A single phase Hbridge inverter is employed in conventional Inductive Power Transfer (IPT) systems as the primary side power supply. These systems may not be suitable for some high power applications, due to the constraints of the power electronic devices and the cost. A highfrequency cascaded multilevel inverter employed in IPT systems, which is suitable for high power applications, is presented in this paper. The Phase Shift Pulse Width Modulation (PSPWM) method is proposed to realize power regulation and selective harmonic elimination. Explicit solutions against phase shift angle and pulse width are given according to the constraints of the selective harmonic elimination equation and the required voltage to avoid solving nonlinear transcendental equations. The validity of the proposed control approach is verified by the experimental results obtained with a 2kW prototype system. This approach is expected to be useful for high power IPT applications, and the output power of each Hbridge unit is identical by the proposed approach.
NOMENCLATURE
Q
_{1}

Q
_{8}
: power MOSFETs.
D
_{1}

D
_{8}
: MOSFET antiparallel diodes.
L_{p}
: equivalent primary coil inductance.
L_{s}
: equivalent secondary coil inductance.
C_{p}
: resonant compensation capacitance for the primary circuit.
C_{s}
: resonant compensation capacitance for the secondary circuit.
M
: mutual inductance between the primary and secondary coils.
R_{L}
: equivalent load resistance.
E
: DC source voltage.
u
_{H1}
: output voltage of the first Hbridge unit.
u
_{H2}
: output voltage of the second Hbridge unit.
u
_{0}
: synthesized voltage of the cascaded multilevel inverter.
i_{p}
: AC current in the primary transmitting coil.
T_{s}
: period of the fundamental period.
Z_{s}
: reflection impedance of the secondary circuit.
(
A
)* : conjugate operation of
A
.
Re(
A
) : real component operation of
A
.
I. INTRODUCTION
Inductive Power Transfer (IPT) technology can deliver power from a power source to a load with no physical contact
[1]
in order to enhance systems’ flexibility. This technology adopts highfrequency electromagnetic coupling
[2]
,
[3]
, resonant inverters
[4]
, power regulation
[5]
and control theory
[6]
. With IPT technology, devices can be applied in contaminated environments to avoid the influences of ice, dirt, moisture and other chemicals. In addition, IPT systems have been used in numerous applications including the wireless charging of biomedical implants
[7]
, mining applications
[8]
, underwater power supply
[9]
and electric vehicles
[10]

[13]
.
For IPT systems, the resonant inverter is one of the most important components, which operates at relatively high frequency (5100 kHz) compared to 50/60Hz. It generates and maintains a highfrequency resonant current in the primary coil, creating a strong magnetic field, which inductively generates an AC voltage of the same frequency in the secondary pickup. Both the secondary and primary circuits are tuned at the resonant inverter’s operating frequency using compensation capacitors. However, the output power capacity of the single phase Hbridge resonant inverter
[14]
employed in conventional IPT systems is limited by the constraints of the power electronic devices and the cost. Therefore, it may not be able to meet some highpower application requirements, such as electric vehicles and rail transit systems.
Multilevel technology has advantages in terms of reducing the voltage stress of switching devices, limiting the changing rate of voltage (du/dt), degrading the harmonic density of output voltage, and increasing the output power capacity. Cascaded Hbridge multilevel inverters are much easier to implement modularization and they have the merits of increased flexibility and fewer electronic devices needed than diodeclamped and flyingcapacitor multilevel inverters. Compared to a single Hbridge inverter, cascaded multilevel inverters have the following advantages: 1) the cascaded structure uses lowvoltage and lowcost power semiconductor devices to enhance a high power output; 2) the cascaded inverters can be built with modularized components which helps to reduce the manufacture cost; 3) the limitation of a single Hbridge inverter power supply can be removed in aspects like heat dissipation, short time overload; and 4) the Total Harmonic Distortion (THD) of the output voltage of the inverter is lower than that of conventional single Hbridge inverter. Therefore, a cascaded Hbridge multilevel inverter is adopted in this paper as the highfrequency source for IPT systems.
