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Power Electronics Group - PEL 1 CCFL Inverters based on Piezoelectric Transformers: Analysis and Design Considerations Prof. Giorgio Spiazzi Dept. Of Information.

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Presentation on theme: "Power Electronics Group - PEL 1 CCFL Inverters based on Piezoelectric Transformers: Analysis and Design Considerations Prof. Giorgio Spiazzi Dept. Of Information."— Presentation transcript:

1 Power Electronics Group - PEL 1 CCFL Inverters based on Piezoelectric Transformers: Analysis and Design Considerations Prof. Giorgio Spiazzi Dept. Of Information Engineering – DEI University of Padova

2 Power Electronics Group - PEL 2 Outline Characteristics of Cold Cathode Fluorescent Lamps (CCFL) Review of piezoelectric effect CCFL inverters based on piezoelectric transformers Design considerations Modeling

3 Power Electronics Group - PEL 3 Cold Cathode Fluorescent Lamp (CCFL) CCFL is a mercury vapor discharge light source which produces its output from a stimulated phosphor coating inside glass lamp envelope. Closely related to neon sign lamps first introduced in 1910 by Georges Claude in Paris Cold cathode refers to the type of electrodes used: they do not rely on additional means of thermoionic emission besides that created by electrical discharge through the tube

4 Power Electronics Group - PEL 4 Cold Cathode Fluorescent Lamp (CCFL)

5 Power Electronics Group - PEL 5 Cold Cathode Fluorescent Lamp (CCFL) The phosphors coating the lamp tube inner surface are composed of Red-Green-Blue fluorescent compounds mixed in the appropriate ratio in order to obtain a good color rendering when used as an LCD display backlight Energy conversion efficiency: Ultraviolet light Visible light

6 Power Electronics Group - PEL 6 Cold Cathode Fluorescent Lamp Lamp v-i characteristic: Lamp length Lamp voltage primarily depends on length and is fairly constant with current, giving a non-linear characteristic. Lamp current is roughly proportional to brightness or intensity and is the controlled variable of the backlight supply.

7 Power Electronics Group - PEL 7 Cold Cathode Fluorescent Lamp These lamps require a high ac voltage for ignition and operation. A sinusoidal voltage provides the best electrical-to-optical conversion. There are four important parameters in driving the CCFL: –strike voltage –maintaining voltage –frequency –lamp current

8 Power Electronics Group - PEL 8 Cold Cathode Fluorescent Lamp Operating a CCFL over time results in degradation of light output. Typical life rating is hours to 50% of the lamp initial output The light output of a CCFL has a strong dependency on temperature Percentage of light output as a function of lamp temperature

9 Power Electronics Group - PEL 9 Cold Cathode Fluorescent Lamp Stray capacitances to ground cause a considerable loading effect that can easily degrade efficiency by 25% Lamp display housing:

10 Power Electronics Group - PEL 10 Current-fed Self-Resonant Royer Converter

11 Power Electronics Group - PEL 11 High voltage transformer

12 Power Electronics Group - PEL 12 Ballast capacitor

13 Power Electronics Group - PEL 13 Self resonant inverter

14 Power Electronics Group - PEL 14 Control of supply current

15 Power Electronics Group - PEL 15 Lamp current measurement

16 Power Electronics Group - PEL 16 Dimming

17 Power Electronics Group - PEL 17 Magnetic and Piezoelectric Transformer Comparison Low cost Multiple sources Single-ended or balanced output Wide range of load conditions (output power easily scaled) Secondary side ballasting capacitor required Reliability affected by the high-voltage secondary winding EMI generation (stray high-frequency magnetic field) Magnetic transformer characteristics

18 Power Electronics Group - PEL 18 Magnetic and Piezoelectric Transformer Comparison Inherent sinusoidal operation High strike voltage (no need of ballasting capacitor) No magnetic noise Small size High cost (but decreasing) Must be matched with lamp characteristics Reduced power capability Single-ended output (balanced output are possible) Few sources Unsafe operation at no load (can be damaged) Piezoelectric transformer characteristics

19 Power Electronics Group - PEL 19 Magnetic and Piezoelectric Transformer Comparison Size comparison

20 Power Electronics Group - PEL 20 Piezoelectric Effect The piezoelectric effect was discovered in 1880 by Jacques and Pierre Curie: –Tension and compression applied to certain crystalline materials generate voltages (piezoelectric effect) –Application to the same crystals of an electric field produces lengthening or shortening of the crystals according to the polarity of the field (inverse piezoelectric effect)

