3.1. Load Matching

The Blumlein PFL consists of three coaxial cylindrical lines filled with de-ionized water as dielectric. The PFL formed by the inner conductor and the middle conductor are usually called the inner line. If the PFL is formed by the middle conductor and the outer conductor, it is called the outer line. The wave impedances of the inner line and outer line are Z ₁ and Z ₂, respectively. So during the Blumlein line discharge into the load, the equivalent circuit of the Blumlein line is shown in Figure 2. Where S is the main switch, CA is the inner line, BD is the outer line, and Z_L is the load. Also, the voltage measured by the resistive divider (A) in Figure 1 is just the output voltage of the HVPT. The voltage at point D in Figure 2 is also just the voltage at the output terminal of HVPT. In order to get the voltage at the output terminal of HVPT, the voltage at point D need to be solved, so a coordinate system is built in Figure 2; the main switch is the zero point. When PFL is charged to a voltage U(U < 0), once the main switch S is closed, the PFL can not stay in equilibrium since the PFL elementary capacitors adjacent to the load Z _L start discharging. So the voltages at the inner line and outer line satisfied the equations

Fig. 2.

(Color online) Equivalent circuit of the BPFL discharge into the load.

(1)$$\left\{\matrix{ - \displaystyle{{\partial u_1 \lpar x\comma \; t\rpar \over \partial x}} = L_i \displaystyle{{\partial i_1 \lpar x\comma \; t\rpar \over \partial t}} \cr - \displaystyle{{\partial i_1 \lpar x\comma \; t\rpar \over \partial x}} = C_i \displaystyle{{\partial u_1 \lpar x\comma \; t\rpar \over \partial t}} } \right. \;\left\{\matrix{ - \displaystyle{{\partial u_2 \lpar x\comma \; t\rpar \over \partial x}} = L_o \displaystyle{{\partial i_2 \lpar x\comma \; t\rpar \over \partial t}} \cr - \displaystyle{{\partial i_2 \lpar x\comma \; t\rpar \over \partial x}} = C_o \displaystyle{{\partial u_2 \lpar x\comma \; t\rpar \over \partial t}} } \right. \cdot$$

In the above equations, L _i, C _i, u ₁(x, t), i ₁(x, t) and L _o, C _o, u ₂(x, t), i ₂(x, t)are inductances of per unit, capacitances of per unit, voltages, and currents of the inner line and outer line, respectively. And the transmission time and impedance of inner and outer line is

(2)$$\left\{\matrix{{\rm \tau} _1 = L\sqrt {L_i C_i } \cr Z_1 = \sqrt {L_i /C_i } } \right. \;\left\{\matrix{{\rm \tau} _2 = L\sqrt {L_o C_o } \cr Z_2 = \sqrt {L_o /C_o } } \right.\; \cdot$$

One should note that if the middle conductor is replaced by a spiral strip middle cylinder, τ ₁ is usually not equal to τ ₂. Here, τ ₁ is usually equal to τ ₂, and it is assumed that

(3)$${\rm \tau} _1 = {\rm \tau} _2 \equiv {\rm \tau} \cdot$$

When t = 0, the initial conditions of Eq. (1) are

(4)$$\left\{\matrix{u_1 \lpar x\comma \; 0\rpar = u_2 \;\lpar x\comma \; 0\rpar = U \cr i_1 \lpar x\comma \; 0\rpar = i_2 \;\lpar x\comma \; 0\rpar = 0 } \right. \cdot$$

And for the characteristic of the PFL, the boundary conditions are

(5)$$\left\{\matrix{x = 0 \to u_i \lpar 0\comma \; t\rpar = 0 \cr x = L \to i_1 \lpar L\comma \; t\rpar = i_2 \lpar L\comma \; t\rpar = i_z \lpar l\comma \; t\rpar \cr \quad \quad \quad \quad u_1 \lpar L\comma \; t\rpar - u_2 \lpar L\comma \; t\rpar = i_z \lpar L\comma \; t\rpar Z_L \cr x = 2L \to i_2 \lpar 2L\comma \; t\rpar = 0 } \right. \cdot$$

So, the load voltage could be solved by Eqs. (1)–(5)

(6)$$\eqalign{U_{ZL}^1 \lpar t\rpar &= \displaystyle{{UZ_L } \over {Z_1 }}\lsqb \lpar {\rm \alpha} - 1\rpar l\lpar t - {\rm \tau} \rpar + \lpar {\rm \alpha} + {\rm \beta} - {\rm \alpha} k - {\rm \alpha} ^2 \rpar \cr &\quad \times l\lpar t - 3{\rm \tau} \rpar + \lpar {\rm \alpha} ^3 - {\rm \alpha} ^2 - {\rm \beta} k - {\rm \alpha} k - 2{\rm \alpha} {\rm \beta} \cr &\quad + {\rm \beta} + k^2 {\rm \alpha} + 2{\rm \alpha} ^2 k\rpar l\lpar t - 5{\rm \tau} \rpar \rsqb + {\rm O}_1 \lpar {\rm \tau} \rpar}$$

