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Full-Text Articles in VLSI and Circuits, Embedded and Hardware Systems

Reinventing Integrated Photonic Devices And Circuits For High Performance Communication And Computing Applications, Venkata Sai Praneeth Karempudi Jan 2024

Reinventing Integrated Photonic Devices And Circuits For High Performance Communication And Computing Applications, Venkata Sai Praneeth Karempudi

Theses and Dissertations--Electrical and Computer Engineering

The long-standing technological pillars for computing systems evolution, namely Moore's law and Von Neumann architecture, are breaking down under the pressure of meeting the capacity and energy efficiency demands of computing and communication architectures that are designed to process modern data-centric applications related to Artificial Intelligence (AI), Big Data, and Internet-of-Things (IoT). In response, both industry and academia have turned to 'more-than-Moore' technologies for realizing hardware architectures for communication and computing. Fortunately, Silicon Photonics (SiPh) has emerged as one highly promising ‘more-than-Moore’ technology. Recent progress has enabled SiPh-based interconnects to outperform traditional electrical interconnects, offering advantages like high bandwidth density, …


Characterization Of Low Power Hfo2 Based Switching Devices For In-Memory Computing, Aseel Zeinati May 2023

Characterization Of Low Power Hfo2 Based Switching Devices For In-Memory Computing, Aseel Zeinati

Theses

Oxide based Resistive Random Access Memory (RRAM) devices are investigated as one of the promising non-volatile memories to be used for in-memory computing that will replace the classical von Neumann architecture and reduce the power consumption. These applications required multilevel cell (MLC) characteristics that can be achieved in RRAM devices. One of the methods to achieve this analog switching behavior is by performing an optimized electrical pulse. The RRAM device structure is basically an insulator between two metals as metal-insulator-metal (MIM) structure. Where one of the primary challenges is to assign an RRAM stack with both low power consumption and …


A Phase Change Memory And Dram Based Framework For Energy-Efficient And High-Speed In-Memory Stochastic Computing, Supreeth Mysore Jan 2023

A Phase Change Memory And Dram Based Framework For Energy-Efficient And High-Speed In-Memory Stochastic Computing, Supreeth Mysore

Theses and Dissertations--Electrical and Computer Engineering

Convolutional Neural Networks (CNNs) have proven to be highly effective in various fields related to Artificial Intelligence (AI) and Machine Learning (ML). However, the significant computational and memory requirements of CNNs make their processing highly compute and memory-intensive. In particular, the multiply-accumulate (MAC) operation, which is a fundamental building block of CNNs, requires enormous arithmetic operations. As the input dataset size increases, the traditional processor-centric von-Neumann computing architecture becomes ill-suited for CNN-based applications. This results in exponentially higher latency and energy costs, making the processing of CNNs highly challenging.

To overcome these challenges, researchers have explored the Processing-In Memory (PIM) …


Design And Analysis Of A Discrete, Pcb-Level Low-Power, Microwave Cross-Coupled Differential Lc Voltage-Controlled Oscillator, Pavin Singh Virdee Sep 2022

Design And Analysis Of A Discrete, Pcb-Level Low-Power, Microwave Cross-Coupled Differential Lc Voltage-Controlled Oscillator, Pavin Singh Virdee

Master's Theses

Radio Frequency (RF) and Microwave devices are typically implemented in Integrated Circuit (IC) form to minimize parasitics, increase precision and tolerances, and minimize size. Although IC fabrication for students and independent engineers is cost-prohibitive, an abundance of low-cost, easily accessible printed circuit board (PCB) and electronic component manufacturers allows affordable PCB fabrication.

While nearly all microwave voltage-controlled oscillator (VCO) designs are IC-based, this study presents a discrete PCB-level cross-coupled, differential LC VCO to demonstrate this more affordable and accessible approach. This thesis presents a 65 mW, discrete component VCO PCB with industry-comparable RF performance. A phase noise of -103.7 dBc/Hz …


Load Modeling And Evaluation Of Leds For Hardware Test Bed Application, Jillian M. Ruff Aug 2022

Load Modeling And Evaluation Of Leds For Hardware Test Bed Application, Jillian M. Ruff

Masters Theses

The lighting industry was revolutionized with the emergence of LED lighting. Over the last 15 years, LED lighting device sales and utilization have grown immensely. The growth and popularity of LEDs is due to improved operation of the device when compared to previous lighting technologies. Efficient performance of the device is critical due to the growth of global energy consumption.

