2.1. Circuit Model of LTD-Based Z-Pinch Accelerators

Figure 1 shows a general schematic of an LTD-based accelerator. The structure in this figure is a four-level transmission line system, other options are available in the model like two or six levels. The electrical pulse is primarily generated by the multi-LTD modules, which are piled up in one to three levels, each LTD module drives an internal water-insulated coaxial transmission line. The impedance profile of the internal transmission line should be matched to the LTD module to maximize the peak forward-going power. In Figure 1, the multi-LTD modules are piled up in two levels and each level is considered identical.

Figure 1: General schematic of the circuit model. The structure in this figure is a four-level transmission line system, other options are available in the model like two or six levels. The center vacuum section from A–D level is simulated by lumped inductances, L _MITL and L _MITLg denote the inductances of the constant impedance section of MITL and constant gap section of MITL, respectively.

The equivalent resistance R _s, inductance L _s, and capacitance C _s of the whole prime power source are determined by (1)–(3), where n _b is the number of LTD bricks within a single LTD cavity, n _c is the number of LTD cavities connected in series within each LTD module, and n _m is the number of LTD modules in parallel. And R _b, L _b, and C _b are the equivalent resistance, inductance, and capacitance of a single LTD brick, respectively. The shunt resistances (4) are also included in the model to account for the ferromagnetic core losses, it is assumed as a constant resistance R _c (∼45 Ω) within each cavity, whose value is much greater than the impedance of a single LTD cavity.

(1) $\begin{array}{}{R}_s=\frac{n_c}{n_b{n}_m}{R}_b,\end{array}$

(2) $\begin{array}{}{L}_s=\frac{n_c}{n_b{n}_m}{L}_b,\end{array}$

(3) $\begin{array}{}{C}_s=\frac{n_b{n}_m}{n_c}{C}_b,\end{array}$

(4) $\begin{array}{}{R}_{\text{shunt}}=\frac{n_c}{n_m}{R}_c.\end{array}$

At the output of the LTD module, each LTD module drives a water-insulated coaxial transmission line, and the coaxial line could be a uniform line or an impedance transformer. Then, these coaxial transmission lines merge into monolithic radial transmission lines (MRTLs). The monolithic radial transmission lines are impedance transformers, which drive the insulator stacks and magnetically insulated transmission lines (MITLs), and ultimately, the physics load at the center of the accelerator.

The input impedance of the coaxial water transmission line is equivalent to the output impedance of the whole LTD system, which is expressed as [Reference Stygar, Cuneo and Headley9].

(5) $\begin{array}{}{Z}_{\text{coax,input}}=Z_{\text{LTD}}=1.1\sqrt{\frac{Ls}{Cs}}+0.8R_s.\end{array}$

It is assumed here that each level of MRTLs is identical. The resistivity ρ _w of the water used to insulate the MRTLs is 3 × 10⁴ Ω·m. And the influence of electrode resistance loss is neglected as it is very slight. The output impedance Z _out of the MRTLs is to be calculated iteratively in the model, and is constrained by several limitations:

1. (1) It is usually required that Z _out should not be smaller than Z _in.

2. (2) The electric field is the highest near the output of the impedance transformer. The minimum AK gap at the outlet of the MRTL should be satisfied with the water insulation criterion [Reference Stygar, Wagoner and Ives17]:

   (6) $\begin{array}{}{E}_{\text{max}}{\tau }_{\text{eff}}^{0.33}=\frac{U_{\text{out}}}{h_{\text{out}}}{\tau }_{\text{eff}}^{0.33}\le 0.113,\end{array}$

   where E _max is the average peak electric field at the output of the MRTL in MV/cm, τ _eff is the temporal width of the voltage pulse at 63% of peak value, which is in μs, U _out and h _out are the MRTL peak output voltage and the AK gap at the output, respectively.

3. (3) The plastic-vacuum interface has the lowest limit on the electric field in the pulsed-power system. Therefore, the AK gap at the output of the MRTL should not exceed the height of the insulator stack.

