J.G. Wohlbier

On the physics of harmonic injection in a traveling wave tube

The physics of signal injection to shape the output spectrum in a traveling wave tube (TWT) is studied using an analytic solution to the approximate nonlinear S-MUSE model and with the large signal code LATTE. The results verify the long-standing conjecture that a frequency canceled by signal injection is composed of a component due to the injected signal and a component due to the nonlinearity of the TWT. Furthermore, the structures of the solutions are exploited to explain and predict behavior for various signal injection schemes. The scenarios studied include second harmonic injection to reduce the second harmonic and enhance the fundamental, multiple harmonic injection to eliminate the sensitive output power dependence on injected phase, second harmonic injection to reduce intermodulation distortion, and multiple signal injection to reduce the intermodulation spectrum. Insights are given regarding the effectiveness of an injection depending on whether the injected signal is within or outside the linear gain bandwidth of the TWT.

Second- and third-order signal predistortion for nonlinear distortion suppression in a TWT

The nonlinearity inherent in the traveling wave tube (TWT) amplifier distorts amplified signals and results in reduced efficiency and bandwidth, limiting its use in communication and electronic countermeasure applications. Signal predistortion is an effective technique for suppressing nonlinear distortion in TWTs that provides high suppression and requires simple circuits for implementation. While conventional predistortion linearizers are based on third-order intermodulation (3IM) injection, this paper proposes using second-order (second-harmonic) signal injection in predistortion circuits. A detailed experimental investigation and comparison of second-order versus third-order signal predistortion is presented. It is observed that both schemes result in suppression of up to 30 dB (55 dBc) for the 3IM distortion products. However, experimental results indicate that second-harmonic signal injection performs better than 3IM in suppressing higher order products. The paper also investigates spatial evolution of the wave spectrum along the TWT axis with and without injection, and sensitivity of the suppression to injected signals amplitude, phase and the fundamental frequency.

A Godunov-like point-centered essentially Lagrangian hydrodynamic approach

We present an essentially Lagrangian hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedron meshes. The scheme reduces to a purely Lagrangian approach when the flow is linear or if the mesh size is equal to zero; as a result, we use the term essentially Lagrangian for the proposed approach. The motivation for developing a hydrodynamic method for tetrahedron meshes is because tetrahedron meshes have some advantages over other mesh topologies. Notable advantages include reduced complexity in generating conformal meshes, reduced complexity in mesh reconnection, and preserving tetrahedron cells with automatic mesh refinement. A challenge, however, is tetrahedron meshes do not correctly deform with a lower order (i.e. piecewise constant) staggered-grid hydrodynamic scheme (SGH) or with a cell-centered hydrodynamic (CCH) scheme. The SGH and CCH approaches calculate the strain via the tetrahedron, which can cause artificial stiffness on large deformation problems. To resolve the stiffness problem, we adopt the point-centered hydrodynamic approach (PCH) and calculate the evolution of the flow via an integration path around the node. The PCH approach stores the conserved variables (mass, momentum, and total energy) at the node. The evolution equations for momentum and total energy are discretized using an edge-based finite element (FE) approach with linear basis functions. A multidirectional Riemann-like problem is introduced at the center of the tetrahedron to account for discontinuities in the flow such as a shock. Conservation is enforced at each tetrahedron center. The multidimensional Riemann-like problem used here is based on Lagrangian CCH work 8,19,37,38,44 and recent Lagrangian SGH work 33-35,39,45. In addition, an approximate 1D Riemann problem is solved on each face of the nodal control volume to advect mass, momentum, and total energy. The 1D Riemann problem produces fluxes 18 that remove a volume error in the PCH discretization. A 2-stage Runge-Kutta method is used to evolve the solution in time. The details of the new hydrodynamic scheme are discussed; likewise, results from numerical test problems are presented.

A point-centered arbitrary Lagrangian Eulerian hydrodynamic approach for tetrahedral meshes

We present a three dimensional (3D) arbitrary Lagrangian Eulerian (ALE) hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedral meshes. The new approach stores the conserved variables (mass, momentum, and total energy) at the nodes of the mesh and solves the conservation equations on a control volume surrounding the point. This type of an approach is termed a point-centered hydrodynamic (PCH) method. The conservation equations are discretized using an edge-based finite element (FE) approach with linear basis functions. All fluxes in the new approach are calculated at the center of each tetrahedron. A multidirectional Riemann-like problem is solved at the center of the tetrahedron. The advective fluxes are calculated by solving a 1D Riemann problem on each face of the nodal control volume. A 2-stage Runge-Kutta method is used to evolve the solution forward in time, where the advective fluxes are part of the temporal integration. The mesh velocity is smoothed by solving a Laplacian equation. The details of the new ALE hydrodynamic scheme are discussed. Results from a range of numerical test problems are presented.

