LETTER • OPEN ACCESS

Maximisation of synchronisability under low
injection power
To cite this article: Hisa-Aki Tanaka et al 2024 EPL 146 52001

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June 2024
EPL, 146 (2024) 52001
doi: 10.1209/0295-5075/ad4414

www.epljournal.org

Maximisation of synchronisability under low injection power
Hisa-Aki Tanaka1(a)and Hiroo Sekiya3, Yoji Yabe1 , Somei Suga1 , Akira Keida1 , Kai Maeda1 , Fumito Mori2

1

The University of Electro-Communications - Tokyo 182-8585, Japan
Kyushu University - Fukuoka 815-8540, Japan
3
Chiba University - Chiba 263-8522, Japan2

received 12 March 2024; accepted in ﬁnal form 26 April 2024
published online 2 July 2024
Abstract – Synchronisability of limit cycle oscillators has been measured by the width of thesynchronous frequency band, known as the Arnold tongue, concerning external forcing. We clarify a fundamental limit on maximizing this synchronisability within a speciﬁed extra low powerbudget, which underlies an important and ubiquitous problem in nonlinear science related toan eﬃcient synchronisation of weakly forced nonlinear oscillators. In this letter, injection-locked
Class-E oscillators are considered as a practical case study, and we systematically analyse theirpower consumption; our observations demonstrate the independence of power consumption in theoscillator from power consumption in the injection circuit and verify the dependency of power
consumption in the oscillator solely on its oscillation frequency. These systematic observations,followed by the mathematical optimisation establish the existence of a fundamental limit on synchronisability, validated through systematic circuit simulations. The results oﬀer insights into theenergetics of synchronisation for a speciﬁc class of injection-locked oscillators.
open accessc 2024 The author(s)Copyright Published by the EPLA under the terms of the Creative Commons Attribution 4.0 International License(CC BY). Further distribution of this work must maintain attribution to the author(s) and the publishedarticle’s title, journal citation, and DOI.

Introduction. – Synchronisation (injection locking)of oscillators to an external forcing (injection signal) ofwidely diﬀerent waveforms is often used to provide meansfor communicating among oscillators or linking between
oscillators and their surrounding environment. In manybranches of science and engineering, methods for eﬃcientsynchronisation, including explosive synchronisation [1],have been developed in recent years because synchronisability plays a prominent role in functioning of engineered
systems and even in the social sciences [2]. The synchronisability is measured by the width of the Arnold
tongue (cf. [3], p. 52); that is, the locking range (synchronous frequency band for the external forcing) in which
the injection-locked state is maintained. During injection
locking, the extra power is consumed through the injection of external forcing and this extra power is a matter ofpractical importance, for instance, in battery-limited electrical oscillators (e.g., heart pacemakers). However, there
seems to be insuﬃcient knowledge on this extra power; as
far as the authors know, even for the widely known van(a) E-mail: htanaka@uec.ac.jp (corresponding author)

der Pol oscillator, the extra power for maintaining injection locking has not been documented, despite of its long
history. Motivated by recent studies conducted from athermodynamic paradigm that clarify the thermodynamic
uncertainty relation of the noisy van del Pol oscillator [4]as well as the energetics of the Stirling heat engine [5,6],
coupled microscale phase oscillators [7], and a synchronising oscillator model [8], this study aims to provide analternative, deterministic viewpoint on the energetics ofinjection locking, which is based on the observation froman existing practical electrical circuit.
Several recent studies have initially articulated the energetics of synchronisation from a stochastic thermodynamical viewpoint. For instance, [8] developed a generalapproach that uniﬁes modelling and theoretical analysis
for circadian systems. Notably, in their mathematicalmodel and theory, the power consumption of the oscillator itself is independent from the extra power maintaining its synchronisation, which is hypothesised to relate to
the amount of adenosine triphosphate (ATP) consumed
throughout the oscillation cycles. Meanwhile, [7] ﬁrst
clariﬁed the generic mechanism intrinsic to the mutually52001-p1