Since the MRTL is usually a nonuniform transmission line, it can be considered as a cascaded multiple segment line, and each segment line has the same electric length but a different impedance. There is a short water-insulated transmission line (water flare) between the MRTL outlet and the insulator stack at each level, which is also simulated by several transmission line elements.

The breakdown limit of the vacuum interface of the insulator stack should be considered in the model. The number and height of insulators and grading rings are conservative in the design to prevent insulator flashover. In the numerical calculations discussed later, with the initial guess of stack voltage and the given heights of the insulator rings and grading rings, the number of them will be determined iteratively to achieve relatively low inductance at this portion, which also determines the height of the insulator stack.

In order to increase the calculation efficiency, the insulator stacks, the vacuum flares, the magnetically insulated transmission lines, the post-hole convolute, and the inner MITL are simulated as several lumped inductances, as demonstrated in Figure 1. An inductance calculation method is embedded in the circuit model, the inductance of each portion can be obtained through iterative calculations with initial parameters generated by upstream circuit elements. This calculation method has been examined and validated by comparing the results from Z, PTS, and Z-300/Z-800 [Reference Gong, Wei, Fan, Sun and Qiu18]. The physics load considered in our simulation is a tungsten wire array for Z-pinch, which is simulated as a time-dependent inductance and a constant pinch resistance. The plasma implosion of the Z-pinch load is simulated by the 0-D model, and the pinch resistance is assumed constant as 5 mΩ/cm [Reference Stygar, Gerdin and Fehl19].

A shunt resistor R _loss is set between the post-hole convolute and the inner MITL to model the electron flow current lost to the anode of the convolute and inner MITL [Reference Jennings, Chittenden and Cuneo20]. R _loss is controlled by a “Z-flow” element, whose value is determined by the vacuum impedances of outer MITLs and the gap closure effects are included.

(7) $\begin{array}{}{R}_{\text{loss}}=Z_{\text{flow}}\sqrt{\frac{\left(I_a+I_c\right)}{\left(I_a-I_c\right)}}.\end{array}$

2.2. Optimization Method

PSOGSA is a hybrid optimization algorithm proposed by Mirjalili and Hashim [Reference Mirjalili and Hashim16], and it balances the advantages of PSO (particle swarm optimization) [Reference Kennedy and Eberhart21, Reference Shi and Eberhart22] and GSA (gravitational search algorithm) [Reference Rashedi, Nezamabadi-pour and Saryazdi23]. In our case, it is expected that with a given peak load current and assumed load implosion time, the algorithm could search for a minimum number of total LTD cavities, which implies how much energy needs to be stored primarily at least by the accelerator’s capacitors to meet the requirements of the experiment, which could make the entire design less expensive and more efficient. Therefore, the fitness function in the optimization is

(8) $\begin{array}{}\text{fitness}=n_c{n}_m.\end{array}$

The basic parameters of the circuit model consist of three parts: LTD prime power source parameters, transmission line system parameters, which include water transmission lines and magnetically insulated transmission lines, and load parameters, as listed in Table 1. These values need to be settled prior to the start of the numerical simulation. Note D _cav, D _inner, and l _cav are the outer diameter and the inner diameter (anode diameter) and length of a single LTD cavity, respectively, Vb is the LTD charge voltage across a single brick, n is the level of the transmission line system, h _p and h _r are the height of a single insulator ring and a grading ring, E _ave is the maximum mean electric field allowed at the insulator stack, Z _MITLs are the vacuum impedances of outer MITLs, and θ ₁ is the angle of the upper cathode of MITL. For load parameters, CR denotes the convergence ratio of the imploding load. With the desired peak load current I _p and load implosion time τ _i , the required total initial mass of the load can be estimated by the scaling relation [Reference Stygar, Ives, Fehl, Cuneo, Mazarakis, Bailey, Bennett, Bliss, Chandler, Leeper, Matzen, McDaniel, McGurn, McKenney, Mix, Muron, Porter, Ramirez, Ruggles, Seamen, Simpson, Speas, Spielman, Struve, Torres, Vesey, Wagoner, Gilliland, Horry, Jobe, Lazier, Mills, Mulville, Pyle, Romero, Seamen and Smelser24, Reference Stygar, Cuneo and Vesey25]:

