The physical phenomenon of amplification in traveling wave tubes can be understood mathematically as a result of system instability or exponentially growing eigenmodes. We study here instability in the uncoupled multi-stream electron beam model, focusing primarily on properties and solutions of the characteristic equation associated with the beam. In particular, we show that in general the zeroes of the characteristic function are distinct. Then, in the two-stream case, we construct a series representation of frequency-dependent solutions to the electron beam characteristic equation near the frequency at which these solutions transition from non-real (unstable eigenmodes) to real (oscillatory eigenmodes).

This dissertation explores the advancement of exceptional points of degeneracy (EPDs) in electrical circuits and their applications in highly sensitive devices, alongside the modeling of traveling wave tubes (TWTs) for microwave amplification. The EPD phenomenon, where two or more eigenmodes coalesce at a critical point, results in systems with heightened sensitivity to perturbations. This work illustrates how EPDs can be achieved using a gyrator-based and simpler circuit without the use of nonreciprocal components. A detailed analysis of various circuit configurations, including series and parallel LC resonators coupled via a gyrator, reveals the potential of these systems to exhibit second- and third-order EPDs. These systems demonstrate exceptional sensitivity, where small changes in capacitance, inductance, or other parameters lead to significant shifts in the resonance frequency, enabling applications such as material characterization and high-performance signal processing.

This dissertation investigates time-modulated systems and their ability to obtain EPDs. Time modulation introduces an additional degree of freedom to systems by periodically varying system parameters, such as damping or capacitance. A dual analogy between mechanical and electrical systems is presented to better understand how energy is dynamically redistributed within these modulated systems. By applying this approach to a mechanical system equipped with a time-modulated damper, the results demonstrate increases in harvested power, making this technique highly advantageous for applications like wireless sensors, remote monitoring devices, and energy-autonomous systems. This dissertation also explores the use of space-time modulation in transmission lines (TL) as a method to directly induce EPDs, a novel approach that can significantly impact microwave circuits and telecommunication systems. By modulating the per-unit-length capacitance of a single transmission line in both space and time, two propagating eigenmodes coalesce at EPD. This research highlights the advantages of this approach, demonstrating how small variations in modulation parameters can lead to large shifts in system behavior, thus providing a powerful tool for next-generation electronic and communication devices.

In the second part of the thesis, the focus shifts to TWTs, which are essential devices for high-power microwave amplification based on linear electron beams. TWTs are widely used in telecommunications, radar, and satellite communication due to their ability to amplify RF signals over a broad frequency range. This research focuses on the inclusion of dispersive slow-wave structures (SWS) and the space-charge effect to better understand and optimize TWT performance. A critical aspect of this work is the introduction of EPD in TWTs by carefully tuning the dispersive properties of the SWS and accounting for electron beam (e-beam) and electromagnetic (EM) wave interactions.

Parametric modeling of serpentine waveguide TWTs is presented as a key advancement in understanding wave propagation and amplification in TWTs. We developed a model for TWTs and applied it to helix TWT and serpentine TWT which are well-known types of TWTs in the industry. This model provides insight into optimizing beam-wave interactions for maximum gain without dealing with the high burden of PIC simulation. Furthermore, the dissertation explores multi-stage TWT designs, introducing the concept of severs—components that stop RF wave propagation at the middle of TWT to prevent back reflections and improve stability. Additionally, the work investigates the small-signal behavior of TWTs, examining how small perturbations in beam current, phase velocity, or input power affect the overall amplification process. Numerical simulations and theoretical models are provided to compare various TWT configurations, offering a clear pathway to design improvements that maximize gain while minimizing energy loss. By bridging the gap between theoretical modeling and practical implementation, this dissertation contributes to the development of more efficient, and high-performing TWTs, making them well-suited to next-generation communication where power, efficiency, and bandwidth are paramount importance.