Hisa-Aki Tanaka et al.
coupled ﬂuctuating phase oscillators with ﬁxed circular
trajectories. Interestingly, as in [8], the power consumption of the oscillator itself is separated and independent of
the extra power for mutual synchronisation (cf. eq. (24)in [7]).
As opposed to these microscale systems in which
stochastic thermodynamics becomes essential, how canthe energetics of synchronisation be developed in deterministic systems? The van der Pol (vdP) oscillator isan example for such deterministic systems, for which vander Pol ﬁrst observed injection locking [9]. The equation
for the vdP oscillator, derived by van der Pol himself, issimple, but the analysis of its power consumption is notstraightforward due to nonlinear resistance in the oscillator. As opposed to the vdP oscillator, Class-E ampliﬁers/oscillators, which have been widely used as power
circuits [10–12], might be exotic to physicists. However,
the Class-E oscillators are suitable for theoretical as wellas numerical analysis of power consumption from the following reasons: i) Their power consumption analysis isstraightforward because all the circuit elements are linear
except for a single transistor; this transistor can be modelled as an ON/OFF switch in the working condition ofthis study. ii) Their nonlinear dynamics has been systematically analysed; a circuit experiment, numerical circuit
simulations, and a phase reduction (to the phase equation
below) have been conﬁrmed to provide consistent results
under various parameter settings [11,12]. This suitability
for analysis motivated us to develop a case study concerning the energetics of synchronisation and a fundamental
limit of synchronisability.
In this letter, we consider the extra power consumption
required to maintain injection locking and address the following two problems that have been beyond the scope of
the previous studies including [11,12].
Question 1 : Can the power consumption of the oscillator itself and the extra power required for synchronisation be separated? Namely, is the extra power
maintaining synchronisation given independently of thepower consumption of the oscillator as it was in theenergetics of [7,8]?
Question 2 : Within a relatively small extra power budget for synchronisation, is it possible to maximise the synchronisability of the oscillator? Similarly, is it possible tominimise the extra injection power to achieve a certain
level of synchronisability? Namely, does a fundamental
limit of synchronisability exist for this particular example?
The answers to these questions are given as Answer 1 andAnswer 2, respectively, at the end of this letter.
The rest of this letter is organised as follows. First,
to answer the above questions, we provide a basic explanation of the circuit. Second, the power consumption in
the oscillator itself and that in the injection circuit added
to it are then systematically analysed through careful circuit simulations. Third, the simulation results demonstrate that these two power consumption sources can betreated separately in the injection-locked state; this en-

Fig. 1: The circuit system analysed in this study. (a) Injectionlocked Class-E oscillator; the Class-E oscillator (main circuit)receives the injection signal (vinj ) through Rinj and Cinj .
(b) Equivalent circuit; the MOSFET is eﬀectively modelled
as an ON/OFF switch. (c) Power ﬂow; the shaded (blank)arrows indicate the power ﬂowing into (out of) the circuit.

ables us to theoretically derive the optimal injection current waveform that maximises the locking range with theaid of the optimisation method in [13–15]. Thus, the fundamental limit of this synchronisability is proven to exist.
Finally, we verify, through systematic, numerical analysis
and experiment, that the theoretically determined optimal injection current is indeed optimal, which validates
the existence of the fundamental limit.
Injection-locked Class-E oscillator. – We show,
in ﬁg. 1, the injection-locked Class-E oscillator [10–12],
which is made up of a Class-E oscillator (the right part inﬁg. 1(a)) and an injection circuit (the left part in ﬁg. 1(a));
ﬁrst, the circuit elements of the Class-E oscillator are as
follows: i) DC-supply voltage VDD , ii) DC-feed inductance
LC , iii) switching devise S; the metal-oxide-semiconductorﬁeld-eﬀect transistor (MOSFET) with rS and gate threshold voltage Vth , iv) shunt capacitance CS , v) series resonant circuit L0 -C0 -R, vi) voltage-dividing capacitances
C1 and C2 , vii) feedback inductance Lf for phase shifting, and viii) resistors Rd1 and Rd2 , which are used tosupply the bias voltage from the gate to the source ofthe MOSFET. These resistances are suﬃciently large tokeep the current through them negligible. In addition, forsimplicity, we set rLC = 0 Ω, rL0 = 0 Ω, and rLf = 0 Ω,
as in [12].
Second, the injection circuit is a simple RC circuit1 ,as shown in ﬁg. 1(a) and (b); we set Rinj = 20 kΩ andCinj = 0.1 μF, similar to the ones in [12], in this study.
1 This circuit conﬁguration is similar to that of the membrane
voltage clamping used to measure the membrane potential of neurons
and cardiomyocytes in which Rinj is also large and Cinj is small [16].