Where ${\rm \alpha} = \displaystyle{{Z_2 + Z_L - Z_1 } \over {Z_2 + Z_1 + Z_L }}\comma \; {\rm \beta} =\displaystyle{{Z_2 + Z_1 - Z_L } \over {Z_2 + Z_1 + Z_L }}\comma \; - k =$$ \displaystyle{{Z_1 + Z_L - Z_2 } \over {Z_2 + Z_1 + Z_L }}$, and O₁ (τ) stands for the remainder, when t < 7τ, O₁ (τ) ≡ 0. l(t) is the step function, and it is defined as

(7)$$l\lpar t\rpar = \left\{\matrix{1 &t \geq 0 \cr 0 &t \lt 0 } \right.$$

When the BPFL is designed as Z ₁ = Z ₂ and the load is matched with the BPFL (Z _L = Z _L + Z ₂), the waveform of load voltage could be obtained by Eq. (6), which is shown in Figure 3.

Fig. 3.

(Color online) Load voltage when the impedance of load is matched with BPFL (τ = 60 ns).

In Figure 3, it is obtained that when the impedance of load is matched with BPFL, the output voltage waveform of IEBA is a square wave, and the amplitude of output voltage is equal to that of charging voltage. Meanwhile, at this kind of condition, once the main switch is closed, according to Eqs. (1)–(5), the voltage at the output terminal of HVPT is

(8)$$U_C^1 \lpar t\rpar = U\lsqb 1 - l\lpar t - 2{\rm \tau} \rpar \rsqb.$$

Obviously, when the load is matched with the BPFL, once the main switch is closed, the the voltage at the output terminal of HVPT turns to zero.

3.2. Load Short

When the load is short, the impedance of load is very small, even zero. Obviously, the load voltage is zero. However, the voltage at the output terminal of HVPT should be recalculated. First, we discuss Z ₁ = Z ₂ and τ₁ = τ₁ = τ₂ ≡ τ. if Z _L ≠ Z ₁ + Z ₂, as the same method, from Eqs. (1)–(4) leads to (Cheng et al., Reference Cheng, Liu, Qian, Zhang and Zhang2009b)

(9)$$U_C^2 \lpar t\rpar = U\lcub 1 - {\rm \gamma} \sum\limits_{n = 0}^\infty {{\rm \zeta} ^n } l\lsqb t - \lpar 4n + 2\rpar {\rm \tau} \rsqb \rcub.$$

Where ${\rm \zeta} = \displaystyle{{Z_L - Z_1 - Z_2 } \over {Z_2 + Z_1 + Z_L }}, {\rm \gamma} = \displaystyle{{4Z_2 } \over {Z_1 + Z_2 + Z_L }}$. So the voltage at the output terminal of HVPT is a damped bipolar oscillation wave, and the peak voltage is determined by ζ and γ. When Z _L→0, γ→2, ζ→ − 1, the amplitude of oscillation is much bigger and the damping of the bipolar oscillating voltage is smaller. If Z _L = 0, this provide a short circuit for the Blumlein line, so ζ = −1, γ = 2, ζ = − 1, γ = 2, the voltage at the output terminal of HVPT is

(10)$$U_C^3 \lpar t\rpar = U\lcub 1 - 2\sum\limits_{n = 0}^\infty {\lpar - 1\rpar ^n } l\lsqb t - \lpar 4n + 2\rpar {\rm \tau} \rsqb \rcub.$$

Figure 4 is the waveform of the voltage at the output terminal of HVPT when the load is short. It clearly shows that the voltage at the output terminal of HVPT is a bipolar rectangular wave with a period of 8τ and duty ratio of 50%. It is obtained that the oscillation frequency is determined by the duration of the BPFL.

Fig. 4.

(Color online) The voltage at the output terminal of HVPT when Z ₁ = Z ₂, Z _L = 0.