The output power of a cascaded inverter can be directly regulated by altering the switching angle with no need for additional DCDC devices. Furthermore, Selective Harmonic Elimination (SHE) methods can be applied in cascaded inverters to eliminate selective harmonic components of the output voltage. However, SHE equations are nonlinear, transcendental equations, so that no solution can be accomplished
[15]
. Homotopy and continuation theory based SHE methods are proposed to solve transcendental equations
[16]

[18]
. However, these approaches need preliminary offline computation of the switching angles, and real time applications fetch the switching angles from precalculated lookup tables stored in the microcontroller’s internal memory.
Multilevel technology has been adopted in IPT systems in
[19]
,
[20]
. A high frequency multilevel IPT system suitable for high power application was described in
[19]
. However, harmonic distortion, power losses, power factor and efficiency were analyzed in detail without consideration of the power regulation or/and selective harmonic elimination. Recently, a MOSFETIGBT multilevel converter, which can reduce the overall costs and the number of harmonic components
[20]
, was proposed for IPT based high frequency fast chargers. This converter needs to solve transcendental equations instead of providing explicit solutions for the SHE equation. A novel modulation strategy and a phase shift modulation algorithm for the multilevel inverters supplying IPT systems were proposed to minimize the switching loss and coil loss
[21]
. However, they ignored the power regulation.
In order to enhance the output power capacity of IPT systems, this paper employs a cascaded multilevel inverter instead of the conventional single Hbridge for supplying IPT systems. The Phase Shift Pulse Width Modulation (PSPWM) method employed in cascaded multilevel inverters is proposed to realize both power regulation and selective harmonic elimination. Explicit solutions against phase shift angle and pulse width are given according to the constraints of the selective harmonic elimination equation and the required voltage avoiding solving nonlinear transcendental equations. Thus, the proposed method is suitable for realtime applications. The active output power of each Hbridge unit is analyzed. They are identical with the proposed approach.
This paper is organized as follows. In Section II, a detailed description of the structure design and the principle analysis of an IPT system based on a cascaded multilevel inverter are given. Section III describes the selfbalancing output power of each Hbridge. The selective harmonic elimination and power regulation method for cascaded multilevel inverters supplying IPT systems is analyzed in detail in Section IV. Then, Section V shows the steadystate results of simulation and experimental systems operating at 2kW. The conclusion is finally drawn in Section VI.
II. STRUCTURE DESIGN AND PRINCIPLE ANALYSIS OF AN IPT SYSTEM BASED ON A CASCADED INVERTER
 A. Structure Design and Principle Analysis of an IPT System Based on Cascaded Inverters
It is wellknown that the larger the number of cascaded multilevel Hbridge inverters employed, the better the power quality of the IPT output voltage becomes. To enhance the quality of the voltage waveform and the output power in a costeffective way, this paper aims to investigate a fivelevel inverter with two cascaded Hbridges instead of more Hbridges. This will be used as an example to demonstrate the findings of this study while considering the cost of the whole system. The structure of an IPT system based on a cascaded multilevel inverter with an SS (seriesseries) tuned circuit is shown in
Fig. 1
. Each Hbridge inverter is connected with an isolated DC voltage source whose voltage may differ from one another. However, they are set to be the same as
E
in this paper. The DC voltage sources can be batteries or rectified voltages according to the required voltage and power.
The schematic of IPT system based on a cascaded multilevel inverter.
Suppose that the series resonant tank on the primary and secondary pickup sides are both tuned to the operating angular frequency
ω
of the Hbridges, then:
In this situation, the reflection impedance
Z_{s}
of the secondary circuit becomes purely resistive.
 B. Staircase Waveform Synthesis with PSPWM
The output of the cascaded multilevel inverter is synthesized by superimposing two identical staircases with different phase shift angles. The synthesized voltage (
u
_{0}
) can be continuously regulated by Phase Shift Pulse Width Modulation (PSPWM) of the two Hbridge units’ voltages (
u
_{H1}
and
u
_{H2}
) in order to meet various power requirements.
To simplify the analysis, a coordinate is established as shown in
Fig.2
. Line
x
is the symmetrical center line of the positive pulse width of the synthesized voltage. The origin is the joint of line
x
and the horizontal axis (
ωt
), while the fundamental angular frequency is (
ω
), and the fundamental period is (
T_{s}
). Consequently, the pulse width of
u
_{H1}
can be expressed by 2
θ_{L}
, measured in radians. The angular difference between the center of
u
_{0}
and the center of
u
_{H1}
is
θ
_{Δ}
. The phase shift angle is denoted by 2
θ_{Δ}
.
