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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 61, NO. 10, OCTOBER 2014

Locking Range Derivations for Injection-Locked
Class-E Oscillator Applying Phase Reduction Theory
Tomoharu Nagashima, Student Member, IEEE, Xiuqin Wei, Member, IEEE, Hisa-Aki Tanaka, Member, IEEE,and Hiroo Sekiya, Senior Member, IEEE

Abstract—This paper presents a numerical locking-range
prediction for the injection-locked class-E oscillator using thephase reduction theory (PRT). By applying this method to theinjection-locked class-E oscillator designs, which is in the field ofelectrical engineering, the locking ranges of the oscillator on any
injection-signal waveform can be efficiently obtained. The locking
ranges obtained from the proposed method quantitatively agreed
with those obtained from the simulations and circuit experiments,
showing the validity and effectiveness of the locking-range derivation method based on PRT.
Index Terms—Injection-locked class-E oscillator, locking range,
phase reduction theory.

I

I. INTRODUCTION

N RECENT years, resonant class-D [1], class-E [2]–[14],and class-DE [15] power oscillators have been widelyused in various applications. These resonant power oscillators
are oscillated by the feedback voltage transformed from theoutput voltage. Class-E and class-DE oscillators can achieve
high power-conversion efficiencies at high frequencies becauseof the class-E zero-voltage switching and zero-derivative
switching (ZVS/ZDS) operations. Because of their high operational efficiency, these oscillators have many applications
such as electric ballasts [6], [7], dc-dc converters [8], [9], andtransmitters for wireless power transfer [10], [11], and wirelesscommunication [12]. In such applications, a limitation of theoscillator is that the free-running frequency usually has a small
error from the specified frequency owing to component tolerances. Therefore, accurately specified frequency operations
are required for practical usage that considers the radio law
and phase noise. The injection locking technique [13], [14],[16]–[33] is one solution for this problem.
In the injection-locked class-E oscillator [13], [14], the injection circuit is added to the class-E free-running oscillator for
oscillation frequency stabilization. When the feedback voltage

Manuscript received February 12, 2014; revised April 15, 2014; accepted
May 12, 2014. Date of publication June 24, 2014; date of current version
September 25, 2014. This work was supported in part by the Grant-in-Aid for
Scientific Research (No. 13J08797, 26289115, 80632009) of JSPS and theTelecommunications Advancement Foundation, Japan. The authors would liketo thank Enago (www.enago.jp) for the English language review. This paper
was recommended by Associate Editor N. M. Neihart.
T. Nagashima and H. Sekiya are with the Graduate School of Advanced
Integration Science, Chiba University, Chiba, 263-8522 Japan (e-mail:nagashima@chiba-u.jp; sekiya@faculty.chiba-u.jp).
X. Wei is with the Department of Electronics Engineering and Computer Science, Fukuoka University, Fukuoka, 814-0180 Japan (e-mail:
xiuqinwei@fukuoka-u.ac.jp).
H.-A. Tanaka is with the Graduate School of Informatics and Engineering,
The University of Electro-Communications, Tokyo, 182-8585 Japan (e-mail:htanaka@uec.ac.jp).
Digital Object Identifier 10.1109/TCSI.2014.2327276

