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Articles 121 - 150 of 254

Full-Text Articles in Electrical and Electronics

Prototyping A Capacitive Sensing Device For Gesture Recognition, Chenglong Lin May 2019

Prototyping A Capacitive Sensing Device For Gesture Recognition, Chenglong Lin

Computer Science and Computer Engineering Undergraduate Honors Theses

Capacitive sensing is a technology that can detect proximity and touch. It can also be utilized to measure position and acceleration of gesture motions. This technology has many applications, such as replacing mechanical buttons in a gaming device interface, detecting respiration rate without direct contact with the skin, and providing gesture sensing capability for rehabilitation devices. In this thesis, an approach to prototype a capacitive gesture sensing device using the Eagle PCB design software is demonstrated. In addition, this paper tested and evaluated the resulting prototype device, validating the effectiveness of the approach.


Synchrophasor-Based Fault Location Detection And Classification, In Power Systems, Using Artificial Intelligence, Hemal Falak May 2019

Synchrophasor-Based Fault Location Detection And Classification, In Power Systems, Using Artificial Intelligence, Hemal Falak

Graduate Theses and Dissertations

With the introduction of sophisticated electronic gadgets which cannot sustain interruption in the provision of electricity, the need to supply uninterrupted and reliable power supply, to the consumers, has become a crucial factor in the present-day world. Therefore, it is customary to correctly identify fault locations in an electrical power network, in order to rectify faults and restore power supply in the minimum possible time. Many automated fault location detection algorithms have been proposed, however, prior art requires topological and physical information of the electrical power network. This thesis presents a new method of detecting fault locations, in transmission as …


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 …


Fabrication And Characterization Of Planar-Structure Perovskite Solar Cells, Guoduan Liu Jan 2019

Fabrication And Characterization Of Planar-Structure Perovskite Solar Cells, Guoduan Liu

Theses and Dissertations--Electrical and Computer Engineering

Currently organic-inorganic hybrid perovskite solar cells (PSCs) is one kind of promising photovoltaic technology due to low production cost, easy fabrication method and high power conversion efficiency.

Charge transport layers are found to be critical for device performance and stability. A traditional electron transport layer (ETL), such as TiO2 (Titanium dioxide), is not very efficient for charge extraction at the interface. Compared with TiO2, SnO2 (Tin (IV) Oxide) possesses several advantages such as higher mobility and better energy level alignment. In addition, PSCs with planar structure can be processed at lower temperature compared to PSCs with …


Straintronic Nanomagnetic Devices For Non-Boolean Computing, Md Ahsanul Abeed Jan 2019

Straintronic Nanomagnetic Devices For Non-Boolean Computing, Md Ahsanul Abeed

Theses and Dissertations

Nanomagnetic devices have been projected as an alternative to transistor-based switching devices due to their non-volatility and potentially superior energy-efficiency. The energy efficiency is enhanced by the use of straintronics which involves the application of a voltage to a piezoelectric layer to generate a strain which is ultimately transferred to an elastically coupled magnetostrictive nanomaget, causing magnetization rotation. The low energy dissipation and non-volatility characteristics make straintronic nanomagnets very attractive for both Boolean and non-Boolean computing applications. There was relatively little research on straintronic switching in devices built with real nanomagnets that invariably have defects and imperfections, or their adaptation …


Energy Efficient Spintronic Device For Neuromorphic Computation, Md Ali Azam Jan 2019

Energy Efficient Spintronic Device For Neuromorphic Computation, Md Ali Azam

Theses and Dissertations

Future computing will require significant development in new computing device paradigms. This is motivated by CMOS devices reaching their technological limits, the need for non-Von Neumann architectures as well as the energy constraints of wearable technologies and embedded processors. The first device proposal, an energy-efficient voltage-controlled domain wall device for implementing an artificial neuron and synapse is analyzed using micromagnetic modeling. By controlling the domain wall motion utilizing spin transfer or spin orbit torques in association with voltage generated strain control of perpendicular magnetic anisotropy in the presence of Dzyaloshinskii-Moriya interaction (DMI), different positions of the domain wall are realized …


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 …


Furnace System Improvements, Kenji K. Anzai, Ethan Fedor, Patrick Zailaa Dec 2018

Furnace System Improvements, Kenji K. Anzai, Ethan Fedor, Patrick Zailaa

Mechanical Engineering

This Final Design Review (FDR) documents the work done by the group to manufacture the system to deliver improvements to the furnace in the Cal Poly microfabrication laboratory (better known as the cleanroom). The document describes the material covered in the Scope of Work (SOW), including background research, customer needfinding, and the problem statement, material associated with concept development and selection for Preliminary Design Review (PDR), material detailing the design and manufacturing plan of the system for Critical Design Review (CDR), and new material detailing the manufacture of the system for FDR.

