On the theory of fluorescent lamp circuits

SCENCE On the theory of fluorescent lamp circuits E. Gluskin, MSc ndexing terms: Electromagnetic theory, Breakdown and gas disrhorges Astract: First, a recently developed theory of fluorescent lamp circuits is reviewed. Then the theory is extended to include the practically important case of the circuits with long-tue (1W 125 W) lamps whose voltage/current characteristics possess strong hysteresis. Power sensitivity of the circuits is the central topic. 1 ntroduction Fluorescent lamp circuits [l-5] are important power consumers [2]. The circuits are strongly ncjnlinear [&lo] and prolematic owing to the physical properties [3, 41 of fluorescent lamps which makes them ad circuit elements. Thus, despite the formal simplicity of commonly used lamp circuits, some of which are shown in Fig. 1, a correct theory of the circuits [lo], which, on the L fluorescent lamd -H C Fig. 1 Commonly used lamp circuits (a) Circuit with two allasts to e replaced y () that done allast (c) Serm connection of two 40 W lamps with an L C allast < a one hand, takes into account their nonlinear specificity and, on the other hand, gives a systematically otained list of formulas permitting the engineering analysis, appears to e not simple. t was explained in Reference 10 why an acceptale staility of circuit operation with respect to the changes in the amplitude of the line voltage (U) can e otained only for an L-C allast, not for an L allast, which contradicts the commonly accepted linear L-R model [4, 51 of the lamp which makes the whole lamp-allast circuit linear, and which makes it even unclear why we need a allast to operate the lamp. The theory of Reference 10 is ased on the study of power (P) Paper 7241A (S3, SE), first received 12th June 1989 and in revised form 9th Feruary 1990 The author is with the Electrical & Computer Engineering Department, Ben-Gurion University of the Negev, PO Box 653, Beer-Sheva 84105, srael EE PROCEEDNGS, Vol. 137, Pt. A, No. 4, JULY 1990 sensitivity defined as dp du dln P - P, 1 k -- _ U dlnu = ~ which is the parameter relevant to oth the power and the nonlinear nature of the circuits. Starting from some of the asic points of Reference 10, we discuss the main circuit theoretic and mathematical details related to the theory of References 10 and 11 and that of Reference 12, where the mathematical statements are proved. At this stage, the lamp is characterised y only one parameter (A) while the allast is considered as a one- or two-port circuit whose structure remains aritrary. Then the theory is extended to the practically important case of 1W125 W lamps whose voltage/ current characteristics differ significantly from those of short (e.g. 40 W) lamps which have een discussed in Reference 10. This is done y means of the introduction of an additional parameter (D) characterising the lamp. 2 Review of some results of the theory ased on the A -model n Reference 10, the current voltage characteristic of the lamp is replaced y the signum function: u,(i) = A sign i That is, the lamp is replaced y an ideal idirectional voltage hardlimiter. This model of the characteristic will e called the A-model. Fig. 2 shows the ideal and, schematically, a realistic ve(i). As a result of the model, if the 0 a 0 v Fig. 2 dealised (a) and realistic schematic volt-ampere charucterlstlc (h) ojthe 40 W.puorescent lamp current is a zero-crossing function, the voltage function of the lamp is a rectangular wave. When we are not considering generalisations of the input voltage function, it is a square wave, ecause the current, although nonsinusoidal, possesses, in practice, the property 20

,. and has two zero-crossings per period T which is the period of the line voltage. The arupt wave (the time function) u,(t) and the input voltage u(t) influence the allast circuit B which may e one-port or a two-port, as shown in Fig. 3. u,(t) causes high harmonic currents in the circuit. the following type: i(t) = L,u(t) - L, u,(t) UL, [(t)- AL, sign i(t) (1) \ A,i \ A 25 - nonlinear theory(l) experimentakl) 2 20-- 10- (universal) experimental ( L - C, WO! 2 w ) A A A Fig. 3 circuit System with (a) series (one-port) and () two-port linear allast The allast circuit is assumed to e linear and time invariant, which corresponds closely with the practical situation. This circuit is suject to a requirement (eqn. 4) which it is unusual to apply to a linear circuit, and which is imposed y the fact