Nonlinear Control Characteristic Research of Self-Oscillating Electronic Ballast

Chenyang Liu, Ping Lin, Fanghua Teng, Zhongchao Zhang, Zhengyu Lu

College of Electrical Engineering, Zhej iang University, Hangzhou, P.R.China

E-mail: chenyang-liu@yahoo . com. cn Tel: 0571-87952887 FAX: 0571-87952887

Abstract-In this paper, the self-oscillating LC series resonant parallel load inverter for electronic ballast applications is investigated from a system point of view. By considering the discharge lamp as a resistor in steady state, the self-oscillating series resonant parallel load inverter with lamp loads can be naturally modeled as a relay system. Precise frequency determination and stability of self-oscillating is analyzed using the methods by describing function method, Tsypkin method (frequency domain) and Hamel method (time domain). The characteristic of these three kind methods is analyzed respectively. The design approach is described and implemented successfully in a 36W fluorescent lamp system.

I. INTRODUCTION

Nowadays, electronic ballasts have been used widely in fluorescent lighting systems. Operating at high frequency, electronic ballast shows small size, lightweight, high luminous efficiency, no flicker, no audible noise and long life [l]. Typical topology used to supply discharge lamps is resonant inverters. These topologies include a high frequency push-pull, half bridge or full bridge inverter followed by a resonant tank circuit. The high frequency inverter generates an alternating high frequency voltage or current wave, and the resonant tank is used to limit the current through the lamp. Nevertheless, based on the considerations of cost and reliability, most of ballasts adopt self-oscillating half-bridge inverters. Such inverters can be classified into two groups: those using a current transformer have a saturable core, generally with power BJT; and those using a current transformer have a linear core, generally with power MOSFET. In designing a lamp with saturable core, large number of variables and factors causes the circuit to be sensitive to its operating environment, and the circuit operating point change with input line voltage and ambient temperature. A mathematical analysis of fluorescent lamp converters of half-bridge MOSFET type is found in [2], in which, the current through the inductor L, is fed back into the gates via a current transformer with linear core and converter into a voltage suitable for driving the MOSFETs into conduction.

Most of the topics in the literature are the circuit topology and loaded-resonant characteristics of inverters [3]-[5]. For a self-oscillating inverter, its switching frequency is determined by itself. As we know, an important phenomenon in control theory and design is the occurrence of limit cycles. The

0-7803-7754-0/03/$17.00 02003 IEEE 475

precise determination of limit cycles is usually performed either by Hamels method [6,7] in the time domain or by Tsypkins method [7,8] in the frequency domain while the describing function method [9] provides an approximate solution. By modeling the self-oscillating circuit with lamp load as a nonlinear relay system, stability of self-oscillating can be analyzed using the methods by describe function, Tsypkin method and Hamel method.

In this paper, a low-Q self-oscillating ballast driven by a series linear transformer is analyzed. Nonlinear relay system control model is established based on the point of view of control system. Aforementioned three methods are used in the design of the LC series resonant parallel load inverter, setting the probable self-oscillating frequency and determining its stability. The characteristic of these three kind methods is analyzed and compared respectively. The design approach is described and implemented successfully in a 36W fluorescent lamp system.

11. SELF-OSCILLATING BALLAST CIRCUIT ANALYSIS

The scheme of self-oscillating ballast is shown in Fig.1. Switch S1 and S2 compose a half-bridge inverter. The resonant tank consists of the resonant inductor Lr, resonant capacitor Cr, dc blocking capacitor C3. The lamp can be characterized as an equivalent resistance R, since the operating frequency is much greater than the thermal time constant of the ionized gas and the current is nearly sinusoidal. The startup circuit consists of R1, C2 and Diac D2. After the power is applied, the capacitor C2 is charged via R1. When the voltage across C2 reaches the breakover voltage of Diac, the Diac conducting and a positive turn-on voltage pulse is applied to the gate of S2. With S2 conducting, any charge remaining across C2 is discharge through D1 preventing further startup pulses. The current through inductor Lr is fed back into the gates via a current transformer with a linear core. This provides the excitation to the gates, so that the oscillation of the converter is perpetuated by its own regenerative feedback means. Fig2 shows the scheme with the function of each section in the self-oscillating ballast [lo]. The switches S1 and S2 are modeled as a hard limit non-linearidad. The input of the hard limit nonlinearity can be splited into two components. The first is the magnetizing inductance of the current transformer and the second is the resonant circuit current reflected to the gate circuit side. On the other hand the output of the hard

limit nonlinearity is the gate drive voltage of switch. When a positive voltage is applied in the gate of switch S1,

the magnetizing current increases linearly and when it equals to the reflected resonant current in the gate drive side, the zener current is zero and the gate voltage changes from positive to negative signal guarantying the switching. Then, this ballast can be designed with a proper nonlinear control theory. In the following section, three kinds of nonlinear control analysis will be used to analyze the operation of this ballast.

