2.1 Gas discharge circuit model

The test capacitive discharge lamp circuit and the equivalent circuit used in this paper is shown in Fig. 1 where v is the voltage source, L is the circuit inductance, R is the lamp resistance, C is the lamp capacitance, and vL is the lamp voltage. The applied voltage waveform is shown in Fig. 2.

2.2 Calculation of lamp resistance and lamp capacitance^[7]

When the voltage of Fig. 2 is applied to the circuit of Fig. 1, equation(1) is established by using Laplace transform.

The attenuation constant $\alpha$ and the angular frequency $\beta$ of the oscillation wave are expressed by the following formula.

From equation(2) and equation(3), the lamp capacitance C and the lamp resistance R are expressed an equation(4) and equation(5), respectively.

In this paper, the frequency and the magnitude(Vo) of the applied rectangular wave voltage(v) are 60kHz and 310V, respectively, with an inductance(L) of 20μH. The lamp size is 1800×70×15mm.

Lamp voltage(vL) and current are measured using an oscilloscope(Tektronix DPO4054). The measured current and voltage waveforms of the discharge lamp in the Fig. 1 are shown in Fig. 3.

2.3 Calculation of dynamic lamp impedance

The dynamic lamp impedance is calculated in the following order: ^[7]

(1) voltage waveform measurement

(2) area calculation

(3) DC average value calculation

(4) AC value calculation (waveform value minus DC average value)

(5) absolute value calculation of AC value

Fig. 4 shows the result of applying the above process to the half-cycle of the voltage waveform in Fig. 3.

Using the data of adjacent points the dynamic impedance of the numbered points is calculated. The value of point n is calculated by using points (n-2), n, and (n+2). Using this method, the range of n for which values can be calculated is 3 to 9.

Repeat the following operations at each point to obtain dynamic impedance^[7].

(6) trend line calculation

(7) damping rate calculation

(8) period calculation of oscillation wave

(9) calculation of dynamic resistance and capacitance

The calculated results are shown in Table 1.

The results of Table 1 are displayed in Fig. 5. The lamp resistance gradually increases up to point 8 and then decreases rapidly, while the lamp capacitance is almost constant up to point 6, and then decrease gradually.

In order to verify the calculated dynamic lamp resistance and capacitance, voltage–current phase difference of the measured waveform and impedance angle are compared.

From the measured lamp voltage and current in Fig. 3, half-cycle waveforms are displayed in Fig. 6. The measured phase difference is calculated using equation (6) in Fig. 6.

Where $\theta_{m}$ is the phase difference at point n, $t_{n'}$ is the time of current waveform at point n’, and $t_{n}$ is the time of voltage waveform at point n’.

The impedance angle $\theta_{c}$ can be calculated by equation (7) using $\beta$, resistance R, and capacitance C in Table 1.

The results of Table 2 are shown in Fig. 7. The absolute value of the measured phase difference appears larger than the absolute value of calculated impedance angle. Interestingly the value of the measured phase difference even exceeds -90 degrees. The reason is that the calculated impedance angle is a value from one time point, while the phase difference is obtained from the measured graph in which the period becomes shorter over time, so the time difference between the peak and valley voltage and current becomes relatively large compared to period. The trend lines of impedance angle and phase difference show a similar trend.