1

Hold-in, pull-in, and lock-in ranges of PLL circuits:rigorous mathematical definitionsand limitations of classical theory.

Leonov G.A., Kuznetsov N.V., Yuldashev M.V., Yuldashev R.V.

Abstract—The terms hold-in, pull-in (capture), andlock-in ranges are widely used by engineers for theconcepts of frequency deviation ranges within whichPLL-based circuits can achieve lock under various ad-ditional conditions. Usually only non-strict definitionsare given for these concepts in engineering literature.After many years of their usage, F. Gardner in the 2ndedition of his well-known work, Phaselock Techniques,wrote “There is no natural way to define exactly any uniquelock-in frequency” and “despite its vague reality, lock-in rangeis a useful concept”. Recently these observations haveled to the following advice given in a handbook onsynchronization and communications “We recommendthat you check these definitions carefully before using them” [1,p.49]. In this survey an attempt is made to discuss andfill some of the gaps identified between mathematicalcontrol theory, the theory of dynamical systems andthe engineering practice of phase-locked loops. It isshown that, from a mathematical point of view, insome cases the hold-in and pull-in “ranges” may notbe the intervals of values but a union of intervals andthus their widely used definitions require clarification.Rigorous mathematical definitions for the hold-in, pull-in, and lock-in ranges are given. An effective solutionfor the problem on the unique definition of the lock-infrequency, posed by Gardner, is suggested.

Index Terms—Phase-locked loop, nonlinear analysis,analog PLL, high-order filter, local stability, globalstability, stability in the large, cycle slipping, hold-inrange, pull-in range, capture range, lock-in range, defi-nition, Gardner’s problem on unique lock-in frequency,Gardner’s paradox on lock-in range.

I. Introduction

THE phase-locked loop based circuits (PLL) arewidely used in various applications. A PLL is es-

sentially a nonlinear control system and its nonlinearanalysis is a challenging task. Much engineering writingis devoted to the study of PLL-based circuits and thevarious characteristics for their stability (see, e.g. a rathercomprehensive bibliography of pioneering works in [2]).An important engineering characteristic of PLL is a set ofparameters’ values for which a PLL achieves lock. In theclassical books on PLLs [3]–[5], published in 1966, suchconcepts as hold-in, pull-in, lock-in, and other frequency

Saint-Petersburg State University, Russia; University of Jyvaskyla,Finland (corr. author e-mail:nkuznetsov239@gmail.com).

IEEE Transactions on Circuits and Systems–I: Regular Papers;DOI 10.1109/TCSI.2015.2476295 (accepted)

ranges for which PLL can achieve lock, were introduced.They are widely used nowadays (see, e.g. contemporaryengineering literature [6]–[8] and other publications). Usu-ally in engineering literature only non-strict definitionsare given for these concepts. F. Gardner in 19791 in the2nd edition of his well-known work, Phaselock Techniques,formulated the following problem [9, p.70] (see also the 3rdedition [6, p.187-188]): “There is no natural way to defineexactly any unique lock-in frequency”. The lack of rigorousexplanations led to the paradox: “despite its vague reality,lock-in range is a useful concept” [9, p.70]. Many years ofusing definitions based on the above concepts has led tothe advice given in a handbook on synchronization andcommunications, namely to check the definitions carefullybefore using them [1, p.49].

In this paper it is shown that, from a mathematicalpoint of view, in some cases the hold-in and pull-in“ranges” may be not intervals of values but a union ofintervals, and thus their widely used definitions requireclarification. Next, rigorous mathematical definitions forthe hold-in, pull-in, and lock-in ranges are given. In ad-dition we suggest an effective solution for the problem ofthe unique definition of the lock-in frequency, posed byGardner.

II. Classical nonlinear mathematical models ofPLL-based circuits in a signal’s phase space

In classical engineering publications various analogPLL-based circuits are represented in a signal’s phasespace (also named frequency-domain [10, p.338]) by theblock diagram shown in Fig. 1.

H(s)

Loop filterPD

VCO

1s L

φ( )φ(θ∆(t)) g(t)

Lg(t)

θ∆(t)θ1(t)

θ2(t)

+

+

-

ωfree

θ2(0)

x(0)

Fig. 1. PLL-based circuit in a signal’s phase space.

1A year later, in 1980, F.Gardner was elected IEEE Fellow “forcontributions to the understanding and applications of phase lockloops”.

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Considering the corresponding mathematical model: thePhase Detector (PD) is a nonlinear block; the phasesθ1,2(t) of the input (reference) and VCO signals are PDblock inputs and the output is a function ϕ(θ∆(t)) =ϕ(θ1(t) − θ2(t)) named a phase detector characteristic,where

θ∆(t) = θ1(t)− θ2(t), (1)

named the phase error. The relationship between the inputϕ(θ∆(t)) and the output g(t) of the linear filter (Loopfilter) is as follows:

x = Ax+ bϕ(θ∆(t)), g(t) = c∗x+ hϕ(θ∆(t)), (2)

where A is a constant matrix, x(t) ∈ Rn the filter state,x(0) the initial state of filter, b and c constant vectors, andh a number. The filter transfer function has the form:2

H(s) = −c∗(A− sI)−1b+ h. (3)

A lead-lag filter [7] (usually H(0) = −c∗A−1b + h = 1,but H(0) can also be any nonzero value when an activelead-lag filter is used), or a PI filter (H(0) is infinite) isusually used as the filter. The solution of (2) with initialdata x(0) (the filter output for the initial state x(0)) is asfollows:

g(t, x0) = α0(t, x(0)) +t∫

0γ(t− τ)ϕ(θ∆(τ))dτ + hϕ(θ∆(t)),

(4)where γ(t − τ) = c∗eA(t−τ)b is the impulse responsefunction of the filter and α0(t, x(0)) = c∗eAtx(0) the zeroinput response (natural response, i.e. when the input ofthe filter is zero). The control signal g(t) adjusts the VCOfrequency to the frequency of the input signal:

θ2(t) = ω2(t) = ωfree2 + Lg(t), (5)

where ωfree2 is the VCO free-running frequency (i.e. for

g(t) ≡ 0) and L the VCO gain. Nonlinear VCO models canbe similarly considered, see, e.g. [12], [13]. The frequencyof the input signal (reference frequency) is usually assumedto be constant:

θ1(t) = ω1(t) ≡ ω1. (6)

The difference between the reference frequency and theVCO free-running frequency is denoted as ωfree

