CCFL Inverters based on Piezoelectric Transformers: Analysis and Design Considerations

1Power Electronics Group - PEL

CCFL Inverters based on Piezoelectric

Transformers: Analysis and Design

ConsiderationsProf. Giorgio SpiazziProf. Giorgio Spiazzi

• Dept. Of Information Engineering – DEI• University of Padova

Outline

• Characteristics of Cold Cathode Fluorescent Lamps (CCFL)

• Review of piezoelectric effect• CCFL inverters based on

piezoelectric transformers• Design considerations• Modeling

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

• 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:Energy conversion efficiency:

Ultraviolet lightUltraviolet lightVisible lightVisible light

Cold Cathode Fluorescent Lamp

Lamp v-i characteristic:Lamp v-i characteristic:

Lamp Lamp lengthlength

Lamp voltage primarily depends on length and is fairly constant Lamp voltage primarily depends on length and is fairly constant with current, giving a non-linear characteristic. Lamp current is with current, giving a non-linear characteristic. Lamp current is roughly proportional to brightness or intensity and is the roughly proportional to brightness or intensity and is the controlled variable of the backlight supply.controlled variable of the backlight supply.

• 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

• Operating a CCFL over time results in degradation of light output. Typical life rating is 20000 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 Percentage of light output as a function of lamp temperatureof lamp temperature

• Stray capacitances to ground cause a considerable loading effect that can easily degrade efficiency by 25%

Lamp display housing:Lamp display housing:

Current-fed Self-

Resonant Royer

Converter

High voltage High voltage transformertransformer

Ballast Ballast capacitorcapacitor

Self resonant Self resonant inverterinverter

Control of Control of supply currentsupply current

Lamp current Lamp current measurementmeasurement

DimmingDimming

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 characteristicsMagnetic transformer characteristics

• 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 characteristicsPiezoelectric transformer characteristics

Size comparisonSize comparison

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)

Piezoelectric Effect• In the 20th 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)

Random orientation Random orientation of polar domainsof polar domains

Polarization using Polarization using a DC electric fielda DC electric field

Residual Residual polarizationpolarization

PolarizationPolarization

Effect of electric field E Effect of electric field E on polarization P and on polarization P and

corresponding corresponding elongation/contraction of elongation/contraction of

the ceramic materialthe ceramic material

Relative increase/decrease in dimension (strain S) in direction of polarization

Residual Residual polarizationpolarization

Disk after polarization

(poling)

Disk compressed

Disk stretched

Applied voltage of

same polarity as poling voltage

Applied voltage of opposite

polarity as poling voltage

Generator and motor actions Generator and motor actions of a piezoelectric elementof a piezoelectric element

Actuator Actuator behaviorbehavior

Transducer Transducer behaviorbehavior

S=sE.T+d.E

D=d.T+T.EWhere:S: Strain [ ]T: Stress [N/m2]E: Electric Field [V/m]s: elastic compliance [m2/N]D: Electric Displacement [C/m2]d: Piezoelectric constant [m/V]

Piezoelectric EffectPolarization

Based on the poling orientation, the piezoelectric ceramics can be design to

function in:

longitudinal mode: P is parallel to THas a larger d33, along the thickness direction when compared to the planar direction

transverse mode: P is perpendicular to THas a larger d31, along the planar direction when compared to the thickness direction

Piezoelectric Transformers (PT)

• In Piezoelectric Transformers, energy is transformed from electrical form to electrical form via mechanical vibration.