52001-p2

Maximisation of synchronisability under low injection power
Third, the circuit parameters in this study are tuned
to almost the same values as those in [12], which wereconﬁrmed to satisfy the zero-voltage switching (ZVS)
and zero-derivative switching (ZDS) conditions. TheseZVS/ZDS conditions, which are essential for reducing
the power consumption in the switching MOSFET eﬀectively to zero, are detailed in sect. (A) of the appendix in
the Supplemental Material Supplementarymaterial.pdf(SM).

from the injection circuit to the main circuit is
Psub = iinj (θ)vf (θ).

(1)

Let Ptot be the sum of the power dissipated by the above
ﬁve resistors in the main circuit, which is equal to thepower supplied to the main circuit (Pmain + Psub ). Then,
Ptot = Pmain + Psub always holds; note that the valuesof Ptot , Pmain , and Psub obtained from numerical circuit
simulations in table B.2 satisfy the relation Ptot = Pmain +Power consumption in each part of the oscilla- Psub with high accuracy, which veriﬁes the validity andtor. – In what follows, ﬁrst the power ﬂow in the circuit is correctness of the circuit simulations.
Next, we consider the remaining power ﬂow in the injecconsidered; we start from analysing each power consumption
circuit (the left part in ﬁg. 1(c)); i) in the injectiontion at the resistor Rinj in the injection circuit in ﬁg. 1(a)lockedstate, the power Pvinj supplied from the injection
and those at the ﬁve resistors R, rg , Rd1 , Rd2 , and rS
voltagevinj to the injection circuit isin the main circuit shown in ﬁg. 1(b) are obtained subsequently. Next, we clarify that the power consumption in
Pvinj = iinj (θ)vinj (θ).
(2)
the main circuit is dependent upon the synchronised oscillation frequency under injection locking, where the os- ii) Similarly, the power consumption Pinj at the resistor
cillation frequency is synchronised to the frequency of the Rinj is
injection signal. To this end, we observe that two metrics
Pinj = i2inj (θ)Rinj .
(3)—the injection power (Pinj in eq. (3)) and total power
consumption in the main circuit (Ptot )— are independent iii) P , which amounts to the power supply P
subvinj miand separate.
nus the power consumption Pinj in the injection circuit,
To begin with, we denote T (= 2π/Ω) s and Ω rad/s as is supplied to the main circuit. Then P
sub = Pvinj − Pinj
the injection-locked oscillation period and the frequency always holds; the values of P
vinj , Pinj , and Psub obtainedof the injection signal, respectively. In addition, ω is the from numerical circuit simulations in tables B.2 and B.3
natural frequency of the free-running oscillator (without of the SM satisfy P− Pinj with high accuracy.
sub = Pv
injection signal) and the frequency detuning Δω ≡ ω − Ω Figure 1(c) summarises theinj