of the free-running class-E oscillator is synchronized with thesmall injection signal, the frequency of the class-E oscillator
can be fixed at the injection signal frequency. The locking range
prediction for the injection-locked resonant power oscillator isvery important and useful for oscillator design [18], [19]. Thisis because the required injection power for the synchronization
can be estimated in the design process.
Analytical expressions of the locking ranges have been previously reported [13] on the basis of Adler’s equation [16]–[22].
From these analytical expressions, the locking ranges of injection-locked class-E oscillator can be approximately estimated.
This analysis is, however, only valid for the case in which the injection signal is a sinusoidal waveform. For the implementation
point of view, it is useful to predict the locking ranges on various injection-signal waveforms. In previous research [14], thelocking ranges for the injection-locked class-E oscillator withvarious injection signal waveforms were numerically obtained.
It was also shown that the locking ranges depend on the injection-signal waveform [14]; however, the numerical derivationsof locking ranges in this study require high computation cost.
The phase reduction theory (PRT) [24]–[31] is an analysis
method developed in the field of physics used for investigating
the synchronization phenomena. By expressing the oscillator
dynamics as functions of phase variables, the synchronization
range can be easily and accurately obtained. The phase sensitivity function (PSF) represents a phase gradient on the limit
cycle of the oscillator, which is the most important functionused for investigating the synchronization ranges in PRT. The
locking range can be obtained from the convolution integral of
the PSF and the function of the injection signal. In previous research [29]–[31], PRT has been applied to the weakly coupledoscillators, such as Sturart-Landau and van der Pol oscillators,
which shows the utility of PRT. Nevertheless, PRT has neverbeen fully utilized in the research field of power electronics. In
this sense, it is important to show that PRT can be applied to thedesign of resonant power oscillators as an application example.
The objective of this study is to present the feasibility oflocking range predictions of the resonant power oscillator by
PRT. To our knowledge, our study here is the first paper to showa systematic design process with respect to the locking rangesof the resonant oscillator, which can be applied to other injection-locked resonant-power-oscillator designs. It is shown that
PRT is quite useful for obtaining the locking ranges of injection-locked class-E oscillators. By applying PRT, it is possible
to obtain the locking ranges on any injection-signal waveform
at a low computation cost. By conducting simulations and circuit experiments, it was confirmed that the locking ranges predicted by the proposed method quantitatively agree with thoseobtained from the simulations and circuit experiments, thereby

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NAGASHIMA et al.: LOCKING RANGE DERIVATIONS FOR INJECTION-LOCKED CLASS-E OSCILLATOR APPLYING PHASE REDUCTION THEORY

Fig. 1. Circuit descriptions. (a) Injection-locked class-E oscillator. (b) Equivalent circuit.

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Fig. 2. Nominal waveforms of free-running class-E oscillator.

demonstrating the validity and effectiveness of the proposed
locking-range-derivation method based on PRT.
II. INJECTION-LOCKED CLASS-E OSCILLATOR
Fig. 1(a) shows the circuit topology of the injection-lockedclass-E oscillator [13], [14]. This circuit consists of the class-E
free-running oscillator [2]–[5] and injection circuit. The class-E, dc-feed infree-running oscillator has dc-supply voltage, MOSFET as a switching device , shunt capacductance, series resonant circuit, voltage-diitanceand, feedback inductanceforviding capacitancesandfor supplying thephase shifting, and resistances
bias voltage across the gate-to-source of the MOSFET, which
are sufficiently large to neglect the current through them.
Fig. 2 shows example waveforms of the free-running class-E
in Fig. 1(a). Generally,
oscillator, particularly forin the oscillator has a highthe resonant circuit
quality factor . Therefore, the current through the resonant
circuit is regarded as a pure sinusoidal current. The feedback
voltage , which is obtained from the output voltage, controls, the MOSFET is in the “on”
the MOSFET. Duringis the gate threshold voltage.
state, as shown in Fig. 2, where, the MOSFET is in
On the contrary, in the case of
the “off” state. During the switch to the off state, the current
produces a pulse-like shapethrough the shunt capacitanceof voltage across the switch, as shown in Fig. 2. The switchinglosses are reduced to zero by the operating requirements of zerovoltage and zero derivative voltage at the turn-on instant, which
are known as class-E ZVS/ZDS conditions [2]–[15], [34] andare expressed as

the original free-running oscillator. Therefore, the circuit design is simple even when the injection-locking technique is applied. For the injection-locked class-E oscillator used in previous studies [13], [14], a small voltage is injected into the
MOSFET gate terminal, as shown in Fig. 1(a). If the feedback
voltage of the class-E free-running oscillator is synchronized, the oscillation frequency is lockedwith the injection signal, which means the frewith the injection-signal frequency. It is easy toquency of the output voltage is fixed withachieve synchronization as the injection-signal power increases.
However, high power injection affects the waveforms of the
feedback voltage and switch-on duty ratio, which yields the design complexity. It is necessary to conduct the total design of thefree-running oscillator and injection circuits for large perturbation. Additionally, low injection-signal power is good from a
power-added efficiency perspective.