•Longitudinal vibration mode– Transverse actuator and Longitudinal

transducer Rosen-type or High-Voltage PT

Three main categoriesThree main categories

•Thickness vibration mode– Longitudinal actuator and Longitudinal

transducer Low-Voltage PT

•Radial vibration mode– Transverse actuator and Transverse

transducer (radial shape preferred)

Equivalent Electric Model

Rosen-type Thick. Vibr. mode Radial Vibr. modeR 0.756199 1.44 6.89991 L 2.464173mH 27H 7.93842mHC 3.57nF 254pF 269.171pFN 35.89 0.47 0.908Ci 81.6216nF 2.305nF 4.60799nFCo 23.85pF 8.911nF 1.62414nFlength=30mm length=20mm radius=10.5mmwidth=8mm width=20mm thickness1=0.76mmthickness=2mm thickness=2.2mm thickness2=2.28mm

Ci-1:N

Voltage GainLoad resistance: 1M, 100k, 10k,

5k, 1k, 500

Rosen-type Piezoelectric Transformer sample

Resonance frequencies

Load resistance: 1M, 100k, 10k, 5k, 1k, 500

Rosen-type Piezoelectric Transformer sample

Input Impedance

Half-Bridge Inverter for PT

PT+UDC

S2C2 iL

Half-Bridge inverter

Soft-Switching Condition

U1 Fundamental components

Half-bridge inverter

Coupling Networks

PT Rosen-type Model

LampHalf-Bridge

inverter

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

Effect of Coupling Inductor Ls on Voltage Conversion Ratio

MPT = UoRMS/UiRMS, Mi = UiRMS/UinvRMS, Mg = Mi MPT

|MPT|{|Mg|{

-1045 50 55 60 65 70 75 80

Io=1mAIo=6mA

|Mgd|Io=1mA

|Mgd|Io=6mA

fsw [kHz]

Udc=13V, Ls=42H

Effect of Coupling Inductor Ls on Input Impedance

Positive input phasePositive input phase

045 50 55 60 65 70 75 80

fsw [kHz]

Io=1mA

Io=6mA

Effect of Coupling Inductor Ls 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 Mg (limitation in switching frequency variation)

Control Characteristics: Variable Frequency

[mARMS]

60 61 62 63 65fsw [kHz]

Udc = 13V

Control Characteristics: Variable dc Link Voltage

[mARMS]

Ls = 42H2

11 12 13 14 15

Udc [V]16

fsw = 65kHz

CNCN11

Ls = 38H

Increasing LIncreasing LSS value causes the gain curve value causes the gain curve IIoo = f(U = f(UAA) to become non monotonic) to become non monotonic

Large-Signal Instability

ILS = [5A/div]

ui = [50V/div]

Io = [10mA/div]

ILS = [5A/div]

ui = [50V/div]

Io = [10mA/div]

Main converter waveforms when Udc is slowly approaching 21V (fsw = 65kHz, Ls = 42H).

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

Effect of Coupling Network on Voltage Conversion Ratio

045 50 55 60 65 70 75 80

fsw [kHz]

}Io=1mA}Io=6mA|MPT|{

|Mgd|Io=1mA

|Mgd|Io=6mA

Io=1mAIo=6mA

CNCN22: L: Lpp=20=20H, CH, CBB=1=1F, UF, Udcdc=30V=30V

Effect of Coupling Network on Input Impedance

045 50 55 60 65 70 75 80

fsw [kHz]

Io=1mA Io=6mA

Effect of Coupling Network on Switch Commutations

• Differently from the series inductor coupling network, now the inductor current iLp 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

Experimental Measurements

iLp io

Trapezoidal PT input voltageTrapezoidal PT input voltage

Charge of input

capacitance

Control Characteristics: Variable Frequency

[mARMS]

64 66 68 70 72fsw [kHz]

Lp = 20H, CB = 1F

Udc = 30V

CNCN22

Control Characteristics: variable dc link voltage

[mARMS]

Lp = 20H, CB = 1F

10 15 20 25 30

Udc [V]35

fsw = 65kHz

CNCN22

• 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

• 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

Duty-cycle Control

onTtcycleduty

Full-Bridge Inverter for PT

pPT+iinv

Full-Bridge inverter

UDC ui

• Switching frequency control• Duty-cycle control• Phase-shift control

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

Resonant Push-Pull Topology

• Variable switching frequency• Voltage gain at PT input• Sinusoidal driving voltage

