A time-domain method for the determination of the injection-locking range of oscillators is presented. The method involves three time dimensions: the first and the second are warped time scales used for the free-running frequency and the external excitation, respectively and the third is to account for slow transients to reach a steady-state regime. The locking range is determined by tuning the frequency of the external excitation until the oscillator locks. The locking condition is determined by analyzing the Jacobian matrix of the system. The method is advantageous in that the computational effort is independent of the presence of widely separated time constants in the oscillator. Numerical results for a Van Der Pol oscillator are presented

Christoffersen, Carlos E.

Condon, Marissa

Xu, Tao

English

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A new method for the determination of the locking range of oscillators

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A New Method for the Determination of theLocking Range of OscillatorsCarlos E. ChristoffersenDept. of Electrical EngineeringLakehead UniversityThunder Bay, CanadaEmail: c.christoffersen@ieee.orgMarissa Condon and Tao XuSchool of Electronic Engineering, RINCEDublin City University,Dublin, IRELANDEmail: {condonm, taoxu}@eeng.dcu.ieAbstract— A time-domain method for the determination of theinjection-locking range of oscillators is presented. The methodinvolves three time dimensions: the first and the second arewarped time scales used for the free-running frequency and theexternal excitation, respectively and the third is to account forslow transients to reach a steady-state regime. The locking rangeis determined by tuning the frequency of the external excitationuntil the oscillator locks. The locking condition is determinedby analyzing the Jacobian matrix of the system. The method isadvantageous in that the computational effort is independent ofthe presence of widely separated time constants in the oscillator.Numerical results for a Van Der Pol oscillator are presented.I. INTRODUCTIONAll free-running oscillators can exhibit the phenomenonof injection locking when perturbed by an external signalwhose frequency is close to the free-running frequency. Thisphenomenon whereby the natural frequency of the oscillatorchanges to be identical to the external frequency is found inall branches of science.In many fields of circuit design, the injection lockingphenomenon is exploited for beneficial purposes. RF Phase-Locked Loops (PLL) are widely used in wireless communica-tion applications such as frequency synthesis. In the feedbackloop of a frequency synthesizer, a frequency prescaler isusually employed to divide the frequency by a fixed number.While static dividers offer a wide bandwidth, Injection LockedFrequency Dividers consume significantly less power [1], [2]and therefore are preferable for low-power applications. Cou-pled array oscillators for phase control in quasi-optical powercombining also involve the technique of injection locking.Thus an accurate and efficient technique for the determinationof injection locking phenomena is important for today’s circuitdesign industry [1], [2], [16], [17]. Various analytical andnumerical approaches have been suggested for predicting thelocking range, e.g. the classic Adler approach in [14] and morerecently [17]–[19].Many early circuits to illustrate the injection-locking phe-nomena in oscillators were built using vacuum tubes [13]. Inthis contribution, the focus is on the accurate determination ofthe locking range of such an oscillator using a multitime scalemodel.Simulation techniques involving multiple timescales haverecently been applied to great effect for circuits with widelyseparated time constants. References [3]–[11] are some ex-amples. Roychowdhury et al. proposed the term MultitimePartial Differential Equation (MPDE) when these techniquesare applied to circuits involving straightforward envelope mod-ulation [3], and Warped MPDE (WaMPDE) for the techniquesused to analyze oscillators or circuits with FM signals [4].An even more general approach for the analysis of oscillatorstermed GeMPDE was also presented [4], [6]. In this paperwe use these names and a similar notation. Mixed frequency-time methods involving Harmonic Balance in the two periodicdimensions have been proposed [10], [11] to analyze PLLcircuits.In this work, a time-domain formulation of the WaMPDE inconjunction with the bisection method is employed to identifythe input frequencies to an oscillator at which locking occurs.Since an injection-locked oscillator is an autonomous systemsynchronized to an external signal, the oscillator autonomoussolution is considered in a warped time scale. The input signalintroduces a second warped time scale. The warped time scalesenable a slow variation of the input and the local frequency. Athird dimension is employed for the transient evolution. Themethod is advantageous in that it is independent of the timeconstants of the oscillator. Only a few transient