A Zerocrossing Analysis

Emanuel Gluskin , Andronov Conf. 2001, vol. 1, 241-250; arXiv:0806.4853v1 [nlin.SI] Appeared in: "Progress in Nonlinear Science", Proceedings of the International Conference dedicated to the 100th Anniversary of A. A. Andronov, Nizny Novgorod, Russia, July 2-6, 2001, Volume 1: Mathematical Problems of Nonlinear Dynamics, pp. 241-250 . A ZEROCROSSING ANALYSIS Emanuel Gluskin Elec. Eng. Dept., Ben-Gurion Univ., Beer-Sheva 84105, and The Acad. Technol. Inst., Holon 58102, Israel. Fax: 03-5026643; email: gluskin@ee.bgu.ac.il We consider a direct representation of a T -periodic tim e function f  t  by means of its zero-crossings (z-cs), t k  s   mod T  , k , s  1 ,2 ,.., m , with even m . The use of the z-cs (simple isolated zeros) as the describing parameters is made possible by a singular model of a strongly nonlinear characteristic of an electrical element. This new method is found helpful in the calculation and synthesis of a practically important system, and deserves attention of the mathematicians. Term ’z-c function’ denotes a continuous f unction with zero average, having only z-cs as its zeros. ’  ’ denotes direct proportionality.  denotes the frequency variable. t k  mod T  are the z-cs of f  t  , which belong to a T -interval. This interval is chosen so that at t 1 the slope of f  t  is positive, thus sign df dt  t k   ≡ sign df dt  t k −    − 1  k  1 . S ≡  t k  .’ 〈 ’i st h e integral average (over the period or the whole t -axis). Z  Z    is impedance [1]. The abbreviation ’z-c’ is used here f or both noun and adjective. n ∈ N . The equation Consider the equation f  t     t  −  L  sign f  t  ,  1  where   t  is a known T -periodic z-c function.  ≥ 0 is parameter. L is some first-order smoothing (integral) operator, generally with oscillatory kernel G  :  L   t  ≡  G ∗   t  , on  − , t  , for an integrable   , which describes the steady-state non-resonant response of a linear circuit to the   t  . If the function on which L acts possesses a nonzero average, L is required to provide a zero average of the resulting functions, i.e. G ∗ 1  0. The requirement of the smoothing nature of L is that  d dt  L       / L o ,a s →  , with some L o  0. As a realistic additional requirement (Section 5), Emanuel Gluskin , Andronov Conf. 2001, vol. 1, 241-250; arXiv:0806.4853v1 [nlin.SI]  d dt L − 1/ L o    O   − d  , d ≥ 2, i.e., d dt L − 1/ L o is a strongly smoothing operator. We also require that 〈  L    0, for any periodic   t  , which means the simplifying assumption that by itself L describes a linear lossless (of a zero average power) subsystem. Eq.(1) is inevitably reduced to an algebraic equation, which makes this integral equation and the role of L unique. The z-cs of   t  are denoted as t k o , k  1, 2, … , m o , S o ≡  t k o  . We consider  as a continuous parameter,  ≥ 0, and the dependence of t k on  means the m apping S o → S    . The basic lemma is that for an interval 0 ≤    c f(t) possesses the z-c features of   t  , which means that though with a continuous increase in  , starting from zero and up to some critical  c , the waveform of f  t  is changed, no z-cs appear or disappear, i.e. m    ≡ m  0   m o . L provides continuity of the functions  t k    in  0,  c  , t k  0    t k o , i.e. t k are ’ evolving ’f r o m t k o , and sign f  t  in the interval  t k , t k  1  equals sign   t  in the interval  t k o , t k  1 o  . We shall prove that with the increase in  , new zeros appear. Thus  c   exists and is the limit for the range of the z-c structural stability [2-4]. t k may be unchanged (unmoved), i.e. t k    ≡ t k o ( S  S o )i n  0,  c  . In this case, sign f  t  ≡ sign   t  , and (1) is solved , f  t     t  −  L  sign   t  . The ’constancy’ of t k may be provided by proper synthesis of t he operator L . This, and also the importance of the ’constancy’ to the power features of a practical circuit, will be seen after turning to a z-c representation of f  t  , which is our