∆ :

ωfree∆ ≡ ω1 − ωfree

2 . (7)

By combining equations (1), (2), and (5)–(7) a nonlinearmathematical model in the signal’s phase space is obtained(i.e. in the state space: the filter’s state x and the differencebetween the signal’s phases θ∆):

x = Ax+ bϕ(θ∆),θ∆ = ωfree

∆ − Lc∗x− Lhϕ(θ∆).(8)

Nowadays nonlinear model (8) is widely used (see, e.g. [7],[14], [15]) to study acquisition processes of various circuits.The model can be obtained from the corresponding model

2 In the control theory the transfer function is often defined withthe opposite sign (see, e.g. [11]): H(s) = c∗(A− sI)−1b− h.

in the signal space (called also time-domain [10, p.329])by averaging under certain conditions [11], [16]–[19], arigorous consideration of which is often omitted (see, e.g.classical books [4, p.12,15-17], [3, p.7]) while their violationmay lead to unreliable results (see, e.g. [20], [21]).

Usually the PD characteristic is an odd function (e.g.a PD realization such as a multiplier, JK-flipflop, EXOR,PFD, and other elements [7]). Note that the PD charac-teristic ϕ(θ∆) depends on the waveforms of the consideredsignals [18], [19]). For the classical PLL with sinusoidal sig-nals and a two-phase PLL we have ϕ(θ∆) = 1

2 sin(θ∆), forthe classical BPSK Costas loop with ideal low-pass filtersand a two-phase Costas loop we have ϕ(θ∆) = 1

8 sin(2θ∆).Classical PD characteristics are bounded piecewise

smooth 2π periodic functions3:ϕ(θ∆(t) + 2πk) = ϕ(θ∆(t)), ∀k = 0, 1, 2...

Thus, it is convenient to assume that θ∆ mod 2π is acyclic variable, and the analysis is restricted to the rangeof θ∆(0) ∈ [−π, π).

For the case of an odd PD characteristic4, system (7) isnot changed by the transformation(

ωfree∆ , x(t), θ∆(t))→

(− ωfree

∆ ,−x(t),−θ∆(t)). (9)Property (9) allows the analysis of system (8) with onlyωfree

∆ > 0 and introduces the concept of frequency deviation|ωfree

∆ | = |ω1 − ωfree2 |.

III. Locked stateThe locked states (also called steady states) of the

model in the signal’s phase space must satisfy the followingconditions:• the phase error θ∆ is constant, the frequency error θ∆

is zero;• the model in a locked state approaches the same

locked state after small perturbations (of the VCOphase, input signal phase, and filter state).

The locally asymptotically stable equilibrium (stationary)points of model (8):

θ∆(t) ≡ θeq + 2πk, x(t) ≡ xeq, (10)are locked states, i.e. satisfy the above conditions5.

Considering the case of a nonsingular matrix A (i.e. thetransfer function of the filter does not have zero poles),the equilibria of (8) (stationary points) are given by theequations

ϕ(θeq) = ωfree∆

L(−c∗A−1b+ h) = ωfree∆

LH(0) ,

xeq = −A−1bϕ(θeq) = −A−1bωfree

∆LH(0) .

(11)

3 If ϕ(θ∆(t)) has another period (e.g. π for the Costas loopmodels), it has to be considered in the further discussion insteadof 2π.

4 There are examples of non odd PD characteristics, where (9) doesnot hold true (see, e.g. BPSK Costas loop with sawtooth signals [18],[19] and others).

5 It can be proved that if the filter is controllable and observable,then only equilibria satisfy locked state conditions, i.e. the filter statex(t) must be constant in the locked state [11].

3

Thus, the equilibria can be considered as a multiple-valuedfunction of variable ωfree

∆ :(xeq(ωfree

∆ ), θeq(ωfree∆ )

). From the

boundedness of the PD characteristic ϕ(θeq) it follows thatthere are no equilibria for sufficiently large |ωfree

∆ | (seeFigs. 2 and 3).

IV. Engineering definitions of stability ranges

The widely used engineering assumption (see Viterbi’spioneering writing [4, p.15]) is that the zero input responseof filter α0(t, x0) does not affect the synchronization ofthe loop. This assumption allows the filter state x(t)to be excluded from the consideration and a simplifiedmathematical model of PLL-based circuit in the signal’sphase space to be obtained from (4) and (5) (see, e.g. [4,p.17, eq.2.20] for h = 0 and [3, p.41, eq.4-26] for γ ≡ 0):

θ∆ =ωfree∆ − L

∫ t

0γ(t− τ)ϕ(θ∆(τ))dτ − Lhϕ(θ∆(t)). (12)

For an example of this one-dimensional integro-differentialequation the following intervals ([3], [4]) are defined: thehold-in range includes |ωfree

∆ | such that model (12) hasan equilibrium θ∆(t) ≡ θeq, which is locally stable (localstability, i.e. for some initial phase error θ∆(0)); the pull-inrange includes |ωfree

∆ | such that any solution of model (12)is attracted to one of the equilibria θeq (global stability, i.e.for any initial phase error θ∆(0)). Thus, the block diagramof the loop in Fig. 1 is usually considered without initialdata x(0) and θ∆(0) (see, e.g. [4, p.17, fig.2.3]).

Viterbi [4] explains the above assumption for the stablematrix A, but considers also various filters with marginallystable matrixes (e.g. a filter – perfect integrator, whereA = 0). At the same time, even for a stable matrix A, theinitial filter state x(0) and α0(t, x0) may affect the acqui-sition process and stability ranges (see, e.g. correspondingexamples for the classical PLL [20] and Costas loops [21]–[24]).

While the above assumption allows introduction ofthe above one-dimensional stability sets, defined only by|ωfree

∆ |, for rigorous study the multi-dimensional stabilitydomains have to considered, taking into account x(0), andtheir relationships with the classical engineering rangeshave to be explained. In [6, p.187] it is noted that theconsideration of all state variables is of utmost importancein the study of cycle slips and the lock-in concept.

V. Rigorous definitions of stability sets

The rigorous mathematical definitions of the hold-in,pull-in, and lock-in sets are now given for the nonlinearmathematical model of PLL-based circuits in the signal’sphase space (8) and corresponding nontrivial examples areconsidered.