Analysis of Small-Signal Instabilities and Modeling

ApproachesExample of high-frequency V –I Example of high-frequency V –I

characteristics characteristics OSRAM L 18W/10OSRAM L 18W/10

Steady-state VRMS-IRMS Characteristic

MATSUSHITA MATSUSHITA FHF32 T-8 32WFHF32 T-8 32W

Negative incremental Negative incremental impedanceimpedance

Positive incremental Positive incremental impedanceimpedance

Modulated Lamp Voltage and Current

Upper trace: iUpper trace: iLampLamp [0.5A/div] Lower trace: u [0.5A/div] Lower trace: uLampLamp [74V/div] [74V/div]

tsinuU2tu sooo tsinUtu moo

mmoo tsinIti

OSRAMOSRAM

L 18W/10L 18W/10

ffmm=200Hz=200Hz ffmm=2kHz=2kHz

ffmm=5kHz=5kHz

tsiniI2ti sooo

Incremental Incremental impedance:impedance:

Lamp Incremental Impedance

Re(ZRe(ZLL))

Im(ZIm(ZLL))

mm= 0= 0 mm= =

zLL s1

s1KZApproximation:Approximation: KKLL< 0, < 0, zz< 0< 0

Right-half plane zeroRight-half plane zero

Lamp Model (Ben Yaakov)

IIo1o1 IIo2o2

UUo2o2

UUo1o1

SRslope

maxoL R

IUR 1oL1oS1o1oSmaxo IRIRUIRU

LRslope

Lamp Model (Ben Yaakov)2

IKR KK22, K, K33 = lamp constants = lamp constants

3oLo IK

The lamp resistance is considered to be dependent The lamp resistance is considered to be dependent on a delayed version of RMS lamp currenton a delayed version of RMS lamp current

3oLqod

odqooqo

Small-signal perturbation:Small-signal perturbation:

Subscript q means quiescent pointSubscript q means quiescent point

Lamp Model (Ben Yaakov)

sIGKRs

sIGIKRs1sIG

IKsIRsU

oL2LqL

sIsGsIs1

1sI oLo

sIsUsZ

0K0Z 2L Lp

Delay:Delay:

Lamp Pspice Model (Ben Yaakov)

Lamp time constantLamp time constant

iioo22 IIoRMSoRMS

iioo=u=uoo/R/RLL

Accounts also for Accounts also for the positive slopethe positive slope

Lamp Model (Do Prado)LPb

L eaR PPLL = Lamp power = Lamp powera, b positive constantsa, b positive constants

21 3 4 5 6Io [mARMS]

RL [M]

1200Uo [VRMS]

21 3 4 5 6Io [mARMS]

Lamp Model (Do Prado)

LooqLqpbPb

ooqpPb

ooqooq pb1iIReeaiIeaiIuU LLLL

LPbL eaR

LoqLqoLqo pIbRiRu

sPIbRsIRsU LoqLqoLqo

sGsUIsIUGsIIsUUsP LooqooqLooqooqL Delay:Delay:

Lamp Model (Do Prado)

bP1bP1

RsGbP1sGbP1

RsIsUsZ

LqLz Pb1

LqLp Pb1

If bPIf bPLqLq>1: >1: zz<0, Z<0, ZLL(0)<0(0)<0

Negative incremental impedance and Negative incremental impedance and RHP zeroRHP zero

Lamp Pspice Model (Do Prado)

RRLL uuoo

uuoo-R-R44iioo

Lamp Model (Onishi)

AA00-A-A44 positive constants positive constantso

L IeAeAAR

Delay:Delay:

oLo II

eAeAAIRUod4od2

odLpIA

21oLqodIIod

oo iReAAeAAiRi

IUu oq4oq2

odqooqo

sIsGsI oLod

sIsGReAAeAAs1RsU oLLpIA

Lqooq4oq2

Lamp Model (Onishi)

sIsUsZ

R Lp oq4oq2 IA

21s eAAeAAR

Negative incremental impedance and Negative incremental impedance and RHP zeroRHP zero

