Resonant LCC Converter for
Low-Profile Applications
A. Pawellek, A. Bucher, T. Duerbaum
Friedrich-Alexander-Universität Erlangen-Nuremberg
Cauerstraße 7
91058 Erlangen
Abstract—In addition to high efficiency of switch mode power
supplies, miniaturization of converters is an increasingly important aspect. In order to meet safety regulations as well as the
design goal of flatness, complicated integrated magnetics are
proposed in literature. Unfortunately, their design is difficult
and the costs of these components are rather high. This paper
focuses on the resonant LCC converter with capacitive output
filter, which is a promising topology with respect to low-profile
designs. The transformer necessary for mains isolation is realized on a ring core, thus regulation requirements are combined
with a low-cost magnetic component. The exact analysis of the
LCC converter in the time domain is presented and the solutions
of the four important modes of the converter are derived. A
prototype was designed in order to demonstrate the feasibility of
the proposed approach with an application scenario typical for
notebook adapters.
I.
INTRODUCTION
Nowadays, switch mode power supplies can be found in
applications for virtually all power classes. The omnipresent
trend within the field of power electronics towards miniaturization is pushing switching frequencies to higher levels in
order to reduce the size of the passive components [1]. Furthermore, high frequency operation is desirable with respect to
a fast converter transient response, which is also an increasingly important aspect. Conventional converters based on
pulse-width modulated topologies are limited in terms of high
switching frequencies, as hard-switching conditions occur,
causing significant switching losses [2]. Therefore, countermeasures such as soft-switching snubbers or advanced PWM
topologies have to be taken against this effect for the purpose
of increasing the switching frequency with these topologies.
Furthermore, the almost square and triangle voltage/current
waveforms of PWM converters have high frequency harmonic
components, which result in a high effort on filtering to fulfill
the requirements of EMI regulations. In addition to that, parasitic components gain influence at high frequencies and have
to be taken into account during the design process.
A more suitable family of converters for high frequency
operation are resonant converters. Their operation allows for
the semiconductor devices to switch under zero-voltage (ZVS)
or zero-current-conditions (ZCS) without any additional efforts. Thus, switching losses are nearly completely avoided
and high switching frequencies can be achieved without sacrificing converter efficiency. The quasi-sinusoidal current wave-
978-1-4244-5794-6/10/$26.00 ©2010 IEEE
forms are EMI friendly and additionally, parasitic elements are
included as a part of the resonant tank.
Resonant converters can be categorized on the one hand by
the number, kind and arrangement of the components in the
resonant tank and on the other hand by the kind of the output
filter. The most simple member of this family is the LC converter, with the tank containing one inductance and one capacitance. Depending on the arrangement of both reactances,
two basic types can be distinguished known as series-resonant
and parallel-resonant converter [3-5]. The series-resonant
converter has good part load efficiency, but cannot regulate
the output voltage in the case of no-load. On the other hand,
the parallel-resonant converter can handle no-load conditions,
with decreased efficiency at low load. In order to combine the
advantages of both converters, resonant converters with a
higher number of resonant elements can be used. A summary
of the arrangement of three and four resonant elements is
given in [6] and [7].
The trend towards miniaturization and low-profile results
in high requirements for the magnetic components. A suitable
resonant converter for low-profile applications is the LCC
converter. By means of a ring core as transformer geometry,
very low converter heights can be realized. The effort to design the magnetic component is reduced and with a classical
wire wound winding, no integrated magnetics are necessary.
Furthermore, ring cores are inexpensive and without an air
gap, fringing field effects regarding additional proximity
losses are avoided.
This paper focuses on the resonant LCC converter with
capacitive output filter. The principle of operation is discussed
in Section II, the analysis of the converter in the time domain
for the four important modes follows. Section IV contains
several evaluations of the derived equations, with special
attention being paid to the boundaries between different
modes as well as the switching behavior. Based on the results
obtained by this analysis, a converter is designed and built in
section V.
II.
OPERATING PRINCIPLE
Fig. 1 illustrates the basic schematic of the LCC converter
with capacitive output filter, driven by a full-bridge configuration. The resonant tank consists of the three reactive components Cs, Ls and Ćp. The actual transformer is represented by a
1309
Io
T1
D1 D4
Ls
Cs
uS(t)
Ui
T4
uCs(t)
iLs(t)
iB(t)
n
T2
Ćp
uĆp
T3
Co
RL
Uo
D2 D3
Figure 1. LCC converter with capacitive output filter
cantilever equivalent circuit, whereas the magnetizing inductance is assumed to be large enough to be neglected. The parallel capacitance is connected to the secondary of the transformer. As a result, the parasitic capacitances of the bridge
rectifier and the leakage inductance of the transformer are a
part of the resonant tank. With a proper design of the magnetic
component, Ls can be integrated within the transformer without the need for an additional series inductor.
