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Full-Text Articles in Engineering

An Engineering Physics Introduction To Electronics For Ece Sophomores, Steve Eugene Watkins Jun 2013

An Engineering Physics Introduction To Electronics For Ece Sophomores, Steve Eugene Watkins

Electrical and Computer Engineering Faculty Research & Creative Works

Electronic devices and circuits are fundamental parts of undergraduate curricula in electrical engineering and computer engineering (ECE). A sophomore-level course that gives a balanced treatment of semiconductor device physics and electronic circuit analysis is described. The course topics are semiconductor physics, diodes, transistors, operational amplifiers, and optoelectronics. The course is a prerequisite for upper-level electronics, semiconductor physics, and semiconductor circuit layout curricula. A stronger link between physical principles and device behavior and an improved sequence in electronics instruction are outcomes.


Surface Waves In Three-Dimensional Electromagnetic Composites And Their Effect On Homogenization, Xiaoyan Y.Z. Xiong, Li (Lijun) Jun Jiang, Vadim A. Markel, Igor Tsukerman May 2013

Surface Waves In Three-Dimensional Electromagnetic Composites And Their Effect On Homogenization, Xiaoyan Y.Z. Xiong, Li (Lijun) Jun Jiang, Vadim A. Markel, Igor Tsukerman

Electrical and Computer Engineering Faculty Research & Creative Works

Reflection and transmission of electromagnetic waves at the boundaries of periodic composites (electromagnetic/optical metamaterials) depends in general on both bulk and surface waves. We investigate the interplay of these two contributions using three-dimensional full-wave numerical simulations and a recently developed non-asymptotic homoge-nization theory. © 2013 Optical Society of America.


A Numerically Efficient Formulation For Time-Domain Electromagnetic- Semiconductor Cosimulation For Fast-Transient Systems, Quan Chen, Wim Schoenmaker, Guanhua Chen, Lijun Jiang, Ngai Wong May 2013

A Numerically Efficient Formulation For Time-Domain Electromagnetic- Semiconductor Cosimulation For Fast-Transient Systems, Quan Chen, Wim Schoenmaker, Guanhua Chen, Lijun Jiang, Ngai Wong

Electrical and Computer Engineering Faculty Research & Creative Works

We report recent progress in developing a numerically efficient formulation for electromagnetic-technology computer-aided design cosimulation for fast-transient computations. The difficulties underlying the currently existing transient formulation stemming from the vector potential-scalar potential (A-V) framework are analyzed. A time-domain electric field-scalar potential (E-V) framework is then developed via equation and variable transformations. This results in better-conditioned systems that are friendly to iterative solutions at fast switching times. Numerical examples show that the proposed E-V solver renders a useful tool for addressing multidomain simulation. © 1982-2012 IEEE.


Economic Modeling Of Compressed Air Energy Storage, Yang Gu, James D. Mccalley, Ming Ni, Rui Bo Apr 2013

Economic Modeling Of Compressed Air Energy Storage, Yang Gu, James D. Mccalley, Ming Ni, Rui Bo

Electrical and Computer Engineering Faculty Research & Creative Works

Due to the variable nature of wind resources, the increasing penetration level of wind power will have a significant impact on the operation and planning of the electric power system. Energy storage systems are considered an effective way to compensate for the variability of wind generation. This paper presents a detailed production cost simulation model to evaluate the economic value of compressed air energy storage (CAES) in systems with large-scale wind power generation. The co-optimization of energy and ancillary services markets is implemented in order to analyze the impacts of CAES, not only on energy supply, but also on system …


Quantum Network Theory Of Computing With Respect To Entangled Flux Qubits And External Perturbation, C. A. Cain, Cheng-Hsiao Wu Apr 2013

Quantum Network Theory Of Computing With Respect To Entangled Flux Qubits And External Perturbation, C. A. Cain, Cheng-Hsiao Wu

Electrical and Computer Engineering Faculty Research & Creative Works

In this work, we attempt to show the differences between traditional qubit-based spintronic methodology for quantum computation and the possible ballistic quantum network implementations. Flux qubits can be considered topologically similar to the persistent currents possessed as the angular momentum in Aharonov-Bohm loops, which can be coupled and thus entangled together. Since entanglement is guaranteed for coupled quantum networks, starting from a point-contacted situation, we first investigate how varying the degree of entanglement strength can affect the superposition of the four possible states for two isolated flux qubits being brought together. In general, the superposition is destroyed once the degree …


Real Time Bridge Scour Monitoring With Magneto-Inductive Field Coupling, Andriy V. Radchenko, David Pommerenke, Genda Chen, Pratik Maheshwari, Satyajeet R. Shinde, Viswa Pilla, Y. Rosa Zheng Mar 2013

