Yu-Kang Lo, Member, IEEE, and Kai-Jun Pai
Abstract—The feedback loop design of a half-bridge resonant inverter based on a piezoelectric transformer (PT) for driving a cold cathode fluorescent lamp (CCFL) is presented in this paper.
In order to stabilize the CCFL driving current and luminance, the PT-based resonant inverter incorporating a feedback compen-sator is designed to track the operating frequency. The dynamic equations and the small-signal model of the presented inverter system are established by using the harmonic approximation and harmonic balance procedures. The feedback compensation is performed by applying the derived small-signal block diagram.
The complete mathematical analysis and design considerations are presented in detail. The experimental results agree with the theoretical predictions and confirm the validity of the proposed design approach.
Index Terms—Cold cathode fluorescent lamp (CCFL) inverter, feedback compensator, harmonic approximation, harmonic balance, piezoelectric transformer (PT).
I. INTRODUCTION
R
ECENTLY, due to the prevalence of digital consumer electronic products, the number of flat display panels (FDPs) has also been proliferating. Among the various FDPs, liquid crystal displays (LCDs) are the most popular for their handy availability and cost-effective performance. Nowadays, LCD panels can be found in a wide array of applications such as in monitors, notebooks, and televisions [1]–[5]. Cold cathode fluorescent lamps (CCFLs) are usually adopted for the back-light sources of the LCD panels. A high-frequency inverter is used to drive the CCFL, which thus determines the visualization performance of the LCD panels. Conventionally, electromag-netic transformers are utilized to provide the electrical isolation and to step up the CCFL driving voltage. However, applications of traditional electromagnetic transforms can be obstructed due to core losses, lower conversion efficiencies, and the difficulty to be manufactured into planiform in high-standard demands.On the other hand, a piezoelectric transformer (PT) features the flat configuration, low power loss, and immunity from the electromagnetic interference. Thus, PTs are used to replace the electromagnetic transformers in some rigid design cases,
Manuscript received November 15, 2006; revised April 11, 2007. This work was supporetd by the National Science Council, Taiwan, R.O.C. under Grant NSC 95-2221-E-011-203-MY2.
The authors are with the Department of Electronic Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan, R.O.C.
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
where the space left for the CCFL inverters is limited [6]–[8].
The principle of a PT is based on the conversion between the mechanical and electrical energies. There are several vibration modes of a PT, such as thickness extensional vibration mode, width-shear vibration mode, and longitudinal vibration mode [9]. The longitudinal vibration mode with multilayer ceramic material possesses a high voltage step-up ratio, which is partic-ularly suitable for driving a CCFL [10].
One well-known approach for stabilizing the lamp current is to compare the sensed lamp current to a current reference by an error amplifier (EA). Then, a control voltage can be produced to modulate the output frequency of a voltage-controlled os-cillator (VCO). However, the bandwidth of a PT is so narrow that the range of the output values of the feedback loop is limited. Therefore, the VCO’s output frequency range must be carefully designed. Otherwise, the CCFL driving current may be fluctuated around the rated value, and the lamp luminance flicker results. The selections of the feedback components are thus crucial for a successful design. Although the half-bridge (HB) resonant inverter incorporating a PT for driving a CCFL backlight module has been presented for several years, the small-signal analysis and compensator design are not clarified.
In this paper, the dynamic behaviors of a PT–CCFL combina-tion are examined and the feedback elements are determined.
The operating frequency of the VCO is adjusted accordingly.
Therefore, the CCFL driving current and lamp power can be maintained even under large parameter variations. The design considerations and the experimental results are also given to confirm the validities of the theoretical analysis and the derived small-signal model.
II. SMALL-SIGNALANALYSIS
The variable-frequency resonant inverter for driving the CCFLs has been proposed to improve the system performance such as dimming control and feedback design [11], [12]. Al-though the feedback compensating scheme has been adopted in many literatures [13]–[16], the small-signal analysis is not presented in detail. In this paper, the harmonic approximation and harmonic balance procedures are applied to derive the dy-namic equations and small-signal model of a PT-based resonant inverter for designing the feedback compensator.