Research done for the SOW resulted in a …


Stability Analysis Of A High-Power Microgrid, David Manuel Carballo Rojas Dec 2018

Stability Analysis Of A High-Power Microgrid, David Manuel Carballo Rojas

Graduate Theses and Dissertations

The objective of this thesis is to perform the modeling and stability analysis of a high-power microgrid with multiple parallel-and grid connected voltage source converters using the system parameters from the high-power microgrid testbed at the National Center for Reliable Electric Power Transmission (NCREPT) at the University of Arkansas in order to identify, minimize, if not eliminate, the potential instabilities that can affect the proper operation of the microgrid testbed. To achieve this objective, the mathematical modeling of the high-power microgrid considering the adverse effects of resonances due to interactions among the converter LCL output filters is presented and analyzed. …


Review Of Recent Nuclear Magnetic Resonance Studies Of Ion Transport In Polymer Electrolytes, Stephen Munoz, Steven Greenbaum Nov 2018

Review Of Recent Nuclear Magnetic Resonance Studies Of Ion Transport In Polymer Electrolytes, Stephen Munoz, Steven Greenbaum

Publications and Research

Current and future demands for increasing the energy density of batteries without sacrificing safety has led to intensive worldwide research on all solid state Li-based batteries. Given the physical limitations on inorganic ceramic or glassy solid electrolytes, development of polymer electrolytes continues to be a high priority. This brief review covers several recent alternative approaches to polymer electrolytes based solely on poly(ethyleneoxide) (PEO) and the use of nuclear magnetic resonance (NMR) to elucidate structure and ion transport properties in these materials.


Thermal And Mechanical Energy Harvesting Using Lead Sulfide Colloidal Quantum Dots, Taher Ghomian Oct 2018

Thermal And Mechanical Energy Harvesting Using Lead Sulfide Colloidal Quantum Dots, Taher Ghomian

LSU Doctoral Dissertations

The human body is an abundant source of energy in the form of heat and mechanical movement. The ability to harvest this energy can be useful for supplying low-consumption wearable and implantable devices. Thermoelectric materials are usually used to harvest human body heat for wearable devices; however, thermoelectric generators require temperature gradient across the device to perform appropriately. Since they need to attach to the heat source to absorb the heat, temperature equalization decreases their efficiencies. Moreover, the electrostatic energy harvester, working based on the variable capacitor structure, is the most compatible candidate for harvesting low-frequency-movement of the human body. …


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 …


Spice Based Compact Model For Electrical Switching Of Antiferromagnet, Xe Jin Chan, Jan Kaiser, Pramey Upadhyaya Aug 2018

Spice Based Compact Model For Electrical Switching Of Antiferromagnet, Xe Jin Chan, Jan Kaiser, Pramey Upadhyaya

The Summer Undergraduate Research Fellowship (SURF) Symposium

A simulation framework that can model the behavior of antiferromagnets (AFMs) is essential to building novel high-speed devices. The electrical switching of AFMs allows for high performance memory applications. With new phenomena in spintronics being discovered, there is a need for flexible and expandable models. With that in mind, we developed a model for AFMs which can be used to simulate AFM switching behavior in SPICE. This approach can be modified for adding modules, keeping pace with new developments. The proposed AFM switching model is based on the Landau-Lifshitz-Gilbert equation (LLG). LLG along with an exchange coupling module is implemented …


Hybrid Perovskite Characterization And Device Applications., Kasun Fernando Aug 2018

Hybrid Perovskite Characterization And Device Applications., Kasun Fernando

Electronic Theses and Dissertations

Hybrid perovskites are a group of materials that has shown a great impact in the field of scientific research in the past decade due to the efficiency gain within a short period of time. Hot casting is one technique that has been producing high efficient and stable solar cells. Electrical transportation of lateral device structure by such film is explored to understand basic properties and predict possible device applications using it. Under dark, memristive ability of the film was explored using various experiments. Unique uni-polar memristor ability was observed. Using the experimental results, a model is hypothesized using the concepts …