As nonrenewable generation fuel is finite, utilities have begun the transition to renewable energy generation. Generation and distribution systems become inherently complex to comprehend and maintain with incorporation of emerging supply and load technologies. With the unprecedented growth of LED …


A Bulk Driven Transimpedance Cmos Amplifier For Sipm Based Detection, Shahram Hatefi Hesari Aug 2022

A Bulk Driven Transimpedance Cmos Amplifier For Sipm Based Detection, Shahram Hatefi Hesari

Masters Theses

The contribution of this work lies in the development of a bulk driven operational
transconducctance amplifier which can be integrated with other analog circuits and
photodetectors in the same chip for compactness, miniaturization and reducing the
power. Silicon photomultipliers, also known as SiPMs, when coupled with scintillator materials are used in many imaging applications including nuclear detection. This thesis discuss the design of a bulk-driven transimpedance amplifier suitable for detectors where the front end is a SiPM. The amplifier was design and fabricated in a standard standard CMOS process and is suitable for integration with CMOS based SiPMs and commercially …


Design Tunneling Transistor And Schottky Junction Solar Cell Using Van Der Waals Semiconductor Heterostructure, Md Azmot Ullah Khan Jul 2022

Design Tunneling Transistor And Schottky Junction Solar Cell Using Van Der Waals Semiconductor Heterostructure, Md Azmot Ullah Khan

LSU Doctoral Dissertations

Transition metal di-chalcogenide (TMDC) materials, being semiconductor in nature, offer Two-dimensional (2D) materials such as graphene and molybdenum disulfide (MoS2) possess unique and unusual properties that are particularly applicable to nanoelectronics and photovoltaic devices. In this dissertation, four different projects have been done that encompass the implementation of these materials to improve the performance of future transistors and Schottky junction solar cells. In chapter 2, an analytical current transport model of a dual gate tunnel field-effect transistor (TFET) is developed by utilizing the principle of band-to-band tunneling (BTBT) and MoS2 as the channel material. Later, using this …


Characterization Of Electrophoretic Deposited Zinc Oxide Nanopartices For The Fabrication Of Next-Generation Nanoscale Electronic Applications, Fawwaz Abduh A. Hazzazi Jul 2022

Characterization Of Electrophoretic Deposited Zinc Oxide Nanopartices For The Fabrication Of Next-Generation Nanoscale Electronic Applications, Fawwaz Abduh A. Hazzazi

LSU Doctoral Dissertations

Several reports state that it is crucial to analyze nanoscale semiconductor materials and devices with potential benefits to meet the need for next-generation nanoelectronics, bio, and nanosensors. The progress in the electronics field is as significant now, with modern technology constantly evolving and a greater focus on more efficient robust optoelectronic applications. This dissertation focuses on the study and examination of the practicality of Electrophoretic Deposition (EPD) of zinc oxide (ZnO) nanoparticles (NPs) for use in semiconductor applications.

The feasibility of several synthesized electrolytes, with and without surfactants and APTES surface functionalization, is discussed. The primary objective of this study …


Machine Learning Applications To Static Timing Analysis, Waseem Mohamed Raslan Jun 2022

Machine Learning Applications To Static Timing Analysis, Waseem Mohamed Raslan

Theses and Dissertations

Modeling complex cell behavior is critical for accurate static timing analysis. Effective current source model, ECSM, and composite current source, CCS, waveform data compression became a necessity to reduce the size of technology files and increase the accuracy of the cell characterization data. We used deep learning nonlinear Autoencoders to compress voltage and current waveforms and compared them with singular value decomposition, SVD, approach. Autoencoders gave ~1.67x compression ratio for voltage waveforms better than SVD approach and gave 45x to 55x better compression ratio compared to other lossless techniques like bz2 and gzip. Autoencoders achieved ~1.7x compression ratio for complex …


Design & Analysis Of Mixed-Mode Integrated Circuit For Pulse-Shape Discrimination, Bryan Orabutt May 2022

Design & Analysis Of Mixed-Mode Integrated Circuit For Pulse-Shape Discrimination, Bryan Orabutt

McKelvey School of Engineering Theses & Dissertations

In nuclear science experiments it is usually necessary to determine the type of radiation, its energy and direction with considerable accuracy. The detection of neutrons and discriminating them from gamma rays is particularly difficult. A popular method of doing so is to measure characteristics intrinsic to the pulse shape of each radiation type in order to perform pulse-shape discrimination (PSD).