Fig. 1 shows the circuit configuration of the PT-based HB resonant inverter for driving a CCFL, including the dc input voltage VDC, power switches S1 and S2, the filter inductance
LO AND PAI: FEEDBACK DESIGN OF A PT-BASED HALF-BRIDGE RESONANT CCFL INVERTER 2717
Fig. 1. PT-based HB resonant inverter for driving a CCFL.
Fig. 2. Equivalent circuits of the PT–CCFL combination: (a) the Rosen-type electrical model, (b) reflected from secondary side to primary side, (c) by transferring the parallel connection to series connection, and (d) the simplified version of (c).
characteristic of a high-selective bandpass filter [6], the dc component and the harmonics are filtered out when the switch-ing frequency of the resonant inverter is near the mechanical resonant frequency of the PT. Therefore, the sinusoidal voltage and current can then be produced to start up and to drive the CCFL. Fig. 2(a) depicts the equivalent circuit model of the PT–CCFL combination under the sinusoidal steady state.
A Rosen-type electrical model for the PT is applied, and the CCFL is represented by its equivalent resistance R. In Fig. 2(a), vsis the fundamental component of the HB inverter output voltage. To simplify the analysis, the circuit elements of the secondary side are reflected to the primary side, as illustrated in Fig. 2(b). The capacitor Coand resistance Roare
Ro= R/N2 (1)
Co= N2Cd2 (2)
TABLE I PARAMETERS IN(8)–(11)
resulting equivalent circuit is illustrated in Fig. 2(c), where Cs and Rscan be calculated as
Cs=1 + ωs2Co2R2o ωs2R2oCo
(3)
Rs= Ro
1 + ωs2Co2R2o (4) where ωsis the switching frequency. From Fig. 2(c), a simpli-fied circuit can be obtained, as shown in Fig. 2(d), where Cx and Rxare
Cx= CsC
Cs+ C (5)
Rx= R + Rs. (6)
According to the aforementioned equations and Fig. 2(d), the procedures for obtaining the small-signal model can be estab-lished as follows.
A. Mesh Equations of the PT-Based Resonant Inverter From Fig. 2(d), the mesh equations in matrix form in the phasor representations are established as follows:
jωsLs+jω1
sCd1
−1 jωsCd1
−1 jωsCd1
1
jωsCd1+jωsL+jω1
sCx+Rx
ILs IL
=
Vs 0
. (7) According to (7), the expressions ILs, IL, VCx, and Vincan be obtained by Cramer’s method as follows:
ILs=(αp1+ βq1) + j(αq1− βp1)
α2+ β2 (8)
IL=(αp2+ βq2) + j(αq2− βp2)
α2+ β2 (9)
VCx=(αp3+ βq3) + j(αq3− βp3)
α2+ β2 (10)
Vin=(αp4+ βq4) + j(αq4− βp4)
α2+ β2 (11)
2718 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 5, OCTOBER 2007
TABLE II
AMPLITUDEVARIABLES IN(17)–(20)
B. State Variables and Dynamic Equations
According to Fig. 2(d), Kirchhoff’s voltage and current laws are used to define the state variables and dynamic equations as follows:
C. Harmonic Approximation Method
The high-order harmonics produced by the resonant inverter can be filtered out by the PT. Therefore, the harmonic approxi-mation equations for the voltage source and the state variables can be derived [17], [18] as follows:
vs= Vsccos ωst + Vsssin ωst (16) iLs= ILcscos ωst + ILsssin ωst (17) vin= Vinccos ωst + Vinssin ωst (18) iL= ILc cos ωst + ILssin ωst (19) vCx= VCcxcos ωst + VCsxsin ωst. (20) In these equations, the variables with the superscript c repre-sent the amplitudes of the cosine term, whereas the variables with the superscript s are the amplitudes of the sine terms.
