (t )  0 move
away from the switching boundary [8]. However, their composition initially results in a growing spiral that
eventually becomes an ellipse representing a limit cycle, which corresponds to the sustainable oscillations
of vC (t ) and iL (t ) shown in Fig. 4.
Equation (2) is now modified to describe the behaviour of vC (t ) for network topology TON during a
generic interval t2  t  t3 and initial conditions vC (t2 )   VC 2n , ( VC 2  0 , n  0,1,2... ) and iL (t2 )  0 ,
Fig. 4. Inductor current and capacitor voltage waveforms for the PRC under VSS control starting from zero initial conditions
( C  10.5 nF , L  8  H , Req  300  , Vg  20V ) .
5
vC ( )  e


2 Req C


(VC 2 n  Vg )


sin d   Vg ,
(VC 2 n  Vg ) cos d 
2 ReqCd




(5)
where   t  t2 .
Expression (5) has its maximum value vCM at instant  c given by

2 1   2VC 2 n 
  ,
d c  tan  
 Vg  VC 2 n (1  2 2 ) 


(6)
vCM  e0 c Vg2  VC22 n  2VgVC 2n (1  2 2 )  Vg ,
(7)
1
2
Under conditions of high Q , it is apparent that   1 , or equivalently, 1  2  1 and d  o , so that
expressions (6) and (7) become
 2VC 2 n 
0 c  tan 1  
 ,
 V  V 
g
C 2n 

(8)
vCM  e0 c (Vg  VC 2n )  Vg .
(9)
If the generic interval t2  t  t3 corresponds to any TON interval of the limit cycle, then the maximum value
vCM will be VC 2 n , and hence


VC 2n  e0 c Vg  VC 2 n1  Vg .
Besides, in the limit cycle the quotient
(10)
2VC 2 n
can be approximated by 2 , and therefore expression (8)
Vg  VC 2 n
becomes
0 c  tan 1 (2 )    2    
(11)
Introducing (11) in (10) yields
1  e 2Vg

1  e 
VC 2 n  Vg
,
(12)
which is exactly the maximum value of the capacitor voltage in steady-state at resonant frequency vCMFHA
considering the first harmonic approximation
vCMFHA 
4

Vg Q 
2Vg

.
Moreover, from (11), the period of the limit cycle will be given by
T  2 c 
2
0
,
(13)
(14)
which is the period of the steady-state sinusoidal waveforms at resonant frequency corresponding to vC (t )
and iL (t ) .
6
Proving the stability of limit cycles in power converters often requires a dedicated involved analysis
leading sometimes to general analytic results [12] or to a mixture of analytical and numerical
demonstrations [13], both types of approach being beyond the scope of this paper. Nonetheless, the
attracting nature of the limit cycle can be illustrated by simulation as shown in Fig. 5, where all the
trajectories reach the limit cycle irrespective of the their initial conditions.
2.2. Minimum Q for self-oscillation
A point of interest for the self-oscillation of the PRC is to establish a minimum Q such that the
system enters the limit cycle when the switching boundary is as defined previously: i L (t )  0 . This is
equivalent to finding whether the current trajectory crosses the zero axis for some t  0 if iL (0)  0 . For
an initial capacitor voltage vC (0)  VC 2n , the current trajectory shown in (4) can be rewritten as follows
iL ( )  e


2 R eq Cd

1
 Vg
cos  

d
 Req


Vg  VC 2 n
Vg 
 Vg
 2  sin   
,

L
2 ReqC 
R

eq


(15)
where the angle   d t is adopted for convenience. It can be shown that VC 2 n  0 , i.e., the start-up
trajectory corresponds to the worst case for which the switching boundary may not be crossed if Q is not
sufficiently large.
Fig. 5. Trajectories reaching the limit cycle from different initial conditions: (1) second quadrant (2) first quadrant (3) fourth
quadrant (4) third quadrant.
7

Assuming e

2 R eq Cd
 1 , the first term of the trajectory
Vg
Req

Vg
Req
Vg 
1 Vg  VC 2 n
 2  sin  can be written as

d  L
2 ReqC 


1

 Vg
(Vg  VC 2 n )
 sin  .
R0 1   2
QR0 1   2 

cos  , is always positive, whereas the
second term
with R0 
L
.
C
It can be seen that
1
R0 1  
2


QR0 1   2
(16)
if   1  Q  0.5 , which was already assumed for an under-
damped response. Thus, this second term is most negative at  
3
, being more negative with increasing
2
values of VC 2 n . As a consequence, a minimum value of Q is required in order to ensure reaching the
switching boundary in the worst case, i.e., during start-up.
Figure 6 shows the start-up trajectory of the current for different values of Q . It can be observed that, for a
large Q , the switching boundary is reached at an angle approximately equal to  , the difference between
the switching angle  and  being equal to      . Linearization of trajectory (4) with a first-order
Fig. 6. Current trajectory of a parallel resonant converter during startup
factors Q  2,
(iL (0)  0,VC  0) for three different quality
3.15, 10 . Steady state current Vg / R (dash-dotted), linearization of the trajectory at    (dotted)
and crossing of the switching boundary iL (t )  0 (circles).
8
Taylor approximation at angle  allows solving for the switching instant  . This first-order Taylor
approximation of the damped sinusoidal terms in (4) can be expressed as

e


2 R eq Cd
cos   e

2 R eq Cd


2 R eq Cd




 1 ,


 2 R eq Cd

(17)