power ﬂow in the circuit deis the diﬀerence between them. The circuit equation, scribed above.
which has been provided in [10], is obtained from stanFinally, as shown in ﬁg. 2 (and ﬁg. B.2 in the SM), thedard nodal analysis for the equivalent circuit illustrated input frequency Ω dependence of the power consumption
in ﬁg. 1(b); all the circuit parameter values are listed in the circuit is analysed using numerical circuit simulain table B.1 in sect. (B) in the SM, which satisfy the tions; we vary Ω centered at Ω/2π = ω/2π = 1.979 MHz,ZVS/ZDS conditions. For the numerical integration of i.e., Δω = 0. For each value of Ω, by controlling thethe circuit equation, we employed the fourth-order Runge- magnitude of i , the power consumption P
inj
inj in the
Kutta method with time step Δt = 2π/(2000ω) ( 3.1 ns) injection circuit is set to a constant value; in eq. (3)in this study. The source code in C used to deﬁne the with R
2−8,
inj = 20 kΩ, by setting iinj  = 4.5 × 10
circuit equation and the circuit parameters is provided in
−4
−8
2
3
Pinj = 9.0 × 10 W (= 4.5 × 10 A × 20.0 × 10 Ω)
the supplementary ﬁle function.c.
always holds. Another essential source code in sect. (C)We start by considering the power ﬂow in the main cirfor generating data in ﬁg. 2 and ﬁg. B.2 is provided in thecuit (the right part in ﬁg. 1(c)). Hereafter, we use thesupplementary ﬁle main.c. In this study, we use the in2π
abbreviation: · ≡ (2π)−1 0 ·dθ with θ ≡ Ωt, and we jected current i
inj for a sine wave, a square wave, a broad
1
deﬁne the norm by ||f ||2 ≡ f 2 (θ) 2 for any function f , bipolar pulse wave, and the optimal waveform (eventuwith ||f ||2 < ∞. Then, in the injection-locked states, the ally obtained as eq. (13)) satisfying i2inj  = 4.5 × 10−8 .
power Pmain supplied from the DC voltage source (VDD ) is The corresponding waveforms are shown in ﬁg. B.1 insect. (B) in the SM. The choice of somewhat broad pulse
1 T(duty ratio 20 (%)) is due to the following reasons: i) The
Pmain =
ic (Ωt)VDD dt = ic (θ)VDD .
T 0magnitude of the injected current needs to be reduced toprotect the circuit in practical circuit environments, andIn contrast, the power consumption Pout (output power)ii) the same broad bipolar pulse has been used in the realat the resistor R iscircuit experiments of [12]. The power consumption of
each resistor in the main circuit for each waveform for
1 T −1 2
R vo (Ωt)dt = R−1 vo2 (θ).
Pout =
Δω = 0 is shown in table B.2 of the appendix in the SM;
T 0
for the respective injection currents of sine, square, pulse,Power dissipation in other resistors rg , Rd1 , Rd2 , and rS is and optimum waveforms, the power consumption in R, rg ,obtained similarly. Furthermore, the power Psub supplied Rd1 , Rd2 , rS and its sum Ptot , power supply Pmain , and52001-p3