Thus, the parasitic stray inductance, which often causes
problems in hard-switched PWM topologies, is incorporated
into the operation of the converter. The full-bridge acts as
inverter stage for the dc input voltage, generating a squarewave voltage with a duty cycle of 50 % which is applied to the
input of the resonant tank. The diode-bridge rectifies the
tank’s output current iB(t). The remaining harmonic frequencies are filtered out by the output capacitance Co. Its value is
assumed to be large enough in order to neglect the remaining
voltage ripple of Uo.
one switching cycle, caused by the input bridge and the output
rectifiers. The transitions of the input bridge are controlled by
a control-IC, whereas the transitions of the bridge rectifier are
dependent on the states of the converter. Therefore, a resonant
converter is a nonlinear, time variant system. Nevertheless, the
intervals between the transitions can be represented by a linear
network. The state variables are the inductor current and the
voltages across the capacitances which have to be periodic
under steady-state conditions. In addition, the analysis can be
reduced to one half of the switching period as the waveforms
of resonant converters are antisymmetric. For this paper analytical steady-state solutions are derived, thus it is necessary to
determine the occurring modes of the converter in advance.
For a wide frequency range, the LCC converter with capacitive output has 4 modes of operation which are of interest.
Each of these modes numbered 1 to 4 consists of a sequence
of three subintervals, each with a different cycle of resonant
and clamped intervals.
For the analysis, the secondary side parallel capacitance Ćp
is referred to the transformer primary side, resulting in the
equivalent capacitance Cp = Ćp/n2 with the transformer turns
ratio n. The equivalent circuits of the converter during the
different subintervals are shown in Fig. 2 and Fig. 3. In order
to perform the analysis of the complete switching cycle, the
waveforms of the state variables for the three subintervals are
required. During interval A, the inductance Ls and the two
capacitances Cs and Cp are resonating with the upfront unknown starting values
iLs t
0 iLs 0
The occurring subintervals can be divided into two major
categories. With the rectifier conducting, the voltage across
the parallel capacitance is clamped to the output voltage with
uĆp(t) = ±Uo and energy is transferred to the output. The transition of the diode bridge is determined by the zero-crossing of
the inductor current iL(t) setting off a resonance with all four
diodes reverse biased. During this resonant subinterval, the
complete inductor current charges the parallel capacitance Ćp
until uĆp(t) reaches the output voltage ±Uo. Due to this discontinuous energy transfer, the bulk output capacitor is stressed
with high rms-currents. However, this is a typical characteristic for resonant converters with capacitive output filter.
Although three resonant elements are present within the
tank of the LCC converter, the behavior of the converter during subintervals with the tank resonating can be described by a
second order differential equation. During subintervals with
clamped parallel voltage, the parallel capacitance is not involved in the resonance. Hence, two different resonant frequencies can be identified, with this converter topology often
referred to as multiresonant converter in literature.
III.
ANALYSIS
uCs t
uCp t
0 uCs 0
Ls
Cs
(1)
iLs(t)
Cp
uCs(t)
Ui
0 uCp 0 .
uCp(t)
Figure 2. Equivalent curcuit, interval A
Ls
Cs
Cp
Ui
±nUo
Figure 3. Equivalent curcuit, interval B (+nUo) and C (-nUo)
The resulting waveforms can be obtained by solving the
differential equation e.g. by using the Laplace transformation.
For a better overview and an easier calculation, the following
base quantities for normalization and abbreviation are introduced:
Several methods exist for the purpose of analysing resonant converters [8,9]. An approximation in the frequency
domain is the First-Harmonic-Approximation (FHA) [9-11]
and the Extended-First-Harmonic-Approximation (eFHA)
[12-14]. An exact mathematical description is the analysis in
the time domain [15]. The steady-state operation of resonant
converters is characterized by several switching events during
1310
U norm
I norm
t norm
Ui
Rnorm
U norm Rnorm
1 Z0
Ls Cs
Z0
F
Z0
2S f 0
Z Z0
1
Ls Cs
(2)
Ls Cs
(3)
f f0
(4)
Q
n 2 RL Z 0
M
U0 Ui
*
]
Cs C p
1 Cs C p
(5)
1 *
(6)
Based on this normalization, the normalized solution for
interval A is given by
1 M
jCs t n mCs t n Cs 0
1 * sin]t n J Ls 0 cos ]t n 1 M
Cs 0
M Cp 0
1 M
Cs 0
1 *>1 cos]t @
n
1 * sin]t n M Cs 0
J Ls 0
mCp t n M Cp 0
M Cp 0 * 1 * >1 cos ]t n @
J Ls 0 *
1 * sin]t n M Cp 0 .