Manufactured solutions for the three-dimensional Euler equations with relevance to Inertial Confinement Fusion ☆

Abstract We present a set of manufactured solutions for the three-dimensional (3D) Euler equations. The purpose of these solutions is to allow for code verification against true 3D flows with physical relevance, as opposed to 3D simulations of lower-dimensional problems or manufactured solutions that lack physical relevance. Of particular interest are solutions with relevance to Inertial Confinement Fusion (ICF) capsules. While ICF capsules are designed for spherical symmetry, they are hypothesized to become highly 3D at late time due to phenomena such as Rayleigh–Taylor instability, drive asymmetry, and vortex decay. ICF capsules also involve highly nonlinear coupling between the fluid dynamics and other physics, such as radiation transport and thermonuclear fusion. The manufactured solutions we present are specifically designed to test the terms and couplings in the Euler equations that are relevant to these phenomena. Example numerical results generated with a 3D Finite Element hydrodynamics code are presented, including mesh convergence studies.

The physics of harmonic injection in a TWT

By studying solutions to the S-MUSE model an approximate 1-D nonlinear multifrequency TWT model, we have identified physical mechanisms for cases of harmonic injection. We highlight the structure of the solutions to S-MUSE, and comment on the mechanism whereby intermodulation products are generated. Then we use the structure of the solutions to identify the mechanism for harmonic injection in a 3IM example and in ECM application.

Injection schemes for TWT linearization

In this paper we investigate various linearization schemes based on signal injection to condition the output spectra. We focus primarily on IM3 (third-order intermodulation) suppression for two-tone drive since the IM3s, being closest to the fundamentals, are the main concern in a communication system. Two primary techniques investigated are second harmonic injection and IM3 injection. These involve injecting amplitude and phase optimized harmonic and IM3 signal respectively, to suppress the inherently generated IM3 at the output of the TWT. We also propose a scheme based on simultaneous injection of both second harmonic and IM3 that offers the prospect of eliminating the need for precise phase control of the former two schemes.

Comparison of transfer curve induced distortions to distortions in nonlinear physical TWT models

Summary form only given, as follows. AM/AM and AM/PM curves are widely used to describe the non-linear behavior of Traveling Wave Tubes (TWTs). Several authors have studied these curves with large signal codes. It has been stated that the phase nonlinearity is related to the slowing down of electrons and the phasing of the charge density with respect to the circuit voltage. Furthermore, in the case of multifrequency inputs, for drive levels restricted to the linear portion of the AM/AM curve, intermodulation distortions are predicted by the phase nonlinearity. In this case again the phase of the charge density with respect to the circuit voltage is shown to be important. However, a direct connection between the distortions generated by the transfer curves and the physical models has not been made. For example, neither the AM/AM nor the AM/PM curves produce distortions about the harmonics of the carrier frequency, which are widely known to exist. Recently we have developed an approximate nonlinear physical model of the TWT, valid prior to saturation, that is analytically solvable. We compare the analytic solution of the AM/PM curve and its role in intermodulation generation to the direct analytic prediction of intermodulation distortions afforded by the solutions to the physical model.

A High-Order Finite-Volume Method for Compressible Flows on Moving Tetrahedral Grids

Arbitrary Lagrangian-Eulerian (ALE) methods incorporate dynamic mesh motion in an attempt to combine the advantages of both Eulerian and Lagrangian kinematic descriptions. They are especially attractive for modelling compressible flows since their moving meshes are able to capture large distortions of the continuum without excessively smearing free surfaces or material/fluid interfaces. It is desirable to combine these ALE descriptions with high-order spatial and temporal discretizations because, for a given accuracy, high-order methods offer the potential to greatly reduce computational costs. However, the application of high-order methods to ALE is complicated by changing mesh geometry and certain stability requirements such as geometric conservation. In addition to these challenges, it is also difficult to obtain accurate high-order discretizations of conservation laws without any unphysical oscillations across discontinuities, especially on multi-dimensional unstructured meshes. One high-order method that was proved to be efficient and robust for static meshes is the central essentially non-oscillatory (CENO) finite-volume method. Here, the CENO approach was extended to an ALE formulation on tetrahedral meshes. The proposed unstructured method is vertex-based and uses a direct ALE approach that avoids the temporal splitting errors introduced by traditional “Lagrange-plus-remap” ALE methods. The new approach was applied to the conservation equations governing compressible flows and assessed in terms of accuracy and computational cost. For all problems considered, which included various idealized flows, CENO demonstrated excellent reliability and robustness. High-order accuracy was achieved in smooth regions and essentially non-oscillatory solutions were obtained near discontinuities. The high-order schemes were also more computationally efficient for high-accuracy solutions, i.e., they took less wall time to achieve a desired level of error than the lower-order schemes.

LATTE/MUSE numerical suite: an open source teaching and research code for traveling wave tube amplifiers

The LATTE/MUSE code is capable of steady-state, multifrequency, 1-D simulations of arbitary TWT amplifier geometries, including velocity tapers and circuit serves, with a substantial amount of parameter and input scanning functionality.