that the linear circuit is included in a strongly nonlinear one, whose power sensitivity need to e determined. The amplitude of the input voltage U, appearing in the definition of k,, in a general situation is the scaling parameter defined as = Ut(t) where [(t) is a wave having the form of u(t), as we use an input variale transformer. The consideration of an aritrary waveform is a generalisation which is helpful for revealing the asic point of the theory, which is a zerocrossing representation of the lamp s current function. Another generalisation relates to the structure of the allast circuit which makes it possile, in particular, to see the importance of the possile oscillatory (L-C) nature of the allast circuit for minimisation of the power sensitivity and to see the universality of the minimisation. Fig. 4 shows k,(u) curves calculated using the formulas of Reference 10 for 40 W lamp circuits with L and L-C allasts. The series L-C allast is such that k, is as low as possile in the theory (which is corrected in Section 5). ts resonant frequency is twice the frequency of the line voltage. We see the significant advantage of the nonlinear theory over the linear one which predicts k, = 2 always. 2.1 Zero-crossing represented current There is an important theoretical connection etween k, and the positions of the zero-crossings of the current function with respect to the time origin (t = 0, which is defined y dt)). The zero-crossing properties of the circuit replace, in some sense, its spectrum properties and are asic, which influences the properties of k,. Zero-crossing of time functions have een considered elsewhere 113-161. However, our results relate to the study of a periodic power system, and they are of a very different character. Let us turn to the equation for the current given y References 1C-12. t is a strongly nonlinear equation of 202 001, 1 140 160 180 200 220 240 260 u/st.v Fig. 4 circuit with L and an L-C allast Experimental and theoretical k,(u) for a single 40 W lamp Here U > 0 and A 2 0 are real-valued parameters, oth of which may e continuously changed in the mathematical investigation. The continuous change in A may e understood as a series connection of very small similar hard-limiting elements. L, and L, are linear integral operators which allow us to represent the contriutions of u(t) and u,(t) separately to the steady-state current, which is possile due to the linearity of B. L,u is required, generally, to e a zero-crossing continuous function (having a finite numer of zero-crossings per period) which allows dt) to e an arupt function. For a series allast L, = L,. For a two-port allast L, # L,. These operators may e presented as (LfXt) = r K(i)f(t - 5) dt using a kernal function K. As we consider only periodic processes, the operators can also e characterised y comparison of the Fourier coefficients of the functions Lf and f using Fourier series for the functions. The relevant mathematical details are discussed in Reference 12 while here their practical meaning is stressed. From eqn. 1, as h = A/U + 0, the zero-crossings t, of i(t) tend to those of Ll[ which we denote as t,. This allows us to know the polarity of the current etween successive points t, for h # 0. This also reveals the preservation of the numer of the zero-crossings of the current per period, for small h. Thus, the function sign i(t) is defined in eqn. 1 y the zero-crossings of i(t), and thus i(t) is represented y means of its zero-crossings. This is a result of the strong nonlinearity of the system which permits many of its properties to e simply revealed. n accordance with the definition of t,, for pure sinusoidal 5, [(t) = sin wt, (L1[)(t) - sin 4 t - ty) (2) Thus, for L,(,, corresponding to a lossless circuit, of? = 42, if the circuit is mainly capacitive, and wty = - ~/2 if it is mainly inductive at o, which actually gives us the polarity of ty also in the cases where [ is not pure sinus- EE PROCEEDNGS, Vol. 137, Pt. A, No. 4, JULY 1990

