A New High-Frequency Fluorescent Lamp Model

Transcription

1 A New High-Frequency Fluorescent ap Model Thoas J. Ribarich and John J. Ribarich International Rectifier Power IC Group 33 Kansas St., El Segundo, CA, Tel.: (31) Fax: (31) as presented at IEEE Industry Applications Society Annual Meeting, St. ouis, Missouri, October 1-16, 1998 Abstract - A new odel has been developed which describes the non-linear resistance of fluorescent laps operating at high-frequency. A parabolic fit has been perfored on actual lap data which allows for siplified solutions to non-linear differential equations describing various fluorescent lap output stages. A lap wavefor analysis has also been done using the odel which shows an elliptical lap current to be optial. I. INTRODUCTION Various odels exist that describe the resistance of a fluorescent lap operating at high-frequency [1]. These include both linear and cubic approxiations. The linear, or resistor, approxiation is given as, where, v = i = v = ir (1) Instantaneous lap voltage [Volts] Instantaneous lap current [Aperes] This odel is siple to use but has liited accuracy. The cubic odel is given as, 3 v = Ai + Bi () This is closer to the actual but ore coplicated to use. An approxiation is needed which is both siple and accurate. The focus of this paper is on a siple and accurate parabolic odel which allows for solutions to non-linear differential equations and provides a rapid ethod for obtaining ballast design paraeters. This paper includes the parabolic odel, optiu lap wavefor analysis, solutions to nonlinear differential equations, and coparison with existing linear solutions. II. THE PARABOIC MODE Using a parabolic function to odel the non-linear lap resistance yields, v = ki (3) Voltage and current easureents taken fro a fluorescent lap operating at high-frequency are plotted (Figure 1) together with the linear, cubic and parabolic odels. ap actual linear parabolic cubic ap Current [Aperes] Figure 1, Non-linear lap resistance and fitted odels for T8/3W lap type running at 5kHz (Plap=3W, R=656 Ohs, k=615, A=78, B=574). The parabola is a good fit for higher current regions of the v-i curve, where ost of the power is consued by the lap. The cubic is linear for low values of current and does not predict the sharp increase in the actual voltage as the current increases. The parabola, however, increases sharper in voltage for a better fit at the higher current values and ay predict higher haronics. Overall, the parabolic is coparable to the cubic and easy to use.

2 III. THE OPTIMUM AMP WAVEFORM The parabolic odel was used to calculate various lap paraeters such as crest factor, peak voltage and current, voltage and current haronics, and power, for the following lap current wavefors: where, I = I (pulse) (4) I = I sin ω t (sine-squared) (5) I = 4 I ft (triangle) (6) I = I sinω t (sinewave) (7) I = I ft (parabola) (8) I = I 1 16 f t (ellipse) (9) I = (rectangle) (1) I I = ap current aplitude [Aperes] The wavefors were selected to transition in shape fro a high, narrow pulse to a short, wide rectangle, in search of an optiu with which the lap should be driven. Driving the lap with a pure sinusoidal current, for exaple, gives a lap voltage and power of the for, V = ki = ki sin wt (11) P = 4 3 ki 3π (1) When plotted (Figure ), the lap voltage has soe zero-crossing distortion and increases sharply as the current increases yielding higher voltage haronics. The current crest factor should be reduced to axiize lap life and the axiu peak current should also be reduced to axiize ballast output stage efficiency. For these reasons, a sinusoidal lap current is not necessarily the optiu. ap tie [us] Vlap Figure, Sinewave lap current and predicted lap voltage for T8/3W lap type running at 5kHz. ilap ap Current [Aps] Driving the lap with a triangular current gives a lap voltage and power of the for, V = ki = 16 ki f t (13) P = 1 3 ki (14) 4 When plotted (Figure 3), peak currents and haronics are higher than the sinusoidal case, and get worse oving in the direction towards a pulse. ap tie [us] vlap ilap Figure 3, Triangular lap current and predicted lap voltage for T8/3W lap type running at 5kHz. A rectangular current yields a crest factor of 1. and iniizes peak lap current. The associated voltage and current haronics, however, are high, tending to show that a squarewave is not an optiu either. In order to best select the optiu, the results fro each wavefor are noralized and plotted (Figure 4) HD3 (Current) Crest Factor HD3 (Voltage) pulse triangle sine ellipse rectangle Figure 4, Optiu lap wavefor analysis suary for T8/3W lap type ap Current [Aps]

