In fast Z-pinches, rise time of drive current plays an important role in development of magneto-Rayleigh-Taylor(MRT) instabilities. It is essential for applications of Z-pinch dynamic hohlraum (ZPDH), which could be used for drivinginertial confinement fusion (ICF), to understand the scaling of rise time on MRTs. Therefore, a theoretical model for nonlinear development of MRTs is developed according to the numerical analysis. It is found from the model that the implosion distance L = r 0 – r mc determines the development of MRTs, where r 0 is the initial radius and r mc is the position of the accelerating shell. The current rise time τ would affect the MRT development because of its strong coupling with the r 0. The amplitude of MRTs would increase with the rise time linearly if an implosion velocity is specified. The effects of the rise time on MRT, in addition, are studied by numerical simulation. The results are consistent with those of the theoretical model very well. Finally, the scaling of the rise time on amplitude of MRTs is obtained for a specified implosion velocity by the theoretical model and numerical simulations.

The Z-pinch dynamic Hohlraum (ZPDH) is a promising indirect-drive approach for inertial confinement fusion. The volume ignition capsule is more robust than the hot-spot ignition capsule for ZPDH due to the fact that the ZPDH radiation drive source has a high energy but low symmetry. Focusing on the ignition design of cryogenic double-shell volume ignition capsules using ZPDH radiation sources, three analytical physical models, including the ablation and implosion model, the shell collision model, and the burn fraction model, are established to quantitatively characterize the relation of capsule parameters. Robust capsule designs are then determined based on these analytical models together with 1D radiation hydrodynamics simulations. The results show that under the 10 ns, 308 eV radiation drive source produced by ZPDH with 50 MA load current, capsules with a large range of parameters can ignite. The fusion yield of the recommended capsule is 16.0 MJ, and the absorbed energy is 1.28 MJ.

In this paper, physical issues of Z-pinch dynamic Hohlraums aimed at ignition are numerically investigated. Three-wave propagation, including the thermal wave, the ablation shock driven by radiation emitted by the nested tungsten wire-array plasma, and the main shock, is found to determine the Hohlraum formation at high currents. Based on requirements of high temperature radiation, three-wave isolation, and a suitable Hohlraum-capsule size ratio, a converter with an initial radius of 5 mm is suggested. As the rise time of the drive current is varied, two kinds of Hohlraum designs are examined. One is to fix the wire-array mass and vary the wire-array radius; the other is to fix the wire-array radius and vary the wire-array mass. In situations of long rise times, the first kind of Hohlraum design should be adopted. Preliminary simulations show that a radiation source with a peak temperature over 308 eV and large enough energy with longer pulse duration is critical for a volume capsule design. Based on the considerations of (1) not underestimating the magneto-Rayleigh–Taylor effect, (2) avoiding the direct shock thermalization on the axis, (3) using of a suitable converter radius, and (4) iteration of dynamic Hohlraum and capsule calculations, a conservative Hohlraum design is proposed. In this Hohlraum design, a radiation pulse with a peak temperature of 312 eV and an efficient time duration of ∼9 ns, which is cut before the main shock arrives at the axis, is produced to drive a two-shell capsule to generate over 10 MJ fusion yield in the case of 50 MA and 100 ns.

Based on the resistive MHD theory, the dispersion relations of instabilities are derived in the liner geometry, where a uniform current flows over the cross section. Both the external axial magnetic field and the electrothermal effect are taken into account. It is found that instabilities act as electrothermal instabilities (ETIs) below a critical wavelength, which is increased by the axial magnetic field. Beyond the critical wavelength, the growth rate increases with the wavelength quickly and approaches the maximum which depends on the gradient scale length of the azimuthal magnetic field. When the axial magnetic field is applied, the m = 1 mode has a higher growth rate than the m = 0 mode although the difference between them is relatively small.

To help design the next generation China Z-pinch driver CZ30, which is a 30 MA driver driven by linear transformer driver (LTD) modules, a new full-circuit simulation code called FCM-CZ30 was established. The code was used to perform simulation of CZ30 coupled with Z-pinch load. The results obtained from FCM-CZ30 are in a good agreement with those obtained from a commercial code called PSpice, which shows that the FCM-CZ30 is a reliable code. The inductance of the LTD system has a significant influence on the rise time of the load current, which is totally different from that in conventional Z-pinch drivers with multistage switches to compress the rise time of the pulse. For a wire-array load with an initial radius of 3.0 cm and a mass per unit length of 15 mg/cm, the peak value and the rise time (10%-90%) of the load current are 33.7 MA and 102.2 ns, respectively.

