Multiphase Resonant Inverters with Common
Resonant Circuit
Mariusz Bojarski,
Dariusz Czarkowski,
and Francisco de Leon
Qijun Deng
Marian K. Kazimierczuk
Hiroo Sekiya
Wuhan University
Wuhan 430072, China
Wright State University
Dayton, Ohio, USA
Chiba University
Chiba, Japan
NYU Polytechnic School of Engineering
Brooklyn, New York, USA
Abstract—A new family of Class D resonant inverters is
proposed in this paper. Multiple identical series resonant inverters are paralleled using intercell transformers to form phasecontrolled multiphase resonant inverter with a common resonant
circuit. Inverters can operate at constant frequency utilizing
phase-shift control to regulate output. A frequency-domain
analysis of the proposed family is performed. An experimental
prototype of a three-phase resonant inverter with common
resonant circuit was built and extensively tested at an output
power of 550 W and switching frequency of 167 kHz.
SH 0
VI
SL 0
ICT0
ICTN-1
SH1
ICT1
SL1
C
L
SHN-1
I. I NTRODUCTION
Resonant inverters can be operated at a constant switching
frequency with phase control [1]. They can obtain zero-voltage
switching (ZVS) conditions over the whole control range
with proper selection of switching frequency as a function
of resonant frequency. Inverters with full-bridge arrangements
of switches and various resonant circuits were described and
analyzed, for instance, in [2]-[4]. The full-bridge arrangement
results effectively in two switching legs, that is, in two phases.
Generalization of this concept leads to paralleling multiple
phases, which is presented in [5].
This paper proposes multiphase structure with a common
resonant circuit. Intercell transformers are introduced to allow
for phase control and to limit circulating currents among
switching legs. The resonant circuit is selected to be a series
resonant one. This series resonant circuit represents a common
load for the phases as shown in Fig. 1(a). The motivation for
such a structure is to obtain a high full-load current through the
resonant inductor at reduced current stresses for the switches.
It is advantageous in such applications as wireless power
transfer.
II. A NALYSIS OF CIRCUIT
A. Circuit Description
A fixed-frequency phase-controlled multiphase Class D
inverter with a common series resonant circuit is shown in
Fig. 1(a). It consists of a dc input voltage source VI , N
switching legs (phases), one resonant inductors L, resonant
capacitor C, N intercell transformers ICT , and an ac load
RL . Each intercell transformer has two windings which are
connected as shown in the figure.
978-1-4799-3432-4/14/$31.00 ©2014 IEEE
(a)
Z0
RL
ICTN-2
SL N-1
ICT0
i0
ICTN-1
i1
V0
iout
Z1
ICT1
C
V1
iN-1
ZN-1
VN-1
ICTN-2
L
+
RL VRL
_
(b)
Fig. 1.
Fixed-frequency phase-controlled multiphase Class D inverter
with common resonant circuit. (a) Circuit. (b) Equivalent circuit for the
fundamental component.
Each switching leg comprises two switches with antiparallel diodes. The switches in all legs are turned on and
off alternately by rectangular voltage sources at a frequency
f = ω/(2π) with a duty cycle slightly smaller than 50%. To
minimize switching losses and EMI, transistors are turned on
when their voltage is zero, yielding zero turn-on switching
loss. However, zero-voltage-switching turn-on of all the transistors can only be achieved for inductive loads of all switching
legs.
B. Assumptions
The analysis of the fixed-frequency phase-controlled multiphase Class D inverter with a common resonant circuit
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presented in Fig. 1(a) is performed under the following simplifying assumptions:
1) The loaded quality factor QL of the resonant circuit is
high enough (e.g., QL > 3) that the currents at the output of
switching legs ik are sinusoidal.
2) The power MOSFETs are modeled by switches with a
constant ON-resistance rDS .
3) The reactive components of the resonant circuit are linear,
time-invariant, and the operating frequency is much lower than
the self-resonant frequencies of the reactive components.
4) All intercell transformer are identical. They are modeled
as transformers with magnetizing inductance Lmag and leakage inductance Lleak .
C. Voltage Transfer Function
In the inverter shown in Fig. 1(a), the switching legs and the
dc input voltage VI form square-wave voltage sources. Since
the currents ik at the switching leg outputs are sinusoidal, only
the power of the fundamental component of each input voltage
source is transferred to the output. Therefore, the square wave
voltage sources can be replaced by sinusoidal voltage sources
which represent the fundamental components as shown in
Fig. 1(b).
These fundamental components are
2kφ (N − 1)φ
−
(1)
vk = Vm cos ωt +
N
N
where k is from 0 to N − 1, magnitude is
2
VI .
(2)
π
and φ is the normalized phase shift in a range from 0 to π
which gives a full control range. The voltages at the inputs of
the resonant circuits are expressed in the complex domain by
(2k−N +1)φ (N −1)φ
j 2kφ − N
j
N
= Vm e
.