Real Time Bridge Scour Monitoring With Magneto-Inductive Field Coupling, Andriy V. Radchenko, David Pommerenke, Genda Chen, Pratik Maheshwari, Satyajeet R. Shinde, Viswa Pilla, Y. Rosa Zheng

Electrical and Computer Engineering Faculty Research & Creative Works

Scour was responsible for most of the U.S. bridges that collapsed during the past 40 years. The maximum scour depth is the most critical parameter in bridge design and maintenance. Due to scouring and refilling of river-bed deposits, existing technologies face a challenge in measuring the maximum scour depth during a strong flood. In this study, a new methodology is proposed for real time scour monitoring of bridges. Smart Rocks with embedded electronics are deployed around the foundation of a bridge as field agents. With wireless communications, these sensors can send their position change information to a nearby mobile station. …


A Hybrid Electromagnetics-Circuit Simulation Method Exploiting Discontinuous Galerkin Finite Element Time Domain Method, Ping Li, Li (Lijun) Jun Jiang Feb 2013

A Hybrid Electromagnetics-Circuit Simulation Method Exploiting Discontinuous Galerkin Finite Element Time Domain Method, Ping Li, Li (Lijun) Jun Jiang

Electrical and Computer Engineering Faculty Research & Creative Works

A hybrid electromagnetics (EM)-circuit simulation method employing the discontinuous Galerkin finite element time domain (DGFETD) method is developed to model single lumped port networks comprised of both linear and non-linear elements. The whole computational domain is split into two subsystems: one is the EM subsystem that is analyzed by DGFETD, while another is the circuit subsystem that is modeled by the Modified Nodal Analysis method to generate a circuit subsystem. The coupling between the EM and circuit subsystems is achieved through a lumped port. Due to the local properties of DGFETD operations, only small coupling matrix equation systems are involved. …


The Serial Resonant Antenna For The Large Field Of View Magnetic Resonance Imaging, Bo Zhu, Ed Xuekui Wu, Patrick Peng Gao, Peng Cao, Lijun Jiang Feb 2013

The Serial Resonant Antenna For The Large Field Of View Magnetic Resonance Imaging, Bo Zhu, Ed Xuekui Wu, Patrick Peng Gao, Peng Cao, Lijun Jiang

Electrical and Computer Engineering Faculty Research & Creative Works

A serial resonant antenna for the large field of view (FOV) magnetic resonance imaging (MRI) is presented. It consists of metallic patches cascaded through lumped capacitors in serial on the top layer of a grounded dielectric substrate. The theoretical analysis show that at the resonant frequency, uniformly distributed current with zero phase delay is produced independent of the antenna length, hence a uniform magnetic field for large FOV MRI can be achieved. Integrated with the L-shaped tunable matching network, the antenna can be tuned easily to operate rigorously at the working frequency of the MRI system. The numerical modeling, physical …


A Novel Fast Solver For Poisson's Equation With Neumann Boundary Condition, Zuhui Ma, Weng Cho Chew, Lijun Jiang Jan 2013

A Novel Fast Solver For Poisson's Equation With Neumann Boundary Condition, Zuhui Ma, Weng Cho Chew, Lijun Jiang

Electrical and Computer Engineering Faculty Research & Creative Works

In this paper, we present a novel fast method to solve Poisson's equation in an arbitrary two dimensional region with Neumann boundary condition, which are frequently encountered in solving electrostatic boundary problems. The basic idea is to solve the original Poisson's equation by a two-step procedure. In the first stage, we expand the electric field of interest by a set of tree basis functions and solve it with a fast tree solver in O(N) operations. The field such obtained, however, fails to expand the exact field because the tree basis is not curl-free. Despite of this, we can retrieve the …


Source Reconstruction Method-Based Radiated Emission Characterization For Pcbs, Ping Li, Li Jun Jiang Jan 2013

Source Reconstruction Method-Based Radiated Emission Characterization For Pcbs, Ping Li, Li Jun Jiang

Electrical and Computer Engineering Faculty Research & Creative Works

A technique based on a source reconstruction method (SRM) is employed to reproduce the same electromagnetic radiated emissions from printed circuit boards (PCB). This equivalent source is composed of both the electric and magnetic currents that are represented by a set of Rao-Wilton-Glisson basis functions. Planar magnetic near field is served as the input information for the SRM. The equivalent source is inversely acquired through the conjugate gradient algorithm that minimizes the norm of the residue. Magnetic field integral equation is utilized to find out the unknown coefficients of the currents. This method is directly extended to model the emissions …