oidal. The equation for t, is otained from eqn. 1 as (LltXtk) = h& sign i(t)xtd (3) which defines t, as functions of h: tk = tk(h) t appears [12] that, for the relevant allast and ((t), there is a continuous interval for h: Oihih' with some h' defined y B and ((t), for which the current is a zero-crossing function and for which the following remarkale property of the zero-crossings may e otained : (L, sign (L,()(tt) = 0 Vk (4) then for the h from the aove interval t,(h) tk(0) = tt, i.e. the zero-crossings of i(t) are those of L,c(t), i.e. those of the current at h = 0, which corresponds to the case of the lamp eing shorted. This means, for instance, that if U is so high that we can connect two similar fluorescent lamps in series, keeping the current zero-crossing, then the mutual positions of the zero-crossings of the current with respect to that of the input voltage are the same as in the case of only one of the lamps connected, for the same allast satisfying eqn. 4. This property of t,, which actually may e otained, is called [11] constancy. Eqn. 4 states a specific correspondence etween the allast circuit and the form of the input voltage wave, which is the unusual requirement of a linear circuit. The importance of the possile constancy of the zero-crossings is revealed when we note that the right-hand side of eqn. 1 is known, and when we turn to the power sensitivity. t appears in the A-model that if the allast is a lossless circuit, then k, 2 1 and k, = 1 (for each permitted h) if, and only if, t, are constant. The constancy of the t, and the achievement of the minimal power sensitivity may e provided only y an L-C allast circuit. t was shown in Reference 10 that, for the input sinusoidal voltage, an L-C series circuit with resonant cyclic frequency w, = 2wp = 4np/T, p = 1, 2, 3,..., is such a circuit. Clearly a parallel connection of such circuits is also a proper allast circuit, which makes it relevant to all other L-C structures having the proper Foster second equivalent representation [17]. 2.2 Some general properties of kp in the A-model Using eqn. 1 we otain, for the case of a lossless linear circuit, where y, = C(L,()(t) sign i(t) dt 1 i(t) sign i(t) dt = UA - y,(t,(h)) T As P is positive, y, is also positive. For the practical fluorescent lamp circuits, it is enough to consider only one zero-crossing, t,, while t, also elongs to the period t2 = t, + T/2. Using the aove expression of k, and finding dt,/du from eqn. 3, we otain 1EE PROCEEDNGS, Vol. 137, Pt. A, No. 4, JULY 1990 f t, is constant, (L15)(tl) = 0 and k, = 1. Otherwise, using the fact that (Lt)(ty) = 0, we otain, for a small h, t is easy to see that, for small h, t, - ty N h, which is ecause of the zero-crossing character of the current function. Thus, for the small h, k, = 1 + O(hz) 3 1 (5) where the positiveness of y, is used. The exact expression for k, otained in Reference 10 for the lossless allast and the sinusoidal c(t), k, = [l - a2h2]-' with some constant 'a', agrees with eqn. 5. f there are power losses in the allast circuit, then terms of order h appear. The important point is that, in the A-model, k, is always a function of h = A/U only. This means that increase in A is equivalent to a decrease in U. This enales us to otain some conclusions using the k,(u) curves for 40 W lamp, shown in Fig. 4. n particular, it appears clear now that two 40 W lamps connected in series (see Fig. 1) cannot operate with an L-allast. ndeed, the corresponding k, should e taken from Fig. 4 at U/$ E 100 V RMS, which gives an unreasonale value in practice. t may e shown [12] that k, 2 1, for the A-model with an aritrary t(t). Thus, if for a lamp, the A-model is good, then no changes in the input function or changes in the structure of the linear allast can make k, significantly smaller than 1. However, there are lamps for which the A-model is not good. 2.3 Lamps of different types The 40 W lamp circuits were carefully tested to confirm the theory of Reference 10. Some other details related to these circuits are given in Section 3. The 38 W, 65 W and 58 W lamps possess more or less similar u,(i) curves [lo]. This suggests that similar estimations (somewhat less precise) may also e otained for the circuits with these lamps. However, in the case of the 100 W (T-12) longtue lamp, which is very important, the form of the u,(i) characteristic differs strongly from that of the 40 W lamp, and this case must e separately considered. Occasionally, the lamp characteristic possesses a specific hysteresis (Fig. 5) which is symmetric in some sense, this permits us to extend the aove analysis to the long-tue lamp circuits relatively easily. 3 Nonlinear theory of the long tue lamps (the A,-D model) To descrie the u,(i) of a 100 W lamp and to otain some corrections also for 40 W lamp circuits, we first introduce two voltage parameters, A, (previously A) and D, which depend on U. This dependence is assumed to e known empirically. t is shown schematically in Fig. 5 how A, and D are defined. There is the important question of whether or not A, and D depend on the type of allast. Experiments show that A, does not depend on whether the allast is L or L-C. There is a prolem with D ecause of some unexpected limitations of U&) near i = 0, for an L-C allast with 40 W lamp (not a 100 W lamp) as shown in Fig. 6. nterpolation of the u,(i) to i = 0 is suggested here, shown in the Figure, which eliminates the prolem, making the cases of L-C and L allasts similar 203

