mathematical models of phase syncronization systems with quadrature and phase-quadrature units

ISSN 0005-1179, Automation and Remote Control, 2008, Vol. 69, No. 9, pp. 1475–1485. c© Pleiades Publishing, Ltd., 2008.Original Russian Text c© G.A. Leonov, 2008, published in Avtomatika i Telemekhanika, 2008, No. 9, pp. 33–43.

DETERMINATE SYSTEMS

Mathematical Models of Phase Syncronization Systems

with Quadrature and Phase-Quadrature Units

G. A. Leonov

St. Petersburg State University, St. Petersburg, RussiaReceived December 10, 2007

Abstract—Integro-differential and differential equations are derived for PLL systems withsquarer and for the Costas loop. The equations are based on special asymptotic methodsof high-frequency harmonic and pulse oscillations analysis. Methods to compute phase detec-tor characteristics are proposed. New classes of such characteristics are introduced for high-frequency pulse clocks.

PACS numbers: 02.30.Yy, 01.60.+q

DOI: 10.1134/S0005117908090038

1. INTRODUCTION

Phase-locked loops (PLL) are widely used in modern radio and circuit engineering [1–24].In [22–24] a PLL description methodology is proposed. This methodology includes three levels of

description: (1) electronic implementation level; (2) phase and frequency input-output relations inflowcharts; (3) differential, integro-differential, and difference equations. The second level involvesthe asymptotic analysis of high frequency oscillations and is essential to correctly derive PLLequations and to pass on to the third level. For example, the key concept of phase detector in thePLL theory arises at the second level. The phase detector characteristics depends on the class ofoscillations in question. If a typical PLL of classical scheme utilizes oscillation multipliers, thenthe phase detector characteristic for harmonic oscillations is also harmonic [1, 24], while for pulseoscillations the characteristic is a continuous piecewise-linear periodic function (providing that theelectronic implementation of the feedback loop is the same) [22–24].

The present paper advances the methodology mentioned above for PLLs with quadrature andphase-quadrature (the so-called Costas loop) units [4, 15]. Phase detector characteristics are com-puted, and differential equations describing noiseless PLL are derived for typical electronic im-plementations. The main result is that the PLL with integrator and harmonic clocks slaves thedouble frequency of the reference clock, while the PLL with pulse clocks slaves the frequency of thereference clock; the Costas loop with harmonic clocks slaves the frequency of the reference clock,while the Costas loop with pulse clocks slaves the half frequency of the reference clock.

All those assertions are strictly proven using the special methodology of high-frequency oscilla-tions asymptotic analysis. Approximate methods of phase detector characteristic computation forharmonic clocks were developed in [4]. It is much more difficult to compute the similar character-istics for pulse clocks.

2. SYSTEMS WITH SQUARER

Consider a flowchart of a PLL with squarer (Fig. 1). Here PS stands for perfect squarer, F forfilter, RC for reference clock, SC for slave clock,

⊗for multiplier.

1475

1476 LEONOV

RC PS F

SC

Fig. 1. System with squarer.

At first consider the case when the RC and SC produce “almost harmonic oscillations” f1(t)and f2(t)

fj(t) = Aj sin(ωj(t)t+ ψj). (1)

Here Aj > 0, ψj are constants, j = 1, 2.The unit

⊗is the oscillation multiplier: its output is u1(t)u2(t) if u1(t) and u2(t) are inputs.

The PS output is u(t)2 when its input is u(t).The relation between the input ξ(t) and the output σ(t) of a linear filter Φ is

σ(t) = aξ(t) + α(t) +t∫

0

γ(t− τ)ξ(τ)dσ.

Here a is a number, γ(t) is the pulse transition function, α(t) is an exponentially damped functionthat depends linearly on the filter’s initial state at t = 0.