(1)

where
and
denote the angulartime with free-running frequency and the state-variation vector,
respectively. In this paper, for simplicity,

The power dissipations caused in the feedback network are
kept small because the current through the feedback network
is much smaller than the output current .
has an error from
Usually, the free-running frequency
the specified frequency. One of the techniques used to solve
this problem is injection locking [13], [14], [16]–[33]. Because
the injection-signal power is low, it is possible to obtain the injection-locked oscillator by just adding the injection signal toIII. PHASE REDUCTION THEORY FOR INJECTION-LOCKED
OSCILLATOR
PRT [24]–[31] is an analytical method developed in the fieldof physics for investigating the synchronization phenomena.
By expressing the oscillator dynamics as functions of phasevariables, the synchronization range can be obtained easily andaccurately.
A. Phase Function
It is considered that a non-perturbed dynamical system canbe expressed as(2)(3)
is periodic inwith period

as follows:(4), and that
It is assumed that (2) has a solution
this solution has a limit cycle in the phase space, as shown in2906

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 61, NO. 10, OCTOBER 2014

B. Time Averaging for Simplification
The phase difference between the perturbed limit cycle andthe external force is(11)
By eliminatingfrom (10) and (11), we can obtain(12)
Fig. 3. Example of oscillator dynamics. (a) Limit cycle. (b)versus.

Fig. 3(a). For the dynamical system, a phase variable is definedasand, the variation of is much
Becauseis regarded as constantslower than that of . Therefore,
during one period of the dynamical system and (12) can be simplified by averaging as follows:

(5)
where

has

-periodicity. The derivative ofis obtained as(13)
C. Locking Range(6)

The perturbed dynamical system of (2) is expressed as(7)
represents the weak perturbation, which expresses
whereis the angular
the injection signal in this paper, and.
time of perturbation with the angular frequency. The
It is also assumed that (7) has a solution
perturbation drives the trajectory away from the limit cycle .
only slightly deviates from the origHowever, the trajectory
inal trajectory because the injection signal is small if the limit
is stable. Therefore, we can define the phaseoncycle
in the neighborhood of , as shown in Fig. 3(a).
From (6) and (7), the phase dynamics of the perturbed dynamical system can be expressed as

When, the original oscillator is synchronized
as a funcwith the weak forcing signal. Fig. 3(b) shows
for , thetion of . If there is at least one solution ofis a periodic
phase difference is locked in the steady state.
function of because and are the -periodicity functions.
has at least one solution for if
As shown in Fig. 3(b),is in the range of
the
(14)
and
are the minimum and maximum values
where. From (14), the locking range is obtained asof(15)
can be obtained by the convolution integral of the
PSF and forcing signal as given in (13). Therefore, the locking
ranges of the injection-locked oscillator can be derived regardless of , particularly injection-signal waveforms, after the
PSF is obtained.
IV. DESIGN OF INJECTION-LOCKED CLASS-E OSCILLATOR
A. Assumptions(8)
from theWhen the perturbation is small, the deviation of, which isoriginal trajectory is also small. Therefore,
included in the right-hand side of (8), can be approximated as(9)
is known as the phase sensitivity function (PSF). By substituting (9) into (8), we can obtain the phase function of theperturbed dynamical system as(10)

For both the free-running class-E oscillator design and thelocking-range derivations, it is necessary to formulate the circuit equations of the injection-locked class-E oscillator, whichcorresponds to (7). The circuit equations are based on the following assumptions.
a) The MOSFET is used as a switching device , which
has zero switching times, infinite off resistance, and on
resistance .
b) The MOSFET has an equivalent series capacitance andequivalent series resistance (ESR) between the gate
and the source of the MOSFET, which areand ,respectively.
c) IRF530 MOSFET is used as a switching device . Table I
gives the IRF530 MOSFET parameters. In this table,,, andwere obtained from the FET manual,NAGASHIMA et al.: LOCKING RANGE DERIVATIONS FOR INJECTION-LOCKED CLASS-E OSCILLATOR APPLYING PHASE REDUCTION THEORY