also for the 40 W lamp. The interpolation is possile ecause we need not to e concerned with what happens to the voltage in the vicinity of the point i = 0, taking t appears to e possile to estimate the voltage function of the lamp as U,([) = A, sign i(t) + D cos o(t - t,) where t, is a zero-crossing of the current and A, = A, + D/4. Fig. 8 shows, schematically, the formation of u,(t) which leads to a realistic waveform, ignoring the vicinity oft, and t,. (Note that we are still using the arupt function to apply the zero-crossing analysis and we are also - "1 200 6 180 AO.D 160-140 - >- Z0-4 40 100-0 > 80- A0 100W.T-12 larnp(thorn) 4Ot 01 1 140 160 180 200 220 240 260 280 unv Fig. 7 A, and A, + D as functions of U for a 100 W (T-12, Thorn) lamp Fig. 5 The lamp characteristic Y Schematic typical form of ujl) for a long-tue lo(t125 W fluorescent lamp, showing the determination of the parameters A, and D. D and, thus. D/A, S much smaller for the 40 W lamp than for a 00 W lamp, see Fig. 2 Experimental ~ ~ ( for 1 ) a 100 W (T-12, Thorn) lamp at line voltage of 237 V RMS a into account the intention of analysing the average power which is weakly influenced y the intervals of the small current. Fig. 7 shows A, and D/4, as experimental functions of U for the 100 W (T-12) lamp. 40W lamp L -C allast - 1 1 '. 1. A Fig. 6 Schematic v,(i) characteristic for a 40 W lamp with an L-C allast relevant to the definition ofa, and D 204 Fig. 8 Schematic formation of the waveform of the voltagefunction of the lamp in the A,-D model EE PROCEEDNGS, Vol. 137. Pt. A, No. 4, JULY 1990

guessing the waveform of u,(t) using the previously U, contrary to the first harmonic. This conclusion is in unknown t,.) good agreement with my recent experiments with different The usage of A, = A, + D/4, instead of A,, ecomes long-tue lamp circuits, supporting the theory. clear if we turn to the mutual positions of the experimen- Because of the assumption of the losslessness of B, tal ue(t) and i(t) waves which are shown in Fig. 9. This L, cos w(t - t,) - sin o(t - t,) (7) Fig. 9 Experimental v&) and i(t) waves for the 100 W lamp where w, is the resonant frequency of a series L-C allast. Note that D is not involved here and the relationship A, = A,(U) is not relevant. The constancy of the zero-crossings does not provide the minimal value of the power sensitivity in the A,-D model. For the 100 W lamp circuits, the resonant frequency of the standard allast, 80 Hz, is very close to that which provides the minimal k,. Although the conditions for the constancy of the zero-crossings are now not the conditions for the minimisation of k, (whose detailed expression is rather complicated in the model), these conditions are useful, as it is easy to distinguish etween the cases of t, constant and nonconstant in the expression for k,, as these conditions are directly connected with the parameters of the allast, and ecause zero-crossings are easily oserved parameters. Figure is otained for an L-C allast, however the situation with an L-allast is very similar. The point t, + T/4, where the cosine function (Fig. 8) is zero, elongs to the region where the current is increasing. t follows from Fig. 5a that the height of the square wave (Fig. 8) should thus e taken larger than A, which corresponds to the maximum of the current. The addition of D/4 to A, is found to e good for any allast. Although the correction to A, is found using an experimental waveform, which makes the method half-empirical, the A,-D analytical representation of the wave permits a valuale qualitative analysis of the circuits. The average value of A, for the 100 W lamp is aout 110 V (rather close to that of the 40 W lamp). However, ecause of the relatively large value of D for the lamp, the average A, appears to e close to 130V, i.e. approximately 20% greater. Taking this into account, we can try to explain the properties of the circuit with 100 W lamps, using the A-model with the replacement of A on A,. Using the fact that k, in the A-model is a function of the parameter A/U (Section 2.2) which should now e replaced y AJU, we simply move the k,(u) curve, shown in Fig. 4, 40 V (this is 0.2 U) to the right. This gives a reasonale estimation for the case of L-allast, when the value of the A-type is very important. However, for the case of L-C allast this estimation is not sufficient (Section S), which means that the influence of the term D cos o(t - t,) is important. 