21 Power Electronics Group - PEL 21 Piezoelectric Effect In the 20 th century metal oxide-based piezoelectric ceramics have been developed. Piezoelectric ceramics are prepared using fine powders of metal oxides in specific proportion mixed with an organic binder. Heating at specific temperature and time allows to attain a dense crystalline structure Below the Curie point they exhibit a tetragonal or rhombohedral symmetry and a dipole moment Adjoining dipoles form regions of local alignment called domains The direction of polarization among neighboring domains is random, producing no overall polarization A strong DC electric field gives a net permanent polarization (poling)

22 Power Electronics Group - PEL 22 Piezoelectric Effect Polarization axis Random orientation of polar domains Polarization using a DC electric field Residual polarization Polarization

23 Power Electronics Group - PEL 23 Piezoelectric Effect Effect of electric field E on polarization P and corresponding elongation/contraction of the ceramic material Relative increase/decrease in dimension (strain S) in direction of polarization Residual polarization E E S P

24 Power Electronics Group - PEL 24 Disk after polarization (poling) Disk compressed Disk stretched Applied voltage of same polarity as poling voltage Applied voltage of opposite polarity as poling voltage Poling voltage Piezoelectric Effect Generator and motor actions of a piezoelectric element

25 Power Electronics Group - PEL 25 Actuator behavior Transducer behavior S=s E.T+d.E D=d.T+ T.E Where: S: Strain [ ] T: Stress [N/m 2 ] E: Electric Field [V/m] s: elastic compliance [m 2 /N] D: Electric Displacement [C/m 2 ] d: Piezoelectric constant [m/V] Piezoelectric Effect Polarization

26 Power Electronics Group - PEL 26 Piezoelectric Effect Based on the poling orientation, the piezoelectric ceramics can be design to function in: longitudinal mode: P is parallel to T Has a larger d33, along the thickness direction when compared to the planar direction transverse mode: P is perpendicular to T Has a larger d31, along the planar direction when compared to the thickness direction

27 Power Electronics Group - PEL 27 Piezoelectric Transformers (PT) In Piezoelectric Transformers, energy is transformed from electrical form to electrical form via mechanical vibration.

28 Power Electronics Group - PEL 28 Piezoelectric Transformers (PT) Longitudinal vibration mode –Transverse actuator and Longitudinal transducer Rosen-type or High-Voltage PT Three main categories

29 Power Electronics Group - PEL 29 Piezoelectric Transformers (PT) Thickness vibration mode –Longitudinal actuator and Longitudinal transducer Low-Voltage PT Three main categories

30 Power Electronics Group - PEL 30 Piezoelectric Transformers (PT) Radial vibration mode –Transverse actuator and Transverse transducer (radial shape preferred) Three main categories

31 Power Electronics Group - PEL 31 Equivalent Electric Model Rosen-typeThick. Vibr. modeRadial Vibr. mode R L mH27 H mH C 3.57nF254pF pF N C i nF2.305nF nF C o 23.85pF8.911nF nF length=30mmlength=20mmradius=10.5mm width=8mmwidth=20mmthickness1=0.76mm thickness=2mmthickness=2.2mmthickness2=2.28mm

32 Power Electronics Group - PEL 32 Voltage Gain Load resistance: 1M, 100k, 10k, 5k, 1k, 500 Rosen-type Piezoelectric Transformer sample Resonance frequencies

33 Power Electronics Group - PEL 33 Load resistance: 1M, 100k, 10k, 5k, 1k, 500 Rosen-type Piezoelectric Transformer sample Input Impedance

34 Power Electronics Group - PEL 34 Half-Bridge Inverter for PT Lamp PT + U DC i inv C1C1 S1S1 S2S2 C2C2 iLiL Half-Bridge inverter uiui + -

35 Power Electronics Group - PEL 35 Soft-Switching Condition T/2 trtr t uiui U DC t / i inv U1U1 Fundamental components Half-bridge inverter

36 Power Electronics Group - PEL 36 Coupling Networks PT Rosen-type Model L C + uiui UoUo + - iLiL R CoCo CiCi - UAUA + + UAUA ioio S1S1 S2S2 C1C1 C2C2 Lamp Half-Bridge inverter 1:n 21 + u inv - Coupling network ZgZg