(3)
V k = Vm e N
Vm =
To calculate voltage across load resistance RL , N voltage
sources with intercell transformers are replaced by a single
voltage source
Vmean =
N
−1
X
k=0
Vk
N
(4)
and the inductor Ls = Lleak /N . The voltages across load
resistance is
VRL =
=
=
Vmean RL
RL + jω(L + Ls ) +
1
jωC
Vm RL Nsinφ
sin φ
N
RL + jω(L + Ls ) +
1
jωC
N
N π RL + jω(L + Ls ) +
The resonant frequency is
ωo =
1
jωC
s
MVI
√
sinφ
2RL sin
φ
VR L
N
h
≡ √
=
2VI
N π RL + jω(L + Ls ) +
(6)
1
jωC
i.
(7)
From (6) and (7) one can see that if L + Ls is constant,
resonant frequency ωo and voltage transfer function MVI are
not affected by leakage inductance Lleak . Moreover, in certain
applications, leakage inductances of intercell transformers can
be used as resonant inductors.
The dc-to-ac voltage transfer function of the actual inverter
is
MVIa = ηI MVI
(8)
where ηI is the efficiency of the inverter [1].
Fig. 2 illustrates |MVI | as a function of normalized phase
angle φ for various numbers of phases N . One can see
that |MVI | is not affected much by the number of phases.
Moreover, value of |MVI | at both ends of regulation range
are identical regardless of the number of phases.
D. Currents
Due to use of intercell transformers, the output currents
from legs of inverters are coupled with those of the neighboring phases. Therefore, one can obtain these currents as a
solution of the following system of equations:
N
−1
X
k=0
i.
1
.
C (L + Ls )
Rearrangement of (5) gives the dc-to-ac voltage transfer function of the inverter
=
sinφ
2VI RL sin
φ
h
=
Fig. 2. Magnitude of the dc-to-ac transfer function of the proposed phasedcontrolled Class D inverter MVI as a function of normalized φ for various
numbers of phases N and the same resonant frequency at QL = 3, ω/ωo =
1.15, Lleak /L = 0.1 and N = 2, 3, 6, 12, 24.
Ik − Ik+1 =
(5)
Ik =
VR L
RL
2(Vk − Vk+1 )
jω(4Lmag + 2Lleak )
where k runs from 0 to N − 2, and
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(9)
(10)
high current amplitude at high values of phase angle φ. One
can see that circulating currents decrease when the magnetizing inductance of intercell transformers Lmag increases.
Fig. 4 shows how the number of phases N affects circulating
currents. One can see that for a high number of phases
N , a higher magnetizing inductance of intercell transformers
is needed to keep the same level of circulating currents.
Otherwise, circulating currents will be large for higher number
of phases which may lead to poor efficiency at high values of
phase angle φ.
Fig. 3. Sum
P of magnitudes of output currents of the phased-controlled Class
D inverter
|Ik | as a function of normalized φ for various values of intercell
transformer magnetizing inductance Lmag . Inductance values are normalized
to the resonant inductance. Currents are normalized to the currents at φ = 0.
Shown for N = 3, QL = 3, ω/ωo = 1.15 and Lmag /L = 0.5, 1, 2, 5, 10.
Fig. 4. Sum
P of magnitudes of output currents of the phased-controlled Class
D inverter
|Ik | as a function of normalized φ for various number of phases.
Currents are normalized to the currents at φ = 0. Shown for QL = 3,
ω/ωo = 1.15, Lmag /L = N and N = 2, 3, 6, 12, 24.
(2k − N + 1)φ
.
N
This solution can be obtained iteratively as
ϕk =
(11)
=
Fig. 5. Minimum and maximum ψk as a function of normalized φ for various
values of intercell transformer magnetizing inductance Lmag . Inductance
values are normalized to the resonant inductance. Shown for N = 3, QL = 3,
ω/ωo = 1.15 and Lmag /L = 0.5, 1, 1.5, 2, 3.
To determine whether the switches are loaded capacitively
or inductively, the impedances seen by the switching legs at
the fundamental frequency are calculated and their angles are
examined. The impedance seen by the voltage source vk is
Zk ≡
k−1
NX
−1−k
X
VRL
−
Im +
[m(IN−m−1 − IN−m )] =
RL
m=0
m=1
TABLE I
PARAMETERS OF EXPERIMENTAL PROTOTYPE
The output current is
Io =
k=0
=
VRL
Ik =
=
RL
sinφ
2VI sin
φ
N
h
N π RL + jω(L + Ls ) +
1
jωC
i.
(14)
III. E XPERIMENTAL RESULTS
k−1
NX
−1−k
2m(VN−m−1 − VN−m )
VR L X
−
Im +
. (12)
RL
jω(4Lmag + 2Lleak )
m=0
m=1
N
−1
X
Vk
= |Zk |ejψk .