Historically, PSD capable systems have been designed with two approaches in mind: specialized analog circuitry, or digital signal processing (DSP). In this work we propose a PSD capable circuit topology using techniques from both the analog and DSP domains. We …


Design Of A Bandgap Voltage Reference, Nicolaus Vail May 2022

Design Of A Bandgap Voltage Reference, Nicolaus Vail

Electrical Engineering Undergraduate Honors Theses

This thesis details the design process of a bandgap voltage reference (BGR) integrated circuit in a 180 nm CMOS process. A BGR provides a constant DC voltage across a range of operating temperatures and supply voltages. By its nature, the circuit is intended as a reference, not to provide current, so the output would be connected to a very high impedance, such as the gate of a transistor. At 27°C, this design provides a 955 mV reference voltage given a nominal VDD of 3 V. From 20°C to 175°C, the output voltage has a variance of 7.2 mV (approximately 0.8%) …


Automatic Testing Of Organic Strain Gauge Tactile Sensors., Brian P. Goulet May 2022

Automatic Testing Of Organic Strain Gauge Tactile Sensors., Brian P. Goulet

Electronic Theses and Dissertations

Human-Robot Interaction is a developing field of science, that is posed to augment everything we do in life. Skin sensors that can detect touch, temperature, distance, and other physical interaction parameters at the human-robot interface are very important to enhancing the collaboration between humans and machines. As such, these sensors must be efficiently tested and characterized to give accurate feedback from the sensor to the robot. The objective of this work is to create a diversified software testing suite that removes as much human intervention as possible. The tests and methodology discussed here provide multiple realistic scenarios that the sensors …


On Improving Robustness Of Hardware Security Primitives And Resistance To Reverse Engineering Attacks, Vinay C. Patil Oct 2021

On Improving Robustness Of Hardware Security Primitives And Resistance To Reverse Engineering Attacks, Vinay C. Patil

Doctoral Dissertations

The continued growth of information technology (IT) industry and proliferation of interconnected devices has aggravated the problem of ensuring security and necessitated the need for novel, robust solutions. Physically unclonable functions (PUFs) have emerged as promising secure hardware primitives that can utilize the disorder introduced during manufacturing process to generate unique keys. They can be utilized as \textit{lightweight} roots-of-trust for use in authentication and key generation systems. Unlike insecure non-volatile memory (NVM) based key storage systems, PUFs provide an advantage -- no party, including the manufacturer, should be able to replicate the physical disorder and thus, effectively clone the PUF. …


Design And Characterization Of Standard Cell Library Using Finfets, Phanindra Datta Sadhu Jun 2021

Design And Characterization Of Standard Cell Library Using Finfets, Phanindra Datta Sadhu

Master's Theses

The processors and digital circuits designed today contain billions of transistors on a small piece of silicon. As devices are becoming smaller, slimmer, faster, and more efficient, the transistors also have to keep up with the demands and needs of the daily user. Unfortunately, the CMOS technology has reached its limit and cannot be used to scale down due to the transistor's breakdown caused by short channel effects. An alternative solution to this is the FinFET transistor technology, where the gate of the transistor is a three dimensional fin that surrounds the transistor and prevents the breakdown caused by scaling …


Fault Modeling And Test Vector Generation For Asic Devices Exposed To Space Single Event Environment, Ahmed Mohamed May 2021

Fault Modeling And Test Vector Generation For Asic Devices Exposed To Space Single Event Environment, Ahmed Mohamed

Theses and Dissertations

This work aims at providing a concise automated flow to predict the effect of Single Event Transients (SETs) on ASIC chips by developing a method to characterize the circuit susceptibility to SET pulses propagation and then generation of the required input vectors that sensitize the victim paths. A new enhanced method for SET electrical propagation modeling is proposed and compared to a previously published analytical model. The method was applied on different standard cells libraries built over XFAB Xh018 technology and verified for accuracy against simulations. The new method showed enhancement in accuracy compared with previous work in literature. Industrial …


Design And Fabrication Of A Microstrip Bandpass Filter In Ltcc, Allison Rucker May 2021