A. Local stability and hold-in setWe now consider the linearization6 of system (8) along

an equilibrium (xeq, θeq). Taking into account (11) andϕ′(θ) := dϕ(θ)/dθ, the linearized system is as follows:(

x

θ∆

)=(

A bϕ′(θeq)−Lc∗ −Lhϕ′(θeq)

)(x− xeqθ∆ − θeq

)(13)

The characteristic polynomial of linear system (8) can bewritten (using the Schur complement, e.g. [11]) in thefollowing form: χ(s) =

(− Lhϕ′(θeq) − s + Lc∗(A −

sI)−1bϕ′(θeq))

det(A−sI), or can be expressed in terms ofthe filter’s transfer function H(s) = a(s)

d(s) , where a(s) andd(s) are polynomials:

χ(s) = −(sd(s) + a(s)Lϕ′(θeq)

). (14)

The characteristic polynomial corresponds to the denom-inator of the closed loop transfer function7.

To study the local stability of equilibria (11), it is nec-essary to check whether all the roots of the characteristicpolynomial (14) for the linearization of model (8) alongthe equilibria (i.e. the poles of the closed loop transferfunction) have negative real parts. For this purpose, atthe stage of pre-design analysis when all parameters of theloop can be chosen precisely, the Routh-Hurwitz criterionand its analogs (see, e.g. Kharitonov’s generalization [30]for interval polynomials) can be effectively applied. Atthe stage of post-design analysis when only the input andVCO output are considered and the parameters are knownonly approximately, various frequency characteristics ofthe loop (see, e.g. Nyquist and Bode plots) and thecontinuation principle can be used (see, e.g, [6], [7]).

If the PD characteristic is an odd function and henceϕ′(θeq) is an even function, from (9) we conclude that

1) there are symmetric equilibria:(xeq(ωfree

∆ ), θeq(ωfree∆ )

)=(− xeq(−ωfree

∆ ),−θeq(−ωfree∆ )

),

2) these symmetric equilibria are simultaneously stableor unstable.

The same holds true for nonstationary trajectories.

Definition 1. A set of all frequency deviations |ωfree∆ | such

that the mathematical model of the loop in the signal’sphase space has a locally asymptotically stable equilibriumis called a hold-in set Ωhold-in.

Thus, a value of frequency deviation belongs to the hold-in set if the loop re-achieves locked state after small per-turbations of the filter’s state, the phases and frequenciesof VCO and the input signals. This effect is also calledsteady-state stability. In addition, for a frequency deviationwithin the hold-in set, the loop in a locked state tracks

6 Here it is assumed that the PD characteristic ϕ(θ∆) is smoothat the point θ∆ = θeq . However, there are PLL-based circuitswith nonsmooth or discontinuous PD characteristics (see, e.g. thesawtooth PD characteristic for PLL [6], the model of QPSK Costasloop [25], and some others [26]–[28]). In such a case care has to betaken of the definition of solutions, the linearization of the model andthe analysis of possible sliding solutions (see, e.g. [29]).

7 Consideration of linearized model (13) allows to avoid the rig-orous discussion of initial states (x(0), θ∆(0)) related to the Laplacetransformation and transfer functions [11].

4

−5 0 5 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

θΔ

x

Fig. 2. Phase portrait for ωfree∆ from the hold-in range. The

system’s evolving state over time traces a trajectory (x(t), θ∆(t)).Trajectories can not intersect. Unstable equilibrium points, such assaddles — black dots, locally asymptotically stable equilibria —green dots, any of which has its own basin of attraction (shadeddomain) bounded by stable saddle separatrices (black trajectoriesgoing to the saddles). There are initial states and correspondingtrajectories (see, e.g. dashed trajectory), which are not attracted toan equilibrium.

small changes in input frequency, i.e. achieves a new lockedstate (tracking process).

In the literature the following explanations of the hold-in range (sometimes also called a lock range [31, p.507],[32, p.10-2], a synchronization range [33], a tracking range[1, p.49]) can be found: “The hold-in range is obtained bycalculating the frequency where the phase error is at itsmaximum” [34, p.171], “The maximum frequency differ-ence before losing lock of the PLL system is called the hold-in range” [8, p.258]. The following example shows thatthese explanations may not be correct, because for high-order filters the hold-in “range” may have holes.

The following example shows that the hold-in set maynot include ωfree

∆ = 0.

Example 1 (the hold-in set does not contain ωfree∆ = 0).

Consider the classical PLL with the sinusoidal PD char-acteristic ϕ(θ∆) = 1

2 sin(θ∆), VCO input gain L = 8, andthe filter transfer function

H(s) = a(s)d(s) = 1 + 0.5s

1 + 0.5s+ 0.5s2 . (15)

From (11) the following equation for equilibria is obtained:12 sin(θeq) = 1

8ωfree∆ . (16)

Applying the Routh-Hurwitz stability criterion8 to thedenominator of the closed loop transfer function (14)

s3 + s2 + s(2 + 4 cos(θeq)) + 8 cos(θeq), (17)

8 For a third-order polynomial χ(s) = a3s3 + a2s2 + a1s + a0,all the roots have negative real parts and the corresponding linearsystem is asymptotically stable if a1,2,3 > 0 and a2a1 > a3a0. Forχ(s) = a4s4 +a3s3 +a2s2 +a1s+a0, all the coefficients must satisfya1,2,3,4 > 0, and a3a2 > a4a1 and a3a2a1 > a4a2

1 + a23a0.

0 10 20 30 40 50−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

θΔ

x

Fig. 3. Phase portrait for ωfree∆ outside the hold-in range: there are

no locally stable equilibria.

the following conditions are obtained:

cos(θeq) > 0, (2 + 4 cos(θeq)) > 0,(2 + 4 cos(θeq)) > 8 cos(θeq).

(18)

Then 0 < cos(θeq) < 12 , and for the locked state the steady-

state phase error (i.e. corresponding to an equilibrium) isobtained

θeq ∈ (−π2 ,−π

3 ) ∪ (π3 ,π

2 ). (19)

Taking into account (16), (19), one obtains the hold-in set

|ωfree∆ | ∈ (2

√3, 4). (20)

The next example shows that the hold-in set may notactually be a range (i.e., an interval) but a union ofintervals, one of which may include ωfree

∆ = 0.