time-stepsare required regardless of how many periods are required tolock the oscillator. Brute force transient simulation would beprohibitive as many thousands of periods are required to lockan oscillator with a high Q factor and each period requiresmany small time steps for accurate simulation. Also for smallinjections close to the free-running frequency, identification oflocking from time-domain waveforms becomes very difficult.Section II reviews the WaMPDE technique. Section IIIdescribes solving for the locking range of an injection-lockedoscillator using the WaMPDE. Section IV illustrates themethod of solution with a simulation of a forced Van DerPol oscillator.II. WARPED MULTITIME PARTIAL DIFFERENTIALEQUATION APPROACHSuppose that an autonomous nonlinear circuit is describedby:x˙ = f (x)+b(t) . (1)tτ10π2(t)1tt n n+1φFig. 1. Relation between τ1, t and φ1(t)where b(t) is the excitation vector, x(t) is a vector of statevariables, and f is a nonlinear function. In the WaMPDE ap-proach [4], the variations of the state variables are decomposedin several time dimensions. Eq. (1) is transformed into thefollowing partial differential equation:p∑i=1(ωi(t)∂xˆ∂τi)+∂xˆt= f (xˆ)+ ˆb(τ1, . . . ,τp, t) , (2)where p+ 1 is the number of time dimensions being con-sidered, and xˆ and ˆb denote the multi-dimensional variablescorresponding to x and b, respectively. τ1, . . . ,τp are warpedtime scales and t is the time scale of the original equation.The local angular frequency in the ith dimension is denoted ωi.The phase (φi) at each warped time dimension is determinedas followsφi(t) =∫ t0ωi(t)dt .It was shown in [4] thatx(t) = xˆ(φ1(t),φ2(t), . . . ,φp(t), t) .Figure 1 shows the relation between τ1, t and φ1(t) for a casewith p= 1. The rectangle with arrows indicates how the PDEin Eq. (2) is integrated along τ1 and t.The WaMPDE can be solved using exclusively time-domainapproaches or mixed time-frequency domain methods. In anycase, periodic boundary conditions must be set for τ1, . . . ,τp.For instance, for τ1 this condition can be expressed asxˆ(τ1,τ2, . . . , t) = xˆ(τ1+2π,τ2, . . . , t) .In addition, in order to obtain a unique solution it is necessaryto impose a ‘phase condition’ for each warped time scale withunknown local frequency [4]. This is because there are infinitesolutions of Eq. (2) that only differ in the phase in the τ1, . . . ,τptime scales. The consequence of this is that after discretization(in frequency or time domain) there are more unknowns thanequations (p extra unknowns if all ωi are unforced). The phasecondition is a set of extra equations that are used to make theJacobian matrix of the system to be full rank. When Eq. (2)is solved in the frequency domain it is a common practice toimpose the phase condition by forcing a particular phase inone of the state variables, e.g. the imaginary part is set to zero.This is equivalent to selecting one of the infinite solutions thatonly differ in the phase. In time domain, imposing the phasecondition is more difficult as the phase of the state variables isnot directly available. The phase condition used in this workis discussed in the following Section.III. DETERMINATION OF THE LOCKING RANGEThe WaMPDE in Eq. (2) is used with three timescales todetermine the locking range of an oscillator. Eq. (2) thus takesthe form:ω(t)∂xˆ∂τ1+ωin∂xˆ∂τ2+∂xˆ∂t = f (xˆ)+b(t) . (3)The oscillator autonomous solution is considered in the τ1time scale. The input signal introduces a second time scale— τ2 and ωin is the input frequency. τ1 and τ2 are warpedtime scales to enable the slow variation of the input andthe local frequencies. A third dimension, t, is employed forreal time. Eq. (3) may be solved with time-domain, mixedtime-frequency domain methods or methods based on otherbasis functions such as wavelets. In this contribution, a time-domain method is presented. The derivatives with respectto the warped variables are evaluated using the five-pointcentered difference formula [20]:∂xˆk∂τ1≈ xk−2−8xk−1+8xk+1− xk+212h , (4)where h is the time-step in the τ1 scale and k denotes the indexin the same scale. The Backward Euler (BE) rule is used inthe t time scale to calculate the transient evolution. One usefulcharacteristic of the BE rule is that it introduces artificialdamping if the time step is large. This helps to attenuateundesired oscillations along the t dimension.The additional equation required to solve for the localfrequency (the ‘phase condition’) is implemented in this workby making one sample of the derivative of one of the statevariables equal to zero:∂xˆ0∂τ1= 0 . (5)This condition is guaranteed to be satisfied by oscillatory andnon-oscillatory solutions if they have a continuous derivative.The derivative only has to be continuous for the state variablewhere the phase condition is imposed.On replacement of the time-domain derivatives in Eq. (3)with Eq. (4), use of the BE rule and the periodic boundaryconditions, the resulting nonlinear algebraic equations may besolved in conjunction with Eq. (5) using Newton’s method[15], [20].The