Table II lists the amplitudes of the state variables in terms of the parameters that appeared in Table I. It should be noticed from Table I that all these amplitudes are functions of the excitation amplitude Vs.
D. Harmonic Balance Procedures
monic balance procedures can be used [19]. First, (17)–(20) are differentiated as follows [17], [18], [20]:
diLs
Substituting (12)–(15) in (21)–(24), the large-signal dynamic equations are obtained as follows:
dILcs
E. Linearized Model
The transfer function of the incremental lamp current versus the switching frequency perturbation is derived for the feed-back design of the PT-based resonant inverter. Introducing the small perturbations (represented by the variables with a cap) to (25)–(32), the dynamic equations of the linearized model can be obtained by Taylor expansion [19] as follows:
d ˆILcs
dt =− 1 Ls
Vˆinc − ωsIˆLss− ILssωˆs (33)
LO AND PAI: FEEDBACK DESIGN OF A PT-BASED HALF-BRIDGE RESONANT CCFL INVERTER 2719 From (33)–(40), the state equations can be derived as dX/dt = AX + Bˆωs, where The output response is the lamp current i, which is equal to vo/NRo. Referring to Fig. 2(b), vois the voltage across Ro, The amplitude of vocan be expressed as [7]
Fig. 3. (a) Circuit diagram and (b) small-signal block diagram of the presented PT-based CCFL inverter.
The small perturbation of vo, as shown in the following, is also decoupled into cosine and sine terms:
ˆ
Thus, the output equation can be expressed as y = ˆi= ˆ
III. FEEDBACKDESIGN
An EA is often applied in the feedback design of switching power supplies and electronic ballasts. Fig. 3(a) illustrates the circuit diagram of the discussed PT-based resonant inverter with the feedback compensator [13], [15]. The rms circuit rectifies the sensed ac lamp current i to a feedback quantity vf. The compensated EA produces an error signal ve to adjust the output frequency of the VCO. Thus, the operating frequency of the HB inverter is optimized to stabilize the lamp current.
The equivalent block diagram concerning the loop gain of the presented system is depicted in Fig. 3(b), where the EA gain is GEA, the VCO gain is Gvco, the gain of the PT-based resonant inverter is represented by Ginv, and the transfer function of the rms circuit is Grms.
A. Open-Loop Gain
In Fig. 3(b), the transfer function of Ginv(s) can be ex-pressed as
2720 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 5, OCTOBER 2007
TABLE III
PRACTICALCIRCUITPARAMETERS INFig. 2(a)
TABLE IV COEFFICIENTS IN(51)
Substituting (42), (43), and (49) in (50), Ginv(s) can be expressed as
Ginv(s) = k× b(s)
a(s) (51)
where k is a constant, and a(s) and b(s) are, respectively, eighth-order and seventh-order polynomials of s. An im-pedance analyzer (HP-4194A) is employed to measure the parameters of the tested PT (EFTU14R0M01A), including Cd2, R, L, C, and Cd1. By substituting the practical circuit parameters, as listed in Table III, in (50), all the coefficients in (51) with descending powers can be calculated by Matlab and are given in Table IV. Although some of these coefficients seem enormous due to divergent values of the circuit elements, (51) can be approximately reduced to a second-order transfer function provided that the dominant poles are below several tens of kiloradians per second. Therefore, only k, b0, a2, a1, and a0matter near the system bandwidth. It is also worth noting that tolerances of the parameters listed in Table III will result in the variations of the coefficients in (51). Nevertheless, the effects on the low-frequency gain and corner frequency are negligible.
Grms(s) can be obtained from Fig. 3(a). It is assumed that Dmis an ideal diode. Rmis 620 Ω and Cmis 0.1 µF. Thus,
Grms(s) = vˆf
ˆi = Rm
sRmCm+ 1 ∼= 107
s + 4.55× 104. (52) The transfer characteristic of the adopted VCO in the experi-ments is shown in Fig. 4. It is evident that the output frequency ωsof VCO is proportional to the input voltage ve. Hence, the transfer function of the VCO can be expressed as
Fig. 4. Transfer characteristic of the adopted VCO.