Example 2 (the hold-in set is a union of disjoint intervals,one of which contains ωfree

∆ = 0). Consider the classi-cal PLL with the sinusoidal PD characteristic ϕ(θ∆) =12 sin(θ∆), the VCO input gain L = 80, and the filtertransfer function

H(s) = 1 + 0.25s+ 0.5s2

1 + 2s+ 2s2 + 2s3 . (21)

From (11) the following equation for the equilibria isobtained:

12 sin(θeq) = 1

80ωfree∆ . (22)

An equilibrium is asymptotically stable if and only if all theroots of polynomial (14):

s(1 + 2s+ 2s2 + 2s3) +K(1 + 0.25s+ 0.5s2) =2s4 + 2s3 + s2(2 + 0.5K) + s(1 + 0.25K) +K,

K = Lϕ′(θeq) = 40 cos(θeq)(23)

have negative real parts. Using the Routh-Hurwitz crite-rion, we obtain

2 + 0.5K > 0, 1 + 0.25K > 0, K > 0,2(2 + 0.5K) > 2(1 + 0.25K),2(2 + 0.5K)(1 + 0.25K) > 2(1 + 0.25K)2 + 22K.

(24)

5

From these inequalities we have

K = 40 cos(θeq) ∈ (0, 12− 8√

2) ∪ (12 + 8√

2,∞),

θeq ∈ (−π2 ,−1.5536) ∪ (−0.9486, 0.9486) ∪ (1.5536, π2 ).(25)

Note that for other values of θeq at least one root ofthe polynomial (23) has a positive real part, making thecorresponding equilibrium unstable. Combining (22) and(25), we obtain the hold-in set

|ωfree∆ | ∈ [0, 32.5) ∪ (39.9942, 40). (26)

Note that in this case, for the values of the VCO input gainL > 24+16

√2 the hold-in set is always a union of disjoint

intervals. For L = 80 the simulation results of transitionprocess in Simulink model 9 in Fig. 4 are shown in Figs. 5–7for the initial data (x(0) = (0; 0; 0.9990), θ∆(0) = 1.5585)and various ωfree

∆ .

100-39.997

free-runningfrequency

filter_output

sin

PDcharacteristic

1s

Integrator

80

Gain

phase_diff

0.5s +0.25s+122s +2s +2s+13 2

Transfer Fcn(with initial states)

100

referencefrequency

1s

Integrator1Subtract

0.5

Gain1

Fig. 4. MatLab Simulink: the signal’s phase space model of theclassical PLL

Related discussion on the frequency responses of loopwith high-order filters can be found in [6, p.34-38, 52-56].

Remark 1. For the first order filters, the set Ωhold-in isan interval |ωfree

∆ | < ωh. For higher order filters, the setΩhold-in may be more complex. Thus, from an engineeringpoint of view, it is reasonable to require that ωfree

∆ = 0belongs to the hold-in set and to define a hold-in range asthe largest interval [0, ωh) from the hold-in set

[0, ωh) ⊂ Ωhold-in

such that a certain stable equilibrium varies continuouslywhen ωfree

∆ is changed within the range10. Here ωh is calleda hold-in frequency (see [3, p.38]).

9Following the above classical consideration, the filter is oftenrepresented in MatLab Simulink as the block Transfer Fcn with zeroinitial state (see, e.g. [35]–[39]). It is also related to the fact that thetransfer function (from ϕ to g) of linear system (2) is defined by theLaplace transformation for zero initial data x(0) ≡ 0. In Fig. 4 weuse the block Transfer Fcn (with initial states) to take into accountthe initial filter state x(0); the initial phase error θ∆(0) can be takeninto account by the property initial data of the Intergator blocks.Note that the corresponding initial states in SPICE (e.g. capacitor’sinitial charge) are zero by default but can be changed manually [40].

10 In general (when the stable equilibria coexist and some of themmay appear or disappear), the stable equilibria can be considered as amultiple-valued function of variable ωfree

∆ , in which case the existenceof its continuous singlevalue branch for |ωfree

∆ | ∈ [0, ωh) is required.

Remark 2. In the general case when there is no symmetrywith respect to ωfree

∆ (see (9)) the hold-in set need not besymmetric and the set ωfree

∆ ∈ Ωhold-in must be consideredin Definition 1.

0 20 40 60 80 100−80

−60

−40

−20

0

20

t

θ∆(t)

0 20 40 60 80 100−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t

g(t)

Fig. 5. ωfree∆ = 3: stable locked state exists.

0 20 40 60 80 100−1000

0

1000

2000

3000

4000

t

θ∆(t)

0 20 40 60 80 100−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

70 70.5−5

0

5x 10−3

t

g(t)

Fig. 6. ωfree∆ = 35: there are no locked states (see also Fig. 3).

0 20 40 60 80 1001.5575

1.558

1.5585

1.559

1.5595

1.56

t

θ∆(t)

0 20 40 60 80 1000.4999

0.5

0.5

0.5

0.5

0.5

t

g(t)

Fig. 7. ωfree∆ = 39.997: stable locked state exists.

B. Global stability (stability in the large) and pull-in setAssume that the loop power supply is initially switched

off and then at t = 0 the power is switched on, and assumethat the initial frequency difference is sufficiently large.The loop may not lock within one beat note, but theVCO frequency will be slowly tuned toward the referencefrequency (acquisition process). This effect is also called atransient stability. The pull-in range is used to name suchfrequency deviations that make the acquisition processpossible (see, e.g. explanations in [3, p.40], [7, p.61]).

To define a pull-in range (called also a capture range[41], an acquisition range [33, p.253]) rigorously, considerfirst an important definition from stability theory.

Definition 2. If for a certain ωfree∆ any trajectory of sys-

tem (8) tends to an equilibrium, then the system with suchωfree

∆ is called globally asymptotically stable (see Fig. 8).

6

−5 0 5 10 15 20 25

−0.2

0

0.2

0.4

0.6

θΔ

x

Fig. 8. Phase portrait for ωfree∆ from the pull-in range: any trajectory

is attracted to an equilibrium (equilibria: green–stable and black–unstable circles); for a sufficiently large initial state of the filter, cycleslipping is possible (see, e.g. dashed trajectory).

We now consider a possible rigorous definition.

Definition 3. A set of all frequency deviations |ωfree∆ | such

that the mathematical model of the loop in the signal’sphase space is globally asymptotically stable is called a pull-in set Ωpull-in.

Remark 3. In the general case when there is no symmetrywith respect to ωfree

∆ the set ωfree∆ ∈ Ωpull-in has to be

considered in Definition 3.