Hisa-Aki Tanaka et al.

Fig. 2: The oscillation frequency dependence of the power consumption in each resistor: (a) R, (b) rg , (c) rS , and its sum(d) Ptot ; (a) is the major part of (d), while both (b) and (c)are minor parts of (d). “Injection-locked” and “out of lock”
surrounded by dotted lines indicate injection-locked and asynchronous states, respectively. In addition, +, ×, ∗, and 
denote the data of sinusoidal, optimal, square, and pulse wave
injection current iinj , as shown in the SM, in ﬁgs. B.1 (a),(b), (c), and (d), respectively. Note: in the injection-locked
state, the power consumption (averaged over T (= 2π/Ω)) does
not depend on the timing of observation because the system
is purely periodic in T . On the other hand, in the out oflock (asynchronous) condition, the system exhibits intermittent phase slips (cf. [3], pp. 53, 54, 81–83) and is no longer
purely periodic, which results in a spread in the time-dependent
values of the power consumptions, as observed in the abovegraphs.

Psub supplied from the injection circuit to the main circuit, and Pvinj supplied by vinj are listed. Furthermore,ﬁg. 2 (and ﬁg. B.2 in the SM) compare, respectively, thepower consumption of each resistor, Psub and Pvinj whenthe value of Ω is varied in and out of the locking range,
centered at Δω = 0. Note that, for each iinj , the injection
locking is directly observed by comparing the frequency ofthe oscillator and the (tunable) frequency Ω of iinj ; thelocking range is uniquely and accurately obtained for eachgiven magnitude of iinj . Although the power consumptionof Rd1 and that of Rd2 are omitted in ﬁg. 2, they are allin the order of μW as shown in table B.2 in the SM; theyshow the same input frequency dependence as the power
consumption of other resistors.
Observations of total power consumption. –While scrutinising the results of tables B.2 and B.3 andﬁg. 2 (and ﬁg. B.2 in the SM), we made the following
observations (O1) and (O2) on the supposition that thefollowing conditions (i), ii), iii)) are met: i) weak forcing(i.e., low injection power Pinj ), ii) Rinj Cinj  (Ω/2π)−1 ,and iii) the ZVS/ZDS conditions (i.e., basic circuit design
constraints mentioned above).
(O1) For given various injection currents iinj under thesame low injection power Pinj (= 9.0×10−4 W above), thepower consumption of each resistor R, rg , Rd1 , Rd2 , andrS in the main circuit, and their sum Ptot do not show
signiﬁcant diﬀerence for all waveforms of iinj as long as
the circuit is injection-locked, as shown in ﬁg. 2.
(O2) The total power consumption Ptot described aboveis speciﬁc to the oscillator itself, which depends only on theoscillation frequency, independent of whether there is injection locking or not. In addition, its oscillation frequency
dependence shows only a relatively small change for thisparticular Class-E oscillator; see ﬁg. C.1 and details insect. (C) of the appendix in the SM.
Viewed from the general theory of limit cycle oscillators
under weak forcings [17,18], the above (O1) and (O2) seem
natural and consistent with it; the oscillation waveform(i.e., oscillation state) of a limit cycle oscillator synchronised to a weak forcing (i.e., a weak perturbation f (Ωt)
in Γ(φ) below of the order O(ε)) will be only slightly deformed by at most O(ε) compared to its original oscillation
waveform without forcings. Then, the observations (O1)
and (O2) can be summarised as follows: i) The power
consumption Ptot of the oscillator itself (the main circuit)in the injection-locked state shows only a slight change
from that of the oscillator without forcing. ii) This slightchange of Ptot is attributed to the oscillation frequency dependence of the power consumption in the oscillator itself.
This is because the main circuit, which oscillates at a frequency diﬀerent from its natural frequency under injection
locking, slightly changes its power consumption depending on its oscillation frequency (for details, see sect. (C)of the appendix in the SM).
From the above i) and ii), under a weak forcing, thepower consumption of the oscillator itself (Ptot in ﬁg. 1(c))
and that of the injection circuit (Pinj in ﬁg. 1(c)), whichis additionally required for the injection locking, are given
independently; for any (small) Pinj , Ptot depends only on
the oscillation frequency, and it is not inﬂuenced by thevalue of Pinj nor by the waveforms of iinj . Therefore, it
is reasonable to analyse the inﬂuence of the input waveform iinj on its locking range only by setting the power
consumption Pinj to a speciﬁc (small) value.
Relationship between injection current and voltage. – To analyse the input waveform dependence of thelocking range, it is useful to obtain an equation relating theinjection current iinj and injection voltage vinj in the injection circuit. First, the circuit equation for the injection
circuit in ﬁg. 1(a) is t−1vinj (t) = Rinj iinj (t) + Cinj
iinj (t )dt + vf (t). (4)0In eq. (4), of interest is the injection-locked state at the
input frequency Ω/2π (as shown in ﬁg. 1(a)); replacing t
in eq. (4) with θ ≡ Ωt, the periodic solutions of iinj (θ),
vinj (θ), and vf (θ) in Fourier series with period 2π are obtained using the harmonic balance method. See sect. (D)of the appendix in the SM for details. As a result, we havethe following concise relation between iinj and vinj :52001-p4−1[vinj (θ) − v Cinj − vf (θ)],
iinj (θ) = Rinj

(5)

