Stability Analysis of Flying-capacitor Linear
Amplifier for Wireless Power Transfer system
Rintaro Kusui1*, Keisuke Kusaka2 and Jun-ichi Itoh1
1 Department of Science of Technology Innovation, Nagaoka University of Technology, Nagaoka, Japan
2 Dept. of Electrical, Electronics and Information Engineering, Nagaoka University of Technology, Nagaoka, Japan*E-mail: kusui@stn.nagaokaut.co.jp
Abstract— In this paper, the stability of a flying-capacitor
linear amplifier (FCLA) for wireless power transfer systems
is analyzed by a small-signal equivalent circuit. A wideband
width controller for FCLA is needed to output the highfrequency voltage for a wireless power transfer system. The
output voltage of the FCLA is controlled by the analog gate
voltage with operational amplifiers on a controller. Due tothis, it is necessary to analyze the stability of FCLA, including
the frequency characteristics of the operational amplifiers.
The maximum stable gain is analyzed with the small-signal
equivalent circuit. Then, the harmonics of the output currentof a 4-series FCLA with an unfolder are analyzed by asimulation. As a result, the harmonic components are
reduced by 40 dB or more than the fundamental component.
Keywords— Flying-capacitor linear amplifier, wirelesspower transfer, stability analysis, small-signal equivalentcircuitI.

INTRODUCTION

In recent years, electric vehicles (EVs) have become
increasingly popular to reduce emissions of greenhousegases such as NOx gas, one of the causes of global
warming. The batteries in electric vehicles have a lower
energy density than those in conventional gasolinepowered vehicles, so they require frequent recharging.
Therefore, wireless power transfer (WPT) systems thatenable easy and safe recharging are being actively studied[1–3]. The WPT systems improve user convenience
because charging is initiated without cumbersome power
cable connections. However, the WPT withelectromagnetic induction transmits power through very
weak magnetic coupling, which causes a large leakage
magnetic field around the transmission coil. Leaky
magnetic fields may cause malfunctions of surrounding
electronic equipment and wireless communication failures.
Therefore, it is necessary to satisfy the regulations set byeach country based on the guidelines [3–4] set by theInternational Special Committee on Radio Interference(CISPR). In particular, a revision of the guideline for WPT
systems is under consideration by CISPR11. Specifically,a reduction of about 30 dB in the 150 kHz to 30 MHz range,
which corresponds to the low-order harmonic component
of WPT systems, is being considered [5]. For this reason,
it is necessary to develop a method to reduce leakage
magnetic fields, especially low-order harmoniccomponents, in WPT systems.
Conventional WPT systems use an inverter that outputs
a square wave voltage as the primary power source [6].

Because the square wave voltage output by the inverter
contains low-order harmonic components, a current also
contains low-order harmonic components that flow on thecoils even if resonant circuits are used. This harmonic
current causes leakage magnetic fields containing
harmonic components to be radiated from the transmission
coil.
Magnetic shielding methods using metals, magnetic
materials, and additional windings have been proposed inorder to reduce leakage magnetic fields [7–9]. In thesemethods, a current flows when the leakage magnetic field
is interlinked to the shielding material, thereby reducing
the leakage magnetic field. Thus, the current in theshielding must be larger to obtain a larger shielding effect,
which causes an increase in losses on the WPT system.
The authors have proposed a flying capacitor linear
amplifier (FCLA) as the primary power supply of a WPTsystem [10–13] to reduce the radiation noise harmonics.
The FCLA allows a significant reduction in leakage fieldharmonics generated by the WPT system. The gate drive
circuits for FCLA are connected to each MOSFET to keepbalancing the flying capacitor (FC) voltages while
controlling the output voltage using a proportional
controller. However, a wideband controller is needed tocontrol the high-frequency voltage used in WPT systems.
In this paper, the bandwidth of the FCLA is analyzed
from the small-signal equivalent circuit of the MOSFET,
and the stability of the FCLA, including the bandwidth ofthe controller, is analyzed. From the analysis results, wedemonstrate by simulation that the FCLA with thedesigned control gain can output high-frequency voltage
stably.
II.