constructive point. The z-c representation of f(t) For the z-c f  t  , sign f(t) is the rectangular-wave f u n c t i o nt ob ee x p r e s s e da sa ne x p l i c i t function of t and t k . Integrating m mutually shifted and inverted combs of  -functions, we obtain sign f  t   D  2  ∑ k  1 m  − 1  k  1 ∑ n  1  sin n   t − t k  n ,   2  / T ,  2  with a constant D .I f D is nonzero, then L is taken such that L  1   0. In view of (2), L sign f  t  depends on t − t k . Thus (1) becomes: f  t     t  −  F  t ,  t s   3  with a known function F of the m  1 arguments t , t 1 , t 2 , … , t m ,a r r a n g e di nt h e m differences t − t s , i.e. f  t  is presented using its z-cs . The equation f  t k   0 obtains the constructive form   t k    F  t k ,  t s  .  4  In accordance with the statements below, this equation has solutions. It is approximately solvable by a linearization, and in som e cases (see example in Section 5) Emanuel Gluskin , Andronov Conf. 2001, vol. 1, 241-250; arXiv:0806.4853v1 [nlin.SI] precisely. Finding t k from (4), we obtain the solution of (1) in the form (3) . In the case of t k independent of  ,  t k  t k o ;   t k o   0, ∀ k  , eq. (4) becomes F  t k o ,  t s o  ≡  L  sign   t k o   0, or F  t k ,  t s  ≡  L  sign f  t k   0, ∀ k .  5  Eq.(5) means that in the case of the ’constancy’ of t k , the set of the z-cs of f  t  ,i . e .o f the jump points of sign f  t  , is included in the set of the zeros (not all of which are z-cs, see Fig.2 below) of  L sign f  t  . (5) is also a condition on the m − 1( f o r s ≠ k ) among the m parameters t k − t s , s  1, 2, . . , m ,a n da l s o T . (Note that t k  1 − t k , k  1, 2, . . , m − 1, give any t k − t s ,a n d t m  1  t 1  T .) Formulated in term s of L , eq.(5) is used below for a circuit synthesis. For non-constant t k , with the precision O    , eq.(4) yields t k ≈ t k o    d  / dt  t k o  − 1 F  t k o ,  t s o  , ∀ k ,  6  to be substituted in (3). The critical parameter  c is found, together with the new (doubled) zero, from the equations f  t   0a n d df / dt  0,   t    c F  t ,  t s   c  , d   t  / dt   c ∂ F  t ,  t s   c  / ∂ t .  7  Eqs.(7) show that the limits for the z-c stability, or the z-c representation (3) of f(t), c an be found using only this representation . These equations are strongly sim plified in the case of constancy of t k , when  c falls from F . Some supporting statements Lemma 1: For any arbitrary T-periodic function   t  ,   0 exists such that for given M  0 and L ,  |  d  L sign   / dt  t k o  |  M. Proof : Since (see Section 5) as →  ,  d dt L  −  / L o     − d   , d ≥ 2, then writing | d dt L  | ≤ | d dt L  −  / L o |  |  |/ L o with   sign  (    1/  ;|  |  1 ) ,w eh a v e|  |/ L o  1/ L o , and | d dt L  −  / L o |  | ∑   n  − d   n  e  int | ≤ ∑ |   n  − d   n  | ≤  ∑ n − 2 d  1/2  ∑ |   n  | 2  1/2   ∑ n − 2 d  1/2 The required  may thus be easily chosen. T he independence of  on the specific form of  is important for Theorem 1 below. Corollary 1:   0 may be found such that for any certain admissable  ,   t  and L ,  |  d  L sign   / dt  t k o  |  k min  |  d  / dt  t k o  |  . Corollary 2: For a small  , and given   t  , the addition of −  L  sign   t  to   t  (see (1)) can not cause any new z-c, i.e. f possesses the z-c features of  . Indeed, since L sign  is lim ited, we can find  so small that  | L sign  | would be smaller than |   t  | at any extreme point of   t  (there is a finite number of such points in Emanuel Gluskin , Andronov Conf. 2001, vol. 1, 241-250; arXiv:0806.4853v1 [nlin.SI] the period), and for  also satisfying the condition of Corollary 1, the addition of −  L  sign   t  cannot cause any new zero, obviously. Theorem 1: For  defined by Corollary 2, eq.