Flexible phase synchronisation control
method using partially unlocking
oscillator arrays

between adjacent oscillators, as well as sufficiently uniform oscillator
characteristics. Under such conditions, a systematic derivation of thephase equation for oscillators can be constructed [3], which eventuallytakes the following form:

H.-A. Tanaka, K. Shimizu, O. Masugata and T. EndoP AjsinðF þ yj  yi Þy_ i ¼ oi þ kj Ai

A phase synchronisation method is proposed for beam-scanning
control, utilising a newly identified phase synchronisation pattern.
The proposed method provides more flexible control compared toconventional methods, even in noisy, non-ideal environments, asconfirmed by systematic simulations and mathematical analysis.

Introduction: Coupled oscillator arrays emerge in a wide range ofengineering issues. Examples include millimetre-wave power-combining
and beam-scanning control systems, central pattern generators (CPG)
in robotics, and Josephson junction arrays. In contrast to theseexamples that utilise the mutually locked (synchronised) state, little
attention has been paid to unlocking states, presumably because few
applications have been sought in such unlocking situations. In beamscanning control systems using coupled oscillator arrays, a linear
phase progression must be maintained across the array for beam
forming (see Fig. 1), and the radiated beam is steered by controlling
the phase difference between the adjacent oscillators. Methods ofcontrolling such phase differences have been proposed and demonstrated, respectively, by Stephan [1] and by Liao and York [2], where
oscillators at both ends of the array (oscillators 1 and N in Fig. 1a) arecontrolled to steer the beam. In [2], antisymmetrical frequency
detuning is applied to the oscillators at the ends of the array. Insuch coupled oscillator-based beam steering, the coupling is loose(weak) when the oscillators are coupled radiatively for instance.
Accordingly, as shown in [2] when a particular scan angle, sayþ12.5 , is required, the frequencies of the end oscillators must be
9.985 and 10.015 GHz, respectively, for instance, while the otheroscillators have a frequency of 10.0 GHz. This suggests the end
oscillators require careful frequency control since the frequency
control resolution for the end oscillators must be within the order ofa few kilohertz.

ð1Þ

where yi, Ai and oi represent the oscillation phase, amplitude and freerunning frequency of the ith oscillator, respectively. Based on the above
assumptions, Ai,j  1 holds and kAj=Ai  k is denoted by Dom. This
Dom is interpreted as the locking range of each oscillator, which isassumed to be small as well as identical for all oscillators. The phase lag
F reflects the signal delay, which cannot be neglected for the case ofradiative coupling. However, if the coupling is constructed from onewavelength waveguides, F is assumed to be 0, and we focus on thiscase. Similar to Liao and York [2], we consider herein the frequency
distribution as: o1 ¼ O þ Do, oN ¼ O  Do, o2, . . . , oN1 ¼ O, where
O and Do denote the common locked frequency and the frequency
detuning, respectively. Hereinafter, Dom is set to 1 without loss ofgenerality, and all numerical integrations are carried out by the fourthorder Runge-Kutta method with a step size of 0.001.
In contrast to the locked state of linear phase progression, weconsider an unlocking state, which is obtained when jDoj > Dom. We
systematically changed Do=Dom from 1.0 to 20.0 and observed thephase relationships. Fig. 2 shows a typical unlocking phase relationshipnumerically observed in (1), where Do=Dom is set to 5.0, and asnapshot of yi þ 1  yi is taken at t ¼ 5000. This example exhibits thefollowing three characteristics observed in these unlocking cases:Fig. 2 Phase progression pattern observed in unlocking system (1)Fig. 1 Coupled oscillator arrays for beam-scanninga Conventional systemb Proposed, partially unlocking system

In this Letter, we consider a counterpart of the injection-locked state,i.e. unlocking states, which emerge quite naturally. In the above
example, if the end oscillators have frequencies of approximately 9.0
and 11.0 GHz, respectively, for instance, then they are unlocking withrespect to the other oscillators. We have analysed such unlocking states,
and found that a robust, exactly linear phase progression is still obtainedin the array, except for a few oscillators near both ends. In the presentstudy, this somewhat counter-intuitive phenomena is systematicallyanalysed both numerically and theoretically, leading to a closed formulaof the phase progression for given frequencies and amplitudes of theend oscillators (or injection signals). Based on this robust phaseprogression in the unlocking state, a method of beam-scanning control
is proposed, which does not require high frequency control resolution.

(i) Oscillators near both ends of the array oscillate at common
frequencies close to O þ Do, respectively, and the other oscillators
oscillate at O.
(ii) Inside the array, a linear phase progression appears, in which thephase difference yiþ1  yi (i ¼ 4, . . . , 21) becomes time-constant and its
fluctuation is always negligibly small. In contrast to the inside of thearray, the oscillators near both ends exhibit a certain amount offluctuation in phase difference yiþ1  yi. The ranges for these fluctuations are shown by as the ‘minimum and maximum phase differences’in Fig. 2.
(iii) In such a phase progression, the phase difference fi  fiþ1(  Df)
is measured as Df  A2inj=Do if the end oscillators have the sameoscillation amplitude A1 ¼ AN  Ainj.
Phase control by unlocking oscillators: Based on these characteristics, an application of the unlocking array is suggested to control thelinear phase progression both by the amplitude Ainj and the detuning
Do at the end oscillators. To consider these observations theoretically,
we introduce a slightly modified version of (1):y_ 1 ¼ O þ Ainj sinððO þ DoÞt  y1 Þ þ sinðy2  y1 Þy_ ¼ O þ sinðy  y Þ þi

