M. Ma (Ed.): Communication Systems and Information Technology, LNEE 100, pp. 235–242. springerlink.com © Springer-Verlag Berlin Heidelberg 2011

A HID Lamp Model in Simulink Based on the Principle of Electric Arc

Xiaohan Guan* and Zhongpeng Li

College of Information Engineering, North China University of Technology

5#, Jinyuanzhuang Road, Shijingshan District, Beijing, China, 100144 gxh@ncut.edu.cn

Abstract. A dynamic conductance model of HID lamp, based on the classical principle of electric arc, will be established in this paper. Followed by the model and corresponding results, this type of HID lamp model will be simulated at low and high frequency in Simulink. The V-I characteristic, the relationship of current and voltage output, also the conductance of HID lamp model will be measured in this paper. It shows that the proposed model faithfully emulates external electrical properties of HID lamp at low and high frequency.

Keywords: HID lamp; model; Simulink; electric arc

1 Introduction

With the development of Computer-Aided Design, simulation tools are more significant in the power electronic systems currently. In the design process of electronic ballasts, the electronic ballast circuit would be seriously considered, and then simulated in order to test the feasibility and optimization of the circuit. Therefore, an equivalent model of HID lamp must be provided to facilitate the simulation.

Matlab/Simulink is the high-performance numerical software of Mathworks Company in United States in the mid-80s in 20th century. After developing within three decades, Matlab has become a basic mathematical tool of mathematical statistics, automatic control theory, dynamic system simulation and many other courses. In fact, there are many articles on HID lamp model in Pspice. However, the model of HID lamp in Simulink is still less. Therefore, in the second section of this paper, a new dynamic conductance model of HID lamp based on the principle of electric arc will be established. The model combined the mathematical model proposed by Cassie and Mayr. This model will be simulated at low and high frequency in Simulink. Simulation results show that the model in this paper can simulate the external electrical characteristics of HID lamp and verify the

* This work is supported by Beijing Education Committee Technology Development Plan

Project (KM200810009011) and Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality PHR201008185.

236 X. Guan and Z. Li

applicability of the model. The model of this paper can provide numerous guidance and reference value for electronic ballasts.

2 Establishment of the New Dynamic Conductance Model

Before a model of HID lamp is established, it is extremely significant to understand some basic knowledge of the HID lamp’s electrical properties, such as different electronic properties [1] of HID lamp at low and high frequency. In the other word, when the low frequency sinusoidal current is treated as input, such as tens of Hertz, lamp voltage will cause the phenomenon of re-ignition. From another view, when the sinusoidal current is changing around zero, the voltage will increase suddenly. This phenomenon is similar as beginning ignition. And with the current increase, the voltage reaches maximum value dramatically and followed by decreasing slowly to the normal level. This phenomenon occurs periodically as the sinusoidal current. So its V-I characteristics show a classic hysteresis phenomenon. When the input current is a high-frequency sine wave, for instance, tens of kHz, current and voltage of lamp are sinusoidal and at the same phase. The V-I characteristics at this time is a linear relationship, however, the slope is not fixed. As it can be seen, HID lamp model should be designed to simulate these two characteristics simultaneously.

Since discharge process of HID lamp is an electric arc, researchers had carried out much study of electric arc followed by lots of results. The study of HID lamp arc can be referenced from the research of switch arc mathematical model. The classical Cassie mathematical model established in 1939 and Mayr mathematical model proposed in 1943, based on simplifications of principal power-loss mechanisms and energy storage in the arc column, have been recognized for many years.

In the Cassie model, Cassie assumed that the current density in an electric arc model is a constant, so the cross-section of the arc varied directly with the arc current. The resistivity and stored energy per unit volume are constants. Based on these basic assumptions, the famous Cassie arc model can be obtained, showed in equation (1). This model has a drawback that the modeled arc cannot be ceased. It describes the behavior of the arc when the current is strong; however it is not suitable for the description of arc characteristics when the arc-current is near to zero.

2

2

ln1

O

d G v

dt Eθ = − (1)

In Mayr’s model, Mayr assumed that the heat loss occurs from the outer arc only. Also the conductance of the arc changed with the energy stored in it. The Mayr mathematical equation is

2ln1

O

d G i

dt P Gθ = − (2)

This equation does allow the arc to cease. With the decrease of conductance G, i2/POG can be still more than unity. Hence, dlnG/dt is positive and the conductance

A HID Lamp Model in Simulink Based on the Principle of Electric Arc 237

continues to decrease until the arc is extinguished. The fitting of model results to measured data is achieved by means of a proper selection of arc parameters like the time constant and the current-dependent cooling power, which are normally taken as a function of arc current and voltage. The modified Mayr model is used in this article, shown as equation (3).