(1) has a T-periodic z-c solution with m  m o . Proof : The iterations f o   , f n   −  L  sign f n − 1  , n  1, 2, . . . c reate T -periodic functions. According to Corollary 2, a certain   0 may be f ound such that each f n will be a z-c function, possessing m o z-cs per period. Consider, for such  , the infinite number of the sets S , given by all of the f n , belonging to a certain period, as the (   m o )-matrix of all of the z-cs. The infinite set (   1  (the first column of the matrix) of all of the first z-cs (’ t 1 ’ for each f n ) has a point of condensation in this finite interval (the period), and similarly for the other m − 1 infinite sets of the ’ t 2 ’, ..., ’ t m ’z - c so f{ f n }. This yields the existence of a T -periodic f  t  having the z-c features of   t  , which is a solution of (1). Theorem 2 : This z-c solution is unique . Proof (by contradiction): Since 〈  L    0, the function   L   t  alternates its polarity. Assume that (1) has two different solutions, f 1 and f 2 . Then f 1 − f 2  −  L  sign f 1 − sign f 2  . Multiplying by  sign f 1 − sign f 2  ,w eo b t a i na n alternative-polarity function in the right-hand side, and the nonnegative expression  | f 1 |  | f 2 |  1 −  sign f 1  sign f 2  in the left-hand side. Remark: By multiplying by sign f , we similarly prove that the equation f  t    L  sign f  possesses only a zero solution. Since with an unlimited increase in  ,( 1 ) becomes the latter equation, this conclusion means that with an unlimited increase in  , f cannot keep the z-c features of  , i.e.  c   . Considering Fig.2 below, one sees, however, that a time shift in one of the sides of f  t    L  sign f  m ay lead to a nonzero solution. Introducing our circuit, we show now that optimization of an important functional of f  t  (eq.(9) below) is obtained for S    ≡ S o , i.e. for constant z-cs. The electrical circuit The theory of the strongly nonlinear fluorescent lamp circuits is an im portant application of the z-c analysis, and the main circuit param eter, power, is subject to a specific optimization, treated here in term s of the z-cs. The lamp’s v-i ( voltage-current ) characteristic (considered in detail in [5,6]) is close to v  i   A sign  i  , where ’ A ’ is a constant ( ≈ 115 volt). This ’hardlim iter’ model introduces the z-cs of the lamp’s current function into the circuit equation that is of the type (1). The lamp’s circuit is shown in Fig.1. Because of the hardlimiter-type characteristic, directly applying the sinusoidal line voltage to the lamp would cause an unlimited current, and a ’ballast’ (sub)circuit B is needed. We consider B to be linear time-invariant , and, for simplification of som e formulae, lossless circuit. It m ay thus include only inductors ( L )a n d capacitors ( C ). At least one inductor (the electrical analog of mass, for series c onnection of the elements) must be present for limiting i  t  . The inductor(s) also sm ooth the waveform of i  t  , which provides the structural stability of the z -cs of i  t  , with respect to a range of Emanuel Gluskin , Andronov Conf. 2001, vol. 1, 241-250; arXiv:0806.4853v1 [nlin.SI] the input parameter U that strongly influences i  t  and the circuit operation. lamp U ξ (t) + - (a) i + - v B B lamp U ξ (t) + - (b) + - v i Figure 1: (a): i  t  is the output current of te 2-port (the two-operator version of B), v  Asign  i  . (b): the 1-port version ( L 1  L 2 in eq.(8)). U (in volts) is the amplitude (scaling factor) of the input voltage function U   t  , |   t  | ≤ 1. U   t  is required here to be periodic and such that number of the z-cs per period of the response function i  t  is finite. The integral character of the linear operators L 1 and L 2 in the circuit equation (8) is inevitably caused by the inductor(s) in B. An important requirement of both   t  and B is that no resonant process in B occurs, i.e. t he f requency spectrum of   t  is completely different from the spectra of the operators that describe B. Despite these limitations, we have significant freedom in the structure of B, for a synthesis of this circuit, required by applications. For the circuit shown in Fig.1(a), the linearity of B results, via the superposition with respect to the voltages on the ports of B, in: i  t   U  L 1   t  − A L 2  sign  i  t  .  