Fig. 5. Simulated and measured frequency responses of Go.
Substituting the small perturbations ˆωsand ˆvein (53) yields the following small-signal transfer function of VCO:
Gvco(s) = ωˆs
ˆ ve
= 3.14× 104. (54) Therefore, the small-signal open-loop gain Go shown in Fig. 3(b) is obtained as
Go(s) = ˆvf ˆ ve
= Ginv(s)× Grms(s)× Gvco(s). (55) By substituting (51), (52), and (54) in (55), Fig. 5 shows that the simulated frequency responses are close to the measured data. In Fig. 5, the simulated low-frequency gain is−3.34 dB and the bandwidth is 5.4 kHz. The measured low-frequency gain is−5 dB, and the bandwidth is 4.7 kHz (using FRA 5096, NF Inc.). It is proven that the harmonic approximation and harmonic balance procedures are both applicable to establish the small-signal model of the PT-based CCFL inverter.
B. Feedback Design
From the simulation results shown in Fig. 5, the low-frequency gain of Go is lower than 0 dB. A
proportional-LO AND PAI: FEEDBACK DESIGN OF A PT-BASED HALF-BRIDGE RESONANT CCFL INVERTER 2721
Fig. 6. Adopted EA compensation circuit.
Fig. 7. Simulated and measured Bode plots of Gx. TABLE V
CIRCUITPARAMETERS ANDSPECIFICATIONS OF THE PT-BASEDCCFL INVERTER
margin (PM) and fast response [21]. The transfer function of the employed EA (LM358), as shown in Fig. 6, is
GEA(s) = vˆe ˆ vf
= Rf
Ri ×s +R1
fCf
s (56)
where the corner frequency regarding the zero is ωz= 1
RfCf
. (57)
The crossover frequency of the PI compensator should be lower than 1/100 of the resonant frequency [22]. Let the corner frequency be equal to the system crossover frequency. At a first
Fig. 8. Waveforms of the PT-based resonant inverter without feedback at (a) 15-V and (b) 20-V dc input voltages.
a 3.34-dB gain is required at the expected system crossover frequency. Thus,
|GEA(jωz)| =
(RfCfωz)2+ 1 RiCfωz
= 1.47. (58)
The value of Ri is arbitrarily set to be 10 kΩ. Substituting ωz= 3405.5 rad / s in (57) and (58), Rf is calculated 10 kΩ and Cf is 28.2 nF. A resistor of 12 kΩ and a capacitor of 33 nF are chosen for Rf and Cf in practice. Therefore, the corner frequency of the EA compensator, which is also the system crossover frequency, is equal to 2525 rad/s. The simulated Bode plots of GEAare depicted in Fig. 7. The calculated and measured frequency responses of the loop gain Gx, which is the product of Goand GEA, are also shown. It is clearly seen that the low-frequency gain increases, the system crossover
◦
2722 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 5, OCTOBER 2007
Fig. 10. Lamp powers at different input dc voltages.