A mathematical expression of the output voltage from a nonuniform transmission line with an arbitrary input pulse was deduced. The first arriving wave, peak power efficiency, and droop of the output voltage were further clarified using analytical method. The transmission characteristics of monolithic radial transmission lines (MRTLs) with different impedance profiles were investigated by 3-D electromagnetic (EM) simulation and it was found that the hyperbolic impedance profile is the best choice for future Z-pinch drivers. The results obtained from 3-D EM simulation are in good agreement with those obtained from the experiments on a scaled-down MRTL.

This paper presents the experimental results of a monolithic radial transmission line (MRTL) that may be used in pulsed power generators and microwave devices. The MRTL with a hyperbolic impedance profile is 508 mm in radius, corresponding to a one-way transit time of 15 ns for the electromagnetic wave. In the experiments, up to twenty identical voltage pulses, 10 ns in FWHM and 2 ns in rise-time, were fed into the MRTL through 20 input BNC connectors that are uniformly distributed along the outer circumference of the MRTL. It was found that the amplitude of the voltage from the output BNC connector located in the center of the MRTL is nearly proportional to the total number of the input branches. The effect of the failure modes on the output voltage was investigated. For the MRTL driven by 20 input branches, while the open-circuit or short-circuit even in one input branch considerably decreases the amplitude of the output voltage, the jitter shorter than 2 ns in 3 input branches makes no obvious effect on the output voltage.

A mathematical expression of the output voltage from a nonuniform transmission line (NTL) with an arbitrary input pulse was deduced. Due to this mathematical expression, two transmission characteristics of NTLs with linear, exponential and Gaussian impedance profiles were further clarified. The first one is that the peak power efficiencies of NTLs with a half-sine input voltage are quantified as functions of Ψ (the ratio of the output impedance to the input impedance of the NTLs) and Γ (the ratio of the pulse width to the one-way transit time of the NTLs). The second one is that the top of an initially rectangle input voltage pulse falls at the terminal of the NTL and that the ratio of the droop to the top of the output voltage is also quantified as a function of Ψ and Γ.

A 21-channel quasi-square-wave nanosecond pulse generator was constructed. The generator consists of a high-voltage square-wave pulser and a channel divider. Using an electromagnetic relay as a switch and a 50-Ω polyethylene cable as a pulse forming line, the high-voltage pulser produces a 10-ns square-wave pulse of 1070 V. With a specially designed resistor-cable network, the channel divider divides the high-voltage square-wave pulse into 21 identical 10-ns quasi-square-wave pulses of 51 V, exactly equal to 1070 V/21. The generator can operate not only in a simultaneous mode but also in a delay mode if the cables in the channel divider are different in length.

In the design of future Z-pinch driver, the monolithic radial transmission lines (MRTL) were used to combine the outputs of many pulse generators to Z-pinch load. In order to investigate the transmission characteristic of the MRTL with different impedance profiles, 3-dimensional electromagnetic simulation has been performed. From the results of 3-dimensional electromagnetic simulation, it was found that the maximum transmitted power efficiency of hyperbolic line is nearly same as that of exponential line. The hyperbolic line may be the best choice for Z-pinch, because it is much easier in fabrication than the exponential line.

The electromagnetic simulation of the monolithic radial transmission lines for future Z-pinch was performed. Focusing on the difference in the maximum transmitted power efficiency between the electromagnetic simulation and the circuit simulation, the monolithic radial transmission lines with different impedance profile (exponential, Gaussian, hyperbolic) were compared. The power efficiency for the exponential line is higher than that for the Gaussian lines and the hyperbolic line, which is similar to that from the circuit simulation. However, all the power efficiencies obtained with the electromagnetic simulation are about 15% lower than those obtained with the circuit simulation, indicating the existence of considerable non-TEM modes and a non-ignorable error in the circuit simulation based on the quasi-TEM mode approximation. In consideration of several monolithic radial transmission lines being stacked together and the flat electrodes required by the stacked lines, the hyperbolic line was compared with the exponential line with several wide radial slots cut on the flat electrodes. While the hyperbolic line has a little bit lower transmitted power efficiency than that of the exponential line, it is much easier in fabrication. For this reason, the hyperbolic line was recommended as the best choice.