Maximisation of synchronisability under low injection power
where v Cinj is a time-independent constant2 . Note thatthe condition Rinj Cinj  (Ω/2π)−1 , as shown in sect. (D)of the appendix in the SM, is naturally satisﬁed in thisstudy, which makes the derivation of eq. (5) simpler.
As a result of eq. (5), when the constant part of iinj
is set to 0, i.e., iinj (θ) = 0, the following relation
holds: vinj  − v Cinj − vf  = 0. This implies thatv Cinj (= vf  − vinj ) is a constant voltage determinedby the diﬀerence in the constant parts of vf and vinj under synchronisation, which is naturally consistent with theinsight obtained from the injection circuit of ﬁg. 1(a).
Proof of fundamental limit by optimal injection
current. – Having observed that the two metrics Pinj
and Ptot are independent (and hence Ptot does not appear
below), and having determined the relation (5) between
iinj and vinj during injection locking, we will now prove
the fundamental limit of synchronisability by directly constructing the optimal input waveform of iinj for a given
Pinj in the following steps (i), ii), iii)).
i) As the ﬁrst step towards this objective, we introduce
the phase reduction method here. In nonlinear oscillators, including Class-E oscillators in which a weak external forcing f (Ωt) is injected, the long-term dynamics of
the injection-locking process can be reduced to the following phase eq. (cf. [13,17,18]): dφ/dt = Δω + Γ(φ), whereΓ(φ) = Z(θ + φ)f (θ), with φ representing the phase
diﬀerence between the oscillator and the external signal,and Δω ≡ ω − Ω denotes the frequency diﬀerence between
the oscillator and external signal. Γ(φ) is a periodic function with period 2π and is called a phase coupling function. The phase sensitivity function Z(θ) is a 2π-periodic
function intrinsic to the oscillator that indicates the inﬂuence (sensitivity) of the external signal on the oscillation
phase.
It is known that this Z can be obtained using various
methods [18]. Speciﬁcally, in the Class-E oscillator shownin ﬁg. 1(a), both iinj (Ωt) and vinj (Ωt) of the injection circuit can be selected as the external forcing f (Ωt) above,
and we write the phase sensitivity function corresponding
to iinj and vinj as ZI (θ) and ZV (θ), respectively. In recent
experimental studies of Class-E oscillators [11,12], vinj has
been treated as the external forcing, simply because vinj
is relatively easily controlled in circuit experiments (compared to controlling iinj ). Hence, [12] introduces the constraint of a given (relatively small) mean square of the2 (in eq. (8)). However, this coninput voltage vinj
straint does not correspond to the power consumption of2 =the circuit Pinj (and Ptot ) since, apparently, vinj
2
iinj Rinj  (= Pinj ).
ii) As the next step toward the fundamental limit, weneed to consider iinj as an external forcing. Now, the objective is to ﬁnd the optimal injection current iopt thatmaximises the locking range among feasible iinj wave-

forms. According to the previous observations (O1) and(O2), if the power consumption of the input circuit Pinj (=i2inj Rinj ) is constrained to a small, constant value (asin eq. (6) below), the total power consumption Ptot of themain circuit under synchronisation is found to be independent of the waveform of iinj . Thus, the optimisation problem of maximising the locking range (≡ L[iinj ] in eq. (9)below), given as a functional of iinj , makes sense and iswell deﬁned under the following constraints 1 and 2 foriinj .
Constraint 1: Pinj is a given constant (= M Rinj ) , i.e.,1iinj 2 ≡ i2inj (θ) 2 = M,
(6)
Constraint 2:

iinj (θ) = 0,

(7)

as opposed to the ones initially assumed in [12]:12vinj(θ) 2 = M,vinj (θ) = 0.

(8)Constraint 2, often called the charge balance constraint,
is required to protect the circuit, in which the net injected charge should be zero. On the other hand, theobjective function L[iinj ] (functional of iinj ), i.e., thelocking range, is given by the diﬀerence between the minimum value Γ(φmin ) and the maximum value Γ(φmax ) of
Γ(φ) (= Z(θ + φ)iinj (θ)) as determined for any feasible
iinj [12]:L[iinj ] = Γ(φmin ) − Γ(φmax )= [ZI (θ + φmin ) − ZI (θ + φmax )]iinj (θ),(9)

where φmax and φmin satisfy the conditions that Δωmax +Γ(φmax ) = 0 and Δωmin + Γ(φmin ) = 0, and Δωmax andΔωmin correspond to the rightmost maximum and leftmost minimum of the locking range for iinj , respectively(cf. [12]). Hereafter, we denote φmin − φmax ≡ Δφ.
iii) As the ﬁnal step, we introduce an additional parameter μ and deﬁne a functional J [iinj ; μ] for maximising
the locking range L[iinj ] under constraints 1 and 2:
J [iinj ; μ] ≡ L[iinj ] + μ [iinj (θ) − 0] = iinj g, (10a)with g(θ) ≡ ZI (θ + Δφ) − ZI (θ) + μ.
(10b)
We note that this method for embedding the constraint 2 (eq. (7)) into the functional J through the additional parameter μ was ﬁrst introduced in [13] and further
developed in [19]; it is useful for solving a certain (convex)
optimisation problem via the Hölder/Cauchy-Schwarz inequality. Note that the constraint 1 (eq. (6)) is also embedded in the process of optimisation (12) below, and theparameters μ and Δφ are ﬁnally determined by solving
eqs. (7), (13) for μ and Δφ.
Thus, ﬁnding the optimal injection current iopt is reduced to solving the following optimisation problem:

2 The relation between i
inj and vinj in eq. (5) can also be obtained by analytically solving the initial value problem for eq. (4)and setting t → ∞, but its calculation is lengthy and omitted here.