Schematic of staircase synthesis method.
The synthesized voltage
u
_{0}
of the cascaded inverter is the superposition of
u
_{H1}
and
u
_{H2}
. Therefore, the synthesized voltage can be described mathematically as:
The output voltage of each Hbridge
u_{Hi}
(t)(
i
=1, 2) is defined by:
When synthesizing two identical staircases with different phase shift angles, there will be two possible situations: (a)
θ
_{Δ}
+
θ_{L}
≤
π
/2 and (b)
π
/2≤
θ
_{Δ}
+
θ_{L}
.
In the first situation (
θ
_{Δ}
+
θ_{L}
≤
π
/2 ), one switching cycle operation of the cascaded inverter is divided into eight submodes as follows. The driving signals of the MOSFETs are shown in
Fig. 3
. The switchmode transitions and the resonant current pathways in the proposed cascaded inverter are illustrated in
Fig. 4
.
Relevant voltage and current operating waveforms.
Mode transitions during one switching cycle.
[
Mode 1
]:
Q
_{2}
,
Q
_{6}
, and
Q
_{7}
switch on, and
D
_{1}
is naturally forward biased. The synthesized voltage of the inverter turns to be –
E
.
[
Mode 2
]: Before the current commutation (mode 2(a)),
Q
_{2}
and
Q
_{6}
switch on, and
D
_{1}
and
D
_{5}
are naturally forward biased. After the current commutation (mode 2(b)),
Q
_{1}
and
Q
_{5}
switch on, and
D
_{2}
and
D
_{6}
are naturally forward biased. The synthesized voltage turns to be zero.
[
Mode 3
]:
Q
_{1}
,
Q
_{4}
, and
Q
_{5}
switch on, and
D
_{6}
is naturally forward biased. The synthesized voltage becomes
E
.
[
Mode 4
]:
Q
_{1}
,
Q
_{4}
,
Q
_{5}
, and
Q
_{8}
switch on, and the synthesized voltage turns to be 2
E
.
[
Mode 5
]:
Q
_{4}
,
Q
_{5}
, and
Q
_{8}
switch on, and
D
_{3}
is naturally forward biased. The synthesized voltage becomes
E
.
[
Mode 6
]: Before the current commutation (mode 6(a)),
Q
_{4}
and
Q
_{8}
switch on, and
D
_{3}
and
D
_{7}
are naturally forward biased. After the current commutation (mode 6(b)),
Q
_{3}
and
Q
_{7}
switch on, and
D
_{4}
and
D
_{8}
are naturally forward biased. The synthesized voltage becomes zero.
[
Mode 7
]:
Q
_{2}
,
Q
_{3}
, and
Q
_{7}
switch on, and
D
_{8}
is naturally forward biased. The synthesized voltage becomes –
E
;
[
Mode 8
]:
Q
_{2}
,
Q
_{3}
,
Q
_{6}
, and
Q
_{7}
switch on. The synthesized voltage becomes –2
E
.
 C. Staircase Waveform Synthesis with PSPWM
In the second situation (
π
/2≤
θ
_{Δ}
+
θ_{L}
), the zero voltage is produced by synthesizing the two voltages of the two opposite voltage levels produced by two Hbridge units (
H
_{1}
and
H
_{2}
) as shown in the blue line of
Fig. 2
. In this case, one Hbridge unit will charge the other Hbridge unit, and the reverse situation will occur in the second half period. This phenomenon influences the stability of the synthesized voltage and will probably damage the DC sources of each Hbridge unit.
One switching cycle operation of the second situation can also be divided into eight submodes, in which Modes 1, 3, 5, 7, and 8 are exactly the same as those of the first situation. However, there are two different modes between the first and the second situation. A detailed analysis is given below:
[
Mode 2
]: Before the current commutation (
Fig. 5
(a)), the DC source of
H
_{2}
charges the DC source of
H
_{1}
through
D
_{1}
,
D
_{4}
,
Q
_{6}
, and
Q
_{7}
. After the current commutation (
Fig. 5
(b)), the DC source of
H
_{1}
charges the DC source of
H
_{2}
through
Q
_{1}
,
Q
_{4}
,
D
_{5}
, and
D
_{7}
. The voltages of the two Hbridges cancel each other out and the synthesized voltage becomes zero.