Integration Of Arbitrary Lumped Multiport Circuit Networks Into The Discontinuous Galerkin Time-Domain Analysis, Ping Li, Li Jun Jiang Jan 2013

Integration Of Arbitrary Lumped Multiport Circuit Networks Into The Discontinuous Galerkin Time-Domain Analysis, Ping Li, Li Jun Jiang

Electrical and Computer Engineering Faculty Research & Creative Works

A hybrid electromagnetic (EM) circuit simulator is proposed for incorporating multiport lumped circuit networks through their admittance matrices into the discontinuous Galerkin time-domain (DGTD) method. The admittance matrix in the Laplace domain can be derived analytically or obtained from network parameters such as -parameters in the frequency domain. To convert frequency-dependent -parameters into the admittance matrix in the Laplace domain, the vector-fitting technique is employed to facilitate the mapping process. The computational domain of interest is split into two subdomains. One is the EM part solved by the DGTD, and another is the circuit part modeled by the basic I-V …


A New Efie Method Based On Coulomb Gauge For The Low-Frequency Electromagnetic Analysis, Xiaoyan Y.Z. Xiong, Li (Lijun) Jun Jiang, Wei E.I. Sha, Yat Hei Lo Jan 2013

A New Efie Method Based On Coulomb Gauge For The Low-Frequency Electromagnetic Analysis, Xiaoyan Y.Z. Xiong, Li (Lijun) Jun Jiang, Wei E.I. Sha, Yat Hei Lo

Electrical and Computer Engineering Faculty Research & Creative Works

To solve the low-frequency breakdown inherent from the electric field integral equation (EFIE), an alternative new form of the EFIE is proposed by using the Coulomb-gauge Green's function of quasi-static approximation. Di®erent from the commonly adopted Lorentz-gauge EFIE, the Coulomb-gauge EFIE separates the solenoidal and irrotational surface currents explicitly, which captures inductive and capacitive responses through electrodynamic and electrostatic Green's functions, respectively. By applying existing techniques such as the loop-tree decomposition, frequency normalization, and basis rearrangement, the Coulomb-gauge EFIE also can remedy the low-frequency breakdown problem. Through comparative studies between the Lorentz-gauge and Coulomb-gauge EFIE approaches from mathematical, physical and …


Reflection-Based Phase-Shifted Long Period Fiber Grating For Simultaneous Measurement Of Temperature And Refractive Index, Jie Huang, Xinwei Lan, Amardeep Kaur, Haining Wang, Lei Yuan, Hai Xiao Jan 2013

Reflection-Based Phase-Shifted Long Period Fiber Grating For Simultaneous Measurement Of Temperature And Refractive Index, Jie Huang, Xinwei Lan, Amardeep Kaur, Haining Wang, Lei Yuan, Hai Xiao

Electrical and Computer Engineering Faculty Research & Creative Works

We report a reflection-based phase-shifted long period fiber grating (PS-LPFG) and demonstrate its capability for simultaneous measurement of temperature and external reflective index (RI). The sensor device comprises a grating directly written by CO2 laser and silver-coated end face. A π-shifted LPFG is presented with two attenuation bands through its reflection spectrum. These two bands have different sensitivity towards temperature and external RI that can be used for simultaneous measurement of the two variables. The experimental results show that this probe-type PS-LPFG performs well in terms of linearity and sensitivity.


Resilience Experiments On The Gpeni Testbed, Egemen K. Çetinkaya, Dongsheng Zhang, Mohammed J. F. Alenazi, Yufei Cheng, Parker Riley, James P. G. Sterbenz Jan 2013

Resilience Experiments On The Gpeni Testbed, Egemen K. Çetinkaya, Dongsheng Zhang, Mohammed J. F. Alenazi, Yufei Cheng, Parker Riley, James P. G. Sterbenz

Electrical and Computer Engineering Faculty Research & Creative Works

No abstract provided.