FLYING CAPACITOR LINEAR AMPLIFIER

A. Circuit Configuration
Figure 1 shows the configuration of a WPT system. The
FCLA is used in the primary power supply. The power
supply circuit consists of an n-series FCLA with n Nchannel MOSFETs (n-MOSFETs), n diodes connected inseries, and the unfolder. An n-series FCLA outputs a fullwave rectification voltage, and by switching polarity withthe polarity switching circuit, a voltage without harmonics
is applied to the transmission coil. The FCLA output only
a positive current by switching the polarity in the unfolder.
The p-MOSFET, which has inferior characteristics

C. Voltage Balancing for Flying Capacitor
Figure 4 shows the state selection of MOSFETs using
phase-shifted carriers in a 4-serial FCLA; the FCLA keeps
balancing of the FC by appropriately selecting theoperating state of each MOSFET. In this system, the stateof each MOSFET is selected by phase-shifted carriercomparison, which is also used in conventional flying
capacitor converters. The frequency of the carrier isasynchronous to the output frequency so that the
combination of the selected operating states changes in
one output cycle to balance the FC voltage for a long
period. If the FC voltage becomes unbalanced due todisturbances, etc., the voltage controller transitions theoperating states to maintain the voltage balance [13].
III.

STABILITY ANALYSIS OF FCLA

A. Open-loop Characteristic of FCLA
Figure 5 shows the small-signal equivalent circuit of the
FCLA. The FCLA is linearized using the transfer
conductance gm and the output resistance ro representing
the channel length modulation to analyze the frequency
response. In addition, parasitic capacitances among drain,

Q1

vg2

Q2

vg(n 1)

Qn 1

vgn

Qn

ifc2

ifcn

VDC C1 C2 Cn 1

vgsn

Vs1
Vs2
Vs(n 1)Cr1 iL1

Vout

ON state

vac1 Lr1

Lr2

Cr2

iload

vac2

R

Fig. 1. FCLA for WPT system.
Vds =Vgs Vth
Active state

大

Vgs

小
Vgs =Vthdrain-source voltage Vds
OFF state
Fig. 2. Operation region of n-MOSFET.

vout* +

va

K

A1

vout

++

1

0

A2

+

+

Sk_on

+

Sk_offA3

MUX

B. Controllor Configuration
Figure 3 shows a block diagram for controlling output
voltage and operating state. The output voltage iscontrolled by a proportional controller. The proportional
controller outputs a gate-source voltage command value.
Each gate driver inputs the gate source voltage command
output by the controller to MOSFET and adds an offset toselect the operation region. FCLA has the cominations ofoperational state which output same voltage and havedifferent current path. Due to this, the flying-capacitor(FC) voltages are balanced by selecting the appropriate
combination of operating state, i.e., current path.

vg1

ifc1

Drain current Id

compared to the n-MOSFET, operates in an active state
when negative current flows. For this reason, the pMOSFET is replaced by a diode, with the addition of an
unfolder. This improves efficiency and output waveform
distortion.
Figure 2 shows the operating region of an n-MOSFET,
which is divided into on, off, and active states according
to the drain-source and gate-source voltages.
Conventional inverters use only the on-state with low
drain-source voltage and the off-state with no current
flow. On the other hand, the FCLA output any continuous
voltage using one of the MOSFETs connected in series in
the active state. Conventional push-pull linear amplifiers
have a low maximum efficiency of 78.5%, making it
difficult to increase transmission power. On the other
hand, the FCLA is driven at high efficiency by connecting
MOSFETs in series, which reduces the applied voltage to
the MOSFETs and the losses caused by the active
operation. An analog voltage must be input between gate
source to use the MOSFETs in active state for FCLA to
output any continus voltage..

vgsk+
Von Voff
2

1
0
1
0

Sk_on
1
1
0

MUX
Sk_off
1
0
1

vgsk
A1
A2
A3

Fig. 3. Block diagram of output voltage controller.