procedure to determine the lower limit of the lockingrange is described next. The process is started by running atransient simulation with the following initial condition: thefree-running oscillation is set along τ1 and no oscillationsalong τ2. The frequency of the excitation is set to a valueknown not to lock the oscillator. Eq. (3), along with theboundary conditions is then solved for several time-steps alongt until the steady-state condition is achieved. In order tominimize the number of time steps necessary to achieve thesteady-state condition (along t), the time step size in thisdimension is exponentially increased. When the differencebetween two steps along t is small enough, the analysis isswitched from a 3-D transient analysis to a 2-D steady-stateanalysis by making the derivatives with respect to t equalto zero. This procedure results in an autonomous solutionalong τ1, if it exists. A similar idea was used in Reference[9] to avoid unstable equilibrium points. At each time-step,but especially when solving for the steady-state condition, theJacobian matrix that is employed in Newton’s method is noted.If the column that corresponds to the local frequency is nearzero, this implies the system is becoming independent of thelocal frequency because there are no more oscillations alongτ1 (to see the proof of this refer to [5]). Thus, an autonomoussolution along τ1 is no longer possible and only the forcedoscillations are present. Locking has happened. Note that thesolution of the locked oscillator is not calculated becauseat this condition the Jacobian matrix is singular and so theNewton iterations are stopped. Thus, the method relies on thefact that the ‘initial guess’ for the Newton method that solvesthe 2-D steady-state solution is very close to the solution.After the steady-state solution is obtained (assuming nolocking), the input frequency (ωin) is progressively incre-mented. For each increment, a transient analysis is performedand followed by a steady-state analysis. If the oscillator isdeemed locked, further analysis may be necessary to determinethe starting point of the locking range to a certain degree ofaccuracy. The bisection method [20] is used in this work.Therefore, the frequency step by which ωin is incrementedis halved and the procedure is re-started with the initialconditions from the last frequency point at which there wasno locking. This is very important because if the systemis initialized with no oscillations along τ1, a non-physicallocked solution may be obtained (in addition to a singularJacobian matrix). This progressive halving of the frequency-step increment is repeated until the lower limit of the lockingrange is attained to a certain tolerance.A similar procedure is subsequently performed to determinethe upper limit of the locking range.IV. SIMULATION OF AN INJECTION LOCKED VAN DER POLOSCILLATORA forced Van Der Pol Oscillator is employed to modela microwave negative oscillator and to illustrate injectionlocking phenomena [12]. The phase dynamics of the Van DerPol Oscillator are the same as the Adler Equation [14] used todescribe the phase dynamics of the injection locked microwavenegative oscillator. The standard Van Der Pol equation has thefollowing form:−Vin j sinωint = ¨Vx− ε(α−V 2x ) ˙Vx+ω20Vx .In this work the following parameters are used: ε = 0.2,α = 1, ω0 = 1 and Vin j = 0.7. This oscillator exhibits widelyseparated time constants when excited with a frequency nearthe edge of the locking range. Fig. 2 shows that when excitedwith a frequency of 0.714 rad/s (which is known to lockthe oscillator), the steady state is not yet achieved after a6200 6210 6220 6230 6240 6250 6260 6270 6280−1−0.8−0.6−0.4−0.200.20.40.60.81Time (s)yFig. 2. Steady state is not yet achieved after 1000 oscillations050246−1.5−1−0.500.511.5τ2(a)τ1y(τ1, τ 2)050246−1.5−1−0.500.511.5τ2(b)τ1y(τ1, τ 2)Fig. 3. Steady state: (a) ωin = 0.714 rad/s and (b) ωin = 0.712 rad/ssimulation time equal to 1000 periods of the free-runningoscillation.A transient analysis of the oscillator using the WaMPDEapproach with three time scales as described in Section III ispresented now. Each step along t involves the calculation of abi-dimensional steady-state problem with a grid with 15 pointsin each dimension, τ1 and τ2. Fig. 3 show the steady statesolutions for two different values of ωin. Only 27 steps alongt were necessary to simulate an elapsed time of 75000 periodsand obtain the locked steady-state of Fig. 3a. The lockingcondition can also be observed in Fig. 4. The Jacobian columnapproaches zero if the oscillator locks. Note that the time toreach the steady state increases as ωin approaches the lockingrange limit.The algorithm to find the locking range is demonstratednow. To determine the lower limit of the frequency range,the initial angular frequency is set to 0.7 rad/s with an initialfrequency step of 0.1 rad/s and a tolerance of 0.001 rad/s.After 64 iterations (this includes rejected time steps) the lowerlocking limit of 0.714 rad/s is determined. Fig. 5 shows thevariation of the forcing frequency and the norm of the Jacobiancolumn as a function of the iteration number. If no locking isdetected, the frequency is increased. Each time that locking isdetected, a lower frequency is tried and the initial conditionsare reset to the last known oscillatory solution. That is thereason for the peaks observed in the norm of the Jacobiancolumn.101 102 103 104 10510−410−310−210−1100101102tNorm of Jacobian ColumnFig. 