(9) $\begin{array}{}{m}_{\text{load}}\propto \frac{I_p^2{\tau }_i^2{h}_{\text{load}}}{r_0^2},\end{array}$

where h _load and r ₀ are the height and initial radius of the Z-pinch load.

Table 1: List of parameters.

There are four major parameters that need to be determined in the circuit simulation with PSOGSA: the number of LTD cavities connected in series n _c, the number of LTD modules in parallel n _m, the electrical length of water-insulated monolithic radial transmission lines T _line, and the radius of the stack-MITL system r _waterflare as listed in Table 2. Their values should be limited to a reasonable range. The upper limit of n _c is totally constrained by the LTD cavity parameters mentioned above and the water insulation criterion:

$\begin{array}{}\text{(10a)}& d=0.5\left(D_{\text{inner}}-D_{\text{cathode}}\right)\times 100,\end{array}$

$\begin{array}{}\text{(10b)}& n_c{Z}_{\text{cav}}=\frac{60}{\sqrt{{\epsilon }_r}}\ln \left(\frac{D_{\text{inner}}}{D_{\text{cathode}}}\right),\end{array}$

$\begin{array}{}\text{(10c)}& \frac{n_c{U}_p}{d}{\tau }_{\text{LTD}}^{0.33}\le 0.113,\end{array}$

where U _p is the peak output voltage (in MV) of a single LTD cavity when connected to a matched load, d is the AK gap in cm,Z _cav is the impedance of a single LTD cavity, D _cathode is the cathode diameter of the LTD module at the outlet, and τ _LTD.is the effective pulse width (in μs) of the voltage pulse at 63% of peak when the single LTD cavity is terminated to a matched load.

Table 2: Parameters that need to be determined from optimization method.

The lower limit of T _line is determined by the number of LTD modules in parallel per level, the electrical length of coaxial transmission line Tcoax, and the outer diameter of the LTD cavity and the stack-MITL system (Note level n= 2, 4, 6):

(11) $\begin{array}{}{T}_{\text{line}}\ge \frac{\left(n_m{D}_{\text{cav}}/\pi n\right)-r_{\text{waterflare}}}{c}\sqrt{{\epsilon }_r}-T_{\text{coax}}.\end{array}$

The full calculation flow chart is illustrated in Figure 2. At the beginning of the calculation, three parts of the basic parameters and the desired peak load current, and load implosion time are inserted into the algorithm, then PSOGSA starts to initiate many sets of values of n _c, n _m, T _line, and r _waterflare within the reasonable range, which are considered as “ particle positions” in PSOGSA. With previous parameters, two initial voltage values U(0) stack, U(0) out and an initial estimated effective pulse width τ(0) eff for the subsequent iterative calculations are generated. U(0) stack includes the initial guesses of peak voltage at the stack at each level. U(0) out is the initial guess of peak voltage at the MRTL outlet (input of the water flare) at the lowest level. τ(0) eff is the effective pulse width of the voltage pulse at 63% of the peak at the MRTL outlet. U(0) stack, U(0) out, and τ(0) eff can be estimated by the following scaling relations:

(12) $\begin{array}{c}{U}_{\text{stack}}^{\left(0\right)}\propto {\left(\frac{I_p}{{\tau }_i}\right)}^{\frac{5}{3}},U_{\text{out}}^{\left(0\right)}\propto n_c{V}_b,\\ {\tau }_{\text{eff}}^{\left(0\right)}\propto \sqrt{L_b{C}_b}.\end{array}$

Figure 2: Full calculation flow chart.