1 ln 1( 1)

( )O i

dG d G vi

G dt dt P P C iθ= = −

+ (3)

G is the instantaneous arc conductance; θ is the arc time constant; EO is the constant steady-state arc voltage; v is the arc voltage; i is the arc current; PO is the constant power loss of temporarily stable; P is the fill pressure of the circuit breaker; Ci is the current constant.

Obviously, the Cassie model and modified Mayr model are not applicable in all cases. But two models are complementary with each other. As the modified Mayr model is more feasible for zero and low arc current region, Cassie model is more suitable for high arc current region. If these two models can be combined reasonably, a more applicable mathematical arc model can be obtained. Therefore, the hypothesis is presented as follow:

1. The time constant in these two models is the same. 2. A complete arc discharge process can be described by the combination of the

Cassie model and modified Mayr model. But the conversion between two models lacks a transition point. It can be assumed that the transition current is I. If the arc current is greater than I, the arc discharge process is described by the Cassie model; otherwise, it is described by the modified Mayr model.

3. This paper supposes that the transition current is continuous. The transition is smooth. It is possible to define a transition factor f, which is an exponential function of the arc current. The transition factor f can be taken as:

2

2exp( )

if

I= − (4)

A new mathematical arc model, equation (5), was got using equations (1), (3) and (4):

2

2

1( 1)(1 ) ( 1)

( )O O i

dG v vif f

G dt E P P C iθ = − − + −

+ (5)

In addition, the arc inherent conductance Gm between the electrodes should be considered in equation (5). Considered equations (4) and (5), the complete model is thus given by

2 2 2 2 22

2 2exp( ) exp( )

( ) mO O O i

dG i i i i iG G G G

dt E E I P P C i Iθ = − − + − − +

+ (6)

238 X. Guan and Z. Li

The equation (6) is the new dynamic conductance mathematics model of HID lamp. A 250W of HID lamp was tested at 50Hz in order to obtain the parameters. The parameters were taken as: θ =2*10-4S, EO=120V, I=0.45A, Gm =1.5*10-8S, PO = 250W. P and Ci were obtained by genetic algorithms and other mathematical calculations. P=4.99bar, Ci =499.8V/bar. According to equation (6), the dynamic conductance model of HID lamp in Simulink was established as Figure 1.

Fig. 1. Dynamic Conductance model of HID lamp in Simulink

3 Simulation of the New Dynamic Conductance Model

The simulation circuit was shown as Figure. 2. According to the HID lamp current in practical work, the input current of simulation model is set to a sinusoidal current that the peak is 1.5A. In order to observe the voltage and current waveforms better, the input of simout1 in Figure. 2 was amplified to certain multiple.

Fig. 2. Simulation Test Circuit of model

A HID Lamp Model in Simulink Based on the Principle of Electric Arc 239

3.1 Simulation of the Model at Low Frequency

First, the conductance of the model at low frequency is observed, which was shown as Figure.3. It shows that the conductance changes periodically with the sinusoidal current. The peak is not fixed, which changes with the peak of the current.

Figure 4 shows the voltage and current waveforms of the model at low frequency. It can be seen that the output voltage of the model is similar to a square wave. Current and voltage are in the same phase, but the voltage has a peak at each zero-crossing point. That is because the phenomenon of re-ignition of HID lamp appears at low frequency. However, as the frequency increases, the peaks gradually become much smaller and then disappear. At this time, the performance of voltage waveform will be improved.

Fig. 3. Conductance at 50Hz Fig. 4. Voltage and Current at 50Hz

Figure 5 is the V-I characteristics of model at low frequency. It shows that the V-I characteristic performed as the classic hysteresis phenomenon, that is, the current reach the maximum, and then voltage reach the maximum as well. After that, both current and voltage are close to zero. Basically, the parameters of HID lamp are changing with this principle regularly at low frequency. All mentioned above are due to the gas thermal inertia in HID lamp. HID lamp at low frequency performed the nonlinear V-I characteristics.