A Wide-Band Equivalent Source Reconstruction Method Exploiting The Stoer-Bulirsch Algorithm With The Adaptive Frequency Sampling, P. Li, Y. Li, Lijun Jiang, J. Hu Jan 2013

A Wide-Band Equivalent Source Reconstruction Method Exploiting The Stoer-Bulirsch Algorithm With The Adaptive Frequency Sampling, P. Li, Y. Li, Lijun Jiang, J. Hu

Electrical and Computer Engineering Faculty Research & Creative Works

Conventional equivalent source reconstruction methods are only operating at a single frequency point each time. However, for wide-band radiators, their computation cost of the source reconstruction increases with the number of frequency points. In this communication, a wide-band equivalent source reconstruction method (WB-SRM) based on the Neville-type Stoer-Bulirsch (SB) algorithm is developed. Since the SB algorithm is a recursive tabular method, it can perform the interpolation over a broadband using a single rational function without any singularity issue. Supported by the adaptive frequency sampling (AFS) scheme, the number of required sampling data is significantly reduced, which leads to much better …


Enhanced-Resolution Folded C-Section Phaser, S. Gupta, Lijun Jiang, C. Caloz Jan 2013

Enhanced-Resolution Folded C-Section Phaser, S. Gupta, Lijun Jiang, C. Caloz

Electrical and Computer Engineering Faculty Research & Creative Works

A new C-section phasers based on folded coupled-line couplers has been proposed for high-resolution analog signal processing (ASP). The folded configuration of the coupled-line coupler provides a tight coupling in a planar configuration between the input and the coupled port, using sections of relatively small coupling values. It subsequently provides a larger group delay swing in the corresponding C-section phaser required for enhanced resolution ASP. The proposed phaser is demonstrated using full-wave simulations. © 2013 IEEE.


The Numerical Steepest Descent Path Method For Calculating Physical Optics Integrals On Smooth Conducting Quadratic Surfaces, Yu Mao Wu, Li (Lijun) Jun Jiang, Wei E.I. Sha, Weng Cho Chew Jan 2013

The Numerical Steepest Descent Path Method For Calculating Physical Optics Integrals On Smooth Conducting Quadratic Surfaces, Yu Mao Wu, Li (Lijun) Jun Jiang, Wei E.I. Sha, Weng Cho Chew

Electrical and Computer Engineering Faculty Research & Creative Works

In this paper, we use the numerical steepest descent path (NSDP) method to analyze the highly oscillatory physical optics (PO) integral on smooth conducting parabolic surfaces, including both monostatic and bistatic cases. Quadratic variations of the amplitude and phase functions are used to approximate the integrand of PO integral. Then the surface PO integral is reduced into several highly oscillatory line integrals. By invoking the NSDP method, these highly oscillatory PO line integrals are defined on the corresponding NSDPs. Furthermore, the critical point contributions for the PO integral are exactly extracted and represented based on the NSDPs. The proposed NSDP …


An Iterative Source Reconstruction Method Exploiting Phaseless Electric Field Data, Ping Li, Lijun Jiang Jan 2013

An Iterative Source Reconstruction Method Exploiting Phaseless Electric Field Data, Ping Li, Lijun Jiang

Electrical and Computer Engineering Faculty Research & Creative Works

Conventional equivalent source reconstruction methods (SRM) require both phase and amplitude information of the acquired field data. However, there are situations where the phase information is not available or impractical to obtain. Hence, the development of SRM using phaseless fields is important. In this paper, a novel iterative SRM based on phaseless electric fields is presented. The reconstructed equivalent current source can be electric, magnetic current, or both. They can fit the physical geometry of the radiator. Electric field integral equation (EFIE) is employed to build the relationship between the reconstructed current source and measured fields. It can precisely reproduce …


Tests For Time Domain Em Solvers For Stability And Towards Passivity, G. Antonini, Albert E. Ruehli, D. Romano, L. (Lijun) Jiang Jan 2013

Tests For Time Domain Em Solvers For Stability And Towards Passivity, G. Antonini, Albert E. Ruehli, D. Romano, L. (Lijun) Jiang

Electrical and Computer Engineering Faculty Research & Creative Works

Stability issues for time domain integral equation based EM solvers have been a problem for a many years. In this paper, we consider ways to test for stability as well as other criteria towards passivity. The tests are designed to monitor the EM solver while the time domain solution is progressing. Hence, the solution can be terminated and improved with a passivization technique when a violation of the criterion has been detected. Also, the tests help to understand the cause of instabilities and results can be used for passivization. © 2013 IEEE.


Large-Scale Electromagnetic Computation For Modeling And Applications, Qing Huo Liu, Lijun Jiang, Weng Cho Chew Jan 2013

Large-Scale Electromagnetic Computation For Modeling And Applications, Qing Huo Liu, Lijun Jiang, Weng Cho Chew

Electrical and Computer Engineering Faculty Research & Creative Works

The papers in this special issue are devoted to the topic of large-scale electromagnetic computation methods for modeling and applicatoins. © 1963-2012 IEEE.