4. Comparison of the norm of the Jacobian column with ωin =0.714 rad/s (continuous line) and ωin = 0.712 rad/s (dashed line)0 10 20 30 40 50 60 700.70.750.8IterationAngular Frequency (rad/s)01020Norm of Jacobian ColumnFig. 5. Local frequency and norm of Jacobian column (dashed line) vs.iteration numberA similar procedure was followed to find the upper lockinglimit which is equal to 1.217 rad/s.V. CONCLUSIONSThe Warped Multitime scale approach is employed to simu-late an injection-locked Van Der Pol oscillator. The techniqueenables a fast and accurate method of determining the lockingrange and is applicable to any kind of oscillator that can bedescribed by a set of differential and algebraic equations sinceno assumptions about the oscillator are made except that it haslocked and unlocked states. This is an important contributionas the determination of the Injection Locked Range for circuitsexploiting oscillator injection locking phenomena is importantin design and analysis work.It was shown that the time to determine the lockingcondition using the proposed method is independent of thepresence of widely separated time constants in the systembeing analyzed.The proposed method works in time domain. Thus adaptivegrids could be used and this should result in a very efficientmethod for strongly nonlinear systems. That and the analysisof more complex oscillators will be the subject of futureresearch.ACKNOWLEDGEMENTThis work is funded by Science Foundation Ireland andScience and Engineering Research Canada (NSERC).REFERENCES[1] T. Xu, Z. Ye, and M. P. Kennedy, “Mathematical analysis of injection-locked frequency dividers,” Proc. NOLTA, Sept. 2006.[2] Z. Ye, T. Xu, and M. P. Kennedy, “Locking range analysis for injection-locked frequency dividers,” Proc. ISCAS, pp.4070-4073, May. 2006.[3] J. S. Roychowdhury, “Analyzing circuits with widely separated timescalesusing numerical PDE methods”, IEEE Trans. Circuits and Systems, vol.48, pp. 578-594, May 2001, pp. 578–594.[4] O. Narayan and J. Roychowdhury, “Analyzing oscillators using multitimePDEs”, IEEE Trans. on circuits and systems – I Fundamental Theory andApplications, Vol. 50, July 2003, pp. 894–903.[5] T. Mei and J. Roychowdhury, “Oscillator-ac: restoring rigour to linearizedsmall-signal analysis of oscillators,” 2004 IEEE/ACM ICCAD Digest, pp.603–608.[6] T. Mei and J. Roychowdhury, “Efficient AC analysis of oscillators usingleast-squares methods,” Proc. Design, Automation and Test in Europe,Vol. 1, March 2006, pp. 1–6.[7] R. Pulch, “Multitime scale differential equations for simulating frequencymodulated signals” Applied numerical math., Vol. 53, 2005.[8] E. Dautbegovic, M. Condon and C. Brennan, “An efficient nonlinearcircuit simulation technique,” IEEE Trans. on Microwave Theory andTechniques, vol. 53, no. 2, Feb. 2005.[9] L. Zhu and C. Christoffersen, “Adaptive harmonic balance analysis ofoscillators using multiple time scales,” 3rd Int. IEEE Northeast Workshopon Circuits and Systems Digest, June 2005, pp. 187–190.[10] S. Sancho and A. Suarez, “Efficient analysis of phase-locked loopsthrough a novel time-frequency approach, based on two envelope transientformulations,” 2003 IEEE MTT Symp. Digest, June 2003, pp. 2153–2156.[11] S. Sancho, A. Suarez and J. Chuan, “General envelope-transient formu-lation of phase-locked loops using three time scales”, IEEE Transactionson Microwave Theory and Techniques, Vol. 52, April 2004, pp. 1310–1320.[12] B. Van Der Pol, “Forced oscillations in a circuit with nonlinear resis-tance,” Phil Msg., Vol. 3 Jan 1927.[13] Jiang, T.Z., Yonghua, L et al., “Injection Locking Phenomena of VanDer Pol Oscillator”, Proc. APMC, 2005.[14] R. Adler, “A study of locking phenomena in oscillators”, Proc. IRE, Vol.34, June 1946.[15] E.L. Allgower and K. Georg, Numerical continuation methods, NewYork, Springer-Verlag, 1990.[16] B.Razavi, “A study of injection locking and pulling in oscillators”, IEEEJournal of Steady-State Circuits’, Vol. 39, No. 9, Sept. 2004.[17] Y. Wan, X. Lai and J. Roychowdhury, “Understanding injection lockingin negative-resistance LC oscillators intuitively using nonlinear feedbackanalysis”, Proc. IEEE 2005 Customs Integrated Conference, pp. 729–732.[18] L. Xiaolue, J. Roychowdhury, “Capturing oscillator injection lockingvia nonlinear phase-domain macromodels,” IEEE Trans. on MicrowaveTheory and Techniques, Vol. 52, Sept. 2004, pp. 2251–2261.[19] F. Schilder and B. B. Peckham, “Computing Arnol’d tongue scenarios,”J. Comput. Phys. 220, 2 (Jan. 2007), pp. 932–951.[20] R. L. Burden and J. D. Faires, Numerical Analysis, Fifth Edition, 1993,PWS Publishing Company, Boston.