IV. EXPERIMENTALRESULTS
The circuit parameters and specifications of the tested PT-based CCFL inverter are listed in Table V. Fig. 8(a) and (b) illustrates the waveforms of the resonant inverter operated at different input voltages without feedback. The VCO’s output frequency fs is fixed at 54.2 kHz. When VDC= 15 V, as shown in Fig. 8(a), vis 710 Vrmsand iis 5.5 mArms. When VDC= 20 V, as shown in Fig. 8(b), it is recorded that vdrops to 636 Vrms and iincreases to 7.07 mArms. Fig. 9(a) and (b) records the waveforms of the resonant inverter at different input voltages with feedback compensation. The operating frequency can be accurately tracked by the EA when the dc input voltage is changed. The VCO’s output frequency changes from 54.2 to 55.5 kHz, whereas the CCFL driving voltage and current are maintained at 710 Vrmsand 5.5 mArms, respectively. Fig. 9(c) shows that the operating frequency can be tracked during a step change of VDCto stabilize and regulate the CCFL driving voltage and current. Fig. 10 depicts the experimental results of the lamp power P under varying input dc voltages. With feedback compensation, the lamp power can be maintained at about 3.9 W. On the other hand, the lamp power will increase over 4.5 W as VDCrises without the feedback mechanism. From these measurements, it is confirmed that the derived small-signal model and the feedback design scheme can be used to effectively track the operating frequency of the PT-based resonant CCFL inverter. In addition, the lamp current, lamp luminance, and lamp power can be stably regulated.
V. CONCLUSION
This paper presents a detailed derivation of the small-signal model for the PT-based resonant CCFL inverter. In addition, the feedback compensation is designed to stabilize the CCFL driving current and lamp power at varying input dc voltages.
The contributions of the presented scheme are summarized as follows. First, a low-profile backlight module can be imple-mented by replacing the electromagnetic transformer with a PT. Second, the harmonic approximation and harmonic balance procedures are both applied to establish the small-signal model.
LO AND PAI: FEEDBACK DESIGN OF A PT-BASED HALF-BRIDGE RESONANT CCFL INVERTER 2723
gain and enough PM for the PT–CCFL inverter. The operating frequency for the PT-based resonant inverter can be tracked even if the bandwidth of a PT is so narrow. Thus, both lamp current and lamp power can be maintained at constant levels.
REFERENCES
[1] C. H. Lin, Y. Lu, H. J. Chiu, and C. L. Ou, “Eliminating the temperature effect of piezoelectric transformer in backlight electronic ballast by apply-ing the digital phase-locked-loop technique,” IEEE Trans. Ind. Electron., vol. 54, no. 2, pp. 1024–1031, Apr. 2007.
[2] C. G. Kim, K. C. Lee, and B. H. Cho, “Analysis of current distribution in driving multiple cold cathode fluorescent lamps (CCFL),” IEEE Trans.
Ind. Electron., vol. 54, no. 1, pp. 365–373, Feb. 2007.
[3] S. K. Kim, H. S. Han, Y. J. Woo, and G. H. Cho, “Detection and regulation of CCFL current and open-lamp voltage while keeping floating condition of the lamp,” IEEE Trans. Ind. Electron., vol. 53, no. 2, pp. 707–709, Apr. 2006.
[4] C. H. Lin, “Digital-dimming controller with current spikes elimination technique for LCD backlight electronic ballast,” IEEE Trans. Ind. Elec-tron., vol. 53, no. 6, pp. 1881–1888, Dec. 2006.
[5] M. S. Lin, W. J. Ho, F. Y. Shih, D. Y. Chen, and Y. P. Wu, “A cold-cathode fluorescent lamp driver circuit with synchronous primary-side dimming control,” IEEE Trans. Ind. Electron., vol. 45, no. 2, pp. 249–255, Apr. 1998.
[6] C. S. Moo, W. M. Chen, and H. K. Hsien, “Electronic ballast with piezoelectric transformer for cold cathode fluorescent lamps,” Proc. Inst.
Electr. Eng.—Electric Power Applications, vol. 150, no. 3, pp. 278–282, May 2003.
[7] C. D. Wey, T. L. Jong, and C. T. Pan, “Design and analysis of an SLPT-based CCFL driver,” IEEE Trans. Ind. Electron., vol. 50, no. 1, pp. 208–
217, Feb. 2003.
[8] J. Diaz, F. Nuno, J. M. Lopera, and J. A. Martin-Ramos, “A new control strategy for an AC/DC converter based on a piezoelectric transformer,”
IEEE Trans. Ind. Electron., vol. 51, no. 4, pp. 850–856, Aug. 2004.