5 k, in the A,-D model gnoring the power losses in the allast, we otain the equation for the lamp power in the A,-D model as 1 1 p = AlU T Yl(tl(h)) + DU T Y,(tl(h)) where and y, = [(L,()(t) sign i(t) dt Y, = [(L,tXt) cos 4 - tl) dt (8) 4 The current function, its frequency harmonics and zeros in thea,-d model To simplify the following discussion, we ignore the influence of the power losses in the allast in most of the following equations, which leads to qualitatively correct conclusions. The equation for the current is now i(t) = UL, ((t) - AL, sign i(t) - DL, cos w(t - t,) (la) Consider asolute values of the harmonic currents otained from this equation. Clearly, the dependence of t, on U influences only the first harmonic which is also influenced directly y U. Thus, for ((t) pure sinusoidal and A, weakly dependent on U, the asolute value of the third, fifth etc. harmonics must e weakly dependent on EE PROCEEDNGS, Vol. 137, Pt. A, No. 4, JULY 1990 The analysis of Section 2.2 related to y, is sufficient for understanding the role of y,. To reveal the roles of y, and D, let us first consider the case of a pure sinusoidal ((t). Then using eqn. 2 we otain, for some small h, and y, N sin w(ty - t,) N h N U- which gives an important estimation for the introduced logarithmic derivative, also in the case of ((t) not eing pure sinusoidal. t is easy to see that sign (ty - t,) = sign ry. This leads to quite different results in the cases of L-C and L allasts. For an L-C allast which is mainly capacitive at w (this is the realistic case) y, > 0, and for L 205

tion of the choke L,, the current spikes charging the capacitor C are very sharp. Thus, the voltage function of C is very close (see Fig. 10 of Reference 18) to a square wave. (The height of the wave is weakly sensitive to the Fig. 13 Basic circuit for the ferroresonance stailiser L, is a strongly nonlinear choke changes in the amplitude of the input voltage uin, and this is also true for the fundamental harmonic of the output voltage, which is the purpose.). This suggests that practical (more complicated and rather prolematic) circuits of the stailisers may and should e effectively calculated using the jump points of the square wave as the defining parameters. These circuits are important and, in particular, are used in the power supply of some computers. They have some prolems with external noise (that of the line) which passes via the feedack sucircuits which are intended to improve the stailisation. 8 References 1 JACK, A.G., and VREHEN, Q.H.F.: Progress in fluorescent lamps, Phillips Tech. Rev., 1986,42, (1 1/12), pp. 342-351 2 CHEN, K.: The energy oriented economics of lighting systems, EEE Trans., 1977, A-13, pp. 62-68 3 GLUSKN, E.: Discussion of the voltage/current characteristic of a fluorescent lamp, EE Proc. A., 1989, 136, (5). pp. 229-232 4 WAYMOUTH, G.F.: Electric discharge lamps (MT Press, Camridge, Massachusetts, 1978) 5 STURM, C.U.: Vorschaltgerete und Schaltungen fur Nieder Spannungs Entladungslampen (Brown-Boveri in Verlag W. Girardet, Essen, 1963) 6 PEECK, S.C., and SPENCER, D.E.: A differential equation for the fluorescent lamp, llumrnation Engineering (Transactions ES), 1968, pp. 157-166 7 HERRCK, P.R.: Mathematical models for high intensity discharge lamps, EEE Trans., 1980, A-16, (5), pp. 648-654 8 LASKOWSK, E.L., and DONOGHUE, J.F.: A model for a mercurv arc lamo s terminal V- ehavior. EEE Trans., 1981, A-17, pp. 419-426 9 POLMAN, J., VAN TONGEREN, H., and VERBEEK, T.G.: Low-pressure gas discharges, Phillips Tech. Rev., 1975, 35, (1 1/12), DD. 321-330 O GLUSKN, E.: The nonlinear theory of fluorescent lamp circuits, nt. J. Electron., 1987.63, (3, pp. 687-705 11 GLUSKN, E.: A contriution to the theory of fluorescent lamp circuits, Proc. EEE, 1988, S CAS, Espoo, pp. 1385-1388 12 GLUSKN, E.: The zerocrossings of periodic functions which appear in a nonlinear equation relevant to electrical