Let’s write the oscillation’s high frequency property in the following form.Consider a large time interval [0, T ] that can be divided into small intervals [τ, τ + δ] (τ ∈ [0, T ])

such that

|γ(t) − γ(τ)| ≤ Cδ, |ωj(t) − ωj(τ)| ≤ Cδ ∀ t ∈ [τ, τ + δ] and ∀ τ ∈ [0, T ], (2)|2ω1(τ) − ω2(τ)| ≤ C1 ∀ τ ∈ [0, T ], (3)

ωj(τ) ≥ R ∀ τ ∈ [0, T ]. (4)

We assume here that δ is sufficiently small with respect to the numbers T , C, C1, while the numberRis sufficiently large with respect to δ : R−1 = O(δ2). The latter condition implies that the functionsγ(t) and ωj(t) are “almost constant” on small intervals [τ, τ + δ], while the functions fj(t) arerapidly oscillating on them. It is clear that these conditions hold for high-frequency oscillations.

Note that condition (3) is the requirement of a PLL with squarer for the difference 2ω1(t)−ω2(t)to be in a certain “acquisition band.” This band ensures the frequency acquisition of the slave clock(asymptotically, as a result of a transition process) to the double frequency of the reference clock.A correctly designed PLL of that type possesses the stability property

limt→+∞(2ω1(t) − ω2(t)) = 0, (5)

whenever |2ω1(t)−ω2(t)| is not greater than a given number (i.e., 2ω1(0)−ω2(0) is in the acquisitionband). Recall that for a conventional PLL (i.e., PLL without squarer) the requirement is

limt→+∞(ω1(t) − ω2(t)) = 0, (6)

instead of (5), and condition (3) is replaced by [24]

|ω1(τ) − ω2(τ)| ≤ C1 ∀ τ ∈ [0, T ]. (7)

AUTOMATION AND REMOTE CONTROL Vol. 69 No. 9 2008

MATHEMATICAL MODELS OF PHASE SYNCRONIZATION SYSTEMS 1477

F

1

f

1

(

t

)

2

g

(

t

)

f

2

(

t

)

F

1

2

θ

1

(

t

)

G

(

t

)

θ

2

(

t

)

PD

Fig. 2. Multiplier and filter. Fig. 3. Phase detector and filter.

Hence, conditions (3) and (7) are the requirements needed to derive PLL equations. Then themethods of stability theory applied to obtained equations [24] yield (5) or (6). Thus one can makethe conclusion that the derived equations describe properly the processes in stable PLLs.

Consider now the flowchart of Fig. 1. We will use the methodologies of [24] to derive the PLLequations.

Consider the flowcharts in Figs. 2 and 3. Let’s call θj(t) = ωj(t)t + ψj the phases of theoscillations fj(t). PD (phase detector or discriminator) is the nonlinear unit with the characteristicϕ(θ), θ(t) = 2θ1(t)−θ2(t). The input ξ(t) and the output σ(t) of the filter Φ1 are related as follows:

σ(t) = α(t) +t∫

0

γ(t− τ)ξ(τ)dτ.

Theorem 1. If conditions (2)–(4) hold, and

ϕ(θ) =14A2

1A2 sin θ,

then, for the same initial conditions of the filter,

|G(t) − g(t)| ≤ Dδ ∀ t ∈ [0, T ],

where D is a number, which does not depend on δ.

Proof. Recall the known formula

(sin β1)(sin β2)2 =12

sin β1 − 14(sin(β1 + 2β2) + sin(β1 − 2β2)). (8)

It is therefore evident that

g(t) −G(t)

=t∫

0

γ(t−s)[A2

1A2(sin(ω1(s)s+ψ1))2 sin(ω2(s)s+ψ2)−ϕ(2(ω1(s)s+ψ1)−ω2(s)s−ψ2)]ds

= A21A2

t∫

0

γ(t− s)[12

sin(ω2(s)s+ ψ2) − 14

sin(ω2(s)s + ψ2 + 2(ω1(s)s+ ψ1))]

ds

at t ∈ [0, T ].Consider the intervals [kδ, (k + 1)δ], where k = 0, . . . ,m, and the number m is such that t ∈

[mδ, (m + 1)δ].It follows from (2) that the following conditions hold at each interval [kδ, (k + 1)δ]:

γ(t− s) = γ(t− kδ) +O(δ), (9)ω2(s) = ω2(kδ) +O(δ), (10)

2ω1(s) + ω2(s) = 2ω1(kδ) + ω2(kδ) +O(δ), (11)

for all s ∈ [kδ, (k + 1)δ].