[9] P. J. M. Smidt and J. L. Duarte, “Powering neon lamps through piezoelec-tric transformers,” in Proc. IEEE PESC, 1996, vol. 1, pp. 310–315.
[10] Texas Instruments Data Manual, Texas Instruments Inc., Dallas, TX, 2002, pp. 18–23. slyt 107.
[11] C. H. Lin, “The design and implementation of a new digital dimming con-troller for the backlight resonant inverter,” IEEE Trans. Power Electron., vol. 20, no. 6, pp. 1459–1466, Nov. 2005.
[12] C. H. Lin and K. J. Pai, “Difference-integral dimming controller for the single-stage back-lighting electronic ballast,” in Proc. IEEE PESC, 2003, vol. 3, pp. 1000–1005.
[13] S. J. Choi, K. C. Lee, and B. H. Cho, “Design of fluorescent lamp ballast with PFC using a power piezoelectric transformer,” IEEE Trans. Ind.
Electron., vol. 52, no. 6, pp. 1573–1581, Dec. 2005.
[14] S. Ben-Yaakov and S. Lineykin, “Maximum power of piezoelectric trans-former HV converters under load variations,” IEEE Trans. Power Elec-tron., vol. 21, no. 1, pp. 73–78, Jan. 2006.
[15] Panasonic Application Note, 2003, Matsushita Electron. Compon. Inc., Piezoelectric Transformers, EFTU14R0M01A.
[16] Texas Instruments Data Manual, Texas Instruments Inc., Dallas, TX, 2002. UCC3977.
[17] O. Ojo and I. Bhat, “Steady-state and dynamic analyses of high-order parallel resonant convertors,” Proc. Inst. Electr. Eng.—Electric Power Applications, vol. 140, no. 3, pp. 209–216, May 1993.
[18] L. Y. Chiu and B. M. Diong, “Small-signal modeling of the current-source parallel resonant DC/DC converter,” in Proc. IEEE CIEP, 2002, pp. 20–34.
[19] E. X. Yang, F. C. Lee, and M. M. Jovanovic, “Small-signal modeling of LCC resonant converter,” in Proc. IEEE PESC, 1992, vol. 2, pp. 941–948.
[20] G. Spiazzi and S. Buso, “Small-signal analysis of cold cathode fluorescent lamp ballasts,” in Proc. IEEE PESC, 2005, pp. 2783–2789.
[21] I. Etxeberria-Otadui, A. L. D. Heredia, H. Gaztanaga, S. Bacha, and M. R. Reyero, “A single synchronous frame hybrid (SSFH) multifre-quency controller for power active filters,” IEEE Trans. Ind. Electron., vol. 53, no. 5, pp. 1640–1648, Oct. 2006.
[22] J. Cardesin, J. M. Alonso, E. Lopez-Corominas, A. J. Calleja, J. Ribas, M. Rico-Secades, and J. Garcia, “Small-signal analysis of a low-cost power control for LCC series–parallel inverters with resonant current mode control for HID lamps,” IEEE Trans. Power Electron., vol. 20, no. 5, pp. 1205–1212, Sep. 2005.
Yu-Kang Lo (M’96) was born in Chia-Yi, Taiwan, R.O.C., in 1969. He received the B.S. and Ph.D.
degrees in electrical engineering from National Taiwan University, Taipei, Taiwan, in 1991 and 1995, respectively.
Since 1995, he has been with the faculty of the Department of Electronic Engineering, National Taiwan University of Science and Technology, where he is currently an Associate Professor and in charge of the Power Electronic Laboratory. His research interests include the design and analysis of a variety of switching-mode power converters and power factor correctors.
Kai-Jun Pai was born in Taipei, Taiwan, R.O.C., in 1981. He received the M.S. degree from National Taiwan University of Science and
Kai-Jun Pai was born in Taipei, Taiwan, R.O.C., in 1981. He received the M.S. degree from National Taiwan University of Science and