Impedance-matched Marx generators

We have conceived a new class of prime-power sources for pulsed-power accelerators: impedance-matched Marx generators (IMGs). The fundamental building block of an IMG is a brick, which consists of two capacitors connected electrically in series with a single switch. An IMG comprises a single stage or several stages distributed axially and connected in series. Each stage is powered by a single brick or several bricks distributed azimuthally within the stage and connected in parallel. The stages of a multistage IMG drive an impedance-matched coaxial transmission line with a conical center conductor. When the stages are triggered sequentially to launch a coherent traveling wave along the coaxial line, the IMG achieves electromagnetic-power amplification by triggered emission of radiation. Hence a multistage IMG is a pulsed-power analogue of a laser. To illustrate the IMG approach to prime power, we have developed conceptual designs of two ten-stage IMGs with $LC$ time constants on the order of 100 ns. One design includes 20 bricks per stage, and delivers a peak electrical power of 1.05 TW to a matched-impedance $1.22\text{−}\Omega$ load. The design generates 113 kV per stage and has a maximum energy efficiency of 89%. The other design includes a single brick per stage, delivers 68 GW to a matched-impedance $19\text{−}\Omega$ load, generates 113 kV per stage, and has a maximum energy efficiency of 90%. For a given electrical-power-output time history, an IMG is less expensive and slightly more efficient than a linear transformer driver, since an IMG does not use ferromagnetic cores.

Figure 1

Cross-sectional view of a ten-cavity linear-transformer-driver (LTD) module. The module is powered by 200 bricks altogether; only 20 are illustrated here.

Figure 2

Cross-sectional view of two LTD cavities. The red arrows indicate the path of parasitic current that flows along the inner surfaces of the walls of each cavity. This current creates a magnetic field within each cavity that diffuses into the ferromagnetic cores and changes their magnetization state. Hence energy is lost to the cores through magnetic-energy diffusion and Ohmic heating. Since the cores absorb energy from the parasitic current, the cores introduce an effective resistance in the path of this current, which reduces its magnitude.

Figure 3

(a) Circuit model of a ten-cavity LTD module. The module drives a ten-segment impedance-matched transmission line that in turn drives an impedance-matched load. The quantity $R_c$ is the effective series resistance of a single LTD cavity, $L_c$ is the effective series cavity inductance, $C_c$ is the series cavity capacitance, $R_{\text{cores}}$ is the effective parallel resistance within a single cavity due to its ferromagnetic cores, $Z$ is the impedance of the first transmission-line segment, $Z_{\mathrm{load}}$ is the load impedance, $\tau$ is the one-way electromagnetic transit time of a single segment, $R_b$ is the effective series resistance of a single brick, $n_b$ is the number of bricks per cavity, $L_b$ is the effective series brick inductance, and $C_b$ is the series brick capacitance. The quantity $Z_c$ is the effective impedance of a single cavity; the expression given by the figure for $Z_c$ is valid when $R_{\text{cores}}\gg [1.1{(L_c/C_c)}^{1/2}+0.8R_c]$ [7]. (b) When each cavity is triggered at time $\tau$ after the cavity immediately upstream is triggered, the circuit of (a) can be simplified as that given by (b). The quantity $R_{\mathrm{LTD}}$ is the effective series resistance of the LTD module, $L_{\mathrm{LTD}}$ is the effective series module inductance, $C_{\mathrm{LTD}}$ is the series module capacitance, and $R_{\text{shunt}}$ is the effective shunt resistance of the module due to its cores. The quantity $Z_{\mathrm{LTD}}$ is the effective impedance of the ten-cavity LTD module; the expression given by the figure for $Z_{\mathrm{LTD}}$ is valid when $R_{\text{shunt}}\gg [1.1{(L_{\text{LTD}}/C_{\text{LTD}})}^{1/2}+0.8R_{\text{LTD}}]$ [7]. When circuit (b) is applicable and the values of $Z$ and $Z_{\mathrm{load}}$ are as given by the figure, the LTD module is impedance matched to the load; i.e., the peak electrical power delivered by the module to the load is maximized.

Figure 4

Cross-sectional view of a ten-stage impedance-matched Marx generator (IMG) with 20 bricks per stage. The IMG is powered by 200 bricks altogether; only 20 are illustrated here.