engineering, J. Franklin nst., 1990, (2/3) 13 LOGAN, B.F.: nformation in the zero-crossings of andpass signals, Bell Syst. Tec. J. April 1977. pp. 488-510 14 KEDEM, B.: Spectral analysis and discrimination y zerocrossings, Proc. EEE, 1986,74, ( ), pp. 1477-1493 15 PRABHU, A., and HARASMHA, R.: Zero-crossings in turulent signals, J. Fluid Mech., 1983, 137, pp. 251-272 16 MARR, D., and ULLMAN, S.: Bandpass channels, zero-crossings, and early visual information processing, J. Opt. Soc. Am., 1979, 69, (7). pp. 914-916 17 VAN VALKENBURG, M.E.: Modern network synthesis (John Wiley and Sons, New York, 19M)), Chap. 5 18 HART, H.P., and KAKALEC, R.J.: The derivation and application of design equations for ferroresonant voltage regulators and regulated rectifiers, EEE Trans., 1971, MAG-7, (), pp. 205-21 1 208 EE PROCEEDNGS, Vol. 137, Pt. A, No. 4, JULY 1990 -

the L allast, to the 40 W circuit, to the right, while that for the L-C allast is lowered. Application of the A,-D model to shorter T-12 lamps is also useful, ut less important. For comparison, A, and D/Ao as functions of U for a 40 W lamp are shown in Figs. 12a and together with the corresponding curves 100 60 * 2oc 06-05- 0 4, 0 4-01 1 120 140 160 180 200 220 240 2M) 280 UlQV U 09t 03-011 01 120 140 160 80 200 220 2LO 260 280 u/nv Fig. 12 Comparison o/d/a, for 100 W and 40 W T-2 lamp circuits in the A,-D model Y Experimental A,(lJ) h Experimental D/A, for the 100 W lamp. Based on our experience with the 100 W lamp, we expect that k,,(u) for L-C allast with wo = 2w is not minimal, and also for the 40 W lamp circuit. ndeed, an average 0.96 instead of 1.15 (Fig. 4) was experimentally otained for wo = 1.6~ for the 40 W lamp circuit. 6 Some other models Based on a more detailed investigation of the lamp's voltage/current characteristics, the following Tale of models of v,(i) may e suggested for use in the calculation and analysis of the circuits. The most general model here EE PROCEEDNGS, Vol. 137, Pf. A, No. 4, JULY 1990 Tale 1 : Models of v,(t) for different lamps 40W (T-12) w,(f)=a, signi(r)+dcosw(t-1,) (D issmall) 36 W (T12) 56 W (T-8) v.(f) =A, sign i(t) - Wi(f) v,(t) =A, sign i(t) 1W125W (T-12) v,(t)=a, signi(t)+dcosw(r-t,)-r'r(f) is u,(t) = A, sign i(t) + D cos w(t - tl) - R'i(t) with small R'imaX + A, and negative resistance R'. For the 36 W lamp, R' is really important, and for the long lamps D is very important. 7 Final comments As the A,-D model is considered in this work, it relates only to the case of two zero-crossings per period separated y the T/2 interval. For a more complicated distriution of t,, we would have to use, in the voltage function of the lamp for each time interval (t,, t,+,), an addition of the type Dk COS Wk(t - tt) with wk dependent on tk+, - t,, instead of the D cos w(t - tl) used aove, to agree with the real voltage/current characteristic. The ovious fact that Dk depends on the maximum value of the current in the interval (t,, t,+j makes the guessing of the voltage wave prolematic in the case of the aritrary set t,. There is, however, an elegant way to introduce the correction to the pure hardlimiter model so that the local maximal current values are taken into account. Namely, instead of eqn. la, the following equation: i(t) + L,[F di/dt + A, sign i(t)] = UL,[(t) with A, > A, and a small inductive constant F, the same for each interval, is introduced. The term with the current derivative may e estimated in a second approximation which leads to results similar to those aove in the simple case of rn = 2. Detailed analysis of this equation is meant to e presented separately. There is still a lack of a formal criterion to determine whether or not the choice of the parameters for the voltage function of the lamp is good enough. However, the numer of these parameters is always small which makes further analytical investigation easier. This is contrary to the often used approach (see References 6 and 7) of finding enough fitting parameters for a very good geometric approximation of an experimental time function y those theoretically (numerically) found, which is of little use for the understanding of the circuit ehaviour. ndeed, the realistic voltage/current characteristic of the lamp is somewhat changed from a circuit with one allast to a circuit with another one (see, for instance, the aove consideration of Fig. 6), thus a set of the fitting parameters which are very good for one circuit may not e good for another circuit. n fact, in Reference 6 good results were otained only for one of the two allasts considered there. Another interesting electric device to the analysis of which the zero-crossing representation of the time functions may e, proaly, applied, is the known ferroresonant stailiser of the asic circuits shown in Fig. 13. Although the very principle of the operation of the stailiser is ased on the strong saturation of the choke L,, the commonly used calculating procedure is ased on linear phase diagrams [18]. Owing to the strong satura- 207

