By Jacek Kudrewicz
Phase-Locked Loops (PLLs) are digital structures that may be used as a synchronized oscillator, a motive force or multiplier of frequency, a modulator or demodulator and as an amplifier of section modulated indications. This e-book updates the equipment utilized in the research of PLLs through drawing at the effects received within the final forty years. Many are released for the 1st time in ebook shape. Nonlinear and deterministic mathematical types of continuous-time and discrete-time PLLs are thought of and their uncomplicated houses are given within the type of theorems with rigorous proofs. The publication indicates very attractive dynamics, and exhibits numerous actual phenomena saw in synchronized oscillators defined by way of whole (not averaged) equations of PLLs. especially chosen mathematical instruments are used the speculation of differential equations on a torus, the phase-plane graphics on a cyclinder, a perturbation conception (Melnikov s theorem on heteroclinic trajectories), crucial manifolds, iterations of one-dimensional maps of a circle and two-dimensional maps of a cylinder. utilizing those instruments, the homes of PLLs, specifically the areas of synchronization are defined. Emphasis is on bifurcations of assorted varieties of periodic and chaotic oscillations. unusual attractors within the dynamics of PLLs are thought of, akin to these chanced on by way of RÃ¶ssler, Henon, Lorenz, may well, Chua and others.
Read or Download Equations of Phase-Locked Loops: Dynamics on the Circle, Torus and Cylinder (World Scientific Series on Nonlinear Science Series a) (World Scientific Series on Nonlinear Science: Series a) PDF
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Additional resources for Equations of Phase-Locked Loops: Dynamics on the Circle, Torus and Cylinder (World Scientific Series on Nonlinear Science Series a) (World Scientific Series on Nonlinear Science: Series a)
6], ). An additional argument is given below. 91) e(0) =p’ dr with a small parameter A. We can seek the solution as a formal power series --p(1+2Asinrcos6’), + 6’(r;p’)= & ( r ) A&(r) + A26’2(r) + . . 91) and comparing terms with the same power of A we obtain the set of equations for all coefficients & ( r )of the series. Let us establish the initial conditions: Oo(0) = p’ and &(o) = 0 for Ic = 1 , 2 , 3 ,... A simple computation gives the Poincari, mapping The first Order continuous-time Phase-Locked Loops e(27r;p ' ) = p' 7rp + 27rp+ A 4p1sin - 1-12 sin(p' + 7rp) + O ( A 2 ) .
K e ~ Dynamics on Circle, Torus and C y l ~ n d e r The geometrical method illustrated by the above examples can be used for ~eterminationof numerous Arnold’s tongues. 77). 5). 8 Fig. 23 Arnold’s tongues of the numbers 1/11, 119, 117, 115, 3/11, 113, 5/13, 317, 519, 315, 7/11, 517, 719, 111 for Eq. 77). 77). Let ,o~(’,L)) be the rotation number of the equation The first order c o n t ~ n u o ~ - Phase-Locked t~~e Loops 49 It is evident that and that ~ 1 ( C , D=) pl(D,C ). 87), we have c--I+ D-1 m P ( P Z , A 2 ) = ;for pz = - 2 and A2 = IC-1- D-11 2(C-1 + D - 1 ) .
That limn+, x, = x*. Defining 34 Equations of Phase-Locked Loops. Dynamics on Circle, Torus and Cylinder ga(zc*)= limn+mga(zn), we obtain the continuous function which maps the interval [0,1] onto itself. Additionally, assuming that ga(z+l) = ga(z)+l, we obtain the function ga defined for all real numbers. Remark. In 1883 Georg Cantor showed an example of a continuous nondecreasing function which maps the interval [0,1] onto itself and which almost everywhere has the derivative equal to zero. This example is important for the fundamentals of mathematical analysis.