In order to analyze the frequency characteristics of the model deeply, the model in this paper was simulated at 5Hz. Simulation results of voltage and current were shown in Figure 6. It can be seen that the dynamic conductance model at 5Hz can reflect the negative incremental impedance characteristics of HID lamp more clearly. In other words, the lamp voltage decreases as the current increases.

240 X. Guan and Z. Li

Fig. 5. V-I characteristics at 50Hz Fig. 6. Voltage and Current at 5Hz

3.2 Simulation of the Model at High Frequency (50 kHz)

The conductance of the model at high frequency was shown in Figure.7 and Figure.8. It can be seen in Figure 7 that conductance of the model gradually increased from zero to a constant value. When the model reached at high frequency in stable state, the equivalent conductance was shown in Figure.8. The conductance fluctuated in small range, and small-scale periodic fluctuations could be ignored, that is, the conductance was regarded as a constant. This fact is coincided with the actual situation of HID lamp. When HID lamp start at high frequency, the resistance is infinite. Otherwise, when it achieves stable state, the equivalent resistance is a constant value.

Fig. 7. Conductance at 50 kHz Fig. 8. Conductance at the stable state

Figure 9 shows voltage and current of the model. It shows that the voltage becomes a sine wave and the peak disappears. The voltage and current are in same phase. This indicates that the model appears a pure resistance at high frequency. Figure 10 shows the V-I characteristics which was a linear relationship, while the slope was not fixed, but changed in certain range. It was verified that resistance with HID lamp could be considered as a variable resistor at high frequency.

A HID Lamp Model in Simulink Based on the Principle of Electric Arc 241

Fig. 9. Voltage and Current at 50 kHz Fig. 10. V-I characteristics at 50 kHz

4 Conclusion

The arc discharge process of HID lamp was described by the classical Cassie model and the modified Mayr model. The new mathematical model was established in this paper followed by the new dynamic conductance model based on the principle of an electric arc. Simulation results showed that the new dynamic conductance model could simulate the main characteristics of HID lamp. The model’s characteristics at low and high frequency are in great agreement with the external characteristics presented by HID lamp. Moreover, the model at a more low-frequency (5Hz) better reflects the negative incremental impedance of HID lamp. Some specific guidance and reference value in future will be provided in the process of electronic ballasts designed.

The Cassie model and modified Mayr model describe the arc characteristics only from the physical concept. Therefore, a description of some complex features about HID lamp, such as the phenomenon of acoustic resonance and stroboscopic, has not yet contained in the new model. Such model establishment will become the following direction of HID lamp model.

References

1. Wang, Y., Wu, W., Wang, J., Yang, L.: Pspice Models of High-Intensity Discharge Lamps. Journal of Shanghai University (Natural Science), 11(6) (2005)

2. Parizad, A., Baghaee, H.R., Tavakoli, A., Jamali, S.: Optimization off Arc Models Parameters Using Genetic Algorithm. In: International Conference on Electric Power and Energy Conversion Systems EPECS 2009, pp. 1–7 (2009)

3. Shvartsas, M., Sam, B.-Y.: A SPICE compatible model of high intensity discharge lamps. In: 30th Annual IEEE Power Electronics Specialists Conference (PESC 1999), pp. 1037–1042 (1999)

4. Wei, Y., Hui, S.Y.R.: A Universal Pspice Model for HID Lamps. In: 37th IAS Annual Meeting,Conference Record of the Industry Applications Conference 2002, pp. 1475–1482 (2002)

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5. Herrick, P.R.: Mathematical models for high-intensity discharge lamps. IEEE Transactions on Industry Applications, 1A 16(5), 648–654 (1980)

6. Wei, Y., Hui, S.Y.R., Chung, H., Cao, X.H.: Genetic algorithm optimized high-intensity-discharge lamp model. Electron. Lett. 38(3), 110–112 (2002)

7. Zissis, G., Damelincourt, J.J., Bezanahary, T.: Modelling discharge lamps for electronic circuit designers: A review of the existing methods. In: Proceedings of the 2001 Industrial Application Society (2001)

8. Anton, J.C.: An equivalent conductance model for high-intensity discharge lamps. In: IEEE Industry Applications Society-Annual Meeting, vol. 2, pp. 1494–1498 (2002)

9. Tseng, K.J.: Dynamic Model of Fluorescent Lamp Implemented in PSpice. In: Proceedings of Power Conversion Conference PCC 1997, vol. 2, pp. 859–864 (1997)