,. allast (or L-C which is mainly inductive at w)y, < 0. Rewriting eqn. 8 as 1 [ A, ;: P,=A,U,,- 1+-- and assuming Dy,/A,, to e small, we use the linearisation n(1 + E) x E and otain k, 1 + k,, + k,, + U - du (9) Here k,, and k,, are the full logarithmic derivatives of A, and y, with respect to U. f the conditions for t, to e constant are satisfied, y, and y2 are independent on U, and thus appears that, ecause of the positiveness of (-yz) for the L-allast, and ecause of the smallness of the (negative) k,, (which is due to the saturation of A,(U) for these - 3 U 1 100 W (T-12,Thd L-c A,(U) mainly decreases with the increase in U (Fig. 7) and thus k,, is negative. n all the relevant cases, y2 < y, which is easily seen using the values of the phase shifts and the general symmetry of the waves involved in the integration. Thus, in accordance with the latter equality, k, < 1 for the L-C allast (for which yz > 0). However, the conditions for the constancy of tk for the l00w lamp circuit with a standard L-C allast are not well satisfied in practice (wo = 1.550 for the common allast). Taking this into account, we otain in the expression for k, two additional terms: = k D Y2 D Y2 + -- [k,, - kyl] E k,, --- A Yl A1 Yl where we used the aove found estimations for k,, and k,,, noting further that k,, is essentially positive ut very small when (wo- 2w)/oO 1. As a result, the addition to k, appears to e negative and for wo # 2w the power sensitivity may e smaller than when tk are constant. The influence of the term k,,, which is nonzero ecause of the nonconstancy oft,, is important at low U, which is explained y the terms of order h2 E U- appearing in an expression for k,,. This causes the steep increase in k, at small U. similar to that (Fig. 4) for a 40 W circuit with L-allast, which is a limiting case of the L-C allast as C - CO. This, finally, qualitatively explains the experimental k(u) curve for the 100 W lamp, shown in Fig. 10. Let us now turn to L-allast. t appears that the L- allast, which is the choke of the standard L-C allast, provides the same lamp power at 240 V line as the L-C allast. (To ignite the lamp it was necessary to ignite it with the L-C allast and then to short the capacitor.) However, such a circuit is not practical for the 100 W lamps, which may e explained y its high-power sensitivity. As is explained in Section 3, an estimation for the k(u) curve may e otained using the A-model with A,, instead of A, which means a simple shift of the k,(u) related to the 40 W lamp. Thus, the otained curve is everywhere higher than that related to the 40 W lamp, which agrees with the experiment (see Fig. 11). For a relatively high U, where AJU is decreased, we have to make some corrections to the k,(u) using the aove equations. t 206 02 O4 Fig. 10 01 1 140 160 BO 200 220 240 260 280 U 1sZ.v Experimental k,(u) for the 100 W lamp (L-C allast) voltages), there is an additional increase in k,(u). As for the highest (2245 V RMS) relevant voltages, k,(u) egan to e an increasing function of U, which is due to the saturation of the choke. As increase in P is clearly associated with the increase in some average current passing through the series circuit, the reason for the saturation of the choke (which is designed to e used in the circuit with much smaller k,) directly follows from the high-power sensitivity of the circuit with the L-allast. Fig. 11 collects the experimental k,(u) curves related to all the cases discussed. Roughly speaking aout the roles of the L and L-C allasts, we can say that, to otain k,(u) for the 100 W circuit, we move the curve related to ix 10Ow.L 4t / 40W.L : 140 160 16 200 220 240 260 280 U JTV Fig. 11 Collected experimental k,(u) data relevant to the aove discussion The continuous lines are drawn to help the eye to follow the data and to separate the groups of points related lo different cases EE PROCEEDNGS, Vol. 137, Pi. A, No. 4, JULY 1990