Operating modes of the switching devices.
[
Mode 6
]: Before the current commutation (
Fig. 5
(c)), the DC source of
H
_{1}
charges the DC source of
H
_{2}
through
D
_{2}
,
D
_{3}
,
Q
_{5}
, and
Q
_{8}
. After the current commutation (
Fig. 5
(d)), the DC source of
H
_{2}
charges the DC source of
H
_{1}
through
Q
_{2}
,
Q
_{3}
,
D
_{5}
, and
D
_{8}
. The voltages of the two Hbridge cancel each other out and the synthesized voltage becomes zero.
To avoid the aforementioned drawbacks,
θ_{L}
and
θ
_{Δ}
should satisfy the following constraint:
III. ANALYSIS OF THE SELFBALANCING OUTPUT POWER
After applying a Fourier Transformation to
u_{Hi}
(t) , the phasor of the k
^{th}
order harmonic is in the form of:
According to (3) and (6), the phasor of the k
^{th}
order harmonic is provided by:
The current of the primary coil can be derived by:
i_{P}
will be in phase with the fundamental synthesized voltage (
u
_{0}
(1) ) when the circuit is tuned at the resonant frequency. The active power of the fundamental output of Hbridge unit
H
_{1}
can be obtained by:
where
and
I_{P}
denote the phasor and Root Mean Square (RMS) of the converter output current.
Similarly, the active power of the fundamental output of
H
_{2}
can be expressed in terms of
θ_{L}
and
θ
_{Δ}
as:
Comparing (9) and (10), it is obvious that
P
_{H1}
=
P
_{H2}
, which means that the active output power of each Hbridge unit is identical. That is to say, the output power of each Hbridge is selfbalanced.
IV. HARMONIC ELIMINATION AND POWER REGULATION METHOD (HEPRM)
 A. Selective Harmonic Elimination
When the k
^{th}
order harmonic of
u_{o}
(
t
) is selected to be eliminated, the harmonic elimination equation is provided according to (7) by:
The solutions of (11) are given by:
Considering the odd multiples of the k
^{th}
order harmonic and according to (12) and(13), the following are obtained:
The two equations above satisfy (11), which eliminates the k
^{th}
order harmonic and the odd multiples of the k
^{th}
order harmonic. Taking the 3
^{rd}
order harmonic elimination as an example, the corresponding 9
^{th}
, 15
^{th}
, etc. order harmonics can be eliminated.
As the lower harmonic components have more influence than the higher harmonic components in some applications, the 3
^{rd}
harmonic component is selected to be eliminated in this paper. Consequently, the following is obtained:
By solving (16) with the constraint of (5), the formula yields
θ
_{Δ}
=
π
/6 or
θ_{L}
=
π
/3 . When
θ
_{Δ}
=
π
/6 or
θ_{L}
=
π
/3 , the 3
^{rd}
order harmonic and the odd triplen harmonics are eliminated.
 B. Power Regulation with the HEPRM
The IPT output power can be described as a function of
U_{o}
(1) as follows:
Therefore, the output power can be regulated by changing the fundamental RMS
U_{o}
(1) of the synthesized voltage. Thus,
U_{o}
(1) can be derived from (7):
Substitute
θ
_{Δ}
=
π
/6 into (18). Then
U_{o}
(1) can be expressed by:
With the restriction of (5):
In this case,
Similarly, when
θ_{L}
=
π
/3 ,
U_{o}
(1) can be expressed by:
With the restriction of (5), there is:
Thus,
U_{o}
(1) ranges over
In a word, the fundamental RMS of the synthesized voltage can be regulated continuously from 0 to
with the elimination of the 3
^{rd}
harmonic as shown in
Fig. 6
. It is worth noting that this synthesized voltage will degrade to a threelevel staircase when the required synthesized voltage drops below
Consequently, the synthesized voltage can be divided into 3 zones according to aforementioned discussion.
θ_{L} and θ_{Δ} against the RMS of the fundamental voltage.
V. SIMULATION AND EXPERIMENTAL VERIFICATION
The performance of the proposed control method is firstly verified via MATLAB/SIMULINK and then realized and tested in an experimental setting. To be consistent, all of the parameters in both the simulation and the experiment are the same as those given in
Table I
.