Figure 5

(a) Circuit model of a ten-stage IMG. The IMG drives a ten-segment impedance-matched transmission line that in turn drives an impedance-matched load. The quantity $R_s$ is the effective series resistance of a single IMG stage, $L_s$ is the effective series stage inductance, $C_s$ is the series stage capacitance, $Z$ is the impedance of the first transmission-line segment, $Z_{\mathrm{load}}$ is the load impedance, $\tau$ is the one-way electromagnetic transit time of a single segment, $R_b$ is the effective series resistance of a single brick, $n_b$ is the number of bricks per stage, $L_b$ is the effective series brick inductance, $C_b$ is the series brick capacitance, and $Z_s$ is the effective impedance of a single stage. (b) When each stage is triggered at time $\tau$ after the stage immediately upstream is triggered, the circuit of (a) can be simplified as that given by (b). The quantity $R_{\mathrm{IMG}}$ is the effective series resistance of the IMG, $L_{\mathrm{IMG}}$ is the effective series IMG inductance, $C_{\mathrm{IMG}}$ is the series IMG capacitance, and $Z_{\mathrm{IMG}}$ is the effective impedance of the ten-stage IMG. When circuit (b) is applicable and the values of $Z$ and $Z_{\mathrm{load}}$ are as given by the figure, the IMG is impedance matched to the load; i.e., the peak electrical power delivered by the IMG to the load is maximized.

Figure 6

Simulated voltage time histories at the outputs of the ten transmission-line segments of the ten-stage IMG illustrated by Fig. 4, as given by the circuit model of Fig. 5. Since $10Z=Z_{\mathrm{load}}$, the voltage at the output of the tenth stage is identical to that at the load.

Figure 7

Simulated electrical-power time histories at the outputs of the ten transmission-line segments of the ten-stage IMG illustrated by Fig. 4, as given by the circuit model of Fig. 5. Since $10Z=Z_{\mathrm{load}}$, the power at the output of the tenth stage is identical to that at the load.

Figure 8

Simulated electrical-power time histories generated by one of the stages of the IMG illustrated by Fig. 4, and the ten-stage IMG. The power generated by a single stage is that which would be delivered by the stage to a matched-impedance load; i.e., a load with impedance $Z_s$. The power generated by the ten-stage IMG is that which would be delivered by the IMG to a matched-impedance load; i.e., a load with impedance $Z_{\mathrm{IMG}}$.

Figure 9

Simulated electrical-power time histories generated by one of the bricks of the IMG illustrated by Fig. 4, and the 200-brick IMG. The power generated by a single brick is that which would be delivered by the brick to a matched-impedance load; i.e., a load with impedance $Z_b$. The power generated by the 200-brick IMG is that which would be delivered by the IMG to a matched-impedance load; i.e., a load with impedance $Z_{\mathrm{IMG}}$.

Figure 10

Simulated electrical-power time histories generated by the ten-stage IMG illustrated by Fig. 4. The red trace assumes the IMG’s impedance-matched transmission line has a stepped impedance profile, as suggested by Fig. 5. The blue trace assumes a piecewise-linear impedance profile, as suggested by the conical center conductor illustrated by Fig. 4.

Figure 11

Cross-sectional view of a ten-stage impedance-matched Marx generator (IMG) with a single brick per stage.

Figure 12

Simulated voltage time histories at the outputs of the ten transmission-line segments of the ten-stage IMG illustrated by Fig. 11, as given by the circuit model of Fig. 5. Since $10Z=Z_{\mathrm{load}}$, the voltage at the output of the tenth stage is identical to that at the load.

Figure 13

Simulated electrical-power time histories at the outputs of the ten transmission-line segments of the ten-stage IMG illustrated by Fig. 11, as given by the circuit model of Fig. 5. Since $10Z=Z_{\mathrm{load}}$, the power at the output of the tenth stage is identical to that at the load.

Figure 14

(a) Circuit model of a generalized three-stage impedance-matched Marx generator (IMG). The quantity $R_s$ is the effective series resistance of a single IMG stage; $L_s$ is the effective series stage inductance; $C_s$ is the series stage capacitance; $Z$ is the impedance of each of the vertical transmission-line segments driven by the IMG, as well as that of the first horizontal segment; ${\tau }_s$ is the one-way electromagnetic transit time of a vertical segment; $\tau$ is the one-way electromagnetic transit time of a horizontal segment; and $V$ is the time-dependent forward-going voltage that is launched by each stage. (b) When each stage is triggered at time $\tau$ after the stage immediately upstream is triggered, the circuit of (a) can be simplified as that given by (b). The quantity $R_{\mathrm{IMG}}$ is the effective series resistance of the IMG, $L_{\mathrm{IMG}}$ is the effective series IMG inductance, $C_{\mathrm{IMG}}$ is the series IMG capacitance, and $Z_{\mathrm{load}}$ is the load impedance. The figure makes clear that all the reflected and backward voltages within a generalized multistage IMG cancel.