52001-p5

maximise J [iinj ; μ],subject to iinj  = 0,iinj 2 = M,

(11)

Hisa-Aki Tanaka et al.
which can be solved with the calculus of variations (as theEuler-Lagrange equation), leading to the same result asthat we obtain below (cf. [14,15,20]). However, we resort
to the following direct approach from eq. (10), which uses
only inequalities, due to its simplicity and clarity:
J [iinj ; μ] = J [iinj ; μopt ] = iinj g ≤ |iinj g| ≡ iinj g 1≤ iinj 2 g 2 = M g 2 .
(12)Here the equality condition: iinj (θ) ∼ g(θ) of the CauchySchwarz inequality iinj g 1 ≤ iinj 2 g 2 is essential. By
iinj ∼ g, the iopt realising the upper bound in eq. (13)must be of the following form (if it exists):
iopt (θ) = M gopt (θ)/ gopt 2 ,Fig. 3: Veriﬁcation of the validity of iopt for injected vopt .
(a) Phase sensitivity function ZI (θ) for injection current iinj .
(b) Observed injection current iopt and corresponding injection
voltage vopt and vf . These are obtained through the numerical circuit simulations, and eq. (5) is veriﬁed to be satisﬁed by
them.

(13)

and this iopt makes all “≤” as “=” in eq. (13), which
implies that iopt is the maximiser of L[iinj ] under constraints (6) and (7). Finally, by substituting eq. (13)into (6) and (7), μopt = 0 and Δφopt = π are uniquelydetermined for a given ZI (θ) in ﬁg. 3(a), following thecareful numerical analysis in [13], sect. 6, which implies
that iopt does exist. We note that the solution Δφopt = π
is proven to always exist (cf. [13], below Theorem 1, p.
6; and the “generic solution” in [14], p. 2). Thus, afterplugging (Δφ, μ) = (Δφopt , μopt ) = (π, 0) in eq. (10b)
we obtain gopt (θ) = ZI (θ + π) − ZI (θ), which implies that
iopt in eq. (13) together with the above gopt is the uniquemaximiser of J [iinj ; μ]; iopt maximises the locking range
L under constraints (6) and (7). In addition, the corresponding injection voltage vopt (θ) (ﬁg. 3(b), green graph),which yields iopt in eq. (13), is uniquely determined by the
relation in eq. (5) (with the numerically obtained v Cinj andvf (θ) in eq. (5), as explained below).
Finally, through numerical circuit simulations, we verify
that the theoretically obtained iopt in eq. (13) is indeed
realised by setting vinj as the theoretically obtained vopt
in the injection circuit in the following manner: First, thecircuit simulations in this study employ the switch modelin ﬁg. 1(b); the timing at which the MOSFET enters the“ON” state is set to θ = 0, and the timing at which the
MOSFET enters the “OFF” state is set to θ = π, forsimplicity. This makes the phase θ in eq. (5) and ZI (θ) is
uniquely ﬁxed on the θ coordinate; on this coordinate, we
observe that under injection locking, iopt , vopt , and vf , asshown in ﬁg. 3(b) via circuit simulations, satisfy eq. (5)with high accuracy. Next, we compare this numericallyobtained vopt with the above theoretically obtained one.
The result shows that they are indistinguishable on thegraph in ﬁg. 3(b), which implies that iopt is indeed realisedin the circuit by injecting vopt .
Veriﬁcation of optimality. – The optimality of theinjection current iopt and the corresponding vopt obtainedin the previous section is validated in the following two
ways. First, to verify that the optimal injection current
iopt indeed maximises the locking range, the value of thelocking range for iopt is compared with that of the locking

Fig. 4: Distribution of locking ranges for 105 randomly generated injected currents of iinj . Pulse, square, sin, and opt
in the ﬁgure correspond to the cases in which iinj has (broadbipolar) pulse, square, sinusoidal, and optimal waveforms inﬁg. B.1, respectively. Their locking ranges are 1.811, 3.590,
4.024 and 4.028 kHz, respectively. Note that the optimal
waveform (opt) signiﬁcantly outperforms the pulse and square
waveforms, while it outperforms the sinusoidal waveform only
slightly.