37 Power Electronics Group - PEL 37 Coupling Networks It is not always possible to find a value for input inductor that guarantees both power transfer and soft switching requirements Less circulating energy as compared to parallel inductor Non linear control characteristics can lead to large signal instabilities Series inductor LsLs CN 1

38 Power Electronics Group - PEL 38 Effect of Coupling Inductor L s on Voltage Conversion Ratio M PT = U oRMS /U iRMS, M i = U iRMS /U invRMS, M g = M i M PT f2f2 70 [dB] |M PT |{ |M g |{ |M i |{ f1f1 I o =1mA I o =6mA |M gd | I o =1mA |M gd | I o =6mA f sw [kHz] U dc =13V, L s =42 H

39 Power Electronics Group - PEL 39 Effect of Coupling Inductor L s on Input Impedance Positive input phase f1f1 60 [dB ] |Z g | f sw [kHz] f2f2 I o =1mA I o =6mA ZgZg 0

40 Power Electronics Group - PEL 40 Effect of Coupling Inductor L s on Voltage Conversion Ratio It introduces an additional voltage gain (frequency dependent) between the RMS value of the inverter voltage fundamental component and the RMS value of the PT input voltage It introduces more resonant peaks in the overall voltage gain M g (limitation in switching frequency variation)

41 Power Electronics Group - PEL 41 Control Characteristics: Variable Frequency I o [mA RMS ] f sw [kHz] 64 U dc = 13V

42 Power Electronics Group - PEL 42 Control Characteristics: Variable dc Link Voltage I o [mA RMS ] L s = 42 H U dc [V] 16 f sw = 65kHz CN 1 L s = 38 H Increasing L S value causes the gain curve I o = f(U A ) to become non monotonic

43 Power Electronics Group - PEL 43 Large-Signal Instability Main converter waveforms when U dc is slowly approaching 21V (f sw = 65kHz, L s = 42 H).

44 Power Electronics Group - PEL 44 Coupling Networks It is always possible to find a value for input inductor that guarantees both power transfer and soft switching requirements Higher circulating energy as compared to series inductor Parallel inductor LpLp CBCB CN 2

45 Power Electronics Group - PEL 45 Effect of Coupling Network on Voltage Conversion Ratio f1f1 50 [dB] f sw [kHz] f2f2 }I o =1mA }I o =6mA |M PT |{ |M g |{ |M i |{ |M gd | I o =1mA |M gd | I o =6mA I o =1mA I o =6mA CN 2 : L p =20 H, C B =1 F, U dc =30V

46 Power Electronics Group - PEL 46 Effect of Coupling Network on Input Impedance f1f1 60 [dB ] |Z g | f sw [kHz] f2f2 I o =1mA I o =6mA ZgZg 0

47 Power Electronics Group - PEL 47 Effect of Coupling Network on Switch Commutations Differently from the series inductor coupling network, now the inductor current i Lp has to charge and discharge also the PZT input capacitance, that is much higher than the switch output capacitances, so that the positive impedance phase is a necessary but not sufficient condition to achieve soft commutations

48 Power Electronics Group - PEL 48 Experimental Measurements Trapezoidal PT input voltage Charge of input capacitance

49 Power Electronics Group - PEL 49 Control Characteristics: Variable Frequency I o [mA RMS ] f sw [kHz] L p = 20 H, C B = 1 F U dc = 30V CN 2

50 Power Electronics Group - PEL 50 Control Characteristics: variable dc link voltage I o [mA RMS ] L p = 20 H, C B = 1 F U dc [V] 35 f sw = 65kHz CN 2

51 Power Electronics Group - PEL 51 Half-Bridge Inverter for PT Square-wave output voltage Switching frequency changes in order to control lamp current Attention must be paid to the resonance frequency change with load Dedicated IC available Frequency Control

52 Power Electronics Group - PEL 52 Half-Bridge Inverter for PT Frequency Control

53 Power Electronics Group - PEL 53 Half-Bridge Inverter for PT Constant switching frequency Asymmetrical output pulses Amplitude of fundamental input voltage component is controlled by the duty-cycle Many control ICs for DC/DC converters can be used Duty-cycle Control t on t uiui U DC TSTS U1U1

54 Power Electronics Group - PEL 54 Full-Bridge Inverter for PT Lamp PT + i inv S1S1 S2S2 iLiL S3S3 S4S4 Full-Bridge inverter U DC uiui + - Switching frequency control Duty-cycle control Phase-shift control