8  Eq.(8) is reduced to (1) by denoting ( L 1   t  as   t  , L 2 as L , A / U as  , i  t  / U as f  t  , a n db yu s i n gt h a ts i g n  i  t   sign  i  t  / U  . Regarding the frequency spectrum of i  t  , we see from (8) that for a sinusoidal   t  ,t h e high harmonics in i  t  are independent of U . Using the z-cs of i  t  as an analytical tool, we now derive some important conclusions regarding the lamp’s power. The lamp’s power and its sensitivity to the variations in U . By the definition of the (average) power, P  〈 iv   〈 iA sign i   A 〈 | i |   AU 〈 | f |  ,a n d using (8) and then that 〈  L    0, we also obtain P  〈 U   t  − A L  sign i  t  A sign i   AU 〈   t  sign f  t  .  9  Since t k    are included in sign f , P  AU     Emanuel Gluskin , Andronov Conf. 2001, vol. 1, 241-250; arXiv:0806.4853v1 [nlin.SI] where   〈  sign f   〈 | f |   0. In practice a very important param eter (function) of such nonlinear circuits is the sensitivity of P to changes in U, expressed in the relative values : K U    ≡  dP / P  /  dU / U   d ln P / d ln U  1  d ln  / d ln U . Since d ln  / d ln U  − 1, we have K U     1 − d ln  / d ln   1 −   /   d  / d  .  10  Theorem 3: K U ≥ 1 , and K U    ≡ 1 iff t k are constant. Proof :F o r  ∈  0,  c  d d   sign f  t   − 2 ∑ 1 m sign  df / dt  t k    t − t k    dt k / d  ,  11  and since sign  df / dt  t k   sign  d  / dt  t k o  , (10) yields K U     1 −   /   d 〈  sign f  / d   1 −   /  〈  d sign f / d    1  2    T  − 1 ∑ 1 m sign  d  / dt  t k o    t k  dt k / d  . For a small  ,   t k    d  / dt  t k o  t k − t k o  ,a n d dt k / d    t k − t k o  /  ; thus K U     1  2   T  − 1 ∑ 1 m |  d  / dt  t k o  |  t k    − t k o  2 ,  12  and it is obvious that K U    ≥ 1 and the minimal (the lowest possible curve) K U    ≡ 1 is obtained iff t k    ≡ t k o .  For non-constant t k ,a n d  close to  c , the power sensitivity may possess unacceptably high values. This depends (see (6)), first of all, on dt k / d   F  t k o ,  t s o    L  sign   t k o  . The range 1  5f o r K U may be easily obtained in the lam p circuits for not very strong variations in U . Contrary to that, if P were the power of a resistor in any linear system , then P  U 2 ,a n d d ln P / d ln U ≡ 2, ∀ U . We see that K U well characterizes the circuit’s nonlinearity, and since in (9)  depends on L , the role of the linear elements (or L )i n K U is unusual ([5,6] for more details). Minimization of K U is an important topic in the theory of the fluorescent lamp circuits. Emanuel Gluskin , Andronov Conf. 2001, vol. 1, 241-250; arXiv:0806.4853v1 [nlin.SI] Example of the synthesis of B that minimizes K U For the common case of   t   −   t  T /2  ,w i t h2z - c sp e rp e r i o d , we obtain f  t   − f  t  T /2  ,a n d t 2 − t 1  T /2  t 2 o − t 1 o . Using the susceptence [1], B    ≡ Im  L e i  t  /  e i  t  (this is sim ply Im  Z    − 1  , B n ≡ B  n   , and Fourier series expansion, we turn the equality F  t k ,  t s  ≡  L sign f  t k   0, k  1, 2, . . , m ,  13  into F  t 1 ,  t 1 , t 2   4  ∑ 1  B 2 n − 1 2 n − 1  0.  14  Since in (14) B 2 n − 1 cannot be all of the same sign, circuit B must be capacitive a t certain frequencies and inductive at others. The simplest appropriate circuit, inductive at high frequencies ,i st h es e r i e s L-C ballast. (Replace B in Fig.1(b) by such a circuit.) For such a B, B n   1 n  C − n  L  − 1 ,  15  and (14) becomes 4  ∑ 1  1  2 n − 1  2 −   o   2 ≡−   o tg   o 2    0,  o   LC  − 1/2 ,  16  where  o is the resonant frequency of the L-C circuit. Eq.