1478 LEONOV

Equations (10) and (11) imply the estimates:

sin(ω2(s)s+ ψ2) = sin(ω2(kδ)s + ψ2) +O(δ), (12)sin(ω2(s)s+ ψ2 + 2(ω1(s)s+ ψ1)) = sin((ω2(kδ) + 2ω1(kδ))s + ψ2 + 2ψ1) +O(δ) (13)

for all s ∈ [kδ, (k + 1)δ].Equations (9), (12), and (13) yield

t∫

0

γ(t− s)[12

sin(ω2(s)s+ ψ2) − 14

sin(ω2(s)s+ ψ2 + 2(ω1(s) + ψ1))]

ds

=m∑

k=0

γ(t− kδ)

(k+1)δ∫

kδ

[12

sin(ω2(kδ)s + ψ2) − 14

sin(ω2(kδ)s + ψ2 + 2(ω1(kδ)s + ψ1))]

ds+O(δ).

Combining (4) and the fact that R is sufficiently large with respect to δ, we obtain the followingestimates:

(k+1)δ∫

kδ

sin(ω2(kδ)s + ψ2)ds = O(δ2),

(k+1)δ∫

kδ

sin(ω2(kδ)s + ψ2 + 2(ω1(kδ)s + ψ1))ds = O(δ2).

This implies that

t∫

0

γ(t− s)[12

sin(ω2(s)s+ ψ2) − 14

sin(ω2(s)s+ ψ2 + 2(ω1(s)s+ ψ1))]

ds = O(δ).

The theorem is proven.The next result is proven the similar way.

Theorem 2. If conditions (2)–(4) hold, and ϕ(θ) =14A2

1A2 sin θ, then

∣∣∣∣∣∣

t∫

0

[f1(s)2f2(s) − ϕ(2θ1(s) − θ2(s))

]ds

∣∣∣∣∣∣≤ Dδ ∀ t ∈ [0, T ],


Consider now the pulse high-frequency oscillations of the form

f1(t) = A1 (1 + sgn sin(ω1(t)t+ ψ1)),f2(t) = A2 sgn sin(ω2(t)t+ ψ2).

(14)

Let’s formulate the analogues of Theorems 1 and 2. Introduce a 2π-periodic function F (θ), whichis also called the phase detector characteristic, in the following form:

F (θ) =

⎧⎪⎪⎨

⎪⎪⎩

2A21A2

(

1 +2θπ

)

at θ ∈ [−π, 0]

2A21A2

(

1 − 2θπ

)

at θ ∈ [0, π].



We assume that condition (7) is satisfied instead of (3).Let’s consider the flowcharts in Figs. 2 and 3 again, where fj(t) are from (14), and there is input

θ1(t) on Fig. 3 instead of 2θ1(t).

Theorem 3. If conditions (2), (4), and (7) hold, then, for the same initial conditions of the filter,

|G(t) − g(t)| ≤ Dδ ∀ t ∈ [0, T ],

where D is a number, which does not depend on δ,

Proof. It is clear that

g(t) −G(t) =T∫

0

γ(t− s)[2A2

1A2(sgn (sin(ω1(s)s+ ψ1) sin(ω2(s)s + ψ2))

+ sgn (sin(ω2(s)s+ ψ2))) − F (θ1(s) − θ2(s))

]ds

at t ∈ [0, T ]Similarly to the proof of Theorem 1, consider the intervals [kδ, (k + 1)δ] and then use (9), (10),

(12), and

ω1(s) + ω2(s) = ω1(kδ) + ω2(kδ) +O(δ). (15)

The estimate follows from (10) and (15):

sin((ω1(s) + ω2(s))s + ψ1 + ψ2) = sin((ω1(kδ) + ω2(kδ))s + ψ1 + ψ2) +O(δ)

for all s ∈ [kδ, (k + 1)δ]. Therefore, the estimate for the integral above is

t∫

0

γ(t− s)[2A2

1A2(sgn (cos(θ1(s) − θ2(s)) − cos((ω1(s)s+ ω2(s))s+ ψ1 + ψ2))