Remark 4. The pull-in set is a subset of the hold-in set:Ωpull-in ⊂ Ωhold-in, and need not be an interval. From anengineering point of view, it is reasonable to require thatωfree

∆ = 0 belongs to the pull-in set and to define a pull-inrange as the largest interval [0, ωp) from the pull-in set:

[0, ωp) ⊂ Ωpull-in,

where ωp is called a pull-in frequency (see [3, p.40]).

Remark 5. If all possible states of the filter are bounded:

x ∈ Xreal (e.g. Xreal = x : cmin < |x| < cmax),

by the design of the circuit (e.g. capacitors have limitedmaximum and minimum charges, the VCO frequency islimited etc.), then in the definition of pull-in set it isreasonable to require that only solutions with x(0) ∈ Xrealtend to the stationary set. Trajectories, with initial dataoutside of the domain defined by x(0) ∈ Xreal (here theinitial phase error θ∆(0) can take any value), need not tendto the stationary set.

For the model without filter (i.e. H(s) = const) thepull-in set coincides with the hold-in set. The pull-inset of PLL-based circuits with first-order filters can beestimated using phase plane analysis methods [42], [43],but in general its rigorous study is a challenging task [4],[12], [17], [44], [45].

For the case of the passive lead-lag filter H(s) =1+sτ2

1+s(τ1+τ2) , a recent work [12, p.123] notes that “the de-termination of the width of the capture range togetherwith the interpretation of the capture effect in the secondorder type-I loops have always been an attractive theoretical

problem. This problem has not yet been provided with asatisfactory solution”. At the same time in [11], [46]–[48]it is shown that the basin of attraction of the stationary setmay be bounded (e.g. by a semistable periodic trajectory,which may appear as the result of collision of unstable andstable periodic solutions), and corresponding analyticalestimations and bifurcation diagram are given.

Note that in this case a numerical simulation may givewrong estimates and should be used very carefully. Forexample, in [40] the SIMetrics SPICE model for a two-phase PLL with a lead-lag filter gives two essentiallydifferent results of simulation with default “auto” samplingstep (acquires lock) and minimum sampling step set to 1m(does not acquire lock — such behaviour agrees with thetheoretical analysis). The same problems are also observedin MatLab Simulink [20], [21], [49], see, e.g. Fig. 9. Theseexamples demonstrate the difficulties of numerical searchof so-called hidden oscillations [48], [50], [51], whose basinof attraction does not overlap with the neighborhoodof an equilibrium point, and thus may be difficult tofind numerically11. In this case the observation of one oranother stable solution may depend on the initial data andintegration step.

g(t)

t

relative tolerance `1e-3`relative tolerance `auto`

Fig. 9. Simulation of two-phase PLL described by Fig. 4 or model(8) [40]: τ1 = 0.0448, τ2 = 0.0185, A = − 1

τ1+τ2, b = 1 − τ2

τ1+τ2,

c = 1τ1+τ2

, h = τ2τ1+τ2

; ϕ(θ∆) = 12 sin(θ∆); ω1 = 10000, ωfree

2 =10000 − 178.9, L = 500. Filter output g(t) for the initial datax0 = 0.1318, θ∆(0) = 0 obtained for default “auto” relative tolerance(red) — acquires lock, relative tolerance set to “1e-3”(green) — doesnot acquire lock.

S. Goldman, who has worked at Texas Instruments over20 years, notes that PLLs are used as pipe cleaners forbreaking simulation tools [54, p.XIII].

11In [52] the crash of aircraft YF-22 Boeing in April 1992, causedby the difficulties of rigorous analysis and design of nonlinear controlsystems with saturation, is discussed and the conclusion is made thatsince stability in simulations does not imply stability of the phys-ical control system, stronger theoretical understanding is required(see, e.g. similar problem with the simulation of PLL in Fig. 9).These difficulties in part are related to well-known Aizerman’s andKalman’s conjectures on the global stability of nonlinear controlsystems, which are valid from the standpoint of simplified analy-sis by the linearization, harmonic balance, and describing functionmethods (note that all these methods are also widely used to theanalysis of nonlinear oscillators used in VCO [12], [13]). However thecounterexamples (multistable high-order nonlinear systems wherethe only equilibrium, which is stable, coexists with a hidden periodicoscillation) can be constructed to these conjectures [48], [53].

7

While PLL-based circuits are nonlinear control systemsand for their nonlocal analysis it is essential to apply theclassical stability criteria, which are developed in controltheory, however their direct application to analysis ofthe PLL-based models is often impossible, because suchcriteria are usually not adapted for the cylindrical phasespace12; in the tutorial Phase Locked Loops: a ControlCentric Tutorial [14], presented at the American ControlConference 2002, it was said that “The general theory ofPLLs and ideas on how to make them even more usefulseems to cross into the controls literature only rarely”.

At the same time the corresponding modifications ofclassical stability criteria for the nonlinear analysis ofcontrol systems in cylindrical phase space were well de-veloped in the second half of the 20th century, see, e.g.[29], [58]–[60]. A comprehensive discussion and the cur-rent state of the art can be found in [11]. One reasonwhy these works have remained almost unnoticed by thecontemporary engineering community may be that theywere written in the language of control theory and thetheory of dynamical systems, and, thus, may not be welladapted to the terms and objects used in the engineeringpractice of phase-locked loops. Another possible reason, asnoted in [61, p.1], is that the nonlinear analysis techniquesare well beyond the scope of most undergraduate coursesin communication theory and circuits design. Note thatfor the application of various stability criteria it is oftennecessary to represent system (8) in the Lur’e form:( ˙x

θ∆

)=(

A 0−Lc∗ 0

)(xθ∆

)+(

b−Lh

)ϕ(θ∆), (27)

wherex = x− xeq = x+A−1bϕ(θeq), ϕ(θ∆) = ϕ(θ∆)− ϕ(θeq),ϕ(θeq) = ωfree

∆ L−1(−c∗A−1b+ h)−1.

See also discussion of some nonlinear methods for theanalysis of PLL-based models in recent books [12], [13],[17], [62].

C. Cycle slips and lock-in rangeLet us rigorously define cycle slipping in the phase space

of system (8).

Definition 4. Iflim supt→+∞

|θ∆(0)− θ∆(t)| > 2π, (28)

it is then said that cycle slipping occurs (see, e.g. dashedtrajectory in Fig. 8).

Here, sometimes, instead of the limit of the difference,the maximum of the difference is considered (see, e.g. [44,p.131]).