A
(9)
(Q = 3; Γ = 1; n = 1; F = 2)
Figure 4. Waveforms mode 1
(10)
1 M Cs 0 # nM >1 cos tn @
J Ls 0 sint n M Cs 0
(11)
mCp t n rnM .
(12)
A. Mode 1
Fig. 4 shows the waveforms of a complete switching cycle
of the state variables for mode 1. In order to obtain a steady
state solution for mode 1, a periodic solution for (7)-(12)
representing the three subintervals in the time domain has to
be found. Based on the waveforms’ symmetry, the values of
the state variables at the end of the third interval are equal
with opposite sign to those at the beginning of the switching
cycle. Thus, a second order system of transcendental equations
is derived with the unknown durations tn1 and tn2.
0 1 cos T1 ^>1 K @ cos T 2 >1 K @`
>
0.5 >1 1
@
1 * @cos T T
0.5 1 1 1 * cos T1 T 2 T 3 1
2
T3 (13)
@
>
@
1 1 * sinT1 T 2 (14)
with
T1
Table I contains the interval sequence of the four modes
for the first half of the switching period. At the beginning of
each mode, the input bridge switches to the positive supply
voltage Ui. After interval B and C, interval A must follow to
charge the parallel capacitance Cp. For the analysis, relative
time variables tn1 to tn3 are introduced, each starting at zero
with the corresponding interval change. As a result, the normalized time variables tn1 and tn2 represent the durations of the
first and the second interval. The duration of the third interval
is the normalized half period less tn1 and tn2.
>
2 sinT 3 1 1 * sinT1 T 2 0
] t n1 ,
T2
tn2 ,
T3
] S F tn1 tn 2 (15)
and the abbreviation
The upper sign in (10)-(12) is valid for subinterval B, the
lower sign for subinterval C.
1 1 * sinT 2 ^>1 K @sinT 3 >1 K @sinT1 `
A
(8)
1 M Cs0 # nM sintn J Ls 0 costn mCs t n C
(7)
Following the same steps for subinterval B and subinterval C (Fig. 3) one obtains
jLs t n TABLE I. INTERVAL SEQUENCE
mode 2
mode 3
mode 4
interval duration
C
A
B
tn1
A
B
A
tn2
B
A
C
π/F - tn1 - tn2
mode 1
A
C
A
K
2FQ1 * S *.
(16)
For a given switching frequency and a given converter
configuration, these durations can be determined by numeric
means. The resulting output voltage for this operation point
can be derived by taking the output current iB(tn) into account
with
M
Q n 2 j B t n tn 2
FQ Sn ³
0
j Ls t n dt n .
(17)
With known values for θ1 to θ2 the voltage transfer ratio
M = Uo/Ui is given by
M
* >nX 1 * @ ^1 cos T1 >1 cos T 2 @
cos T 2 cos T1 T 3 1 1 * sinT 2 (18)
˜ >sinT1 sinT 3 sinT1 T 3 @`
with
X
1 1 * sinT 2 sinT 3 T1 cos T 2 cos T 3 T1 1 .
(19)
B. Mode 2
Fig. 5 illustrates the waveforms of the state variables with
the interval sequence according to Table I over a switching
period for mode 2. Following the same argumentation as in
mode 1, two equations for the unknown interval durations are
obtained. The first equation
1311
cos T1 >1 cos T 2 cos S F T 2 ] 1 1 * sinT 2 sinS F T 2 ] @
1
˜ ^>1 1 K @ >1 1 K @cos T 2 `
(20)
delivers θ1 as a function of θ2 with
T1
T2
t n1 ,
]t n 2 ,
T3
S F tn1 tn 2 .
ZCS
ZVS
(21)
Substitution of (20) in
0 1 1 * sinT 2 cos T 3 cos T 2 sinT 3 sinT1 ,
(22)
yields a single transcendental equation for θ2. With a valid
solution for θ2, the dependent variable θ1 can be calculated
with (20). The voltage transfer ratio M for mode 2 is then
given by
M
X
* cosT1 >cosT1 1@ >nX 1 *@
(23)
1 1 * sinT 2 sinT 3 T1 (24)
cos T 2 cos T 3 T1 1 .