Design And Fabrication Of A Microstrip Bandpass Filter In Ltcc, Allison Rucker

Electrical Engineering Undergraduate Honors Theses

The goal of the project was to design and fabricate a bandpass filter with a center frequency of 25GHz with a 2GHz bandwidth. The first step was to do the calculation to design a bandpass filter to meet these specifications along with the properties of the DupontTM GreenTapeTM 9K7. HFSS was then used to verify the results from the initial calculations. There was a significant error between the two results, so more tweaking was done to the calculations to get a better center frequency. After a final design was decided, the fabrication process started. Low-Temperature Co-Fired Ceramics (LLTC) …


Enercon Station Vacuum Pump Replacement, Clint Hembree, Jared D'Amico, Connor Moore, Paul Jeffrey Fontenot, Sydnee Castello, J.J. Clements Apr 2021

Enercon Station Vacuum Pump Replacement, Clint Hembree, Jared D'Amico, Connor Moore, Paul Jeffrey Fontenot, Sydnee Castello, J.J. Clements

Senior Design Project For Engineers

This details the progress of the ENERCON pump replacement project as completed by the Kennesaw State University interdisciplinary senior design group. This project is a two-semester capstone effort for the engineering program at Southern Polytechnic School of Engineering, overseen by Dr. McFall during Fall 2020 and Dr. Khalid during Spring 2021 semesters. The 2020-2021 KSU Interdisciplinary Senior Design team was tasked with completing an Engineering Change Package (ECP) for existing vacuum pumps at ENERCON Station. The mechanical, electrical, and civil students worked together, performing evaluations on existing plant systems to ensure the plant could support the new vacuum pumps. By …


Design Of An Ultra-Wideband Frequency-Modulated Continuous Wave Short Range Radar System For Extending Independent Living, Toai-Chi Nguyen Apr 2021

Design Of An Ultra-Wideband Frequency-Modulated Continuous Wave Short Range Radar System For Extending Independent Living, Toai-Chi Nguyen

Master's Theses

Falls in the disabled and elderly people have been a cause of concern as they can be immobilized by the fall and have no way to contact others and seek assistance. The proposed frequency modulated continuous wave (FMCW) short range radar (SRR) system, which uses ultra-wideband (UWB) signals can provide immediate assistance by monitoring and detecting fall events. The unique characteristics of this system allow for a frequency-based modulation system to carry out triangulation and sense the location of the fall through the usage of a continuous chirp signal that linearly sweeps frequency. This project focuses on the development, design …


Design, Development, And Evaluation Of Customized Electronics For Controlling A 5-Dof Magneto-Rheological Actuator Collaborative Robot, Ziqi Yang Jan 2021

Design, Development, And Evaluation Of Customized Electronics For Controlling A 5-Dof Magneto-Rheological Actuator Collaborative Robot, Ziqi Yang

Electronic Thesis and Dissertation Repository

In recent years, Magneto-Rheological (MR) fluids has been used in various fields such as robotics, automotive, aerospace, etc. The most common use of the MR fluids is within a clutch-like mechanism, namely an MR clutch. When mechanical input is coupled to the input part of the MR clutch, the MR clutch provides a means of delivering this mechanical input to its output, through the MR fluids. The combination of the mechanical input device and the MR clutch is called an MR actuator. The MR actuator features inherently compliance owing to the characteristic of the MR fluids while also offering higher …


Device-Level Predictive Modeling Of Extreme Electromagnetic Interference, Nishchay H. Sule Jul 2020

Device-Level Predictive Modeling Of Extreme Electromagnetic Interference, Nishchay H. Sule

Electrical and Computer Engineering ETDs

Radio Frequency (RF) interference is a prominent issue for modern electronic devices. As device size and supply power shrink to meet the on-going demand for compact and complex Integrated Circuits (ICs), their susceptibility to external noise coupling to the input or power supply increases significantly. One such type of noise that acts upon a system to be considered is Extreme Electromagnetic Interference (EEMI). Previous works done to understand and evaluate the impact of EEMI onto a system or sub-system have been conducted on a statistical or empirical analysis level, which has led to complex and convoluted analysis, that requires significant …


Switching Trajectory Control For High Voltage Silicon Carbide Power Devices With Novel Active Gate Drivers, Shuang Zhao Aug 2019

Switching Trajectory Control For High Voltage Silicon Carbide Power Devices With Novel Active Gate Drivers, Shuang Zhao

Graduate Theses and Dissertations

The penetration of silicon carbide (SiC) semiconductor devices is increasing in the power industry due to their lower parasitics, higher blocking voltage, and higher thermal conductivity over their silicon (Si) counterparts. Applications of high voltage SiC power devices, generally 10 kV or higher, can significantly reduce the amount of the cascaded levels of converters in the distributed system, simplify the system by reducing the number of the semiconductor devices, and increase the system reliability.