Properties Of Matter, Mike Jackson, Holly Haney Jul 2018

Properties Of Matter, Mike Jackson, Holly Haney

High School Lesson Plans

Students will investigate the relationship(s) between thermal and electrical properties of matter. First, students will use a multimeter and temperature probe to investigate the relationship between electrical resistance and temperature of an electrical resistor composed of metals. They will then graph collected data to analyze the relationship and draw a conclusion as to their relationship. They will then perform the same investigation on a thermal resistor made of a semiconducting substance and analyze that collected data. Finally, using ClaimEvidence-Reasoning (CER) structure, students will use their experimental evidence to state the similarities and differences between the electro-thermal properties of metals and …


Voltage-Controlled Deposition Of Nanoparticles For Next Generation Electronic Materials, Subhodip Maulik May 2018

Voltage-Controlled Deposition Of Nanoparticles For Next Generation Electronic Materials, Subhodip Maulik

LSU Doctoral Dissertations

This work presents both a feasibility study and an investigation into the voltage-controlled spray deposition of different nanoparticles, namely, carbon nanotubes (CNTs), as well as molybdenum disulfide (MoS2) and tungsten disulfide (WS2) from the transition metal dichalcogenides (TMDCs) family of materials. The study considers five different types of substrates as per their potential application to next-generation device electronics. The substrates selected for this research were: 1) aluminum as a conducting substrate, 2) silicon as a semiconducting substrate, 3) glass, silicon dioxide (SiO2), and syndiotactic poly methyl methacrylate (syndiotactic PMMA) as insulating substrates.

Since the …


Evaluation Of Terahertz Imaging For Breast Cancer Detection Using Image Morphing, Tanny Andrea Chavez Esparza May 2018

Evaluation Of Terahertz Imaging For Breast Cancer Detection Using Image Morphing, Tanny Andrea Chavez Esparza

Graduate Theses and Dissertations

This thesis proposes the use of a mesh morphing algorithm for the quantitative evaluation of terahertz (THz) images. This work differs from traditional evaluation methods based on qualitative evaluation because it provides a fair and quantitative measurement of the THz imaging system's performance. The objective of the algorithm is to match the alignment, shape, and resolution of the THz and reference pathology images. Therefore, the proposed morphing method provides a pathology reference for a pixel-by-pixel evaluation of the region classification in the THz image. To achieve this, the morphing algorithm aligns the images using the Pearson's correlation coefficient and reshapes …


Challenges In Flexible Microsystem Manufacturing : Fabrication, Robotic Assembly, Control, And Packaging., Danming Wei May 2018

Challenges In Flexible Microsystem Manufacturing : Fabrication, Robotic Assembly, Control, And Packaging., Danming Wei

Electronic Theses and Dissertations

Microsystems have been investigated with renewed interest for the last three decades because of the emerging development of microelectromechanical system (MEMS) technology and the advancement of nanotechnology. The applications of microrobots and distributed sensors have the potential to revolutionize micro and nano manufacturing and have other important health applications for drug delivery and minimal invasive surgery. A class of microrobots studied in this thesis, such as the Solid Articulated Four Axis Microrobot (sAFAM) are driven by MEMS actuators, transmissions, and end-effectors realized by 3-Dimensional MEMS assembly. Another class of microrobots studied here, like those competing in the annual IEEE Mobile …


Application Of Modular Multilevel Converters (Mmc) Using Phase-Shifted Pwm And Selective Harmonic Elimination In Distribution Systems, Thouhidul Islam May 2018

Application Of Modular Multilevel Converters (Mmc) Using Phase-Shifted Pwm And Selective Harmonic Elimination In Distribution Systems, Thouhidul Islam

Graduate Theses and Dissertations

Reducing the size and weight of a power electric system is a prodigious challenge to researchers as the development of the latest technologies emerge in the field of electrical engineering. A similar urge is there to develop a light-weight mobile power substation (MPS) to use in the electric power distribution systems during emergency conditions. This thesis proposes a power electronics based solution using the modular multilevel converter (MMC) topology to design the MPS system. The market-available power semiconductor devices are analyzed and suitable devices are selected to design the system. The phase-shifted pulse width modulation (PS-PWM) and selective harmonic elimination …


Fabrication And Characterization Of Organic-Inorganic Hybrid Perovskite Solar Cells, Hojjatollah Sarvari Jan 2018

Fabrication And Characterization Of Organic-Inorganic Hybrid Perovskite Solar Cells, Hojjatollah Sarvari

Theses and Dissertations--Electrical and Computer Engineering

Solar energy as the most abundant source of energy is clean, non-pollutant, and completely renewable, which provides energy security, independence, and reliability. Organic-inorganic hybrid perovskite solar cells (PSCs) revolutionized the photovoltaics field not only by showing high efficiency of above 22% in just a few years but also by providing cheap and facile fabrication methods.