2 R eq Cd
,
e
sin    e
and then, the switching angle can be derived as
(18)



Ld 
2 R eq Cd
1  e
.
(19)


Req 


This approximation remains valid only for sufficiently small values of  . If a rather arbitrary upper bound
of  is chosen, e.g.   
4
, the switching angle will then satisfy


 
Ld 
2 R eq Cd
1  e

,
 4
Req 


whose exponential term can be linearized to yield
(20)

 

(21)
1  1 
  .
2
R
C

4
eq
d 

Since L > 0 and C > 0, this expression can be simplified to obtain an inequality that depends solely on Q
Ld
Req
 Q 2  2  4 4Q 2  1  0 ,
(22)
which can be reduced to Q  3.15 .
This result, while approximated, represents a valid lower bound for Q . Equal or larger values of Q
guarantee the self-oscillation and allow the converter to reach the limit cycle described in the previous
subsection.
3. Extension to other resonant converters
A similar self-oscillating mode can be induced in the series resonant converter (SRC) whose power
stage is depicted in Fig. 7a and whose corresponding limit cycle generation is shown in Fig. 8a. The VSS
control law also employs the sign of iL (t ) to switch between the two network topologies. The prediction of
both amplitude and period of the limit cycle can be carried out using the procedure of Section 2. Moreover,
inducing a self-oscillating mode in higher order resonant converters using the same VSS principle for the
sign of input inductor current is a relatively simple task but the corresponding analytical description is a
complex work. Besides, the phase-plane representation in terms of the state variables does not provide any
9
insight on the limit cycle-generation because more dimensions are required, which would imply three
dimensional descriptions in some cases or a sort of phase-plane representations considering all possible
combinations of state variable pairs.
To visualize the limit cycle generation, it is proposed a type of phase-plane representation consisting in
describing the sum of all normalized inductive currents versus the sum of all normalized capacitive
voltages, i.e. J L    R0 / Vg  iL and M C   vCi / Vg where m and n are the number of inductors and
n
m
i 1
i
i 1
capacitors respectively. Thus, this approach could be used in all resonant converters irrespective of the
number of reactive elements because in steady-state operation all currents and voltages are sinusoidal
waveforms of the same frequency and adding separately inductor currents and capacitor voltages will
result in two waveforms of the same frequency but of different amplitude and phase. Therefore, the
corresponding phase-plane representation will be an ellipse. This idea is used to describe the limit cycle
generation in the SRC, LCC, LLC and LCLC resonant converters whose equivalent circuits based on first
harmonic approach are depicted in Figs. 7a, 7b, 7c and 7d and their corresponding representations of J L
versus M C are shown in Figs. 8a, 8b, 8c and 8d respectively.
The main conclusion of the extension of the reported technique to other resonant converters is that
applying the VSS control law to the input inductor current imposes a zero phase-shift between the input
inductor current and the first harmonic of the input voltage. Due to this fact, the self-oscillating converters
with the proposed VSS control mechanism can be considered as resonant loss-free resistors, which are
particularly suitable for high-frequency, high-efficiency and low EMI applications [14].
a)
b)
c.)
d)
Fig. 7. Equivalent circuit based on first harmonic approach of other resonant converters (a) SRC (b) LCC (c) LLC (d)
LCLC
10
a)
c)
b)
d)
Fig. 8. Limit cycle generation (a) SRC (b) LCC (c) LLC (d) LCLC.
4. Prototype design and experimental results
An experimental prototype of the PRC with the converter parameters used in the simulations of Figs.25 and the VSS control based on the sign of the inductor current has been implemented. Fig. 9 shows the
circuit scheme of both power stage and VSS controller.
The power stage is made up of an H-bridge circuit and a resonant tank. The H-bridge employs 4
MOSFETs, which are activated by a driver stage based on the IC ADUM7234. The resonant tank has been
built with a ceramic (C0G) capacitor of 10.5 nF and an air-gap-based inductor of 8 H , this implying a Q
of 11.5 for a theoretical resonance frequency of 570 kHz and an equivalent load resistance of 300  . The
VSS controller employs the LT1116 comparator to establish the change of polarity of the input voltage in
agreement with the zero-crossings of the inductor current, which is sensed by a current transformer (CST)
with a transformer ratio 1:8. The output signal of the comparator activates a circuit adding a dead time to
avoid short-circuits during transitions. This stage, which is composed by two logic ports, is then connected
to the driver circuitry described above. The experimental prototype can be seen in Fig. 10 and the design
values are shown in Table 1.
11
A first step in the design is the selection of the inductance L and capacitance C such that the
corresponding satisfies the lower bound for the required range of load resistances. It is worth noting that,
due to the high switching frequency constraint, the capacitor and inductance values are also restricted to a
certain set of values. Next, the current sensor is implemented as an AC transformer following the design