A study of locking phenomena in oscillators”,

Adaptive harmonic balance analysis of oscillators using multiple time scales,” 3rd Int.

An efﬁcient nonlinear circuit simulation technique,”

Analyzing circuits with widely separated timescales using numerical PDE methods”,

Analyzing oscillators using multitime PDEs”,

Capturing oscillator injection locking via nonlinear phase-domain macromodels,”

Computing Arnol’d tongue scenarios,”

Efﬁcient AC analysis of oscillators using least-squares methods,”

Efﬁcient analysis of phase-locked loops through a novel time-frequency approach, based on two envelope transient formulations,”

Forced oscillations in a circuit with nonlinear resistance,”

General envelope-transient formulation of phase-locked loops using three time scales”,

Injection Locking Phenomena of Van Der Pol Oscillator”,

Locking range analysis for injectionlocked frequency dividers,”

Mathematical analysis of injectionlocked frequency dividers,”

Multitime scale differential equations for simulating frequency modulated signals”

Numerical Analysis, Fifth Edition,

Numerical continuation methods,N e w York,

Oscillator-ac: restoring rigour to linearized small-signal analysis of oscillators,”

study of injection locking and pulling in oscillators”,

Understanding injection locking in negative-resistance LC oscillators intuitively using nonlinear feedback analysis”,

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A new method for the determination of the locking range of oscillators

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