THE CONFIGURATION OF THE IPT SYSTEM
THE CONFIGURATION OF THE IPT SYSTEM
 A. Simulation Results
To verify the validity of the proposed control method, a simulation model of the cascaded twoHbridge inverter was constructed in Matlab/Simulink. Detailed configuration information is given in
Table I
.
Zone 1 (
θ
_{Δ}
=
π
/6,0≤
θ_{L}
≤
π
/6 ): the inverter outputs a synthesized voltage over
and appears as a threelevel voltage waveform as shown in
Fig. 7
(a). The 3
^{rd}
order harmonic and the odd triplen harmonics have been completely eliminated, and its fundamental RMS complies with (14) (the experimental values are slightly smaller than the theoretical ones, because there are internal resistances in the power switches). The Total Harmonic Distortion (THD) is rather high (up to 75.32%) under such an operating condition.
The simulation waveform of voltage and current of primary side and secondary side, synthesized voltage harmonic analysis when θ_{Δ}=π/6, θ_{L}=π/9 .
The simulation waveform of voltage and current of the θ_{Δ}=π/6, θ_{L}=π/4 . cascaded inverter when
Zone 2 (
θ
_{Δ}
π
/6,
π
/6≤
θ_{L}
π
/3 ): the amplitude of the synthesized voltage ranges over
and is a fivelevel staircase waveform. In such an operating condition, the THD is relatively low (up to 33.12%) as shown in
Fig. 10
.
The simulation waveform of voltage and current of the cascaded inverter when θ_{Δ}=π/10, θ_{L}=π/3 .
Harmonic density against the synthesized voltage. Input voltage is 50V dc.
Zone 3 ( 0≤
θ
_{Δ}
≤
π
/6,
θ_{L}
=
π
/3 ): the synthesized voltage is a fivelevel staircase ranging in the segment of
In such a case, the THD is rather low (up to 16.73%) as shown in
Fig. 10
.
The THD of the 3
^{rd}
, 5
^{th}
and 7
^{th}
harmonics against
U_{o}
(1) are shown in
Fig. 10
. The THD is low if
U_{o}
(1) is relatively high. However, the THD becomes relatively high if
U_{o}
(1) decreases to a relatively low level. In addition, the 3
^{rd}
order harmonic remains at a low level when the synthesized voltage varies.
 B. Experimental Results
Applying the system configuration in
Table I
, an IPT system is established based on a cascaded multilevel inverter with a TMS320F28335 as the controller and an IRF3710 as the power switch. The primary and secondary coils are wound with Litz wire and the distance between the two coils is set to be 50mm. The experimentation setup is shown in
Fig. 11
. Because of the limited capacity of the MOSFET, the maximum output power of the IPT system is around 2kW. Correspondingly, the rated power of each Hbridge unit is approximately 1kW.
The experimentation setup of a 2 kW IPT system. (a)Experimental setup. (b)Lateral view of the coils. (c)Top view of the coils.
In
Fig. 12
, the inverter output degrades to a threelevel staircase. The 3
^{rd}
order harmonic and its odd triplen harmonics have been completely eliminated. However, the 5
^{th}
and 7
^{th}
order harmonic distortions remain high (up to 25% and 15%, respectively), and
U_{o}
(1) is rather low (24V). The inverter output power is approximately 180W (each of the Hbridge units,
H
_{1}
and
H
_{2}
, has an output power of 90W). The load of the secondary circuit consumes about 120W and the IPT system has an efficiency of around 75%. In
Fig. 13
and
Fig. 14
, the synthesized voltages appear to be fivelevel staircases, and the 3
^{rd}
order harmonic and their odd triplen harmonics are also eliminated with a low THD (35% and 16.91%, respectively). In the two cases, the values of
U_{o}
(1) are set to be 43V and 70V, respectively. Correspondingly, the inverter output powers are 600W (the output power of both Hbridge units,
H
_{1}
and
H
_{2}
, are approximately 300W) and 1600W (800W for each Hbridge unit). The power consumption of the secondary loads are 480W and 1400W, which yield efficiencies of 80% and 87.5%, respectively.
The waveform of voltage and current of primary and secondary sides, synthesized voltage harmonic analysis when θ_{Δ}=π/6, θ_{L}=π/10 .