12For example, in the classical Krasovskii–LaSalle principle onglobal stability the Lyapunov function has to be radially unbounded(e.g. V (x, θ∆)→ +∞ as ||(x, θ∆)|| → +∞). While for the applicationof this principle to the analysis of phase synchronization systemsthere are usually used Lyapunov functions periodic in θ∆ (e.g.V (x, θ∆) in Remark 8 is bounded for any ||(0, θ∆)|| → +∞), andthe discussion of this gap is often omitted (see, e.g. patent [15] andworks [55]–[57]). Rigorous discussion can be found, e.g. in [11], [29].

Definition 4’ If

supt>0|θ∆(0)− θ∆(t)| > 2π, (29)

it is then said that cycle slipping has occurred.Note that, in general, Definition 4’ need not mean

that finally (after acquisition) condition (28) can not befulfilled.

Sometimes, the number of cycle slips is of interest.

Definition 5. If

2kπ < lim supt→∞

|θ∆(0)− θ∆(t)| < 2(k + 1)π, (30)

it is then said that k cycle slips occurred.

A numerical study of cycle slipping in classical PLLcan be found in [63]. Analytical tools for estimating thenumber of cycle slips depending on the parameters of theloop can be found, e.g. in [11], [58], [64].

The concepts of lock-in frequency and lock-in range(called also a lock range [65, p.256], a seize range [66,p.138]), were intended to describe the set of frequencydeviations for which the loop can acquire lock within onebeat without cycle slipping. In [3, p.40] the following defi-nition was introduced: “If, for some reason, the frequencydifference between input and VCO is less than the loopbandwidth, the loop will lock up almost instantaneouslywithout slipping cycles. The maximum frequency differencefor which this fast acquisition is possible is called the lock-infrequency”.

However, in general, even for zero frequency deviation(ωfree

∆ = 0) and a sufficiently large initial state of filter(x(0)), cycle slipping may take place (see, e.g. dashedtrajectory in Fig. 10, left). Thus, considering of all statevariables is of utmost importance for the cycle slip analysisand, therefore, the concept lock-in frequency lacks rigor forclassical simplified model (12) because it does not take intoaccount the initial state of the filter. The above definitionof the lock-in frequency and corresponding definition ofthe lock-in range were subsequently in various engineeringpublications (see, e.g. [67, p.34-35], [68, p.161], [69, p.612],[70, p.532], [71, p.25], [1, p.49], [14, p.4], [72, p.24], [73,p.749], [74, p.56], [54, p.112], [7, p.61], [66, p.138], [75,p.576], [8, p.258]).

The loop model (8) has a subdomain of the phasespace, where trajectories do not slip cycles (called a lock-indomain), for each value of ωfree

∆ . The lock-in domain is theunion of local lock-in domains, each of which correspondsto one of the equilibria and has its own shape (see, e.g.shaded domain in Fig. 10, left defined by correspondingseparatrices). The shape of the lock-in domain significantlyvaries depending on ωfree

∆ . In [4, p.50]) a lock-in domain iscalled a frequency lock. Some writers (e.g. [44, p.132], [76,p.355]) use the concept lock-in range to denote a lock-indomain.

In general, taking into account nonuniform behavior ofthe lock-in domain shape, Gardner wrote “There is nonatural way to define exactly any unique lock-in frequency”[9, p.70], [6, p.188].

8

−5 0 5 10−0.06

−0.04

−0.02

0

0.02

0.04

0.06

θ∆

x

−5 0 5 10−0.06

−0.04

−0.02

0

0.02

0.04

0.06

θ∆

(black) positive ω∆free

(red) negative ω∆free

x

−5 0 5 10−0.06

−0.04

−0.02

0

0.02

0.04

0.06

θ∆

x

Fig. 10. Phase portraits for the classical PLL with the following parameters: H(s) = 1+sτ21+s(τ1+τ2) , τ1 = 4.48 · 10−2, τ2 = 1.85 · 10−2,

L = 250, ϕ(θ∆) = 12 sin(θ∆), and various frequency deviations. Black color is for the system with positive ωfree

∆ = |ω|. Red is for the systemwith negative ωfree

∆ = −|ω|. Equilibria (dots), separatrices pass in and out of the saddles, local lock-in domains are shaded (upper blackhorizontal lines is for ωfree

∆ > 0, lower red vertical lines is for ωfree∆ < 0). Left subfig: ωfree

∆ = 0; middle subfig: ωfree∆ = ±65; right subfig:

ωfree∆ = ±68. .

Below we demonstrate how to overcome these problemsand rigorously define a unique lock-in frequency and range.

We now consider a specific ωfree∆ and denote by

Dlock-in(ωfree∆ ) the corresponding lock-in domain. Such a

domain exists for any |ωfree∆ | ∈ Ωhold-in because at least

the equilibria are contained in this domain. For a setωfree

∆ ∈ Ω we consider the intersection of correspondinglock-in domains (see, e.g. the intersections of local lock-indomains for various ωfree

∆ = ±|ω| in Fig. 10 — domainsshaded both by red vertical and black horizontal lines):

Dlock-in(Ω) =⋂

ωfree∆ ∈Ω

Dlock-in(ωfree∆ ).

Definition 6. A lock-in range is the largest interval [0, ωl)such that for any |ωfree

∆ | ∈ [0, ωl) the mathematical modelof the loop in the signal’s phase space is globally asymptoti-cally stable (i.e. [0, ωl) ⊂ [0, ωp)) and the following domain

Dlock-in((−ωl, ωl)

)=

⋂|ωfree

∆ |<ωl

Dlock-in(ωfree∆ ).

contains all corresponding equilibria:(xeq(ωfree

∆ ), θeq(ωfree∆ )

)∈ Dlock-in

((−ωl, ωl)

).

We call such domain Dlock-in = Dlock-in((−ωl, ωl)

)a uni-

form lock-in domain (uniform with respect to (−ωl, ωl)),ωl is called a lock-in frequency (see [3, p.40]).

Various additional requirements may be imposed on theshape of the uniform lock-in domain Dlock-in, e.g. it has tocontain the line defined by x ≡ 0 (see, e.g. [8, p.258]) or theband defined by |x| < cmax. If instead of global stabilityin the definition of the pull-in set we consider stabilityin the domain defined by Xreal, then we require thatthe intersection Dlock-in

⋂Xreal contains all corresponding

equilibria.