U(0) out and τ(0) eff determine the minimum AK gap at the MRTL outlet with (6), as the peak voltage at the MRTL outlet at the lowest level is the largest among all levels, this AK gap in other levels will also meet the requirement given by (6), then MRTL output impedance Z _out is obtained with this AK gap and r _waterflare by:

(13) $\begin{array}{}{Z}_{\text{out}}=\frac{60h_{\text{out}}}{n\times \sqrt{{\epsilon }_r}\times r_{\text{waterflare}}}.\end{array}$

The value of U(0) stack, r _waterflare, and E _ave are inserted into the inductance calculation method mentioned above, the number of insulators and grading rings are determined by U(0) stack, E _ave, and h _p. The height of the insulator stack for each level is therefore obtained. Then, the geometry and circuit elements for water flares can be determined to connect the MRTL and the stack. Then with Z _MITLs, θ ₁, r _waterflare, the height of the stacks, preset vacuum flare radial distance, minimum AK gaps allowed in MITLs, and other basic settled parameters, the geometry of each MITL level and the inductance of each portion in the vacuum stack-MITL system are distinguished and calculated [Reference Gong, Wei, Fan, Sun and Qiu18].

At this point, all circuit elements needed in the model are completed. The circuit model is run for the first time and generates three new values as U(1) stack, U(1) out, and τ(1) eff. The loop continues until the differences between these values obtained from two adjacent iteration loops are within the limit of error (<0.05 MV for voltages, <5 ns for pulse width), which means these three values converge. After all of the candidate positions are calculated in the model, the circuit simulation is finished in the first big iteration of PSOGSA. The current best position and fitness are found. The PSOGSA algorithm will continue to calculate the particle masses, gravitational forces, particle accelerations, and update sets of velocities and positions to search for better solutions until the fitness value remains unchanged.

4.1. Influence of Maximum Allowed Mean Electric Field

In this section, the influence of the maximum mean electric field E _ave allowed at the insulator stack to the required minimum initial energy storage is investigated. The equivalent resistance R _b, inductance L _b, and capacitance C _b of a single LTD brick are 0.3 Ω, 200 nH, and 50 nF, respectively. Each LTD cavity is a right-circular annulus with an outer diameter of 2.3 m, an inner diameter of 0.91 m, and a length of 0.22 m, which contains 23 LTD bricks. At the output of each LTD module, a 90 ns uniform coaxial water-insulated transmission line is connected to provide 180 ns transit-time isolation between adjacent LTD modules. The height of insulator rings and grading rings are 5 and 0.8 cm, respectively. A four-level transmission line system is adopted, and the vacuum impedances of four MITLs are 2, 2, 3, and 3 Ω, and θ ₁ = 15°. The Z-pinch load is a wire array with an initial radius of 2 cm and a length of 2 cm, the initial inductance of the load is 1.1 nH, and the convergence ratio is assumed as 10:1. The target peak load current is 30 MA and the implosion time is 157 ns, hence the total initial load mass is chosen as ∼20.7 mg with (9). Figure 3 shows the minimum LTD cavity number versus the maximum allowed mean electric field E _ave under different charge voltages. It is seen that the total number of cavities decreases with the increase in E _ave, and it is obvious that the number of cavities can be greatly reduced with larger charge voltages. As E _ave increases from 100 to 200 kV/cm, the number of insulators needed decreases with the increase of E _ave, the total initial inductance L _center of the stack-MITL system is consequently reduced from ∼14.4 to ∼11.6 nH, the LTD cavity number is reduced by ∼14%. Note L _center also includes the inductance of the water flares. In these sets of simulations, the MRTL electrical length and radius of the stack-MITL system are free values within reasonable ranges as illustrated in Table 2. Their influences are further studied in Section 4.2 and Section 4.3.

Figure 3: Minimum LTD cavity number versus the maximum allowed mean electric field E _ave under different charge voltages.