range for the randomly generated, injected current waveform; speciﬁcally, for iinj (that satisﬁes the constraints (6)and (7)) of 100000 ﬁfth-order Fourier series3 with Fouriercoeﬃcients generated by uniform random numbers, thedistribution of the locking ranges is obtained from thephase equation (ﬁg. 4). As can be seen in ﬁg. 4, the locking range for iopt is wider than that for any other input
waveform, although the locking range for iopt and that forthe sinusoidal iinj are quite close.
Second, the consistency between iopt in this study andthe previous result in [12] is further veriﬁed; the optimal∗, previously obtained in [12] (frominjection voltage vopt
vopt (θ) = M gopt (θ)/||gopt 2 , similar to eq. (13)) for theexternal forcing vinj , has almost the same shape as thatof iopt obtained in this study, because ZI in this study
and ZV in [12] have almost the same shape. Namely, the∗in [12] under constraint (8) implies the
optimality of vopt
optimality of iopt under constraints (6) and (7). For moredetail, see section (E) in the SM.
3 Note that the ﬁfth-order Fourier series for i
inj are suﬃcient forthis analysis, because ZI in this study has small magnitude in theirhigher harmonics, which implies higher harmonics in the injection
current iinj does not aﬀect Γ in the phase equation: dφ/dt = Δω +Γ(φ).

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Maximisation of synchronisability under low injection power
Discussion. – Although this study is conducted forthe injection-locked Class-E oscillator for which power
consumption analysis is easily carried out, the obtainedframework can be useful for analysing other injectionlocked oscillators; for example, the injection-locked CMOS
ring oscillators [21] as well as the vdP oscillator [9], witha similar injection circuit in this study. This is because
observations (O1) and (O2) are expected for those oscillators as general properties of synchronous oscillators withweak forcings [17,18], which implies such a speciﬁc class of
injection-locked oscillators is amenable to the frameworkof this study. However, we need to be cautious in using our method for the following technical reasons; ﬁrstly,
the power consumption in the injection circuit needs tobe reliably estimated (or controlled to be a small value),
which implies a relatively simple injection circuit (suchas the linear circuit in this study) would be a reasonablechoice. Secondly, the separability of the (low) injection
power Pinj and the total power consumption Ptot in theoscillator needs to be veriﬁed (as in (O1)), which requires a
reliable estimation of Ptot including power consumptionsin the nonlinear elements; the transistor working as the
ON/OFF switch in this study would be an ideal instance.
Finally, as a future study, it is theoretically as well as practically interesting to consider the implications of the pioneering framework in [22] for our particular circuits, where
the feature size of the transistors is scaled below several
nm. In addition, an extension toward energy propagation
on networks [23,24] from our framework is open for future
study.
Conclusion. – This study veriﬁes that the power consumption of the Class-E oscillator and the extra power
required in the injection circuit can be separated, andshows that the locking range can be maximised under theconstraint of a weak power consumption of the injection
circuit. Speciﬁcally, the following ﬁndings were obtained
respectively for Question 1 and Question 2 raised at the
beginning of this letter, which characterise the fundamental limit of synchronisability for this particular oscillator.
Answer 1 : In the injection-locked Class-E oscillator ofthis study, the power consumption of the oscillator itself
and the extra power consumption in the injection circuit
are independent and can be treated separately as far as
the injection locking is maintained.
Answer 2 : Under the condition: (Rinj Cinj )−1  Ω/2π,
for a given low injection power, the locking range can be
maximised by designing the optimal injection current iopt(of eq. (13)), obtained through the optimal injection voltage vopt due to the one-to-one correspondence between iinj
and vinj according to eq. (5). It is also immediately obtained from eq. (12) that for this iopt , under a given locking range (= L[iinj ] = J [iinj ] of eq. (12) since iopt satisﬁes
iopt  = 0), the injected power (∼ M of eq. (12)) is minimised. This is because all “≤” become “=” in eq. (12)which implies iopt minimises M .

∗∗∗This work was supported in part by JSPS KAKENHI (JP24K03008). FM acknowledges JSPS KAKENHI(JP22K03453) for ﬁnancial support.
Data availability statement: All data that support theﬁndings of this study are included within the article (andany supplementary ﬁles).
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