+ sgn (sin(ω2(s)s+ ψ2))) − F (θ1(s) − θ2(s))

]ds

=m∑

k=0

γ(t− kδ)

⎡

⎢⎣

(k+1)δ∫

kδ

[2A2

1A2(sgn cos(θ1(kδ) − θ2(kδ)) − cos((ω1(kδ) + ω2(kδ))s + ψ1 + ψ2)

)

+ sgn (sin(ω2(kδ)s + ψ2))]ds − F (θ1(kδ) − θ2(kδ))δ

⎤

⎥⎦ +O(δ). (16)

Let’s show that (2), (4), and (7) yield the estimate

2A21A2

(k+1)δ∫

kδ

sgn [cos(θ1(kδ) − θ2(kδ)) − cos((ω1(kδ) + ω2(kδ))s + ψ1 + ψ2)]ds

= F (θ1(kδ) − θ2(kδ))δ +O(δ2). (17)

Toward this aim, denote θ = θ1(kδ) − θ2(kδ) and note that the interval [kδ, (k + 1)δ] can bedivided into small intervals of length 2π/(ω1(kδ) + ω2(kδ)) such that the integral in question canbe easily calculated over those intervals:

2πF (θ)ω1(kδ) + ω2(kδ)

.


1480 LEONOV

This implies the estimate (17). Similarly, it can be shown that

(k+1)δ∫

kδ

sgn (sin(ω2(kδ)s + ψ2))ds = O(δ2).

The assertion of Theorem 3 follows from this estimate and (17).The following result is proven the similar way.

Theorem 4. If conditions (2), (4), and (7) hold, then∣∣∣∣∣∣

t∫

0

(f1(s)2f2(s) − F (θ1(s) − θ2(s))

)ds

∣∣∣∣∣∣≤ Dδ ∀ t ∈ [0, T ],


3. COSTAS LOOP

Consider now the flowchart of a system with phase-quadrature tracking loop (the Costas loop)(Fig. 4). The notation of Fig. 4 is the same as that of Fig. 1; 90◦ is the appropriate phase shift.

SC F

90˚RC

Fig. 4. Costas loop.

90˚

F

1

f

2

(

t

)

g

(

t

)

f

1

(

t

)

Fig. 5. Three multipliers and a filter.

F

1

2 θ

1

(

t

)

G

(

t

)

2

θ

2

(

t

)

PD

Fig. 6. Phase detector and filter.

Consider once again the case of high-frequencyharmonic and pulse signals fj(t). While (2) and (4)are supposed to be satisfied, condition (3) is re-placed by (7) for the signals (1) and by

|ω1(τ) − 2ω2(τ)| ≤ C1 ∀ τ ∈ [0, T ] (18)

for the signals (14).We will establish the asymptotic equivalence of

the flowcharts in Figs. 5 and 6. In order to obtaintheorems analogous to Theorems 1 and 2, we usethe following formula instead of (8):

(sin β1)(

sin(

β1 − π

2

))

(sin β2)2

= −14

sin 2β1 +18(sin 2(β1 + β2) + sin 2(β1 − β2)).

To get theorems analogous to Theorems 3 and 4 weuse the formula

(sin β1)(

sin(

β1 − π

2

))

(sin β2)

= −(cos(2β1 − β2) − cos(2β1 + β2)).

Applying the same scheme of proof as for Theorem 1, we obtain the following result for signalsof the form (1).

Theorem 5. If conditions (1), (2), (4), and (7) hold, and

ϕ1(θ) =18A2

1A22 sin θ,

then, for the same initial conditions of the filter,

|G(t) − g(t)| ≤ Dδ ∀ t ∈ [0, T ],

where D is a number, which does not depend on δ, and ϕ1(θ) is the PD characteristic.



The next result is analogous to Theorem 2 for the signals of the form (1).