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where

TABLE I
IRF530 MOSFET MODEL USED IN DESIGN

(20)
(21)
and

where
is the permissible maximum gate-source
voltage.
andwere measured by HP 16047A at
MHz.
, , andhave an ESR,d) All of the inductances, and, respectively.
e) The shunt capacitanceincludes the MOSFET
drain-to-source parasitic capacitances.
f) All of the passive elements including switch-on resistance, MOSFET parasitic capacitances, and ESRs of inductances work as linear elements.
g) The MOSFET turns on at
and turns off at
inthe free-running operation.
From above assumptions, we have an equivalent circuit of theinjection-locked class-E oscillator, as shown in Fig. 1(b).
B. Circuit Equations
From the equivalent circuit in Fig. 1(b), the circuit equations
are expressed as the equation at the bottom of the page.
In (16),is the equivalent resistance of the switch. According to the assumption a) and f),
is expressed as
for
for
Now,

and

(17)

are defined as

(18)
and
(19)

Using

and

, (16) is expressed as

(22)
(23)

which corresponds to (7).
In this paper, numerical waveforms are obtained by solving
the differential equations with the forth-order Runge-Kutta
method.
C. Free-Running Class-E Oscillator Design
The design specifications for the design example are given
as follows: operating frequency, input voltage, output quality factor, output, RMS voltage across the output resistanceresistance, ratio of the output resonant inductance to the dc-feedinductance,,,. Because
and
,
and
and
. Additionally,,, andwere obtained as given in
Table I, and all of the ESRs of the inductances are zero in the design process. The class-E free-running oscillator for achieving
the class-E ZVS/ZDS conditions can be obtained, as given in
Table II, by following design methods previously reported [5],[8]. It should be noted thatis given for the free-running oscillator design.
Fig. 4 shows the numerical and experimental waveforms of
the free-running class-E oscillator, and Table II gives the measurement results. All experimental element values were measured using a HP4284A LCR meter, and the voltages and cur-

(16)

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TABLE II
DESIGN VALUES OF INJECTION-LOCKED CLASS-E OSCILLATOR

Fig. 5. Phase shift due to impulse perturbation at

Fig. 6.

.

of the designed injection-locked class-E oscillator for.

Fig. 4. Waveforms of designed injection-locked class-E oscillator. (a) Numerical waveforms. (b) Experimental waveforms.

rents were measured using a Agilent 3458A multimeter. The
parasitic capacitance of the IRF530 MOSFET was estimated to
be 270 pF, which was obtained from a previous study [34]. Fig. 4
and Table II show that the experimental results quantitatively
agreed with the numerical predictions. In the free-running oscillator, the free-running frequency is 1.0077 MHz, which wasmeasured by the multimeter.
V. DERIVATION OF LOCKING RANGESIn this section, the locking ranges for the injection-lockedclass-E oscillator are estimated on the basis of PRT.
A. Definition of Impulse Sensitivity Function and Linear
Response Region
For deriving the locking ranges, it is necessary to obtain the
PSF as stated in Section III because
is obtained by the
convolution integral of the PSF and the injection signal. The
PSF can be regarded as the impulse response of the phase [18],[20], [21], [23], [28]. In the proposed locking-range derivation
method, the PSF is obtained numerically.
It is assumed that the free-running class-E oscillator is in thesteady state because the phase in (7) is defined in the steadystate. An impulsive voltage perturbation can be injected at any
phase of the one-cycle oscillation. Fig. 5 shows an exampleof the trajectory when the impulse perturbation is injected at

Fig. 7. Impulse sensitivity function (ISF) of the designed injection-lockedclass-E oscillator.