4.2. Influence of Electric Length of MRTL

In this section, the charge voltage is fixed to ±100 kV, then the upper limit of LTD cavities connected in series should not exceed 29 according to (10a)–(10c). The maximum mean electric field E _ave allowed at the insulator stack is set to 100 kV/cm, other parameters remain the same as in Section 4.1. Table 4 shows the influence of MRTL electrical length when r _waterflare is fixed to 2 m. The minimum number of LTD cavities decreases with the increases in T _line because the power transport efficiency of MRTL increases with the increase of T _line. But the electrical length should not be lengthened too much, because the influence of water dielectric losses will be more remarkable when T _line is getting longer, and the improvement to power transport efficiency will be inconspicuous. The longer MRTL also requires a larger diameter of the entire accelerator and increases the cost.

Table 4: Minimum LTD cavity number varies with different MRTL electrical lengths.

4.3. Influence of the Outer Radius of Stack-MITL System

The individual influence of the outer radius of the stack-MITL system is investigated when the electrical length of MRTL is fixed to 180 ns, as shown in Table 5 (Note Eave is still 100 kV/cm). It is seen that the minimum number of cavities decreases dramatically first in the given range of r _waterflare and then becomes insensitive to r _waterflare. Because the total initial center inductance L _center decreases at the beginning as the stack inductance is reduced with the increase in r _waterflare, and the MRTL output impedance keeps decreasing with the increase in r _waterflare, which improves the impedance matching in the MRTL (the ratio of Z _out/Z _in decreases) and increases the power transport efficiency. But when r _waterflare is large enough, the center inductance will increase slowly with the same variation trend of peak stack voltage (Table 5 shows the peak stack voltage at the lowest level), because the increase of inductance in outer MITLs becomes more significant, which limits the performance of the whole system.

Table 5: Minimum LTD cavity number varies with different radius of the stack-MITL system.

4.4. Influence of the Inner Diameter of the LTD Cavity

In this section, the influence of the inner diameter (anode diameter) of the LTD cavity is investigated. The number of LTD bricks per cavity (n _b) is different with different cavity inner diameters. T _line and r _waterflare are free values within reasonable ranges as illustrated by Table 2. The charge voltage is fixed to ±100 kV, as the D _inner increases from 0.8 to 1.1 m, the upper limit of n _c increases from 23 to 40 according to (10a)–(10c), which makes it possible for more cavities connected in series within each module. The minimum number of LTD cavities versus target peak load currents with different D _inner is shown in Figure 4. The total number of cavities decreases with the increase in D _inner, as more LTD cavities can be connected in series within each module and fewer modules are arranged in parallel, the input impedance of MRTL increases, which also improves the impedance matching in the MRTL. And with a higher target peak load current, the reduction of cavity number is more remarkable with the increase in D _inner. For example, the cavity number is decreased by 950 (total LTD bricks are reduced by 9462) if the inner diameter of the cavity increases from 0.8 to 1.1 m for a 40 MA target peak current, as detailed in Table 6.

Figure 4: Minimum LTD cavity number versus peak load currents with different inner diameters of the LTD cavity.

Table 6: Simulation results with different inner diameters of the LTD cavity for a 40 MA target peak load current.

The methods mentioned above can be used to optimize the Z-300/Z-800 design. First, when the electrical length of MRTL and the radius of the stack-MITL system are free values, simulation results will be better than in Table 3, as indicated by the second and fourth row in Table 7. If the inner diameter increases from 0.8 to 0.9 m for Z-300 (number of bricks per cavity remains unchanged), the upper limit of n _c becomes 51, the cavity number is further reduced to 2565, as indicated by the third row. If the inner diameter increases from 1.3 to 1.4 m for Z-800, the upper limit of n _c becomes 84, the cavity number is reduced to 4686, as indicated by the fifth row.

Table 7: Simulation results for Z-300 and Z-800 conceptual designs with more free variables and a larger inner diameter of cavity.