Theorem 6. If conditions (1), (2), (4), and (7) hold, and

ϕ1(θ) =18A2

1A22 sin θ,

then∣∣∣∣∣∣

t∫

0

(

f1(s)2f2(s)f2

(

s− π

2

)

− ϕ1(2(θ1(s) − θ2(s))))

ds

∣∣∣∣∣∣≤ Dδ ∀ t ∈ [0, T ],


Introduce the phase detector characteristic P (θ) for the signals of the form (14): P (θ) =−A2F (θ), where F (θ) is the PD characteristic used in Theorems 3 and 4. Replace the input2θ1(t) by θ1(t) in Fig. 6.

Theorem 7. If conditions (2), (4), (14), and (18), hold, then, for the same initial conditions ofthe filter,

|G(t) − g(t)| ≤ Dδ ∀ t ∈ [0, T ].

The next theorem is analogous to Theorem 4.

Theorem 8. If conditions (2), (4), (14), and (18), hold, then∣∣∣∣∣∣

t∫

0

(

f1(s)2f2(s)f2

(

s− π

2

)

− P (θ1(s) − 2θ2(s)))

ds

∣∣∣∣∣∣≤ Dδ ∀ t ∈ [0, T ].

4. DIFFERENTIAL EQUATIONS OF THE SYNCHRONIZATION SYSTEM

Consider the value

θj(t) = ωj(t) + ωj(t)t.

For a correctly designed (globally stable) PLL the value ωj(t) damps exponentially:

|ωj(t)| ≤ Ce−αt. (19)

Here C and α are positive constants, which do not depend on t. Therefore, ωj(t)t is sufficientlysmall with respect to the number R (see condition (4)).

From the above reasoning it can be deduced that the following approximate equation holds:

θj(t) = ωj(t). (20)

In the PLL theory this equation is usually considered to be exact for the same reasons [24].The control law of the slave clock is usually assumed to be linear [4–6]:

ω2(t) = ω2(0) + Lg(t). (21)


1482 LEONOV

Here ω2(0) is the initial frequency of the slave clock, L is a number, g(t) is the control signal, whichis the filter output.

It follows from (21) that

θ2(t) = θ2(0) + ω2(0)t+ L

⎛

⎝a

t∫

0

ξ(τ)dτ +t∫

0

α(τ)dτ +t∫

0

τ∫

0

γ(τ − s)ξ(s)dsdτ

⎞

⎠ , (22)

where ξ(t) is the filter input.From Theorems 2, 4, 6, 8 we get that the value

t∫

0

ξ(τ)dτ

can be asymptotically replaced by the values

t∫

0

ϕ(2θ1(τ) − θ2(τ))dτ,t∫

0

F (θ1(τ) − θ2(τ))dτ,

t∫

0

ϕ1(2θ1(τ) − 2θ2(τ))dτ,t∫

0

P (θ1(τ) − 2θ2(τ))dτ.

From Theorems 1, 3, 5, 7 we obtain that the values

t∫

0

γ(t− s)ξ(s)ds

are asymptotically equivalent to the values

t∫

0

γ(t− s)ϕ(2θ1(s) − θ2(s))ds,t∫

0

γ(t− s)F (θ1(s) − θ2(s))ds,

t∫

0

γ(t− s)ϕ1(2θ1(s) − 2θ2(s))ds,t∫

0

γ(t− s)P (θ1(s) − 2θ2(s))ds.

Making these asymptotic substitutions in (22) and supposing that the reference clock is highlystable (i.e., ω1(t) ≡ ω1(0)), we obtain the following integro-differential equations of PLLs in ques-tion.

For the PLL with squarer and harmonic clocks:

d

dt(2θ1(t) − θ2(t))

• + L

⎛

⎝aϕ(2θ1(t) − θ2(t)) + α(t) +t∫

0

γ(t− τ)ϕ(2θ1(τ) − θ2(τ))dτ

⎞

⎠

= 2ω1(0) − ω2(0). (23)

For the PLL with squarer and pulse clocks:

d

dt(θ1(t) − θ2(t))

• + L

⎛

⎝aF (θ1(t) − θ2(t)) + α(t) +t∫

0

γ(t− τ)F (θ1(τ) − θ2(τ))dτ

⎞

⎠

= ω1(0) − ω2(0). (24)



RC F

SC

PD

β

2

(

t

)

β

1

(

t

)

Fig. 7. PLL with squarer and Costas loop.