-o,050.05

+

I 0.005 0.01

:

Vd

-

D3 Cr

D1 N B

TIA Lr Lamp

c3 -- --

Fig.1 Self-oscillating half-bridge LC series resonant ballast

(d ) (C)

Fig2 Self-oscillating ballast function diagram

111. NONLINEAR CONTROL ANALYSIS

A. Describe Function Method

The describe function (DF) method is identical to a harmonic balance approach, where only the first harmonic is balanced, but was developed in a way more suitable for use in feedback control. The DF, N, of a nonlinear element was defined as the ratio of the fundamental output to the magnitude of an applied sinusoidal input. Thus, when using the DF in analysis the high harmonics produced by sinusoidal inputs are neglected. DF method can be conveniently used to determine the existence of self-sustained oscillations as well as to determine stability. The system shown in Fig.2 can be reduced in the block diagram of Fig.3. The nonlinearity of the relay system is defined by DF.

To investigate the possibility of the limit cycle oscillation in the system of Fig.3, the characteristic equation of the system is defined by: 1 + NG( jw) = 0 (1)

I - m l Fig.3 Reduced block diagram

In equation (l), the linear transfer finction G(j a ) can be deduced as:

G ( S ) = K n G f ( S ) - G M ( S ) (2)

Vd n Where K = - , Vd is input DC bus voltage. n = 2,

is the tum ratio of current transformer. Gf(s) is transfer function of the resonant inverter:

2v* ns

(3)

1 And GM(S) is: G M ( s ) = - LMS

Typically, this is done using a Nyquist diagram shown in Fig.4, where the loci G(j 0) and -1/N are plotted, and any intersection of the loci gives the amplitude and frequency of a possible limit cycle. The stability of the limit cycle is normally obtained from the direction of crossing of two loci using the criterion due to Leob [l 11.

476

As shown in Fig.4, there is a possible operation point of the loci that presents a possible operating point A for input data specification. Assuming that the system initially operates at point A, with a self sustained oscillating of amplitude of Iz, and frequency f. Due to a small disturbance, the amplitude Iz is slightly decreased, and the system operation point move from A to C through the origin line. If GO 0) encircles C, the point is unstable, and Iz will increase. Therefore, the operation point will move toward A, but if the converter is disturbed and Iz amplitude increase, with the point moved to the point B, the amplitude of Iz decrease since B is not encircled by GO 0). Thus, the self sustained oscillating associated with A is a stable point. In the stable operation point A has the equation:

Im(G(jw))=Im(KnGf(jo)-(L,jw)-') =O (4) The magnetizing inductance of current transformer can be obtained as:

( 5 ) [(a2 - b ) 2 + J

mnK[a2w - w(b - w 2 ) J L, = L

And the relay constant value in Fig.2 (a) is

n nV, E=-

2wL, During the late'40 and early '50, relays were finding

increasing application in control systems as a relatively cheap and reliable power amplifier. It was realized that, unlike other continuous nonlinear elements, the output from a relay, once it had switched, became independent of the input. This led to the development by Tsypkin in Russia [8] and Hamel in France [6] of techniques for the accurate evaluation of limit cycles in relay systems. Both started by assuming a periodic form for the relay output, with Tsypkin working in the frequency domain and Hamel in the time domain to produce the same solution. In Fig.1, the current through resonant inductor is defined as the input of relay system and the voltage across the resonant tank Vs is defined as the output of the relay system. Then, the relay system diagram of the self-oscillating ballast is shown in Fig.5. The relation between voltages applied in the resonant inverter Vs and inductor iL is shown in Fig.6.

I I

Fig.5 Relation system diagram of the self-oscillating ballast

Based on Fig.5, the necessary conditions self-oscillating will be delivered:

0' 0

< O ,,L (7)

t

Fig.6 Waveform of resonant voltage and current

Where Ts is the switching period, due to a symmetry of inductor current waveform, the above condition become

B. Tsypkin s Method

The Tsypkin hnction is defined by

The condition of (8) becomes

ImT(w) = -E, Re T ( o ) > 0 (10) The admittance function of LC series resonant parallel load

circuit is . U

J - ZO

( l - u 2 ) + j - yi, =

U

=!I (11) Then, the Tsypkin hnction can be deduced as:

O t 1

'0.6

'0.8

for 0 0.5

Re(T(o , 3 1 0 ) ) Fig.7 Tsypkin locus of self-oscillating ballast

477

?l k=l

j Im

478

Frequency (KHz) Q

DF 50.11 50.50 (K=lO)

50.10 (K=lOO) 50.0O(K=lOOO)

[; 1 2 ) Q2 l.:r [ 13 Q2 1:r p2:= p1 := - + - *- - - - .- Tsypkin Hamel 50.00 The Hamel function can be deduced as [14]:

Relay Constant

0.57

o.57

0.57

IV. EXPERIMENTAL RESULTS

A prototype for TLD 36W lamps was built based on above analysis. The lamp voltage is 95Vrms and lamp resistance is 310Q. The switching frequency is selected to be SOkHz, and the output voltage is SSOVrms, input voltage is 11OVac. The zener diode D3-D6 is 1N4742 with V2=12V. Resonant inductor L1-0.7mH and resonant capacitor C ~ 1 5 n F . The tum ratio of current transformer is 1:2:2 with magnetizing inductance of 220uH. Fig.9 shows the tested result,

1 ) RefA: 2) Ref B

500 mVolt 5 us 50 Volt 5 us

Fig.9 a) Resonant current and voltage

M POS: -10.40~5 l T e w ! o l * * * x w , , i , a a 7

: : : : : : ; : : 1 ........................ r . . . . ..................... i

M 1 0 . 0 ~ ~ ~ CH1 f

. . . . i . i

Fig.9 b) Gate drive voltage

y v v v v y 1) RefA: 5 0 Volt 10 us 12) RefB: 5 Volt 10 us

Fig. 9 c) Lamp voltage and current

Fig. 9 a) shows the tested resonant current and voltage OSNdiv, SOV/div and Susldiv); Fig. 9 b) shows the drive waveform (lOV/div, and lOus/div); Fig. 9 c) shows the tested lamp current and voltage waveform (OSNdiv, SOVIdivand 10 uddiv).

V.CONCLUSION

In this paper, nonlinear relay system control model is established for self-oscillating electronic ballast based on the point of view of control system. Three relay system theories including DF method; Tsypkin method and Hamel method are applied to analyze the self-oscillating operation. It can be found that these methods are precisely calculated the probable self-oscillating frequency and determining its stability. The characteristic of these three kind methods is analyzed and compared respectively. The design is implemented successhlly in a 36W fluorescent lamp system. The experiment result proved that these methods are useful in the design of self-oscillating ballast.

REFERENCES

[l] E. E. Hammer, and T.K. Mcgowan, Characteristics of various F40 Fluorescent system at 6OHz and high frequency, IEEE Trans. Ind. Appl., V0121, no.1, pp16-21, Jan./Feb. 1985 [2] L. R. Nerone, A mathematical Model of the Class D Converter for Compact Fluorescent Ballasts. IEEE Tran. On PE, vol 10, No 6, Nov.1995 [3] M. K. Kazimierczuk, and W. Szaraniec, Electronic ballast for fluorescent lamp ballast, IEEE Trans. Power Electronics, Vo1.8, No.4, pp.386-395, Oct., 1993 [4] R.L Steigenvald, A comparison of half-bridge resonant converter topologies, IEEE Trans. Power Electronics, Vo1.3, No.2, pp.174-182, April, 1988 [ 5 ] D. Tadesse, F.P. Dawson and S.B. Dewan, A comparison of power circuit topologies and control techniques for a high frequency ballast, IEEE Power Electronic Specialists Conf. Rec., 1993, pp.2341-2347 [6] B. Hamel, Contribution a Ietude mathematique des systemes de reglage par toutou-rien, CEMVI7, 1949 [7] Francesco Paoletti, etal, A Cad Tool for Limit Cysle Prediction in Nonlinear Systems, IEEE Trans. on Education, vo1.39, No.4, Nov.1996. [SI Ya. Z. Tsypkin, Relay Control System. Combridge, U. K. Cambridge Univ. Press, 1984 [9] A. Gelb and W.E.Vander Velde, Multiple-Input Describing Function and Nonlinear System Design. New York McGraw-Hill, 1968. [IO] Pavao, R.K.; Bisogno, F.E.; Seidel, A.R. and do Prado, R.N, Self-oscillating electronic ballast design based on the point of view of control system, IASO1 pp. 21 1 -217. [ l l ] J. M. Loeb, Recent Advances in Nolinear Servo Theory, in Olderburger R.ed., Frequency Response, Macmillan, New York, pp265-268 [I21 Chin. Chang, Joseph. Chang and Gert. W. Bruning Analysis of the Self-oscillating Series Resonant Inverter for Electronic Ballasts, IEEE Trans. on PE, vo1.14, No.3, May.1999. [13] R.N.do Prado, A. R. Seidel, F.E.Bisogno and M. A. Dalla Costa A Design Method for Electronic Ballasts for Fluorescent Lamp. IEEE IASOO pp. 2279-2284. [14] Chin Chang, Gert W. Bruning Self-oscillating Ballast Analysis Using the Relay Systems Approach, IEEE Trans. on IA, ~01.37, No.1, Jan.2001. [I51 M.A. Dalla Costa, A. R. Seidel, F.E.Bisogno and R. N. do Prado Self-Oscillating Dimmable Electronic Ballasts, IEEE IECONOI pp.1038- 1043.

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