C. Mode 3 and Mode 4
On the first look, the interval sequence for all four modes
appears to be different. Nevertheless, the modes can be divided into two groups with a characteristic sequence of subintervals, with interval A as the resonant interval and interval B
and C as the clamp interval:
Mode 1& Mode 3
Mode 2 & Mode 4
resonance Ö clamp Ö resonance
clamp Ö resonance Öclamp
Thus, the steady state of mode 3 and mode 4 can also be described by means of the derived equations of mode 1 and
mode 2 respectively. With the output diode bridge conducting
in the opposite direction, a negative sign for the voltage transfer ratio M in (18) and (23) is obtained due to the reverse flow
direction of iB(t).
IV.
EVALUATION
Fig. 6 shows the voltage transfer ratio M given by (18) and
(23). A turns ratio of n = 8 and a capacitor ratio of Γ = 10 is
assumed. Additionally, the area with ZVS and ZCS is highlighted. In order to provide the specified output voltage under
full load conditions with Q = Qmin at the minimum input
C
A
B
(Q = 2; Γ = 1; n = 1; F = 1.35)
Figure 5. Waveforms Mode 2
Figure 6. Voltage ratio M = Uo/Ui versus normalized switching
frequency F for n = 8, Γ = 10
voltage (M = Mmax), a suitable value for the turns ratio n and
parameter Γ must be chosen. Attention must be paid to the
peak value of the Qmin curve which must be higher than the
specified maximum voltage conversion ratio in order to fulfill
the specification.
Fig. 7 illustrates the switching behavior as a function of
the normalized switching frequency F and the normalized load
resistor Q. Additionally, the area of operation of an exemplary
converter design with 0.095 < M < 0.152 corresponding to the
input voltage range and a minimum load resistor of Qmin = 2 at
full load is highlighted. According to the figure, the converter
is operating under ZVS conditions for the complete load
range. A negative series inductor current is necessary at the
beginning of the switching cycle when the input bridge
switches to the positive supply voltage. The current flowing in
the negative direction charges the parasitic capacitances of the
MOSFETs, allowing them to switch under zero-voltage conditions. The boundary between ZVS and ZCS is reached if the
inductor current crosses zero exactly at the beginning of the
switching cycle. Therefore, a condition exists to calculate the
switching behavior as a function of the load and the switching
frequency (Fig. 7).
The appearing mode depends on the load resistor and the
switching frequency. Starting with the converter in mode 1
and reducing the load resistor, the clamped interval C is increased at the expense of the first interval in order to transfer
more power to the output. Therefore, the duration tn1 is reduced until it vanishes at the boundary between mode 1 and
mode 2. The same considerations apply to the boundary of
mode 3 and mode 4. It can be shown, that mode 1 occurs only
for frequencies F > ζ and mode 4 only for frequencies F < 1.
Thus, the boundary of mode 3 and mode 2 is suited in the
frequency range 1 < F < ζ, where the third interval vanishes.
The mode boundaries are shown in Fig. 8. The operation area
in this picture is the same as in Fig. 7. For no-load conditions
with Q → ∞ mode 2 vanishes, with the boundaries of mode 1
and mode 3 approaching one another. The vertical asymptotic
is situated at the normalized switching frequency F = ζ with
ζ = 3.3 for the discussed example. Together with the ZVS
boundary of Fig. 7 it can be stated that ZVS is obtained for
F > ζ under all load conditions. In order to obtain a regulated
output voltage, the switching frequency has to be controlled
for different load situations. Therefore the operating area has
to be limited to mode 2 and mode 1 if ZVS is to be guaranteed. Table II summarizes the switching behavior of the four
1312
Figure 9. Winding structure
For the transformer, a ring core TN32/19/13 [16] is used. To
meet the requirements of the 10 mm height constraint, the
height of the core is reduced down to 7 mm. Fig. 9 shows the
arrangement of the primary side and the secondary side on the
core. A rather large distance between primary and secondary
is on the one hand necessary in order to fulfill safety requirements, on the other hand high values for Ls can be expected.
Figure 7. Switching behavior as function of normalized load resistor Q
and normalized switching frequency F
TABLE II. SWITCHING BEHAVIOUR OF THE FOUR MODES
mode 1
mode 2
mode 3
mode 4
ZVS
ZVS
ZVS, ZCS
ZCS
The transformer data is listed in Table IV. The values of the
leakage inductance Ls, the magnetizing inductance Lh and the
transformer turns ratio n are based on the cantilever-model of
a two winding transformer. An additional degree of freedom
for optimization is the capacitance ratio Γ. A capacitor ratio of
Γ = 11 is used as an optimal value. With denormalization, the
values of the capacitors are
Figure 8. Mode boundaries
modes. Additional modes exists with frequencies F < ζ/(1+ ζ),
which are characterized by more than three subintervals.