So at Z _L → 0, there will be a bipolar oscillating voltage at the output terminal of HVPT, which is disadvantageous for the operation of the HVPT, because surface flashover occurred unexpectedly at low fields under the bipolar oscillating voltage, Miller et al. (Reference Miller, Prestwich, Poukey, Epstein, Freeman, Sharpe, Tucker and Shope1981) and Adler et al. (Reference Adler, Miller, Prestwich and Smith1983) found this phenomenon. Also the critical flashover voltage decreases with the increment of the angle frequency of the oscillating voltage, and for the same critical flashover voltage, the flashover distance increases with the increment of the applied voltage (Pai & Zhang, Reference Pai and Zhang1995). When the impedance of load becomes small or zero, this makes it easier for surface flashover along the insulation materials to form and decrease the lifetime of the HVPT. Hence, in order to avoid the surface flashover, the insulation materials of the HVPT need to be improved and the impedance of load should not be too small.

Also, if Z ₁ ≠ Z ₂, Z _L ≠ Z ₁ + Z ₂. The voltage at the output terminal of HVPT could be obtained from Eqs. (1)–(4)

(11)$$U_C^{\rm 4} \lpar t\rpar = U\lcub 1 - {\rm \gamma} \sum\limits_{n = 0}^{\infty} \sum\limits_{\,j = 0}^m \displaystyle{{\lpar n - j\rpar ! \over \,j!\lpar n - 2j\rpar !}} {\rm \xi}^{n - 2j} {\rm \zeta}^j\, {\bf \cdot} l\lsqb t - \lpar 2n + 2\rpar {\rm \tau} \rsqb \rcub.$$

In Eq. (11), m = n/2 for n being an even number and m = (n − 1)/2 for n being an odd number. ${\rm \xi} = \displaystyle{{2\lpar Z_1 - Z_2 \rpar } \over {Z_2 + Z_1 + Z_L }}$. Obviously, when Z _L → 0, the voltage at the output terminal of HVPT is still a damped bipolar oscillation wave, but there may be γ > 2, so the amplitude of oscillation is much bigger and the damping of the bipolar oscillating voltage is smaller.

3.3. Load Open

When the load is open, the impedance of load is infinite. In other words, the impedance of load is much larger than the impedance of BPFL. We just discussed the ideal equivalent circuit of the BPFL discharge into the load (see Fig. 3). Once the main switch is closed, the load voltage and the voltage at the output terminal of HVPT could be obtained from Eq. (6) and Eq. (8), respectively, when τ₁ = τ₂ ≡ τ, Z ₁ = Z ₂Z _L ≫ Z ₁ + Z ₂.

The ideal load voltage and the voltage at the output terminal of HVPT are shown in Figure 5, it is obtained that the voltage at the output terminal of HVPT is a constant valve, and which is just the charging voltage. Meanwhile, the load voltage is a periodic rectangular wave with a period of 4τ and duty ratio of 50%, and the amplitude of the load voltage is twice the charging voltage.

Fig. 5.

(Color online) Ideal load voltage and voltage at the output terminal of HVPT when τ₁ = τ₂ ≡ τ, Z ₁ = Z ₂Z _L ≫ Z ₁ + Z ₂.

However, when the load is open, grounding inductor and resistor of the resistive divider (B) in Figure 1 has a great effect on the load voltage and the voltage at the output terminal of HVPT. Once the main switch (ideal switch) is closed, the actual discharging circuit of IEBA is shown in Figure 6.

Fig. 6.

(Color online) After the main switch is closed, the discharging circuit of IEBA when the load is open.

As shown in Figure 6, C_P is the primary energy-storage capacitance, and R_ST and L_ST are the resistance and inductance of primary winding circuit, HVPT are modeled by XForm2 in the circuit, R_RD1 (tens of kilohm) is the resistance of resistive divider (A) in Figure 1, R_BMO (megohm level) is the bleeder resistance of outer line, L_MO (less than microhenry) and C_MO is the inductance and capacitance of outer line, respectively, L_D is the grounding inductor, C_IO and R_BIO is capacitance and bleeder resistance of the output pulse forming line, respectively, and C_IO is much less than C_MO, R_RD2 (about 2 kilohm) is the resistance of resistive divider (B) in Figure 1, Z_L is the load of IEBA, when the load is open, Z_L is about megohm level. So according to the circuit and value of the parameters, the discharging circuit could be simplified as Figure 7.

Fig. 7.

(Color online) Simplified discharging circuit of IEBA when the load is open.