source and gate are added to the small-signal equivalentcircuit of the MOSFTE to represent the frequency
response. Since the FCLA is based on a source follower
circuit with the class-B operation, the equivalent circuit
has the same form as the source follower circuit. In thecase of a source follower circuit, DC voltage is ignored insmall-signal circuits. On the other hand, in the case of theFCLA, the DC voltage is represented as a disturbancecomponent because the DC voltage appears to vary withthe range of output voltage. Based on these assumptions,the transfer function GFCLA from the input voltage vin
between the gate and source terminals to the output voltage
vout is given by

GFCLA =K n 2 (1 − sTh )(1).

s + 2n s + n 22

where each coefficient K, Th, n,  are expressed by (2–
5).
g rR
K= mo
(2)
ro + RTh =cgd

OFF state
0.75
Active state
0.5
0.25

1
rg

ro R
(CgsCds + CdsCgd + Cgd Cgs )ro + R

ON state(4)
S1 S2 S3 S4ro R(1 + rg gm ) Cgd + Cds + rg (Cgd + Cgs )ro + R =rRrg o ( CgsCds + CdsCgd + Cgd C gs )ro + R

Fig. 4. State selection by comparing with phase-shifted carrier for 4-series

(5)

FCLA.

ig

K pGFCLA
1 + K pGFCLA

=

K c

2
nc

(1 − sTh )s 2 + 2 cnc s + nc2

(6).
where each coefficient is expressed by equations (7–9).
Kc =

K pK
1+ K pKnc = n 1 + K p Kc =2 − K p KnTh
2 1+ K pKGvin

ig d cgd

rg

cgs

vgs

D
id s

ro

id

cds

vdc

id = gmvgs

S
vo ut

R

Fig.5.Small signal equivalent circuit model of FCLA.
vout*

+

Gamp a

Kp

vgs

vout

FCLA

Gamp b

Fig. 6. Control block diagram with op-amp for detection and amplify.
100

Gai n (dB)

Kp_l im = 2843

50
Kp = 100

Kp = 100

Kp = 1-50
0Phase (deg)B. Closed-loop Characteristic of FCLA without limited
bandwidth for operational amplifiers
Figure 6 shows a block diagram of the FCLA voltage
control system with the operational amplifiers which has
limited bandwidth. The FCLA requires an analog voltage
to be applied between the gate and source of each
MOSFET according to the output voltage or current to use
the MOSFETs in an active state. Thus, the control circuit
and gate drive circuit are implemented as analog circuits
using operational amplifiers. For this reason, it isnecessary to analyze the system including the operational
amplifier bandwidth in order to analyze the stability of the
FCLA voltage control. In Fig. 6, Gamp indicates thetransfer function of the op-amps, the number of op-amps
in the amplification and drive sections and the number of
op-amps in the detection part. First, the control responseis evaluated when the ideal op-amp (Gamp = 1) is used. If
the proportional gain of feedback is Kp, the transfer
function Gc from the command value vout* to the output
voltage vout is given by
Gc =Output voltage ref.

(3)

ro

n =

Threshold for OFF state
Threshold for ON state1

Kp = 100-45
Kp = 1

Kp_l im

Kp = 10

-90
-135
-180
102

103

104

105

106

Frequency (kHz)Fig. 7. Close-loop characteristic with ideal op-amps.

(7)
(8)
(9)

Equation (6) shows that the closed-loop characteristicsof the FCLA with proportional control is expressed as thesum of a first-order phase lead element and a second-orderphase delay element. The damping ratio c shown in (9)must be positive for stable control of FCLA. Thus, theproportional gain Kp_lim of the stability limit is expressedas (10), and the control bandwidth nc_max at that time isobtained as equation (11).
2
K p _ lim =
(10)
KnTh
nc _ max = n 1 +

2
nTh

(11)

Figure 7 shows the closed-loop characteristics when the
proportional gain is varied. From Fig. 7 and equations (8)and (9), it can be seen that the natural angular f requency

c increases as the gain is increased. On the other hand,increasing the gain makes the braking coefficient csmaller, resulting in an oscillatory response.