Phase synchronisation in injection-locked oscillator arrays: For
weakly coupled quasi-optical oscillators, York [3] developed asystematic reduction of model equations. In this Section, we follow
this reduction and explain how the linear phase progression is realisedin the coupled oscillator array. Contrary to previous studies, weconsider herein an injection-unlocking state, and analyse the phaserelationship in the oscillator array. We also assume a weak couplingiþ1sinðyi1  yi Þ;ið2  i  N  1Þy_ N ¼ O þ Ainj sinððO  DoÞt  yN Þ þsinðyN 1  yN Þð2Þ

where the end unlocking oscillators in (1) are replaced by the externali(ODo)t. This modification is shown schematiinjection signals A1,Ninj ecally in Fig. 1b. Thus, the modification is not essential to the phase

ELECTRONICS LETTERS 7th June 2007 Vol. 43 No. 12

relationship considered here, and by this modification the analysis of (2)becomes much more tractable, as follows. First, a new variable fi  yi Ot is introduced to (2), which yields:
f_ 1 ¼ A1inj sinðDot  f1 Þ þ sinðf2  f1 Þ;
f_ ¼ AN sinðDot  f Þ þ sinðff ÞNinjNN 1f_ i ¼ sinðfiþ1  fi Þ þ sinðfi1  fi Þ;Nð2  i  N  1Þ ð3Þ

For the long-term, slow movement of the phase relationship fi  fiþ1,
we reduce (3) by averaging the fast moving terms Ainjsin(Dot  f1) andAinjsin(Dot  fN), assuming Dot as the fast variable. In this averaging, a somewhat technical calculation is possible, using a nonlinear
transformation of variables. This result is mathematically validated forlarge Do limit. Owing to lack of space, we omitted details. Moregeneral results will be reported elsewhere. Finally, (3) is averaged toyield:2ðA1inj Þþ sinðf2  f1 Þ2Do2ðANinj Þþ sinðfN 1  fN Þ;
f_ N ¼ 2Do
f_ i ¼ sinðfiþ1  fi Þ þ sinðfi1  fi Þ
f_ 1 ¼

ð4Þ

Interestingly, (4) takes the form of (1), and the phase progression
Df  fi  fiþ1 is explicitly given as!
A2inj
ð5Þ
where Ainj  A1inj ¼ ANinj
Df ¼ sin12Do
which explains the observed phase progression inside the array.
Simulation results: Based on the closed formula (5) of the phaseprogression, a new beam-scanning control method is proposed. First,
we check the consistency of the simulation results from (3), (4) andthe closed formula of Df (5). Fig. 3 shows typical examples of thesethree data obtained for Do ¼ 5.0 and 10.0 (in a normalisedfrequency), where they are verified to be in good agreement. Also,from systematic simulations, it is observed that the consistency
becomes better as Do increases. However, when Do becomes
small, say, 0.1, the consistency decreases because f_ 1  Do (or f_ N  Do) is no longer large enough for validating the averaging results.
Thus, the numerical simulations and the analytical results (4) and (5)suggest that the proposed method stably controls the linear phaseprogression by tuning Ainj and=or Do.

The issues of robustness in the proposed method, i.e. noise immunity
and the effect of asymmetric detuning frequencies, are considered.
Extensive numerical simulations using random initial conditions arecarried out according to [4], assuming the same white noise in bothcases. Results suggest that the method of Liao and York [2] and the
proposed method have approximately the same noise immunity under theabove condition.
The effect of asymmetric detuning frequencies has also been considered numerically, both for the method of [2] (1) and for the proposed
method (4). Results show that the amounts of distortion in the linear
phase progression are approximately the same for both methods (datanot shown due to lack of space). This suggests that both methods haveapproximately the same robustness as the asymmetric detuning frequencies. However, the proposed method effectively controls the phaseprogression by tuning the injection amplitude. In the above example
using asymmetric detuning d, the proposed method tunes the amplitudes A1inj and=or ANinj according to (5), resulting in perfect linear phaseprogression.
Conclusions: We have proposed a novel control method for a linear
phase pattern in coupled oscillator arrays, which provides more
flexible and possibly robust control ability in coupled oscillatorswith small locking ranges. The phase pattern is clearly described in(5) and its validity and usefulness is confirmed by systematicsimulations.
# The Institution of Engineering and Technology 2007
21 February 2007
Electronics Letters online no: 20070094
doi: 10.1049/el:20070094
H.-A. Tanaka and O. Masugata (Department of Electronic Engineering, The University of Electro-Communications (UEC), 1-5-1 Choufugaoka, Choufu-shi, Tokyo 182-8585, Japan)
K. Shimizu and T. Endo (Department of Electronics and Communication, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki
City 214-8571, Japan)
References
1 Stephan, K.D.: ‘Inter-injection-locked oscillators for power combining
and phased arrays’, IEEE Trans. Microw. Theory Tech., 1986, MTT-34,pp. 1017–1025
2 Liao, P., and York, R.A.: ‘A new phase-shifterless beam-scanningtechnique using arrays of coupled oscillators’, IEEE Trans. Microw.
Theory Tech., 1993, MTT-41, pp. 1810–1815
3 York, R.A.: ‘Nonlinear analysis of phase relationships in quasi-opticaloscillator arrays’, IEEE Trans., 1993, MTT-41, pp. 1799–1809
4 Chang, H.-C., et al.: ‘Phase noise in coupled oscillators: theory andexperiment’, IEEE Trans., 1993, MTT-41, pp. 1799–1809

Fig. 3 Comparison of phase progression Df obtained by (3), (4) and (5)a Do ¼ 5.0
b Do ¼ 10.0

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