In Figure 7, C _MO = τ₂/Z ₂ and considering that $\displaystyle{1 \over {2R_{RD2} C_{MO} }} \lt \displaystyle{1 \over {\sqrt {L_D C_{MO} } }}$, the voltage at the output terminal of HVPT is

(12)$$U_C^4 \lpar t\rpar = \displaystyle{U \over {\rm \omega} } \times e^{ - {\rm \delta} t} \lsqb {\rm \omega} Cos\lpar {\rm \omega} t\rpar - {\rm \delta} Sin\lpar {\rm \omega} t\rpar \rsqb.$$

Where ${\rm \omega} = \sqrt {{\rm \omega} _0^2 - {\rm \delta} ^2 } \comma \; {\rm \omega} _0 = \displaystyle{1 \over {\sqrt {L_D C_{MO} } }}\comma \; {\rm \delta} = \displaystyle{1 \over {2R_{RD2} C_{MO} }}$. In Eq. (12), it is obvious that the voltage at the output terminal of HVPT is not a constant value when the load is open. The voltage at the output terminal of HVPT is a damped bipolar oscillation wave, and the oscillation frequency is decided by the parameter of outer line, grounding inductor, and resistance of resistive divider (B). Then, submitting these parameters into Eq. (12), the voltage at the output terminal of HVPT and its oscillation frequency could be obtained. Of course, the load voltage will be changed with the voltage at the output terminal of HVPT, and the load voltage is

(13)$$U_{ZL}^2 \lpar t\rpar = R_{RD2} \times i_{R_{RD2} } \lpar t\rpar = L_D \displaystyle{{di_{L_D } \lpar t\rpar } \over {dt}}.$$

Where $i_{R_{RD2} } \lpar t\rpar$ is the current of resistive divider (B), $i_{L_D } \lpar t\rpar$ is the current of grounding inductor. Then to clearly shown the effect of the grounding inductor on the voltage at the output terminal of HVPT and load voltage, PSPICE circuit analysis was used to simulate the accelerator based on HVPT and BPFL when the load is open. Figure 8 shows schematically the full circuit model. The whole circuit is connected to components built in PSPICE component storage or modeled by the authors according to the performance of the device.

Fig. 8.

(Color online) PSPICE circuit schematic used to model the IEBA.

In Figure 8, T_mid_inner and T_mid_outer are connected in parallel to simulate the BPFL. And in the simulation, supposing τ₁ = τ ₂ ≡ 60 ns, Z ₁ = Z ₂ = 6 Ω and Z _L = 100 MΩ, The output voltage of the HVPT can be obtained at different grounding inductor. Figure 9 shows the simulation results.

Fig. 9.

(Color online) The voltage at the output terminal of HVPT for different grounding inductor when the load is open.

It is shown that the main switch is closed at time of 5 µs, and then the voltage at the output terminal of HVPT is a damped bipolar oscillation wave. At different grounding inductor, the oscillation wave is difference. Also, according to Figure 9, the oscillation frequency of the voltage at the output terminal of HVPT could be obtained by Fourier analysis. Meanwhile, submitting the parameters that are used in the PSPICE simulation to Eq. (12), the oscillation frequency also could be calculated theoretically.

In Table 1, as the increment of the inductance of the grounding inductor, the oscillation frequency is decreased. Also, the theoretical results are almost the same as that of PSPICE simulation. So, the grounding inductor has a great effect on the load voltage and the voltage at the output terminal of HVPT when the load is open. Obviously, once the main switch is closed, the load voltage is twice the charging voltage of BPFL. However, as the oscillation of the voltage at the output terminal of HVPT, the load voltage would be an oscillation waveform.

Table 1.

Theoretical calculated and PSPICE simulated oscillation frequency of the voltage at the output terminal of HVPT

Figure 10 shows the typical load voltage at load open, it clearly shows that once the main switch is closed, the amplitude of the load voltage is twice the charging voltage of BPFL, which is similar with that in Figure 5. Subsequently, the voltage at the output terminal of HVPT is decreased, which causes the amplitude of the load voltage to be reduced. The amplitude of the load voltage is changed with the oscillation of the voltage at the output terminal of HVPT.

Fig. 10.

(Color online) PSPICE simulated load voltage and voltage at the output terminal of HVPT for load open when L _D is 60 µH.

To further verify the correctness of the theoretically results, an experiment was performed on an IEBA based on spiral type Blumlein line, the transmission time and impedance of inner line are 70 ns and 6.4 Ω, the outer line are 76 ns and 4 Ω. The experimental waveform of load voltage and the voltage at the output terminal of HVPT are shown in Figure 11.

Fig. 11.

(Color online) Experimental load voltage and voltage at the output terminal of HVPT for load open when L _D is 60 µH.