For the Costas loop with harmonic clocks:

d

dt(2θ1(t) − 2θ2(t))

•

+2L

⎛

⎝aϕ1(2θ1(t) − 2θ2(t)) + α(t) + 2t∫

0

γ(t− τ)ϕ1(2θ1(τ) − 2θ2(τ))dτ

⎞

⎠

= 2ω1(0) − 2ω2(0). (25)

For the Costas loop with pulse clocks:

d

dt(θ1(t) − 2θ2(t))

•

+2L

⎛

⎝aP (θ1(t) − 2θ2(t)) + 2α(t) + 2t∫

0

γ(t− τ)P (θ1(τ) − 2θ2(τ))dτ

⎞

⎠

= ω1(0) − 2ω2(0). (26)

The flowcharts of PLLs corresponding to these equations are presented in Fig. 7.Here β1(t) = 2θ1(t) β2(t) = θ2(t) for(23); β1(t) = θ1(t), β2(t) = θ2(t) for (24); β1(t) = 2θ1(t),

β2(t) = 2θ2(t) for (25); β1(t) = θ1(t), β2(t) = 2θ2(t) for (26).When the transfer function a + W (p) of the filter is not degenerate, i.e., when its numerator

and denominator don’t have common zeros, Eqs. (23)–(26) are equivalent to the following systemof differential equations [9–13, 24]:

z = Az + bψ(σ),σ = c∗z + ρψ(σ).

(27)

Here A is a constant (n × n)-matrix, b and c∗ are constant n-vectors, ρ is a number, ψ(σ) is a2π-periodic function such that

ρ = −aL, W (p) = L−1c∗(A− pI)−1b (for (23) and (24));

ψ(σ) = ϕ(σ) − 2ω1(0) − ω2(0)L(a+W (0))

, σ = 2θ1 − θ2 (for (23));

ψ(σ) = F (σ) − ω1(0) − ω2(0)L(a+W (0))

, σ = θ1 − θ2 (for (24));

ρ = −2aL, W (p) = (2L)−1c∗(A− pI)−1b (for (25) and (26));

ψ(σ) = ϕ1(σ) − ω1(0) − ω2(0)L(a+W (0))

, σ = 2θ1 − 2θ2 (for (25));

ψ(σ) = P (σ) − ω1(0) − 2ω2(0)2L(a+W (0))

, σ = θ1 − 2θ2 (for (26)).


1484 LEONOV

Theory of global stability is well developed for system (27) with nonlinearities of the typedescribed above [9–13, 24]. It can be applied to obtain different constructive conditions for esti-mates (19). These conditions allow to establish correctness of (27) as PLL description.

Theorems 1–8 show that the deterministic (noiseless) description of PLL with squarer and Costasloop does not require auxilliary filters, which are usually introduced [4, 15]. The center filter takesthe functions of auxilliary filters. It is possible to implement proportional-integrating filters (whena �= 0), for the slave clock itself has some filtering properties (Theorems 2, 4, 6, 8).

5. CONCLUSION

The obtained integral estimates for high-frequency oscillations allow to determine characteristicsof phase detectors for the PLL and the Costas loop. While the phase detector characteristics areharmonic in the case of harmonic clocks, they are continuous piecewise-linear periodic functions inthe case of pulse clocks. The analysis of these characteristics reveals that the PLL with integratorand harmonic clocks slaves the double frequency of the reference clock, while the PLL with pulseclocks slaves the frequency of the reference clock; the Costas loop with harmonic clocks slaves thefrequency of the reference clock, while the Costas loop with pulse clocks slaves the half frequencyof the reference clock.

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This paper was recommended for publication by B.T.Polyak, a member of the Editorial Board


mathematical models of phase syncronization systems with quadrature and phase-quadrature units

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