guidelines given in [9]. The transformer ratio is chosen such that the voltage applied across the comparator
input is always within its specified limits.
Table 1 VSS self-oscillating PRC parameters set
Parameter
Vg
Cin
S1-4
L
C
R
CST
Rsens
RDT
DDT
Driver
Q
fsw
Component
Value
12 V
X7R
11 µF
IPB200N15NS3
Air-core Inductor 8.3 µH
C0G/NP0
10.5 nF
Resistor
300 Ω
Transformer
1:8
SMD Resistor
5Ω
Potentiometer
< 500 Ω
SDM03U40
ADUM 7234
11.5
570 kHz
Fig. 9. Implementation diagram of the parallel resonant converter with the proposed VSS-based controller.
12
B
A
C
D
Fig. 10. Experimental prototype: (a) resonant tank, (b) current sensor and comparator, (c) driver and (d) dead time circuit.
a. Experimental results
Figure 11a illustrates the resonant tank experimental behaviour in steady-state in open-loop
operation, i.e., when the H-bridge is activated by an external signal which results in a 10V square wave
periodic waveform of 570 kHz in the input voltage , this yielding an efficiency of 90 % for 100 W supply
to the load resistance. Steady-state waveforms of the PRC in self-oscillating mode are shown in
Fig. 11b with an oscillation frequency of 547 kHz while supplying 60 W to the load with an efficiency of
89 %. The corresponding experimental generation of the limit cycle for open loop and self-oscillating
mode is illustrated in Figs. 12a and 12b respectively.
It has to be pointed out that, in this case, the phase shift of current and voltage is not 90º and that
the phase displacement between the inductor current and the first harmonic of the input voltage is not 0º
either. These discrepancies with the open-loop operation are mainly due to a 176 ns delay in the
a)
b)
Fig. 11. Experimental waveforms of the PRC (a) open-loop (b) closed-loop (self-oscillation)
13
MOSFETs activation with respect to the current zero-crossing illustrated in Fig. 11, which eventually
leads to an oscillation below the resonance frequency, a Q value of 10.86 and a decrease of output power.
This delay is the result of accumulating the corresponding delays in the response of the comparator, logic
gates and driver stage, which introduce 13 ns, 37 ns and 126 ns respectively.
a)
b)
Fig. 12. Experimental generation of the limit cycle in the PRC (a) open-loop (b) closed-loop (self-oscillation)
Fig. 13. Delay between inductor current zero-crossing and activation of the MOSFETs
5. Conclusions
The generation of a limit cycle in a self-oscillating high- Q PRC under a VSS control law has been
explained and the amplitude and frequency of the limit cycle calculated. The control mechanism is simple
and it is based on the change of sign of the inductor current. A minimum value of Q guaranteeing the
14
self-oscillating behaviour has been established. The VSS control has been extended to SRC and to three
high-order resonant converters, whose limit cycle generation has been verified by simulation. The
phase-plane description of total normalized inductive current versus total normalized capacitive voltage
has been proposed to explain the limit cycle generation of high order converters. A common fact to all
converters in self-oscillating operation is their nature of loss-free resistors due to the unitary power factor
they exhibit in steady-state operation, and to their nature of lossless circuits in the connection between the
input port and the load.
Experimental results in self-oscillating operation are in good agreement with the theoretical predictions.
A limit cycle with an oscillation frequency below the resonance has been obtained experimentally, this
resulting in a decrease of output power with respect to the open-loop behaviour due to a delay in the
MOSFETs activation originated in the control circuitry. Compensating the effect of the delay by
modifying the control law with the inclusion of a capacitor voltage term, i.e., by considering a switching
boundary of the type i L  kvC is in progress.
Results confirm that the VSS control law implemented is able to operate at the maximum resonant
frequency, provided that the minimum Q criterion is satisfied. Potential applications of this research are
induction heating [15] or LED driving [16], for the elementary resonant converters. Also, the applicability
of this controlling technique in high-order resonant converters suggests its use in some types of plug-in
[17, 18] and contactless [19] battery chargers, some efficient lighting systems [20, 21], and PFC
applications [22].
Besides, the use of self-oscillating resonant converters can be explored in the power electronics
interface of photovoltaic systems [23] or in fuel cell-based power distribution architectures [24] with the
aim of introducing efficient structures with simple control mechanism to facilitate converters
interconnection.
6. Acknowledgments
The research leading to these results has received funding from: (i) the People Programme (Marie
Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA
grant agreement n° 626117, (ii) from the Spanish Ministry of Economy and Competitiveness under grants
DPI2012-31580/BES-2013-063288 and CSD2009-00046, and also (iii) from the Generalitat de Catalunya,
Beatriu de Pinòs programme under Award BP-B00047.
15
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16
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