3
GFCLAGamp

1 + GFCLAG

5
amp

+

vout

n

amp s + amp ) + K p K  (1 − sTh ) 
2 5

2

amp

2
n

Kp = 10Gain (dB)50

Kp = 1000

Kp = 1
with ideal op-amp

-50
-100

Kp = 1
with non-ideal op-amp-1500Phase (deg)

-90
-180
-270
-360
-450

-540
101

102

105

106

Kp = 100

Kp = 100-20

Kp = 1
with ideal op-amp

-40
-60

10
amp

(13).
Figure 9 shows the open-loop and closed-loop
characteristics of the system shown in equation (13) when
the gain is varied in (a) and (b), respectively. Figure 9(a)shows that the phase changes significantly due to the
influence of the operational amplifier and that the phasemargin decreases even at low gain. Figure 9(b) shows thatthe bandwidth is similar with and without the op-amp for
low gains where the margins are high, and the bandwidthis sufficiently far from the op-amp bandwidth. It can also
be confirmed that even the stable gain without the op-amp
is unstable due to the decrease in the stability margin.
From the above results, it is confirmed that the
bandwidth of the FCLA can be designed by separating the
bandwidth of the FCLA from that of the op-amp
sufficiently, rather than the transfer characteristics without
considering the op-amp.

103
104
Frequency (kHz)(a) Open-loop characteristic20

Kp = 1
with non-ideal op-amp-80
Kp = 1000Phase (deg)( s + 2 s +  )( s + 22nsourceState selecteFig. 8. Circuit diagram of voltage controller and gate driver.
100

6
K p K n2amp(1 − sTh ) ( s 2 + 2 ampamp s + amp )2Gate drive vg s

Detetion

2 2

=

gate

+
vbias

(12).

s 2 + 2 ampamp s + amp 2

where amp is the natural angular frequency of theoperational amplifier and amp is the damping factor of theoperational amplifier.
Figure 8 shows the control circuit and gate drive circuit
used in this system. The control circuit consists of adetection section, a command value input section, an
amplification section, a state selection section, and a drive
section. Of these, the detection, amplification, and drive
sections affect the control characteristics. Two operational
amplifiers are connected to the detection stage, two to theamplification stage, and one to the drive unit. Therefore,
the transfer function that takes into account the bandwidthof the operational amplifiers is expressed by Equation(13).
G camp =+Gain (dB)amp2

Amplifing+vout *

C. Closed-loop Characteristic of FCLA with limited
bandwidth for op-amps
In this section, the stability of the FCLA including
operational amplifiers with limited bandwidth is analyzed.
The operational amplifier is expressed as a second-order
low-pass filter, so the transfer function becomes (12). In
this system, a current feedback operational amplifier isused, and the bandwidth can be set regardless of the gainof the amplifier circuit by appropriate selection of the
feedback resistor. From the above, all operational
amplifiers constituting the controller are expressed by thetransfer function in equation (12), and the gain is allcollectively denoted as Kp.
Gamp =

Reference input

Kp = 1
with ideal op-amp-180

Kp = 1
with non-ideal op-amp

-360

-540
102

Kp = 10

103

104
105
106
Frequency (kHz)(b) Close-loop characteristic
Fig. 9. Bode diagram of small signal equivalent circuit for FLCA.

IV.

SIMULATION RESULT

Figure 10 shows the operating waveforms of the source
follower circuit. Figure 10 shows the source voltage andgate-source voltage simulated by a linearized small-signal
equivalent circuit of a MOSFET and a SPICE model of a
MOSFET, respectively. The command value is a sine