The waveform of voltage and current of primary and secondary sides, synthesized voltage harmonic analysis when θ_{Δ}/6, θ_{L}=π/5 .
The waveform of voltage and current of primary side and secondary side, output voltage harmonic analysis when θ_{Δ}=π/12, θ_{L}=π/3 .
In this experiment, various values of
θ
_{Δ}
and
θ_{L}
are adopted to verify the performance of the HEPRM. A comparison of the RMS of the experimental and theoretical synthesized voltages is shown in
Fig. 15
. The trends of the changes in the experimental and theoretical values are similar. However, the theoretical values are slightly smaller than the experimental ones because of the internal resistance of the power switches and the equivalent circuit resistance. When the synthesized voltage increases (the current also increases), the experimental values differ greater from the theoretical values.
The curve of experimentation and theoretical value of the fundamental voltage’s RMS against θ_{L} and θ_{Δ} .
VI. CONCLUSION
This paper proposed a HEPRM method for IPT systems to enhance the power capacity of these systems, eliminate selective order harmonics, and simultaneously regulate the IPT output power. This is achieved by changing the phase shift angle and pulse width according to the explicit relation of the two quantities. In this analysis, a fivelevel twoHbridge inverter is taken as an example to provide the explicit expression of the phase shift angle and pulse width. The proposed method is verified with simulation and experimental results. These results allow some conclusions to be made. When compared with traditional IPT systems based on a single phase Hbridge inverter, the system has the following characteristics:
1) The proposed system enhances the IPT output power when using the switches with the same capacity.
2) The system employs the staircase synthesis method instead of highfrequency modulation. Thus, it is applicable to the highfrequency inverters employed in IPT systems, and each cascaded Hbridge unit has the power selfbalancing characteristic.
3) The system can eliminate selective order harmonics of the output voltage. The explicit expression of the phase shift angle and pulse width can be obtained without solving higher transcendental equations. As a result, the computational complexity is reduced and realtime control is facilitated.
4) The system eliminates the 3rd order harmonic and the odd triplen harmonics. It also continuously regulates the output power without an additional DCDC converter.
5) The system’s synthesized voltage appears to be a threelevel staircase with a low fundamental RMS and a high THD. Meanwhile, the synthesized voltage appears to be a fivelevel staircase with a large fundamental RMS and a low THD. This means that the characteristics of the proposed method are suitable for high power applications.
This work was supported by Scientific R&D Program of China Railway Corporation (2014J013B) and National Natural Science Foundation of China (Grant No. 51507147).
BIO
Yong Li was born in Chongqing, China, in 1990. He received his B.S. degree in Electrical Engineering and Automation from Southwest Jiaotong University (SWJTU), Chengdu, China, in 2013, where he is presently working toward his Ph.D. degree in the School of Electrical Engineering. His current research interests include wireless power transfer, and modular multilevel converters supplying IPT systems for high power applications.
Ruikun Mai was born in Guangdong, China, in 1980. He received his B.S. and Ph.D. degrees from the School of Electrical Engineering, Southwest Jiaotong University (SWJTU), Chengdu, China, in 2004 and 2010, respectively. He was with AREVA T&D U.K. Ltd., from 2007 to 2009. He was a Research Associate at The Hong Kong Polytechnic University, Hung Hom, Hong Kong, from 2010 to 2012. He is presently an Associate Professor in the School of Electrical Engineering, Southwest Jiaotong University. His current research interest includes wireless power transfer and its application in railway systems; power system stability and control; and phasor estimator algorithms and their application in PMUs.
Mingkai Yang was born in Chengdu, China, in 1990. He received his B.S. degree in Electronics and Information Engineering from Southwest Jiaotong University (SWJTU), Chengdu, China, in 2012, where he is presently working toward his Ph.D. degree. His current research interests include inductive power transfer and power conversion in rail transit applications.
Zhengyou He was born in Sichuan, China, in 1970. He received his B.S. and M.S. degrees in Computational Mechanics from Chongqing University, Chongqing, China, in 1992 and 1995, respectively. He received his Ph.D. degree in Electrical Engineering from Southwest Jiaotong University, (SWJTU), Chengdu, China, in 2001. He is presently working as a Professor at Southwest Jiaotong University. His current research interests include signal process and information theory applied to power systems, and the application of wavelet transforms in power systems.
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