(16) gives  o  2  ,4  , . . . In the practice of the circuits, 2  ,o re v e nas o m e w h a t smaller value should be used, in order not to increase too much (with respect to the simplest possible, purely inductive B), harmonic currents of the frequencies 3  ,5  ,.. . The condition  o ≲ 2  is only one of two possible conditions on the two elements L and C . The other condition is that P should be of the nominal value for a nominal value of U . The function ( L sign f  t  is shown, for  o  2  , in Fig.2. We see the role of  o ,a n d note that the set of z-cs of ( L sign f  t  cannot coincide with S . The latter can be proved in the general case by contradiction; | 〈 sign f L sign f  | would t hen be a large value, but L is such that 〈  L    0. t 1 t Figure 2: ( sign f )( t ) and ( Ls i g nf ) for the L-C-lamp circuit, for  o  2  . Emanuel Gluskin , Andronov Conf. 2001, vol. 1, 241-250; arXiv:0806.4853v1 [nlin.SI] For a more com plicated   t  ,i . e .f o r m o  4 ,6 ,..., B w o u l d b e r e q u i r e d t o i n c l u d e correspondingly more elements, in order t o ensure K U ≡ 1. For a ny S o aB( o r L ) can be synthesized, using not more that m o elements of the L and C types, such that the requirements for K U and P are satisfied. Expanding, in the general case, the relevant polynomial fraction B n ,d e s c r i b i n gB ,i n t o partial fractions of the type (15), we can represent B as a parallel connection of the simple L − C series circuits, with different  o . According to the expansion of (15) by the powers of 1/ n ,  1 n  C − n  L  − 1  − 1 n  L  1    o n   2  ...  , we have, in the general case, as n →  (or →  ) B n  − 1 n  L o  1  O  n − 2  ,o r B     − 1  L o  1  O   − 2  . If linear resistive elements, causing power losses, were included in B, then an O   − 2  -term would replace O   − 3  in the more general equality  d dt L − 1/ L o    O   − d  , d ≥ 2, which was used in Lem ma 1. Final remarks For the circuit with the series L-C ballast, the equation for i  t  may be written as L di dt  A sign i  t   1 C  − t i    d   U   t  .  17  The nonlinearity of (17) is obvious when using sign  , but not when using  sign i  given by series (2) . The nonlinearity in (2) is due to the fact that the shifting param eters t k belong to the unknown solution . Following [7], we call such a ”hidden” nonlinearity a ’ z-c nonlinearity ’. There are other important examples of this ’z-c nonlinearity’ [7-12], forming a class of dynamic problems where the c onstructive role of the z-cs of a priori unknown functions, and the appearance of the z-cs as time-shifts, are typical. Thus, it is shown in [8] that a linear oscillatory equation (as (17) when A  0); with a sinusoidal right-hand side, for a function x  t  , with the additional ’reflecting’ condition  dx / dt  t k    −  dx / dt  t k −  , where t k are zeros of x  t  , may lead to a chaotic x  t  . Much earlier work [9] shows that a ’shift-nonlinearity’, with the shift defined by the trajectory of a particle in cyclotron, leads to a typical nonlinear resonance. A relevant nonlinear analysis of also stochastic processes is suggested in [10] and discussed in [11]. In g eneral, since z-cs are easily observable, detectable, and suitable for control parameters, the z-c nonlinearity is very important. Finally, a standard ’dynamical systems’ formulation may be given to our problem. Considering L as a solution operator for an LTI equation of the type Emanuel Gluskin , Andronov Conf. 2001, vol. 1, 241-250; arXiv:0806.4853v1 [nlin.SI] ∑ 0 r max a r d 2 r dt 2 r y  t     t  , using the ’phase variables’ x 1  y , x 2   x 1 , ... , x r   x r − 1 , we replace (1) by  x  f  x , t  18  with sign x 2  t  included in the right-hand side. Some systems with discontinuous right-hand sides have already been investigated, e.g. in the known works by N.N. Bautin on clock synchronization, and t he known book by A.F. Filipov. However this is the first time that such a practically important system as the fluorescent lamp circuit (such circuits consume i n total about 20% of all electrical power generated) has been analyzed using the discontinuity in (18). Acknowledgements I am grateful to M. Lifsic (M.S. Lifshitz), S. Shnider, V. Gotlib, B-Z. Shklyar and the late S.G. Mikhlin for discussions. 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