wave of 85 kHz with an offset of half of the DC voltage,for Class-A operation. From Figure 10, it can be seen thatthe output voltage is equal for both models. The gatesource voltage input to output the same voltage is also
approximately the same, indicating that the linearized
small-signal equivalent circuit is valid. This is because thesmall-signal equivalent circuit ignores the small-signal
equivalent circuit.
Figure 11 shows the circuit configuration of the FCLA
to be simulated. The simulation circuit is a 4-serial FCLA
with a polarity switching circuit and a resistor connected
as a load. 160 V DC voltage is set so that the applied
voltage per MOSFET is about 40 V since the withstandvoltage of the MOSFET is 60 V. The output voltage
command was a full-wave rectified voltage with anamplitude of 160 V and an output frequency of 85 kHz.
Figure 12 shows the operating waveforms. The
simulation is simulated using a second-order LPF so thatthe number of operational amplifiers is the same as thatused in the system. Figure 12 shows that the system
operates stably with the gain analyzed by the small-signal
circuit model. However, near the zero-crossing of the fullwave rectification voltage, the output voltage cannot beadequately controlled and is distorted. This is because avery high control bandwidth is required at the point where
the output voltage becomes discontinuous.
Figure 13 shows the output current harmonics analysis
results of the polarity switching circuit. When applied toa WPT system, it is the current that causes the leakage
magnetic field. Therefore, the output current harmonics of
the unfolder are analyzed. In the simulation, a voltage
with an amplitude of 160 V is applied to the resistance ofthe load24. Therefore, the RMS value of the fundamental
current is 4.71 A (13.5 dBA). From Figure 13, it can be
confirmed that the fundamental component of the current
is 4.71 A. The harmonic component can be reduced bymore than 40 dB.

Vs_spice

Gate-source voltage

Vgs_linear
Vgs_spice

2.4
2.3

5ms
Fig. 10. Operation waveform of class-A source follower with small-signal
equivalent model and MOSFET SPICE model.

ifc1
ifc2
ifcn

vg1

Q1

vg2

Q2

vg3

Q3

vg4

VDC

=160 V

C1 C2

C3

vgs4

MOSFET:
IPB016N06L3

Q4

Vout

vac

R = 24

Fig. 11. Simulation circuit of 4-sereies FCLA with unfolder.
FCLA output voltage and output voltage command

200
160
120
80
40
0

Vout *

Vout
Unfolder output voltage

200
100

va c

0
-100
-200

Unfol der output current

8
6
4
2
0
-2
-4
-6
-8

EXPERIMENTAL RESULT

Figure 14 shows the configuration of the test circuit.
The test circuit is a source follower equivalent to a singleseries FCLA. The DC voltage is set to 30 V because thewithstand voltage of the MOSFET is 60 V. The load is 24
 resistors. The output voltage command is a full-wave
rectified voltage with an amplitude of 30 V and 20 kHz
which is used in WPT systems.
Figure 15 shows the waveforms of the gate-source
voltage, the output voltage of the source follower, andthe output voltage command of the test circuit. This
result shows that the linear amplifier outputs the 20 kHz
full-wave rectified voltage following the command value
by feedback control. The output voltage is continuous
without distortion or oscillation. From these results, it isconfirmed that the controller and GDU for the proposedsystem enable the linear amplifier to stably output thehigh-frequency voltage used in the WPT system.

Vs_line ar

2.5

iac

3ms
Fig. 12. Operation waveform of FCLA with Unfolder.
20

Output current iac [A]V.

Source voltage

40
30
20
10
0
2.6

0

13.5 dBA at Fundamental components-44.3dB

-20
-40
-60
-80
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Number of orderFig. 13. Harmonics analyze of output current.

VI.

CONCLUSION

In this paper, the WPT system with a flying capacitor
linear amplifier is proposed as a primary power supply to
reduce the harmonics of radiated emissions. The FCLA
applies a harmonic-free voltage to the transmission coils
by using MOSFET in active-state. Because of this, theproposed system significantly reduces the harmonics ofthe radiated emissions generated from the transmission
coil. The configuration of the controller and GDU used in
the proposed system is described and their stability
analysis is presented. Furthermore, it is confirmed that thesource follower is equivalent to single-series FCLA
output the high-frequency voltage used for the WPTsystem.
In future work, the verification result of a WPT systemwith 16-series FCLA will be provided. The results of theharmonics analysis of the primary side current will bepresented.

Voltage controller and GDU

Vin

Vref +K

Regioncontrollor

vin

Vgs

Vout

VDC

R

Vbias
Fig. 14. Experimental circuit configuration.

Gate-source voltage Vgs [1V/div]

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1-series FCLA output voltage Vout [10V/div]00

Output voltage command Vout* [0.5V/div]

10 ms/div0
Fig. 15. Operation waveform.