If RIf RSS>0: >0: zz<0, Z<0, ZLL(0)<0(0)<0

Lamp Pspice Model (Onishi)

IIoRMSoRMS

UUoo=R=RLLiioo

Control Problem

sIsUsZ z

An Impedance with a RHP zero cannot be driven An Impedance with a RHP zero cannot be driven directly by a voltage source, since its current directly by a voltage source, since its current

transfer function will contain a RHP poletransfer function will contain a RHP pole

Series Impedance Lamp Ballast

++UUSS(s)(s)

ZZLL--

UUoo(s)(s) FB

IIoo(s)(s)

TTFF must satisfy Nyquist stability criterion must satisfy Nyquist stability criterion

Example of Instability

Series inductor coupling network

LSInverter

PT Lamp

ffoscosc 6kHz 6kHz

Example of Instability

ILp = [1A/div]

Io = [2mA/div]

fosc=6.45kHz

Parallel inductor + dc blocking capacitor Parallel inductor + dc blocking capacitor coupling networkcoupling network

Phasor Transformation [11]

tj SetXetx

A sinusoidal signal x(t) can be represented by a time A sinusoidal signal x(t) can be represented by a time varying complex phasor , i.e.: varying complex phasor , i.e.: tX

Example: AM signalExample: AM signal tcosxX2tx sM

sinjcosxX2tX M

Phasor Transformation

Example: FM signalExample: FM signal xtcosX2tx sM

xsinjxcosX2tX M

Inductor phasor transformation:Inductor phasor transformation: tudt

tdiL LL

SS etUeetIedtdL

LStjL SSS etUeetIje

Inductor phasor transformation:Inductor phasor transformation: tudt

tdiL LL

tUtILj

L LLSL

++LL iiLL

--uuLL ++LL

-- tIL

jjSSLL

Capacitor phasor transformation:Capacitor phasor transformation: tidt

tduC CC

tItUCj

C CCSC

CC iiCC

--uuCC ++CC

-- tIC

1/j1/jSSCC

Generalized Averaging Method [13]

A waveform x(A waveform x(••) can be approximated on the interval ) can be approximated on the interval [ t-T, t ] to arbitrary accuracy with a Fourier series [ t-T, t ] to arbitrary accuracy with a Fourier series

representation of the form:representation of the form:

sTtjkk

setxsTtx s s (0, T], (0, T], T2

tx k = time-dependent complex Fourier series coefficients = time-dependent complex Fourier series coefficients calculated on a sliding window of amplitude Tcalculated on a sliding window of amplitude T

dsesTtxT1tx

sTtjkT

Generalized Averaging Method

The analysis computes the time evolution of these The analysis computes the time evolution of these Fourier series coefficients as the window of length T Fourier series coefficients as the window of length T

slides over the waveform x(slides over the waveform x(••). The goal is to ). The goal is to determine an appropriate state-space model in determine an appropriate state-space model in which these coefficients are the state variableswhich these coefficients are the state variables

txdxT1tx 0

Classical state-space averaging theory:Classical state-space averaging theory:

The average value coincides with the The average value coincides with the Fourier coefficient of index 0!Fourier coefficient of index 0!

Application to Power Electronics

tu,txfdt

u(t) = periodic function of time with period Tu(t) = periodic function of time with period T

Let’s apply the generalized averaging method to a Let’s apply the generalized averaging method to a generic state-space model that has some periodic generic state-space model that has some periodic

time-dependence: time-dependence:

tu,txfdt

Let’s compute the Let’s compute the relevantrelevant Fourier Fourier coefficients of both sides: coefficients of both sides:

Differentiation Property tjk

This relation is valid for constant frequency This relation is valid for constant frequency ss, , but still represents a good approximation for but still represents a good approximation for

slowly varying slowly varying ss(t) (t) k

ktu,txf

kksk tu,txftjk

Transform of Functions of Variables

?tu,txf k

A general answer does not exist unless function f A general answer does not exist unless function f is a polynomial. In this case, the following is a polynomial. In this case, the following

convolutional relationship can be used:convolutional relationship can be used:

iikk yxyx

where the sum is taken of all integers i.where the sum is taken of all integers i.