55 Power Electronics Group - PEL 55 Full-Bridge Inverter for PT Constant switching frequency Amplitude of fundamental input voltage component is controlled by phase shifting the inverter legs No DC voltage applied to PT Dedicated control IC Phase-Shift Control

56 Power Electronics Group - PEL 56 Resonant Push-Pull Topology

57 Power Electronics Group - PEL 57 Resonant Push-Pull Topology Variable switching frequency Voltage gain at PT input Sinusoidal driving voltage

58 Power Electronics Group - PEL 58 Analysis of Small-Signal Instabilities and Modeling Approaches Example of high-frequency V –I characteristics OSRAM L 18W/10

59 Power Electronics Group - PEL 59 Steady-state V RMS -I RMS Characteristic MATSUSHITA FHF32 T-8 32W Negative incremental impedance Positive incremental impedance

60 Power Electronics Group - PEL 60 Modulated Lamp Voltage and Current Upper trace: i Lamp [0.5A/div] Lower trace: u Lamp [74V/div] OSRAM L 18W/10 f m =200Hz f m =2kHz f m =5kHz Incremental impedance:

61 Power Electronics Group - PEL 61 Lamp Incremental Impedance Re(Z L ) Im(Z L ) m = 0 m = 0 m = m = Approximation: K L < 0, z < 0 Right-half plane zero

62 Power Electronics Group - PEL 62 Physical Explanation Original first-order differential equation describing the gas discharge behavior proposed by Francis [1] : P L = Lamp power y L = 1/z L = lamp conductance A, B positive constants Steady-state:

63 Power Electronics Group - PEL 63 Physical Explanation Small-signal perturbation: Substituting into the Francis equation:

64 Power Electronics Group - PEL 64 Physical Explanation No negative incremental impedance nor RHP zero Small-signal incremental impedance:

65 Power Electronics Group - PEL 65 Physical Explanation Modified first-order differential equation: f(i o ) = monotonically decreasing function of i o y L = 1/z L = lamp conductance A, B positive constants Steady-state:

66 Power Electronics Group - PEL 66 Physical Explanation Small-signal perturbation: Substituting into the modified Francis equation:

67 Power Electronics Group - PEL 67 Physical Explanation Small-signal incremental impedance: Negative incremental impedance and RHP zero

68 Power Electronics Group - PEL 68 Lamp Model (Ben Yaakov) I o1 I o2 U o2 U o1

69 Power Electronics Group - PEL 69 Lamp Model (Ben Yaakov) K 2, K 3 = lamp constants The lamp resistance is considered to be dependent on a delayed version of RMS lamp current Small-signal perturbation: Subscript q means quiescent point

70 Power Electronics Group - PEL 70 Lamp Model (Ben Yaakov) Delay:

71 Power Electronics Group - PEL 71 Lamp Pspice Model (Ben Yaakov) uouououo Lamp time constant + - io2io2io2io2 I oRMS 2 i o =u o /R L

72 Power Electronics Group - PEL 72 Accounts also for the positive slope Lamp Model (Do Prado) P L = Lamp power a, b positive constants I o [mA RMS ] R L [M ] U o [V RMS ] I o [mA RMS ]

73 Power Electronics Group - PEL 73 Lamp Model (Do Prado) Small-signal perturbation: Subscript q means quiescent point Delay:

74 Power Electronics Group - PEL 74 Lamp Model (Do Prado) If bP Lq >1: z <0, Z L (0)<0 z <0, Z L (0)<0 Negative incremental impedance and RHP zero

75 Power Electronics Group - PEL 75 Lamp Pspice Model (Do Prado) PLPLPLPL RLRLRLRL uouououo u o -R 4 i o ioioioio Lamp time constant

76 Power Electronics Group - PEL 76 Lamp Model (Onishi) A 0 -A 4 positive constants Small-signal perturbation: Subscript q means quiescent point Delay:

77 Power Electronics Group - PEL 77 Lamp Model (Onishi) Negative incremental impedance and RHP zero If R S >0: z <0, Z L (0)<0 z <0, Z L (0)<0

78 Power Electronics Group - PEL 78 Lamp Pspice Model (Onishi) I oRMS RLRLRLRL U o =R L i o Lamp time constant

79 Power Electronics Group - PEL 79 Control Problem An Impedance with a RHP zero cannot be driven directly by a voltage source, since its current transfer function will contain a RHP pole