Ik
Fig. 5 depicts principal arguments ψk as functions of φ. One
can see that only for limited values of Lmag /L all ψk are
within the range of 0o to 180o . This indicates that all inverters
are loaded by inductive loads.
(N − k)Ik =
=
E. Boundary Between Capacitive and Inductive Load
(13)
Fig. 3 and Fig. 4 illustrate output currents of each of N P
legs of
the inverter by showing a sum of current magnitudes
|Ik |.
Fig. 3 shows how intercell transformer magnetizing inductance
Lmag affects circulating currents which are responsible for
Parameter
Value
Unit
Parameter
Value
Unit
N
3
–
L
20
µH
fo
146
kHz
Lmag
24
µH
f
167
kHz
Lleak
2.6
µH
VI
240
V
C
57
nF
RL
1.62
ohm
PO
550
W
To verify experimentally the proposed circuit, a three-phase
phase-controlled resonant inverter with a common resonant
circuit was built. Its parameters and component values were
selected to provide inductive type of load for each leg of
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Fig. 6.
Experimental setup for the three-phase inverter.
Fig. 8.
Current magnitudes |Ik | as a function of normalized φ for
the prototype. Solid lines show calculated values, x’s show corresponding
experimental results.
Fig. 7. Measured voltage and currents waveforms at the output of switching
legs. Channels 1, 2 and 3 are output voltages on switching legs of phases 1,
2 and 3. Channel 4 is the current at the output of switching leg for phase 1.
Measurements were performed for normalized φ = 30o .
the inverter in the whole range of the control phase-shift. A
summary of the design is presented in Table I.
For the inverter prototype, six IRGP4063D IGBT transistors
were used. The inverter was loaded with a full-bridge currentdriven rectifier with resistive load. The rectifier was built using
four DSEI2X101-06A diodes. The dc load was 2 ohms which
results in 1.62 ohms of the corresponding ac load RL . The
experimental setup is presented in Fig. 6. The measured waveforms are shown in Fig. 7. Experimental results are compared
with calculations in Fig. 8 and Fig. 9. For calculations, the
efficiency of the inverter was assumed as 90%. The differences
between experimental and theoretical results are caused by
voltage drops on rectifier diodes and IGBTs, which were not
take into account in calculations.
IV. C ONCLUSIONS
In this paper, a new topology of a resonant inverter is proposed. Frequency domain analysis of the circuit is performed.
Experimental results shows that theoretical analysis is correct.
Application of a multiple number of phases provides a possibility of using many smaller but faster switches. This allows
the inverter to operate at high power with high frequency.
The proposed topology will be further investigated in terms
of topology variations and control strategies. One possible
topology variation is the resonant capacitance distribution. In
that case each switching leg has a separate resonant capacitor
Fig. 9. ψk as a function of normalized φ for the prototype. Solid lines show
calculated values, x’s show corresponding experimental results.
in series with intercell transformers. Then, each capacitor has
a value of 1/N of a single resonant capacitor of the presented
topology. This may help to reduce stresses of resonant capacitors and eliminate a possible dc component of the current
flowing between switching legs. Other topology variations may
include application of series-parallel types of resonant circuits.
The proposed topology introduces also a new possibility
of control. In this paper, symmetrical phase control was
investigated. Another possibility given by the topology is an
asymmetrical phase control. In this case, there are N − 1
independent control inputs for phase control. This opens a way
for control strategy optimization. In the next research step, a
combined phase and frequency control will be investigated.
This type of control should result in significant efficiency improvement, especially in wireless power transfer applications.
R EFERENCES
[1] M. K. Kazimierczuk and D. Czarkowski, Resonant Power Converters,
2nd ed., Ch. 11, Wiley Interscience, 2011
[2] D. Czarkowski and M. K. Kazimierczuk, “Single–capacitor phase–
controlled series resonant converter,” IEEE Trans. Circuits Syst., vol.
CAS–40, pp. 381–391, June 1993.
[3] D. Czarkowski and M. K. Kazimierczuk, “Phase–controlled series–
parallel resonant converter,” IEEE Trans. Power Electronics, vol. PE–8,
pp. 309–319, July 1993.
[4] S. Zheng and D. Czarkowski, ”Modeling and Digital Control of a PhaseControlled Series-Parallel Resonant Converter,” IEEE Trans. Industrial
Electronics, vol. 54, pp. 707-715, April 2007.
[5] C. Braas, F.J. Azcondo, R. Casanueva, ”A Generalized Study of Multiphase Parallel Resonant Inverters for High-Power Applications,” Circuits
and Systems I: Regular Papers, IEEE Transactions on, vol. 55, no. 7, pp.
2128-2138, Aug. 2008.
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