Lamp dynamic Model

tytitidt

tuyGtueAAtyti

oLotyA30

L IeAAR

Only the negative slope in the UoRMS-IoRMS curve is modeled

y(t) is lamp RMS current squared

Generalized averaged lamp model

Considering that lamp voltage uo(t) and current io(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.

00ooL0

1o0L1o0L1o

yiidtyd

uyGuyGiuyGuyGi

21o1o1o1o1o1o1o0oo i2ii2iiiiii

Non-linear large-signal lamp model

,oL,oyA3o

yii2dt

uGueAA

uo = uo+juo io = io+jio

The fundamental component amplitude of the lamp current is:

2o1o ii2i2

Each complex variable is decomposed into real and imaginary part:

Comparison between complete model and fundamental

component model

0 0.2 0.4

Time [ms]

t Io [

7Fundamental component

Completemodel

Step change of the lamp RMS current from 4 to 6mARMS

Small-signal lamp modelConsidering small-signal perturbations around an

operating point:

2o4 II2A

43Sq eAAR

2o4o4 II2A

II2eAA

,oL,oyA3o

yii2dt

uGueAA

oooooLqLo

oSqLqLqoLqo

yiUiUG4dtyd

URG1GuGi

oooo2o

o iIiII4iIiI

o2o1o ii2i2

Lamp current fundamental component amplitude

Ballast dynamic model

titiC1

tututRiL1

tutsinsignUL1

1o1i1L1Ls

ujdtud

iiC1uj

uiRL1ij

u2jUL1ij

only the complex Fourier coefficients of indexes +1 are considered

2jtsinsign 1s

Ballast small-signal model

oiLsLLs

C1ˆUu

ˆUudtud

CiˆUu

iiC1ˆUu

uiRL1ˆIi

unuuiR

L1ˆIi

uU2L1ˆIi

LuˆIi

Complete ballast Complete ballast model:model:

xCzuBxAx

Tss Uˆu

TooCiLs yuuuiiz

Large-signal and small-signal model comparison

UUss amplitude step variation (-5%) amplitude step variation (-5%)

0 100 200 300 400 500 600 700 800 900 1000

Time [s]

Large-signalnon linear model

Small-signallinear model

0 100 200 300 400 500 600 700 800 900 1000

Time [s]

Large-signalnon linear model

Small-signallinear model

Instability analysis

1 2 3 4 5 6 7 8

Lamp current Io [mARMS]

Unstable

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

49.6 49.7 49.8 49.9 50-8

Time [ms]

t Io [

Instability analysis

1 2 3 4 5 6 7 8

Lamp current Io [mARMS]

Unstable

]) Ls=10H

Ls=15HLs=20H

Ls=28H

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 Ls (L = 100krad/s)

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

References1.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.664-669.

2.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. 36-41.

3.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.29-35.

4.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.1135-1141.

5.Ray L. Lin, Eric Baker and Fred C. Lee, “Characterization of Piezoelectric Transformers”, Proceedings of Power Electronics Seminars at Virginia Tech, Sept. 19-21, 1999, pp. 219-225.

6.E. Deng, S. Cuk, “ Negative Incremental Impedance and Stability of Fluorescent Lamps,” IEEE Applied Power Electronics Conf. Proc. (APEC), 1997. pp.1050-1056.

7.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.1486-1492.

References8.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. 2000.

9.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. 1531‑1536.

10.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.1103-1110.

11.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.201-211.

12.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), 2001.

13.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.251-258.

14.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, 2002.

15.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.1001-1005.

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