V.
DESIGN
An application scenario typical for notebook adapters was
chosen with an input voltage range of 250 V - 400 V and an
output voltage of 19 V. The dc-bus voltage is generated by a
preregulated ac-dc stage with power factor correction and
minimized conducted emissions, in order to fulfill the requirements of EMI regulations. The LCC converter provides a
maximum continuous output power of 100 W. An important
design-goal was a height constraint of 10 mm. Based on the
results obtained by the methods of chapter III, a converter with
the specifications according to Table III was designed and
built.
Cs = 13.6 nF = 6.8 nF + 6.8 nF
(25)
Cp = 66 nF = 33 nF + 33 nF.
(26)
In each case, two capacitors are parallelled, in order not to
exceed the maximal current through the capacitors. Fig. 10
shows the prototype of the designed LCC converter. Normal
litz wire is used for the transformer windings to eliminate hf
losses. To meet safety regulations, triple isolated wire would
be necessary. This modification only has a small effect on the
transformer data and thus, no significant influence on the
design process can be expected. The denormalized voltage
transfer function is shown below in Fig. 11. Additionally, the
area of operation is highlighted. The maximum switching
frequency of 500 kHz is reached in the case of no load with
the maximum input voltage. The minimum frequency of
133 kHz is obtained at full load conditions together with the
minimum input voltage of 250 V. A wide frequency span is
necessary to control the output voltage through the whole load
For the input a half-bridge configuration was chosen, reducing the amplitude of the driving square-wave voltage to
one half of the nominal DC input voltage. A turns ratio of
n = 7 is chosen as design goal for the transformer together
with an inductance value of Ls above 150 µH in order to get
less reactive currents.
TABLE III. CONVERTER SPECIFICATION
Ui,min = 250 V – 400 V
input voltage range
Uo= 19 V
output voltage
Po = 100 W
output power
fmax = 500 kHz
maximum switching frequency
hmax = 10 mm
maximum converter height
Core:
Material:
Primary side
Secondary side
Ls = 192 µH
TABLE IV. TRANSFORMER DATA
TN32/19/13, reduced to 7 mm
3F3
Np = 63
Ns = 9
35 x 0,1 mm
90 x 0,1 mm
Lh = 5830 µH
n=7
20 mm
Figure 10. Prototype
1313
RP = 113.8 mΩ
RS = 9.35 mΩ
and the switching behavior as well as design parameters such
as rms-values can be calculated. It has been shown that ZVS is
tank is designed properly. In the last section of the paper, a
prototype was designed based on a specification typical for
notebook adapters. A ring core was chosen for the magnetic
component with a classical wire wound structure, thus expensive integrated magnetic components are avoided. The prototype reaches a converter height of 10 mm, but also flatter
designs are possible.
VII. REFERENCES
[1]
Figure 11. Voltage transfer function
and input voltage range. In spite of the full bridge
rectification, the converter reaches an efficiency up to 92 % at
the optimum operating point. The corresponding waveforms
for the state variables at the operating point given in Table V
are shown in Fig. 12.
VI.
[2]
[3]
CONCLUSION
One of the main trends driving the development of power
electronics is miniaturization. The family of resonant converters are attractive for high switching frequency operation as
the size of the passive components can be reduced without
sacrificing converter efficiency. Offering virtually lossless
switching without any additional effort and including parasitic
components into the design, they are promising candidates
compared to traditional hard-switching PWM converters. This
paper presented the three-element multiresonant LCC converter with capacitive output filter, which is suitable for lowprofile applications. The converter was analysed in the time
domain and four modes of operation were identified which
occur for a wide frequency range. The steady waveforms of
converter can be calculated by solving a second order transcendental equation system (mode 1 and mode 3) or by
solving a single nonlinear equation (mode 2 and mode 4).
Based on the voltage conversion ratio, the mode boundaries
guaranteed throughout the complete load range if the resonant
TABLE V. OPERATING POINT
Ui = 300 V
input voltage:
Po = 83 W
output load:
f = 190 kHz
switching frequency:
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
Figure 12. Waveforms at the operating point
1314
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URL: www. ferroxcube. com/prod/assets/tn321913.pdf.