80 Power Electronics Group - PEL 80 Series Impedance Lamp Ballast + U S (s) ZBZBZBZB ZLZLZLZL - U o (s) I o (s) T F must satisfy Nyquist stability criterion

81 Power Electronics Group - PEL 81 Example of Instability Series inductor coupling network LSLS LSLS Inverter PT Lamp uiui uiui isis isis + - f osc 6kHz

82 Power Electronics Group - PEL 82 Example of Instability I Lp = [1A/div] I o = [2mA/div] f osc =6.45kHz Parallel inductor + dc blocking capacitor coupling network

83 Power Electronics Group - PEL 83 Phasor Transformation [11] A sinusoidal signal x(t) can be represented by a time varying complex phasor, i.e.: Example: AM signal

84 Power Electronics Group - PEL 84 Phasor Transformation Example: FM signal Inductor phasor transformation:

85 Power Electronics Group - PEL 85 Phasor Transformation Inductor phasor transformation: + L iLiLiLiL - uLuLuLuL + L - j S L

86 Power Electronics Group - PEL 86 Phasor Transformation Capacitor phasor transformation: + C iCiCiCiC - uCuCuCuC + C - 1/j S C

87 Power Electronics Group - PEL 87 Generalized Averaging Method [13] A waveform x() can be approximated on the interval [ t-T, t ] to arbitrary accuracy with a Fourier series representation of the form: s (0, T], = time-dependent complex Fourier series coefficients calculated on a sliding window of amplitude T

88 Power Electronics Group - PEL 88 Generalized Averaging Method The analysis computes the time evolution of these Fourier series coefficients as the window of length T slides over the waveform x(). The goal is to determine an appropriate state-space model in which these coefficients are the state variables Classical state-space averaging theory: The average value coincides with the Fourier coefficient of index 0!

89 Power Electronics Group - PEL 89 Application to Power Electronics u(t) = periodic function of time with period T Lets apply the generalized averaging method to a generic state-space model that has some periodic time-dependence: Lets compute the relevant Fourier coefficients of both sides:

90 Power Electronics Group - PEL 90 Differentiation Property This relation is valid for constant frequency s, but still represents a good approximation for slowly varying s (t)

91 Power Electronics Group - PEL 91 Transform of Functions of Variables A general answer does not exist unless function f is a polynomial. In this case, the following convolutional relationship can be used: where the sum is taken of all integers i.

92 Power Electronics Group - PEL 92 Lamp dynamic Model Only the negative slope in the U oRMS -I oRMS curve is modeled y(t) is lamp RMS current squared

93 Power Electronics Group - PEL 93 Generalized averaged lamp model Considering that lamp voltage u o (t) and current i o (t) are, with a good approximation, sinusoidal waveforms, we can take into account only the complex Fourier coefficients corresponding to indexes +1 and –1 (actually only one of the two coefficients is necessary), while for the variable y(t), only the index 0 coefficient is considered, since we are concerned with its dc value.

94 Power Electronics Group - PEL 94 Non-linear large-signal lamp model u o = u o +ju o i o = i o +ji o The fundamental component amplitude of the lamp current is: Each complex variable is decomposed into real and imaginary part:

95 Power Electronics Group - PEL 95 Comparison between complete model and fundamental component model Time [ms] Lamp current I o [mA RMS ] 7 Fundamental component model Complete model Step change of the lamp RMS current from 4 to 6mA RMS

96 Power Electronics Group - PEL 96 Small-signal lamp model Considering small-signal perturbations around an operating point: where: Lamp current fundamental component amplitude

97 Power Electronics Group - PEL 97 Ballast dynamic model only the complex Fourier coefficients of indexes +1 are considered

98 Power Electronics Group - PEL 98 Ballast small-signal model Complete ballast model:

99 Power Electronics Group - PEL 99 Large-signal and small-signal model comparison U s amplitude step variation (-5%) u ipk [V] Time [ s] Large-signal non linear model Small-signal linear model u opk [V] Time [ s] Large-signal non linear model Small-signal linear model

100 Power Electronics Group - PEL 100 Instability analysis Lamp current I o [mA RMS ] Unstable max (Re[ ]) L =120krad/s L =100krad/s L =80krad/s L =60krad/s Plot of the highest real part of the system eigenvalues as a function of the RMS lamp current for different values of the lamp time constant L =1/ L Time [ms] Lamp current I o [mA RMS ]