Remark 6. In the general case when there is no symmetrywith respect to ωfree

∆ we have to consider a unsymmetricalinterval containing zero in Definition 6.

Similarly, we can define an extension of the lock-in range:Ωlock-in ⊃ [0, ωl), called a lock-in set (however, in general,such an extension may be not unique).

In other words, the definition implies that if the loopis in a locked state, then after an abrupt change of ωfree

∆within a lock-in range [0, ωl), the corresponding acquisitionprocess in the loop leads, if it is not interrupted, to a newlocked state without cycle slipping.

Finally, our definitions give Ωlock-in ⊂ Ωpull-in ⊂ Ωhold-in.If there is a certain stable equilibrium varies continuouslywhen ωfree

∆ is changed within the hold-in, pull-in, and lock-in ranges (see Footnote 10), then

[0, ωl) ⊂ [0, ωp) ⊂ [0, ωh)

which is in agreement with the classical consideration (see,e.g. [67, p.34], [69, p.612], [7, p.61], [66, p.138], [8, p.258]).

D. Approximations of the lock-in range of the classical PLLFor the case of the classical odd PD characteristic (see

Fig. 10), taking into account that equilibria are propor-tional to the frequency deviation (see (11)) and using thesymmetry

(xeq(ωl), θeq(ωl)

)= −

(xeq(−ωl), θeq(−ωl)

), we

can effectively determine ωl. For that, we have to increasethe frequency deviation |ωfree

∆ | step by step and at eachstep, after the loop achieves a locked state, to changeωfree

∆ = ω abruptly to ωfree∆ = −ω and to check if the loop

can achieve a new locked state without cycle slipping. Ifso, then the considered value |ωfree

∆ | belongs to Ωlock-in. Ifωfree

∆ =0 belongs to Ωpull-in, then it is clear that 0 belongsto Ωlock-in (see Fig. 10, left). The limit value ωl is definedby the case in Fig. 10, middle. At the next step when avalue |ωfree

∆ | = |ω| > ωl is considered, for ωfree∆ = −|ω| the

trajectory from the initial point, corresponding to a stableequilibrium for ωfree

∆ = |ω| (see Fig. 10, right: red trajectoryoutgoing from a black dot), is attracted to an equilibriumonly after cycle slipping. In other words [77], for this case:

9

The lock-in range is a subset of the pull-in range suchthat for each corresponding frequency deviation the lock-in domain (i.e. a domain of the loop states, where fastacquisition without cycle slipping is possible) contains bothsymmetric locked states (i.e. locked states for the positive

−5 0 5 10−0.06

−0.04

−0.02

0

0.02

0.04

0.06

θΔ

x

(black) positive ω∆free

(red) negative ω∆free

Fig. 11. Phase portrait. Separatrices, equilibria and correspondinglocal lock-in domains (shaded): upper black is for ωfree

∆ = 61.5, lowerred is for ωfree

∆ = −61, 5. The uniform lock-in domain is approximatedby the band between two blue horizontal lines: |x| ≤ 0.0110..

and negative value of the difference between the referencefrequency and the VCO free-running frequency).

In Fig. 10, middle the set Dlock-in: contains all equilib-ria xeq(ωfree

∆ ) for 0 ≤ |ωfree∆ | < ωl. However for some

non-equilibrium initial states from the band defined byx : |x| < |xeq(ωl)| (phase error θ∆ takes all possiblevalues), cycle slipping can take place. For example, see thepoints to the left and to the right of the black equilibriumstates (i.e. for ωfree

∆ = |ωl| > 0), lying above the redseparatrix (i.e. for ωfree

∆ = −|ωl| < 0), correspond to thered trajectories (i.e. for ωfree

∆ = −|ωl| < 0), which areattracted to an equilibrium only after cycle slipping. Toapproximate the Dlock-in by a band, ωl can be slightlydecreased to cut the above points. In Fig. 11 the banddefined by Xlock-in = x : |x| < |xeq(ωl)|, ωl < ωl iscontained in Dlock-in and for any initial state from the bandthe corresponding acquisition process in the loop leads, ifit is not interrupted, to lock up without cycle slipping.Such a construction is more laborious and requires rigorousanalysis of the phase space or exhaustive simulation.

Remark 7. If we define (see, e.g. [78, p.92]) cycle slippingby the interval of maximum length 2π instead of 4π inDefinition 4: i.e. lim supt→∞ |θ∆(0)− θ∆(t)| > π, then forany |ωfree

∆ | > 0 a distance between neighboring unstable andstable equilibria and a phase deviation of the correspondingunstable saddle separatrix may exceed π (see, e.g. Fig. 11).Thus, the lock-in range may contain only |ωfree

∆ | = 0.

Remark 8. If the filter – perfect integrator can be imple-mented in considered architecture, the loop can be designedwith the first order PI filter having the transfer functionH(s) = 1+sτ2

sτ1. Equations of the loop in this case become

−5 0 5 10−0.06

−0.04

−0.02

0

0.02

0.04

0.06

θΔ

x

(black) positive ω∆free

(red) negative ω∆free

Fig. 12. Phase portraits for the classical PLL with the following pa-rameters: H(s) = 1+0.0225s

0.0633s , L = 250, and ωfree∆ = ±47. Separatrices,

equilibria and corresponding local lock-in domains (shaded): upperblack is for ωfree

∆ = 47, lower red is for ωfree∆ = −47. The uniform

lock-in domain is approximated by the band between the two bluehorizontal lines: |x| ≤ 0.0119.

x = 1τ1ϕ(θ∆), θ∆ = ωfree

∆ − Lx− Lτ2τ1ϕ(θ∆), (31)

or equivalently

θ∆ = −L 1τ1ϕ(θ∆)− Lτ2

τ1ϕ′(θ∆)θ∆. (32)

Here the equilibria are defined from the equations

ϕ(θeq) = 0, xeq = ωfree∆ L−1.

Because model (32) does not depend explicitly on ωfree∆ ,

the hold-in and pull-in ranges are either infinite or empty.Note, that the parameter ωfree

∆ shifts the phase plane ver-tically (in the variable x) without distorting trajectories,which simplifies the analysis of the uniform lock-in domainand range (see Fig. 12). If the transfer function H(s) ofa high order filter has the term sr with r ∈ N in the de-nominator, then instead of equilibria we have a stationarylinear manifold: ϕ(θeq) = 0, c1x1

eq + . . .+ crxreq = −ωfree

∆L .