However, the gate drivers for high voltage SiC devices are not available on the market. Also, the characteristics of the third generation 10 kV SiC MOSFETs …


Model Development And Assessment Of The Gate Network In A High-Performance Sic Power Module, William Austin Curbow May 2019

Model Development And Assessment Of The Gate Network In A High-Performance Sic Power Module, William Austin Curbow

Graduate Theses and Dissertations

The main objective of this effort is to determine points of weakness in the gate network of a high-performance SiC power module and to offer remedies to these issues to increase the overall performance, robustness, and reliability of the technology. In order to accomplish this goal, a highly accurate model of the gate network is developed through three methods of parameter extraction: calculation, simulation, and measurement. A SPICE model of the gate network is developed to analyze four electrical issues in a high-speed, SiC-based power module including the necessary internal gate resistance for damping under-voltage and over-voltage transients, the disparity …


Experimental Study And Modeling Of The Gm-I Dependence Of Long-Channel Mosfets, Michael Fong Cheng Mar 2019

Experimental Study And Modeling Of The Gm-I Dependence Of Long-Channel Mosfets, Michael Fong Cheng

Master's Theses

This thesis describes an experimental study and modeling of the current-transconductance dependence of the ALD1106, ALD1107, and CD4007 arrays. The study tests the hypothesis that the I-gm dependence of these 7.8 µm to 10 µm MOSFETs conforms to the Advanced Compact Model (ACM). Results from performed measurements, however, do not support this expectation. Despite the relatively large length, both ALD1106 and ALD1107 show sufficiently pronounced ‘short-channel’ effects to render the ACM inadequate. As a byproduct of this effort, we confirmed the modified ACM equation. With an m factor of approximately 0.6, it captures the I-gm dependence with sub-28% maximum error …


Investigation Of Radiation-Hardened Design Of Electronic Systems With Applications To Post-Accident Monitoring For Nuclear Power Plants, Qiang Huang Feb 2019

Investigation Of Radiation-Hardened Design Of Electronic Systems With Applications To Post-Accident Monitoring For Nuclear Power Plants, Qiang Huang

Electronic Thesis and Dissertation Repository

This research aims at improving the robustness of electronic systems used-in high level radiation environments by combining with radiation-hardened (rad-hardened) design and fault-tolerant techniques based on commercial off-the-shelf (COTS) components. A specific of the research is to use such systems for wireless post-accident monitoring in nuclear power plants (NPPs). More specifically, the following methods and systems are developed and investigated to accomplish expected research objectives: analysis of radiation responses, design of a radiation-tolerant system, implementation of a wireless post-accident monitoring system for NPPs, performance evaluation without repeat physical tests, and experimental validation in a radiation environment.

A method is developed …


S.A.V.E. M.E., Taylor Davis, Kelly Nicole O'Neill, Adrianna M. Dunlap, Parsa Esshaghi Bayat Jan 2019

S.A.V.E. M.E., Taylor Davis, Kelly Nicole O'Neill, Adrianna M. Dunlap, Parsa Esshaghi Bayat

Williams Honors College, Honors Research Projects

S.A.V.E. M.E. stands for Submerged Automated Vehicular Elevation Minor Extraction or alternatively a Home Swimming Pool Rescue Device. The objective of this project is to design and prototype a system that will make unattended swimming pools through detecting a victim’s presence, deploying a means to save the victim, and alerting others nearby of the situation. This system encompasses sensors and devices within the pool and an alarm system outside of the pool. Upon detection of a sufficiently sized object entering the pool when the system is armed, a device will maneuver to the victim and deploy a flotation device that …


Investigating The Effect Of Detecting And Mitigating A Ring Oscillator-Based Hardware Trojan, Lakshmi Ramakrishnan Oct 2018

Investigating The Effect Of Detecting And Mitigating A Ring Oscillator-Based Hardware Trojan, Lakshmi Ramakrishnan

Electrical Engineering Theses and Dissertations

The outsourcing of the manufacturing process of integrated circuits to fabrications plants all over the world has exposed these chips to several security threats, especially at the hardware level. There have been instances of malicious circuitry, such as backdoors, being added to circuits without the knowledge of the chip designers or vendors. Such threats could be immensely powerful and dangerous against confidentiality, among other vulnerabilities.

