2.1 Gas discharge circuit model

The gas discharge lamp circuit model used in this paper is shown in Fig. 1, and the test voltage waveform is shown in Fig. 2.

2.2 Model circuit analysis using Laplace transform

In Fig. 3, the voltage can be assumed since $i_{L}(0)=0$, $i_{R}(0)=0$ and $i_{C}(0)=0$. Therefore, it has an the initial condition of $v_{2}(0)=0$, $v_{2}'(0)=0$.

Equation(1) is established by using the Laplace transform of Fig. 3 considering the initial valuesof $v_{2}$.

Equation(1) is transformed into equation(2).

The voltage source waveform in Fig. 2 is expressed as equation(3).⁽⁴⁾

The Laplace transform of one period of the voltage source waveform is obtained by equation(4).

Using equation(4), the Laplace transform of the voltage source waveform is represented by equation(5).

By substituting equation(5) into equation(2), it is summarized as equation.(6)

In order for $v_{2}$ to oscillate, equation(7) must have an imaginary solution, so the condition of equation(8) must be met.⁽⁵⁾

The solution of equation(7) is represented by equation.(9)

In the case of $s=\alpha +- j\beta$ ($\alpha$ and $\beta$ are real numbers, $\beta\ne 0$), $\alpha$ is calculated by equation(10), and $\beta$ is calculated by equation(11).

Using $\alpha$ and $\beta$, equation(11) becomes equation(12).

Equation(13) is used to analyze the damped oscillating waveform.

The inverse Laplace transform of equation(13) becomes equation.(14)

Where, $S_{a}=\dfrac{\alpha}{\alpha^{2}+\beta^{2}}$, $T_{a}=-\dfrac{\beta}{\alpha^{2}+\beta^{2}}$, $A=s H_{1}(s)|_{s=0}=LC$.

Equation(14) is transformed into equation(15).

Substituting equation(15) into equation(12) to obtain equation.(16)

In equation(16), the Laplace transform of the first half-period waveform is represented by equation.(17)

Equation(18) can be obtained by inverse Laplace transform of equation.(17)

In equation(18), $\alpha$ is the damping coefficient and $\beta$ is the angular frequency of the oscillation wave.

2.3 Calculationofimpedanceof discharge lamp

The impedance of the discharge lamp is calculated in the following order:

① voltage waveform measurement

② area calculation

③ calculate DC average value

④ calculate the AC value (waveform value minus DC average value)

⑤ absolute value calculation of AC value

⑥ trend line calculation

⑦ damping rate calculation

⑧ average period calculation of oscillation wave

⑨ calculation of effective resistance and effective capacitance

In this paper, a voltage v is 60kHz, 310V square wave, and inductance L is 20μH is applied for the circuit presented in Fig. 1.

Fig. 4 is a measurement of the current and voltage wave forms of the discharge lamp in the Fig. 1 circuit.

In Fig. 4, the time and voltage of the peaks and valleys are measured on the half cycle of the voltage waveform. The area of ​​the graph is obtained by using the trapezoidal formula, and the area is divided by time to obtain the average value of voltage.

Fig. 5 shows the peaks and valleys of the measured voltage waveform, and the average voltage value, 310V, of the measured voltage waveform. The calculated average value of the voltage should be the same as the voltage of the applied square wave as shown in equation.(18)

Fig. 6 shows the waveform with the average value (DC component) removed from the measured voltage waveform.

The waveform in Fig. 6 shows an exponential multiply sine wave function form ($v=A e^{\alpha t}\sin\omega t$). In order to obtain the attenuation coefficient α, the absolute value of the waveform in Fig. 6 is taken, and the trend line is calculated by using the first three values. Fig. 7 shows this process.⁽⁶⁾

It is $\alpha =-5.365\times 10^{5}$ calculated from the trend line.

In Fig. 6 the average period is calculated by using the time from the first peak to the second valley. The calculated average period is 1.37μs. From the above relation, it is calculated as $\beta =4.59\times 10^{6}$ by equation.(19)

From equation(10) and equation(11), the circuit capacitance C is calculated as equation(20), and the circuit resistance R is calculated as equation(21).

In the circuit used in this paper, the circuit inductance L is 20μH. Therefore the circuit capacitance C is calculated as 2.35nF and the circuit resistance R is calculated as 397Ω.

In order to verify the calculated resistance and capacitance values ​​of the discharge lamp, Fig. 1(b) circuit was simulated by using LTspice.

Fig. 9 shows the simulation result of Figure 8 circuit. It can be seen that the simulation waveform has a similar trend to the actual waveform in Fig. 4.

When the measured voltage waveform and the simulated voltage waveform are compared quantitatively, a difference between the two can be seen.