. After injecting the pulse waveform, the waveforms return to the steady state via transient response. However, a phaseshift
remains, as shown in Fig. 5. In particular, the phaseshift
is expressed as a function of the injection phase .
This function is known as the impulse sensitivity function (ISF)[18], [20], [21], [23], [28]. The ISF is apparently equivalent to
the PSF because the PSF represents the phase gradient on thelimit cycle of the oscillator. When the ISF is determined
numerically, the injection-pulse widthand heightare important because the phase shift depends on the impulse-perturbation injection phasein addition to the pulse shape. If thepulse is narrow and low, the phase shift is in proportion to thepulse area. The region whereis in proportion tois defined as the linear response region (LRR). When the ISF is

NAGASHIMA et al.: LOCKING RANGE DERIVATIONS FOR INJECTION-LOCKED CLASS-E OSCILLATOR APPLYING PHASE REDUCTION THEORY

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Fig. 8. Experimental, calculated (phase reduction theory; PRT), and numerical locking ranges. (a) Sinusoidal-wave injection. (b) Rectangular-wave injection. (c)Triangular-wave injection.

obtained in this region, it is uniquely determined by normalizing
the phase shift by the pulse area.
In this paper,
is obtained from
(24)
where

is the ISF.

B. Numerical Derivation of ISF
Fig. 6 shows
andis

as a function of the impulse amplitude for. In particular, the injection voltage

for
(25)
for other
The pulse-shape waveforms were injected at the instant ofswitch turn-on, and the phase shifts through thetransient state with 5000 periods was plotted. It was confirmedthat 5000 periods are sufficiently long for the oscillator to
be in the steady state. It is seen from the extended region in
Fig. 6 that the phase shift varies linearly with the pulse height.
From this figure, the impulse signal that has,
particularly, andis used for the
ISF derivation.
Fig. 7 shows the ISF of the designed class-E oscillator. The
ISF is defined as the phase shift by applying a unit impulse as afunction of injection point. It is ensured that the phase shift is in
proportion to the pulse area because it is in the LRR, as shownin Fig. 6. Therefore, the ISF is obtained from

Fig. 9. Experimental and numerical output power and power conversion efficiency in the locking range of the sinusoidal-wave injection at.

C. Locking Ranges of Designed Oscillator
In this paper, the locking ranges of the class-E oscillator for
three types of injection signals, which are sinusoidal, rectangular, and triangle waveform signals, are predicted. The injection signals are expressed as1) sinusoidal wave
(27)2) rectangular wave

(26),
It is seen in Fig. 7 that a discontinuous point appears atparticularly switch turn-off instant, because there was a jump in
the switch current at the turn-off instant, as shown in Fig. 2. In
at, particularly turn-on instant, was contincontrast,
uous, as shown in Fig. 7. This is because the class-E ZVS/ZDS
conditions, which are given in (1), were achieved. Because of
these conditions, both the switch current and voltage waveforms
were continuous at
.

for
for

(28)

and
3) triangle wave
for
for
for
where

,

, and

(29)
are the peak values of each waveforms.

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 61, NO. 10, OCTOBER 2014

be adapted to any practical oscillators with many variables.
In addition, the accurate locking ranges can be obtained byformulating circuit model in detail. 2) Locking ranges on any
injection-signal waveform can be efficiently obtained because
the ISF is independent of the injection-signal waveform.
Fig. 9 shows the output power and power conversion efficiency as a function of frequency ratio for the sinusoidal-wave
injection at. For obtaining
the numerical results in Fig. 9, the ESRs of the inductances
given in Table II were considered. In this figure, it is shownthat a power conversion efficiency of more than 93% can be
kept in all locking ranges. This is because the injection signal
does not affect the switch-voltage waveform, and the class-E
ZVS/ZDS conditions can be achieved. Fig. 10 shows the numerical and measured waveforms for
andat. It is shown in the figure that the class-E
ZVS/ZDS conditions were satisfied in both numerical and measured switch-voltage waveforms. The figure also shows thatthe oscillation frequencies were locked with the injection-signal
frequencies in both Fig. 10(a) and (b). Additionally, it was confirmed that the phase shift between
and , particularly
in Fig. 10(a), is different from that in Fig. 10(b).
VI. CONCLUSION

Fig. 10. Waveforms at.