A mathematical expression of the output voltage from a nonuniform line was deduced by investigating the transient response of a cascaded multiple-section line to an input voltage of half-sine shape that is close to the wave shape of the pulse voltage for Z-pinch. The correctness of the mathematical expression was verified by comparing the result from the mathematical expression with that from circuit simulation using PSPICE code. The high-pass and pulse-compression characteristics of a nonuniform line were further clarified by the theoretical analysis of the analytical expressions.

The electromagnetic simulation of the monolithic radial transmission lines for future Z-pinch was performed. Focusing on the difference in the maximum transmitted power efficiency between the electromagnetic simulation and the circuit simulation, the monolithic radial transmission lines with different impedance profiles (exponential, Gaussian, hyperbolic) were compared. The power efficiency for the exponential line is higher than that for the Gaussian lines and the hyperbolic line, which is similar to that from the circuit simulation. However, all the power efficiencies obtained with the electromagnetic simulation are about 15% lower than those obtained with the circuit simulation, indicating the existence of considerable non-TEM modes and a non-ignorable error in the circuit simulation based on the quasi-TEM mode approximation. In consideration of several monolithic radial transmission lines being stacked together and the flat electrodes required by the stacked lines, the hyperbolic line was compared with the exponential line with several wide radial slots cut on the flat electrodes. While the hyperbolic line has a little bit lower transmitted power efficiency than that of the exponential line, it is much easier in fabrication. For this reason, the hyperbolic line was recommended as the best choice.

A mathematical expression of the output voltage from a nonuniform line was deduced by investigating the transient response of a cascaded multiple-section line to an input voltage of half-sine shape that is close to the wave shape of the pulse voltage for Z-pinch. The correctness of the mathematical expression was verified by comparing the result from the mathematical expression with that from circuit simulation using PSpice code. Based on this mathematical expression, a code was wrote using MATLAB to calculate the output voltage from a nonuniform line. To make our code friendly to users, a graphical user interface was designed using MATLAB.

The circuit simulation of nonuniform transmission lines that were under consideration for the next generation of pettawatt-class z-pinch drivers was performed. It was confirmed that the power-transport efficiency of the designed radial transformer with an exponential impedance profile is higher than that with Gaussian profiles. Based on the Fourier's theorem and the principle of superposition, it was found that there are considerable low-frequency components for an input voltage of half-sine wave with an angular frequency of 14MRad/s. Compared to the exponential transformer, the Gaussian transformers amplify the low-frequency components to a lower extent, which makes the difference in power efficiency between the two types of the transformers. The radial transformer also serve as a passive high-pass filter and the Gaussian transformer may give better performance than that the exponential transformer does if the signal transport rather than the power transport is concerned.

Disclaimer This report is provided by the Texas Engineering Experiment Station (TEES). The information provided in this report is intended to be the best available information at the time of publication. TEES makes no claim or warranty, express or implied that the report or data herein is necessarily error-free. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not constitute or imply its endorsement, recommendation, or favoring by the Energy Systems Laboratory or any of its employees. The views and opinions of authors expressed herein do not necessarily state or reflect those of the Texas Engineering Experiment Station or the Energy Systems Laboratory. ACKNOWLEDGEMENT This report was based on a review of numerous valuable works. Special thanks to Mr. Bernard Nagengast for providing information about textbooks, to Prof. Larry Degelman for many valuable insights and to Dr. Mohsen Farzad for providing the history of Carrier Handbook and HAP software. EXECUTIVE SUMMARY This report provides a detailed literature review on the history of building peak heating and cooling load and annual energy use calculation methods from the 1800s to the present. Building annual energy use calculations include: forward methods, data-driven methods and simulation methods. The report is organized as follows:  Section 1 describes the introduction of this report, including the history of the related sciences, computer developments and a brief history of ASHRAE information about U.S. commercial buildings including the distribution and age are covered in this section as well.  Section 2 details the history of peak heating and cooling load calculation methods.  Section 3 presents building annual energy use calculation methods, where the basic concepts are introduced and the most popular calculation methods are reviewed.  Section 4 includes a summary followed by the related references.