3 The ellipse (Figure 5) proves to be ore of an optiu wavefor with an acceptable crest factor, lower peak currents and lower voltage haronics due to a ore sinusoidal lap voltage. 5.3 for which a solution exists of the for, V in i( t) = tanh k kv in t (16) ap vlap 15 ilap tie [us] ap Current [Aps] The linear and non-linear solutions for the lap current are plotted (Figure 7) over one-half of a coplete cycle during steady state. Fro the plot it is seen that the non-linear solution rises sharper than the linear solution. This results in a slightly lower peak current for the non-linear case. A slight difference in phase can also be seen, which, for a given lap power, will result in a higher operating frequency for the non-linear solution. Figure 5, Elliptical lap current and predicted lap voltage for T8/3W lap type running at 5kHz IV. NON-INEAR SOUTIONS The parabolic odel allows for closed-for solutions of non-linear differential equations describing different fluorescent lap output stages. A typical output stage consists of an in series with the lap (Figure 6). ap Current [Aperes] R Figure 7, inear and non-linear solutions for the series - ap output stage. ap type: T8/3W The lap voltage is also plotted (Figure 8), which shows a higher peak voltage and higher haronics for the non-linear case due to the non-linear lap resistance. Figure 6, Series -lap output stage Assuing the lap has already been ignited and is running at noinal power, and using the odel, the following non-linear differential equation results: di dt + ki = V in (15)

4 5 15 i 1 q = e k C kc k q () ap Figure 8, inear and non-linear solutions for the series - ap output stage. ap type: T8/3W Another popular output stage consists of an -Clap series configuration (Figure 9). C where, q = charge stored in capacitor C These are new solutions and are not known to have been discovered before. Using (18), () and basic circuit analysis, the inductor current and lap voltage are again plotted (Figures 1 and 11) for the linear and non-linear solutions. Fro the plots, a slight difference in the peak current can still be seen, but the phase difference is now negligible. The lap voltage solutions show even higher peak voltage differences and still higher haronics in the non-linear case I ap R Current [Aperes] Figure 9, Series -C-lap output stage If the cubic odel is ipleented, the result is a nonlinear differential equation of the for, di dt + Ai + Bi + c idt = V in 3 1 (17) The solution is not known for this Rayleigh/Van der Pol type equation. Using the parabolic, however, the following non-linear differential equation and solution results, di dt + ki + c idt = V in 1 kv C di in + 1 = + ki + ln dt kc kc di + 1 dt or, (18) (19) Figure 1, inear and non-linear solutions for the Series -Clap output stage. ap type: T8/3W Volts [Voltage] V ap Figure 11, inear and non-linear solutions for the Series -Clap output stage. ap type: T8/3W

5 The traditional output stage consisting of an in series with a parallel lap and C (Figure 1) has also been solved I ap 1 R C Current [Aperes] Figure 1, Series- parallel lap-c output stage A atheatical solution exists which when plotted (Figures 13, 14 and 15) shows soe interesting results. Here a bup occurs in the capacitor current (Figure 13) for the non-linear solution. This is due to the non-linear lap resistance decreasing as the lap current approaches zero. The decrease in lap resistance gives an increase in the capacitor current and therefore a sharp increase in the lap current near zero. This causes the non-linear lap current to go to a lower peak than the linear solution, for a given lap power. This results in a lap current which appears elliptical in shape, giving rise to a ore sinusoidal lap voltage (Figure 15) as predicted earlier (Figure 5). -.6 Figure 14, inear and non-linear solutions for the series- parallel ap-c output stage. ap type: T8/3W V ap Current [Aperes] I C I Figure 13, inear and non-linear solutions for the series- parallel ap-c output stage. ap type: T8/3W Figure 15, inear and non-linear solutions for the series- parallel ap-c output stage. ap type: T8/3W V. CONCUSIONS The parabolic odel is siple, accurate and has greatly siplified lap wavefor analysis. The parabolic odel has also provided new closed-for solutions to non-linear differential equations therefore reducing ballast design tie. Future iproveents include expanding the odel to include diing, in which k changes as a function of lap power.

6 REFERENCES [1] U. Mader and P. Horn, A Dynaic Model for the Electrical Characteristics of Fluorescent aps, IEEE IAS 199, pp [] S.. Salas and Einar Hille, Calculus: One and Several Variables, John Wiley and Sons, Canada, 198. [3] T. Ribarich, J. Ribarich, A New Model for High-Frequency Ballast Design, in IEEE- IAS Conf. Rec., 1997, pp [4] T. Ribarich, J. Ribarich, A New Control Method for Diable High-Frequency Electronic Ballasts, in IEEE-IAS Conf. Rec., 1998.