for (a)and (b)can be obtained numerically by the convolution integralof the ISF shown in Fig. 7 and the injection signal. For the numerical calculations, the 50th order Fourier expansion of the ISF
is applied to the convolution integral calculations with the injection waveforms, which provides an accurate approximation.
Fig. 8 shows the locking ranges of the designed injection-locked class-E oscillator for each injection waveform. It
is shown in (24) thatvaries linearly with the injection
signal. Therefore, the theoretical locking ranges increase inproportion to the peak voltage of the injection voltages. The
numerical results in Fig. 8 were obtained by the calculation
algorithm in [14]. It is shown in Fig. 8 that the theoreticallocking ranges quantitatively agree with the experimental andthe numerical ranges, regardless of injection-signal waveforms, which validates the effectiveness and usefulness of thelocking-range predictions based on PRT. Because (15) and (16)are normalized by the free-running frequency, it can be statedthat the locking ranges in Fig. 8 are independent of the specified
operating frequency.
Fig. 8(a) also shows the analytical locking range from a previous study [13] based on Adler’s equation [16]. It can be statedthat the proposed locking-range prediction method gives moreaccurate locking ranges than the values from the analyticalmethod [13]. Compared with the analytical derivation methodof locking ranges [13], [32], [33], the numerical derivation
method presented in this paper includes the following advantages: 1) When the circuit equations are formulated, the locking
ranges can be estimated following the similar calculations steps
presented in this paper. In particular, the proposed method canThis paper has presented the numerical derivation methodof the locking range for the injection-locked class-E oscillator
using PRT. It is possible to obtain the locking ranges on any
injection-signal waveform accurately from PRT with low computation cost. The predicted locking ranges from the proposed
locking-range derivation method quantitatively agree with thoseobtained from the simulations and circuit experiments, whichshows the validity of this method.
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2911

Tomoharu Nagashima (S’11) was born in Saitama,
Japan, on February 4, 1989. He received the B.E. and
M.E. degrees from Chiba University, Chiba, Japan, in2011 and 2012, respectively. He is currently working
toward the Ph.D. degree at Chiba University.
His current research interest is in resonant dc/dc
power converters, dc/ac inverters, and wireless powertransfer systems.

Xiuqin Wei (S’10–M’12) was born in Fujian, China,
on December 7, 1983. She received the B.E. degreefrom Fuzhou University, China, in 2005, and the
Ph.D. degree from Chiba University, Japan, in 2012.
Since April 2012, she has been with Fukuoka University, Fukuoka, Japan, and she is now an Assistant Professor at the Department of Electronics Engineering and Computer Science. Her research interests include high frequency power amplifiers.

Hisa-Aki Tanaka (S’93–M’95) received the B.E.
degree in electrical engineering and the M.E. and
D.E. degrees in electronics and communicationengineering from Waseda University, Tokyo, Japan,
in 1990, 1992, and 1995, respectively.
From 1996 to 1997, he was a visiting researcher
with EECS Department, UC Berkeley. From 1997
to 2001, he was with the Sony Computer Science
Laboratory. In 2001, he joined the Department ofElectronic Engineering, Faculty of Electro-Communications, University of Electro-Communications,Tokyo, Japan, as an Associate Professor. His current research interests are insynchronization issues in computing and communication systems as well as
nonlinear physics.
Dr. Tanaka received the 1995–1997 Research Fellowships of the Japan Society for the Promotion for Young Scientists. He is the holder of twelve patents.
He was the recipient of the 2007, 2008, and the 2010 Best Paper Award respectively from the Telecommunications Advancement Foundation.

Hiroo Sekiya (S’97–M’01–SM’11) was born inTokyo, Japan, on July 5, 1973. He received the B.E.,
M.E., and Ph.D. degrees in electrical engineeringfrom Keio University, Yokohama, Japan, in 1996,
1998, and 2001, respectively.
Since April 2001, he has been with Chiba University, Chiba, Japan, and now he is an Associate
Professor at Graduate School of Advanced Integration Science. From February 2008 to February
2010, he was with the Department of ElectricalEngineering, Wright State University, Dayton, OH,
USA as a Visiting Scholar. His research interests include high-frequency
high-efficiency tuned power amplifiers, resonant dc/dc power converters, dc/ac
inverters, and digital signal processing for wireless communications.
Dr. Sekiya is a member of the Institute of Electronics, Information, and Communication Engineers (IEICE) of Japan, the Information Processing Society ofJapan (IPSJ), and the Research Institute of Signal Processing (RISP), Japan.