Defense mechanisms against such attacks have been probed and defense techniques have been developed. But with the passage of time, attack techniques have improved immensely as well. From directly observing the inputs or outputs, adversaries …


Skybridge-3d-Cmos: A Fine-Grained Vertical 3d-Cmos Technology Paving New Direction For 3d Ic, Jiajun Shi Jul 2018

Skybridge-3d-Cmos: A Fine-Grained Vertical 3d-Cmos Technology Paving New Direction For 3d Ic, Jiajun Shi

Doctoral Dissertations

2D CMOS integrated circuit (IC) technology scaling faces severe challenges that result from device scaling limitations, interconnect bottleneck that dominates power and performance, etc. 3D ICs with die-die and layer-layer stacking using Through Silicon Vias (TSVs) and Monolithic Inter-layer Vias (MIVs) have been explored in recent years to generate circuits with considerable interconnect saving for continuing technology scaling. However, these 3D IC technologies still rely on conventional 2D CMOS’s device, circuit and interconnect mindset showing only incremental benefits while adding new challenges reliability issues, robustness of power delivery network design and short-channel effects as technology node scaling. Skybridge-3D-CMOS (S3DC) is …


Analog Signal Processing Solutions And Design Of Memristor-Cmos Analog Co-Processor For Acceleration Of High-Performance Computing Applications, Nihar Athreyas Jul 2018

Analog Signal Processing Solutions And Design Of Memristor-Cmos Analog Co-Processor For Acceleration Of High-Performance Computing Applications, Nihar Athreyas

Doctoral Dissertations

Emerging applications in the field of machine vision, deep learning and scientific simulation require high computational speed and are run on platforms that are size, weight and power constrained. With the transistor scaling coming to an end, existing digital hardware architectures will not be able to meet these ever-increasing demands. Analog computation with its rich set of primitives and inherent parallel architecture can be faster, more efficient and compact for some of these applications. The major contribution of this work is to show that analog processing can be a viable solution to this problem. This is demonstrated in the three …


Parameters Affecting The Resistivity Of Lp-Ebid Deposited Copper Nanowires, Gabriel Smith Jan 2018

Parameters Affecting The Resistivity Of Lp-Ebid Deposited Copper Nanowires, Gabriel Smith

Theses and Dissertations--Electrical and Computer Engineering

Electron Beam Induced Deposition (EBID) is a direct write fabrication process with applications in circuit edit and debug, mask repair, and rapid prototyping. However, it suffers from significant drawbacks, most notably low purity. Work over the last several years has demonstrated that deposition from bulk liquid precursors, rather than organometallic gaseous precursors, results in high purity deposits of low resistivity (LPEBID). In this work, it is shown that the deposits resulting from LP-EBID are only highly conductive when deposited at line doses below 25μC/cm. When the dose exceeds this value, the resulting structure is highly porous providing a poor conductive …


An Exact Analysis For Four-Order Acousto-Optic Bragg Diffraction Which Incorporates Both Incident Light Angle And Sound Frequency Dependencies, Adeyinka Sunday Ademola May 2017

An Exact Analysis For Four-Order Acousto-Optic Bragg Diffraction Which Incorporates Both Incident Light Angle And Sound Frequency Dependencies, Adeyinka Sunday Ademola

Electrical Engineering Theses

This thesis extends the prior work which produced an exact solution to the four-order acousto-optic (AO) Bragg cell with assumed fixed center frequency and with exact Bragg angle incident light. The extension predicts the model that incorporates the dependencies of both the input angle of light and the sound frequency. Specifically, a generalized 4th order linear differential equation (DE), is developed from a simultaneous analysis of four coupled AO system of DEs. Through standard methods, the characteristic roots, which requires solving a quartic equation, is produced. Subsequently, a derived system of homogeneous solutions, which absorbs the roots obtained using …