In this dissertation, fabrication of PSCs in both ambient air conditions and environmentally controlled N2-filled glove-box are studied. Several characterization methods such as SEM, XRD, EDS, Profilometry, four-point probe measurement, EQE, and current-voltage measurements were employed to examine the quality of thin films …


Designing And Testing 3-D Printed Wafer-Box With Embedded Pzt Sensors To Identify The Shape Effect On Energy Harvesting, Ahmad Jami Safayet Jan 2018

Designing And Testing 3-D Printed Wafer-Box With Embedded Pzt Sensors To Identify The Shape Effect On Energy Harvesting, Ahmad Jami Safayet

College of Graduate Studies: Theses & Dissertations

Piezoelectric energy has been recently paid attention in the field of alternative energy. Day by day the traditional energy sources including Coal tar and oils are becoming scarce. People are heading to an alternative energy source to meet the future energy demand. Piezoelectric energy is one of the competitive energy sources compared to the conventional renewable energy sources including solar, wind, and geothermal power and so on. This energy production method bears enormous research potential because it can be used as the roadway for a new method of power generation. This research project aimed to identify which shaped wafer-box produced …


Flexible Custom Electric Skateboard, Caleb Adam Kelsay Dec 2017

Flexible Custom Electric Skateboard, Caleb Adam Kelsay

Electrical Engineering

At Cal Poly traveling long distances only accessible by foot such as across campus or from the car park to class can be tiring and time-consuming. While not physically demanding, the trek consumes time in our tight college life schedule. To improve our time efficiency this project provides a relatively lightweight, powered solution to quickly and effectively travel long distances while maintaining full control and safety even on hills, a current limitation of traditional skateboards.

This electric skateboard includes the battery capacity to travel all around campus on a single charge with a compact and flexible skateboard design that provides …


Modeling Of Thermally Aware Carbon Nanotube And Graphene Based Post Cmos Vlsi Interconnect, K M Mohsin Nov 2017

Modeling Of Thermally Aware Carbon Nanotube And Graphene Based Post Cmos Vlsi Interconnect, K M Mohsin

LSU Doctoral Dissertations

This work studies various emerging reduced dimensional materials for very large-scale integration (VLSI) interconnects. The prime motivation of this work is to find an alternative to the existing Cu-based interconnect for post-CMOS technology nodes with an emphasis on thermal stability. Starting from the material modeling, this work includes material characterization, exploration of electronic properties, vibrational properties and to analyze performance as a VLSI interconnect. Using state of the art density functional theories (DFT) one-dimensional and two-dimensional materials were designed for exploring their electronic structures, transport properties and their circuit behaviors. Primarily carbon nanotube (CNT), graphene and graphene/copper based interconnects were …


Design Of Dc-Link Vscf Ac Electrical Power System For The Embraer 190/195 Aircraft, Eduardo Francis Carvalho Freitas, Nihad E. Daidzic Sep 2017

Design Of Dc-Link Vscf Ac Electrical Power System For The Embraer 190/195 Aircraft, Eduardo Francis Carvalho Freitas, Nihad E. Daidzic

Journal of Aviation Technology and Engineering

A proposed novel DC-Link VSCF AC-DC-AC electrical power system converter for Embraer 190/195 transport category airplane is presented. The proposed converter could replace the existing conventional system based on the CSCF IDGs. Several contemporary production airplanes already have VSCF as a major or backup source of electrical power. Problems existed with the older VSCF systems in the past; however, the switched power electronics and digital controllers have matured and can be now, in our opinion, safely integrated and replace existing constant-speed hydraulic transmissions powering CSCF AC generators. IGBT power transistors for medium-level power conversion and relatively fast efficient switching are …


Nanostructure Evolution Of Magnetron Sputtered Hydrogenated Silicon Thin Films, Dipendra Adhikari, Maxwell M. Junda, Sylvain X. Marsillac, Robert W. Collins, Nikolas J. Podraza Aug 2017

Nanostructure Evolution Of Magnetron Sputtered Hydrogenated Silicon Thin Films, Dipendra Adhikari, Maxwell M. Junda, Sylvain X. Marsillac, Robert W. Collins, Nikolas J. Podraza