In Figure 11, the voltage at the output terminal of HVPT is still an oscillation wave, which is similar with that in Figure 10. Also, the amplitude of the first main pulse of load voltage is about twice that of the voltage at the output terminal of HVPT. Meanwhile, the amplitude of the load voltage is also changed with the oscillation of the voltage at the output terminal of HVPT. However, there is a little difference on the characteristics of load voltage. Comparing with the waveform in Figure 10, the amplitude of load voltage in Figure 11 is damped quickly. In fact, the main switch is not an ideal switch, so during the experiments, the resistance of the switch channel can not be ignored, and energy may be dissipated in the main switch. Then, adding the resistance of the switch channel into Figure 8, the waveform of load voltage can be recalculated in Figure 12.

Fig. 12.

(Color online) PSPICE simulated load voltage at difference resistance of main switch when load open.

In Figure 12, it is obtained that the load voltage at difference resistance of switch channel is difference. For ideal condition, R _S = 0.001 Ω the amplitude of the load voltage is damped very slowly, which is similar with that in Figure 10. However, when the resistance of switch channel is up to 0.1 Ω, even to 0.5 Ω, the amplitude of the load voltage is damped quickly, and the waveform is similar with that in Figure 11.

One should note that the voltage at the output terminal of HVPT is oscillation waver when the load is short and open, but the oscillation frequency is difference. For example, the period of the oscillation wave is about 8τ (480 ns) and 5 µs when the load is short and open, respectively. Of course, the effect on the HVPT is also difference, and the output voltage has a greater effect on the HVPT when the load is short.

3.4. Surface Flashover in the Vacuum Chamber

Vacuum chamber is one of the important parts of IEBA, and it is used to place the load. However, if surface flashover occurs along the insulator that is used to support the inner cylinder of the BPFL, the operating characteristics of IEBA could be changed. So, the operating characteristics of the IEBA should be investigated when the surface flashover occurs in the vacuum chamber. In most conditions, the surface flashover does not occur, because of that the insulator is designed to avoid surface flashover occur. So to research the operating characteristics of IEBA when the surface flashover occurs, a sample should be used to simulate the surface flashover that occurs along the insulator, as shown in Figure 13.

Fig. 13.

(Color online) Schematic of the vacuum chamber.

In Figure 13, the material of the sample is high density polyethylene (HDPE), and the sample is place between electrodes. The gap spacing could be adjusted by changing the electrode and dimension of the sample. So the surface flashover could be controlled by the load voltage and gap spacing. Figure 14 shows the typical waveform of load voltage and surface flashover current at different gap spacing.

Fig. 14.

(Color online) Typical waveform of flashover at voltage of 180 kV, (a) No sample, (b) D = 100 mm, (c) D = 170 mm, (d) D = 210 mm.

In Figure 14a, it was shown that, before adding the sample, the load voltage was a typical quasi-square wave with voltage of 180 kV and FWHM of 180 ns, and the surface flashover current measured by a Rogowski coil in Figure 13 is zero. After adding the sample, surface flashover occurs, but the time at which the surface flashover occurs is not the same. If the pre-breakdown delay time is defined as the time from the load voltage rise to the load voltage drop due to surface breakdown (or flashover current starts) (see Fig. 14), it is obtained that the delay time is increased with the increment of the gap spacing. Also, one should be noted that the FWHM of the load voltage is changed, before adding the sample, the FWHM is about 180 ns. However, after adding the sample with gap spacing of 100 mm, the FWHM is only 98 ns (Fig. 14b) and the flashover current is very high, which is up to 34 kA. When the gap spacing is increased to 170 mm, the FWHM is also increased to 160 ns (Fig. 14c), and the flashover current is decreased to 21 kA. In other words, the surface flashover in the vacuum chamber will affect the FWHM of the output voltage. But if the surface flashover occurs after the main pulse of the output voltage, there is little effect on the FWHM of the output voltage, such as Figure 14d, the FWHM is also up to 170 ns, and the flashover current is further decreased to 10 kA.

According to the above, one conclusion could be obtained, if the flashover occurred at the flat part of the main pulse, the value of the peak flashover current was very high, and which will decrease the FWHM of the output voltage. However, if the flashover occurred after the main pulse, there was little effect on the output voltage of IEBA. So, during the operation of the IEBA, it must be avoided that the flashover occurred at the flat part of the main pulse.

When the surface flashover occurs, the impedance of the load is less than 1 Ω. So the characteristic of the voltage at the output terminal of HVPT is similar with that of load short. Once the surface flashover occurs in the vacuum chamber, the voltage at the output terminal of HVPT would be a damped bipolar oscillation wave. In Cheng et al. (Reference Cheng, Liu, Zhang, Hong and Qian2011), we have reported the results. In a word, the output voltage of IEBA and the voltage at the output terminal of HVPT are all affected by the surface flashover in the vacuum chamber, so which should be avoided during the operation of IEBA.