101 Power Electronics Group - PEL 101 Instability analysis Lamp current I o [mA RMS ] Unstable max (Re[ ]) L s =10 H L s =15 H L s =20 H L s =28 H Plot of the highest real part of the system eigenvalues as a function of the RMS lamp current for different values of the coupling inductor L s ( L = 100krad/s)

102 Power Electronics Group - PEL 102 Conclusions Piezoelectric transformers represent good alternative to magnetic transformers in inverters for CCFL Different inverter topologies and control techniques must be compared in order to find the best solution for a given application Large-signal as well as small-signal instabilities can arise due to the dynamic lamp behavior

103 Power Electronics Group - PEL 103 References 1.Ray L. Lin, Fred C. Lee, Eric M. Baker and Dan Y. Chen, Inductor-less Piezoelectric Transformer Electronic Ballast for Linear Fluorescent Lamp IEEE Applied Power Electronic Conference (APEC), 2001, pp Chin S. Moo, Wei M. Chen, Hsien K. Hsieh, An Electronic Ballast with Piezoelectric transformer for Cold Cathode Fluorescent Lamps Proceedings of IEEE International Symposium on Industrial Electronics (ISIE), 2001, pp H. Kakehashi, T. Hidaka, T. Ninomiya, M. Shoyama, H. Ogasawara, Y. Ohta, Electronic Ballast using Piezoelectric transformer for Fluorescent Lamps IEEE Power Electronics Specialists Conference Proc. (PESC), 1998, pp Sung-Jim, Kyu-Chan Lee and Bo H. Cho, Design of Fluorescent Lamp Ballast with PFC using Power Piezoelectric Transformer IEEE Applied Power Electronic Conference Proc. (APEC), 1998, pp Ray L. Lin, Eric Baker and Fred C. Lee, Characterization of Piezoelectric Transformers, Proceedings of Power Electronics Seminars at Virginia Tech, Sept , 1999, pp E. Deng, S. Cuk, Negative Incremental Impedance and Stability of Fluorescent Lamps, IEEE Applied Power Electronics Conf. Proc. (APEC), pp S. Ben-Yaakov, M. Shvartsas, S. Glozman, Statics and Dynamics of Fluorescent lamps Operating at High Frequency: Modeling and Simulation, IEEE Trans. On Industry Applications, vol.38, No.6, Nov./Dec. 2002, pp

104 Power Electronics Group - PEL 104 References 8.S. Ben-Yaakov, S. Glozman, and R. Rabinovici, Envelope simulation by SPICE compatible models of electric circuits driven by modulated signals, IEEE Trans. Ind. Electron., vol. 47, pp. 222–225, Feb S. Glozman, S. Ben-Yaakov, Dynamic interaction analysis of HF ballasts and fluorescent lamps based on envelope simulation, IEEE Trans. Industry Application, vol. 37, Sept./Oct. 2001, pp Y. Yin, R. Zane, J. Glaser, R. W. Erickson, Small-Signal Analysis of Frequency-Controlled Electronic Ballast, IEEE Trans. On Circuits and Systems, - I: Fund. Theory and Applications, vol.50, No.8, August 2003, pp C. T. Rim, G. H. Cho, Phasor Transformation and its Application to the DC/DC Analyses of Frequency Phase-Controlled Series Resonant Converters (SRC), Trans. On Power Electronics, Vol.5. No.2, April 1990, pp J.Ribas, J.M. Alonso, E.L. Corominas, J. Cardesin, F. Rodriguez, J. Garcia-Garcia, M. Rico-Secades, A.J. Calleja, Analysis of Lamp-Ballast Interaction Using the Multi-Frequency-Averaging Technique, IEEE Power Electronics Specialists Conference CDRom. (PESC), R. Sanders, J. M. Noworolski, X. Z. Liu, G. Verghese, Generalized Averaging Method for Power Conversion Circuits, IEEE Trans. On Power Electronics, Vol.6, No.2, April 1991, pp M. Cervi, A. R. Seidel, F. E. Bisogno, R. N. do Prado, Fluorescent Lamp Model Based on the Equivalent Resistance Variation, IEEE Industry of Application Society (IAS) CDROM, Onishi N., Shiomi T., Okude A., Yamauchi T., "A Fluorescent Lamp Model for High Frequency Wide Range Dimming Electronic Ballast Simulation" IEEE Applied Power Electronic Conference (APEC), 1999, pp


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