Electrical & Computer Engineering Faculty Publications

Hydrogenated silicon (Si:H) thin films have been prepared by radio frequency (RF) magnetron sputtering. The effect of hydrogen gas concentration during sputtering on the resultant film structural and optical properties has been investigated by real time spectroscopic ellipsometry (RTSE) and grazing incidence x-ray diffraction (GIXRD). The analysis of in-situ RTSE data collected during sputter deposition tracks the evolution of surface roughness and film bulk layer thickness with time. Growth evolution diagrams depicting amorphous, nanocrystalline and mixed-phase regions for low and high deposition rate Si:H are constructed and the effects of process parameter (hydrogen gas concentration, total pressure and RF power) …


Design & Evaluation Of A Hybrid Switched Capacitor Circuit With Wide-Bandgap Devices For Dc Grid Applications, Joshua L. Stewart Jul 2017

Design & Evaluation Of A Hybrid Switched Capacitor Circuit With Wide-Bandgap Devices For Dc Grid Applications, Joshua L. Stewart

Electrical and Computer Engineering ETDs

ABSTRACT

As technologies advance, the rate at which renewable power sources, such as solar photovoltaic (PV) and wind, are being added to the power grid is increasing. Typically, PV power plants require large inverters for direct current to alternating current (DC-AC) power conversion, as well as large transformers to step up voltages to the grid voltage. Offshore wind farms and large PV power plants in remote locations often aggregate power on a DC bus in order to improve efficiency and reduce the cost of power conversion hardware within the energy complex. However, the power must still be converted to AC …


Ionic Thermoelectric Paper, Fei Jiao, Ali Naderi, Dan Zhao, Joshua Schlueter, Maryam Shahi, Jonas Sundström, Hjalmar Granberg, Jesper Edberg, Ujwala Ail, Joseph W. Brill, Tom Lindström, Magnus Berggren, Xavier Crispin Jun 2017

Ionic Thermoelectric Paper, Fei Jiao, Ali Naderi, Dan Zhao, Joshua Schlueter, Maryam Shahi, Jonas Sundström, Hjalmar Granberg, Jesper Edberg, Ujwala Ail, Joseph W. Brill, Tom Lindström, Magnus Berggren, Xavier Crispin

Physics and Astronomy Faculty Publications

Ionic thermoelectric materials, for example, polyelectrolytes such as polystyrene sulfonate sodium (PSSNa), constitute a new class of materials which are attracting interest because of their large Seebeck coefficient and the possibility that they could be used in ionic thermoelectric SCs (ITESCs) and field effect transistors. However, pure polyelectrolyte membranes are not robust or flexible. In this paper, the preparation of ionic thermoelectric paper using a simple, scalable and cost-effective method is described. After a composite was fabricated with nanofibrillated cellulose (NFC), the resulting NFC–PSSNa paper is flexible and mechanically robust, which is desirable if it is to be used in …


A Hardware One-Time Pad Prototype Generator For Localising Cloud Security, Paul Tobin, Lee Tobin, Michael Mckeever, Jonathan Blackledge Jun 2017

A Hardware One-Time Pad Prototype Generator For Localising Cloud Security, Paul Tobin, Lee Tobin, Michael Mckeever, Jonathan Blackledge

Conference papers

In this paper, we examine a system for encrypting data before storing in the Cloud. Adopting this system gives excellent security to stored data and complete control for accessing data by the client at different locations. The motivation for developing this personal encryption came about because of poor Cloud security and doubts over the safety of public encryption algorithms which might contain backdoors. However, side-channel attacks and other unwanted third-party interventions in Cloud security, probably contribute more to the poor security record history. These factors led to the development of a prototype for personalising security locally which defeats cryptanalysis. The …


Chaos-Based Cryptography For Cloud Computing, Paul Tobin, Lee Tobin, Michael Mckeever, Jonathan Blackledge Jun 2017

Chaos-Based Cryptography For Cloud Computing, Paul Tobin, Lee Tobin, Michael Mckeever, Jonathan Blackledge

Conference papers

Cloud computing and poor security issues have quadrupled over the last six years and with the alleged presence of backdoors in common encryption ciphers, has created a need for personalising the encryption process by the client. In 2007, two Microsoft employees gave a presentation ``On the Possibility of a backdoor in the NIST SP800-90 Dual Elliptic Curve Pseudo Random Number Generators'' and was linked in 2013 by the New York Times with notes leaked by Edward Snowden. This confirmed backdoors were placed, allegedly, in a number of encryption systems by the National Security Agency, which if true creates an urgent …


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 …