For the classical PLL with the filter’s transfer functionH(s) = β+αs

s it can be analytically proved that the pull-in range is theoretically infinite. Some needed explanationsare given by Viterbi [4] using phase plane analysis. But,even in such a simple case, rigorous phase plane analysisis a complex task (e.g. [79], the proof of the nonexistenceof heteroclinic and first-order cycles is omitted in [4]).The rigorous analytical proof can be effectively achievedby considering a special Lyapunov function [11], [55],[79]: V (x, θ∆) = 1

2(x − ωfree

∆L

)2 + 2βL sin2 θ∆

2 ≥ 0 and

10

V (x, θ∆) = −hβ sin2 θ∆ ≤ 0. Here it is important thatfor any ωfree

∆ the set V (x, θ∆) ≡ 0 does not contain thewhole trajectories of system (31) except for equilibria.

E. Initial and free-running frequencies of VCONote that in the above Definitions 1, 3, and 6 the hold-

in, pull-in, and lock-in sets are defined by the frequencydeviation, i.e. by the absolute value of the differencebetween VCO free-running frequency (in the open loop)and the input signal’s frequency: |ωfree

∆ | = |ω1 − ωfree2 |.

The VCO free-running frequency ωfree2 is different from the

VCO initial frequency ω2(0): ω2(0) = ωfree2 + g(0), where

g(0) = c∗x(0) + hϕ(θ∆(0)) is the initial control signal,depending on the initial states of the filter x(0) and theinitial phase difference θ∆(0).

It is interesting that for simplified model (12) withh = 0 (see eq. 2.20 in the classic reference [4]) theabsolute value of the initial difference between frequencies|θ∆(0)| = |ω∆(0)| = |ω1 − ω2(0)| is equal to the frequencydeviation |ωfree

∆ | = |ω1 − ωfree2 |. Following such simplified

consideration in engineering literature the concept of an“initial frequency difference” can be found to be in useinstead of the concept of a “frequency deviation”: see, e.g.[3, p.44] “If the initial frequency difference (between VCOand input) is within the pull-in range, the VCO frequencywill slowly change in a direction to reduce the difference”,[80, p.1792] “The maximum frequency difference betweenthe input and the output that the PLL can lock within onesingle beat note is called the lock-in range of the PLL”,[1, p.49] “Whether the PLL can get synchronized at all ornot depends on the initial frequency difference between theinput signal and the output of the controlled oscillator.” Ingeneral, the change of ωfree

2 to ω2(0) may lead to wrongresults in the above definitions of ranges because in thecase of x(0) 6= 0, h 6= 0 or non-odd function ϕ(θ∆) for thesame values of ω2(0) the loop can achieve synchronizationor not depending on the filter’s initial state x(0), the initialphase difference θ∆(0), and ωfree

2 . See the correspondingexample.

Example 3. Consider the behavior of model (8) forthe sinusoidal signals (i.e. ϕ(θ∆) = 1

2 sin(2θ∆)) and thefixed parameters: ω∆ = 100, H(s) = (1+sτ2)

1+s(τ1+τ2) , τ1 =0.0448, τ2 = 0.0185, L = 250. In Fig. 13 the phase portraitof system (8) is shown. The blue dash line consists ofpoints for which the initial frequency difference is zero:ω∆(0) = θ∆(0) = 0. Despite the fact that the initialfrequency differences of all trajectories outgoing from theblue line are the same (equal to 0), the green trajectorytends to a locked state while the magenta trajectory cannot achieve this.

VI. CONCLUSIONSThis survey discussed a disorder and inconsistency in

the definitions of ranges currently used. An attempt ismade to discuss and fill some of the gaps identified betweenmathematical control theory, the theory of dynamical sys-tems and the engineering practice of phase-locked loops.

0 2 4 6 8 10−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

θΔ

x

(blue) zero initial frequency diffirence ω (0)

(green) tends to a locked state(magenta) tends to infinity

Δ

Fig. 13. Phase portrait for ωfree∆ = 100. Blue dash curve corresponds

to the set defined by θ∆(0) = 0. Initial points of the green (upper) andmagenta (lower) trajectories correspond to the same initial frequencydifference ω∆(0) = 0.

Rigorous mathematical definitions for the hold-in, pull-in,and lock-in ranges are suggested. The problem of uniquelock-in frequency definition, posed by Gardner [9], is solvedand an effective way to determine the unique lock-infrequency is suggested.

ACKNOWLEDGEMENTSThis work was supported by the Russian Scientific

Foundation (project 14-21-00041) and Saint-PetersburgState University. The authors would like to thankRoland E. Best, the founder of the Best EngineeringCompany, Oberwil, Switzerland and the author of thebestseller on PLL-based circuits [7] for valuable discussion.

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Gennady Leonov received his Candidate de-gree in 1971 and Dr.Sci. in 1983 from Saint-Petersburg State University. In 1986 he wasawarded the USSR State Prize for develop-ment of the theory of phase synchronization forradiotechnics and communications. Since 1988he has been Dean of the Faculty of Mathemat-ics and Mechanics at Saint-Petersburg StateUniversity and since 2007 Head of the Depart-ment of Applied Cybernetics. He is member(corresponding) of the Russian Academy of

Science, in 2011 he was elected to the IFAC Council. His researchinterests are now in control theory and dynamical systems.

Nikolay Kuznetsov received his Candidatedegree from Saint-Petersburg State Univer-sity (2004) and PhD from the University ofJyvaskyla (2008). He is currently Deputy Headof the Department of Applied Cyberneticsat Saint-Petersburg State University and Ad-junct Professor at the University of Jyvaskyla.His interests are now in dynamical systemsstability and oscillations, Lyapunov exponent,chaos, hidden attractors, phase-locked loopnonlinear analysis, nonlinear control systems.

E-mail: nkuznetsov239@gmail.com (corresponding author)

Marat Yuldashev received his Candidatedegree from St.Petersburg State University(2013) and PhD from the University ofJyvaskyla (2013). He is currently at Saint-Petersburg University. His research interestscover nonlinear models of phase-locked loopsand Costas loops, and SPICE simulation.

Renat Yuldashev received his Candidatedegree from St.Petersburg State University(2013) and PhD from the University ofJyvaskyla (2013). He is currently at Saint-Petersburg University. His research interestscover nonlinear models of phase-locked loopsand Costas loops, and simulation in MatLabSimulink.