Magneto-Electric Dipole Antenna Based On Differentially-Excited Composite Right/Left-Handed (Crlh) Transmission Lines, S. Gupta, Lijun Jiang, C. Caloz Jan 2013

Magneto-Electric Dipole Antenna Based On Differentially-Excited Composite Right/Left-Handed (Crlh) Transmission Lines, S. Gupta, Lijun Jiang, C. Caloz

Electrical and Computer Engineering Faculty Research & Creative Works

A new magneto-electric (ME) dipole antenna array is proposed which provides a combined electric- and magnetic-dipole response in a planar configuration. The proposed structure is based on Composite Right/Left-Handed (CRLH) transmission lines, which is differentially fed and operated in the zeroth-order regime to provide maximal gain for the combined radiators. The basic principle of the proposed ME-dipole antenna is discussed and demonstrated with full-wave simulation results. © 2013 IEEE.


Skin-Effect Loss Models For Time- And Frequency-Domain Peec Solver, Albert E. Ruehli, Giulio Antonini, Li (Lijun) Jun Jiang Jan 2013

Skin-Effect Loss Models For Time- And Frequency-Domain Peec Solver, Albert E. Ruehli, Giulio Antonini, Li (Lijun) Jun Jiang

Electrical and Computer Engineering Faculty Research & Creative Works

A challenging and interesting issue for the solution of large electromagnetic problems is the efficient, sufficiently accurate modeling of the broadband skin-effect loss for conducting planes and 3-D shapes. The inclusion of such models in an electromagnetic (EM) solver can be very costly in compute time and memory requirements. These issues are particularly important for the class of signal, power, and noise integrity (NI) problems. In this paper, we concentrate on partial element equivalent circuit (PEEC)-type methods which are suitable for the solution of this class of problems. Progress has been made recently in the design of skin-effect models. The …


Pulsed Line Source Response Of A Thin Sheet With High-Contrast Dielectric And Conductive Properties - A Time-Domain Analysis, Adrianus T. De Hoop, Ling Ling Meng, Li (Lijun) Jun Jiang Jan 2013

Pulsed Line Source Response Of A Thin Sheet With High-Contrast Dielectric And Conductive Properties - A Time-Domain Analysis, Adrianus T. De Hoop, Ling Ling Meng, Li (Lijun) Jun Jiang

Electrical and Computer Engineering Faculty Research & Creative Works

A methodology is presented for analytically modeling the reflection and transmission of line-source excited pulsed electromagnetic fields at a thin, planar layer with high-contrast dielectric and conductive properties. Closed-form analytic time-domain expressions are derived for the field components in a 2-D setting via an extension of the Cagniard-DeHoop method. The response to the power exponential source and the power exponential monocycle pulse were studied using the newly proposed method. © 2013 IEEE.


Finite-Width Feed And Load Models, Yat Hei Lo, Li (Lijun) Jun Jiang, Weng Cho Chew Jan 2013

Finite-Width Feed And Load Models, Yat Hei Lo, Li (Lijun) Jun Jiang, Weng Cho Chew

Electrical and Computer Engineering Faculty Research & Creative Works

We demonstrate a new method of applying the feed model for the method of moments (MoM) formulation for the electric field integral equation (EFIE). The model is based around a previously reported magnetic ribbon current model which is accurate and allows for a finite width of the feed port. However, with proper approximations, one can reduce the formulation such that the magnetic field operator can be removed in order to simplify computations arising from the curl of the dyadic Green's function and its singularities. We show here that the new feed model can also be used to model a lumped …


A Systematic Electromagnetic-Circuit Method For Emi Analysis Of Coupled Interconnects On Dispersive Dielectrics, Min Tang, Jiaqing Lu, Junfa Mao, Lijun Jiang Jan 2013

A Systematic Electromagnetic-Circuit Method For Emi Analysis Of Coupled Interconnects On Dispersive Dielectrics, Min Tang, Jiaqing Lu, Junfa Mao, Lijun Jiang

Electrical and Computer Engineering Faculty Research & Creative Works

This paper presents a systematic electromagnetic-circuit method for efficient simulation of coupled interconnects on dispersive dielectrics in the presence of electromagnetic interference. The dispersive substrate of coupled interconnects is characterized by the multiterm Debye or Lorentz formula. According to the differential equations derived from the equivalent circuit modeling of dispersive media, the full-wave finite-difference time-domain algorithm is utilized to capture accurate field in a dispersive substrate. The effects of incident field, under the quasi-TEM assumption, are then represented by the distributed sources along transmission lines. An efficient algorithm based on the waveform relaxation with transverse partitioning (WR-TP) is employed for …


No-Boundary Thinking In Bioinformatics Research, Xiuzhen Huang, Barry Bruce, Alison Buchan, Clare Bates Congdon, Carole L. Cramer, Steven F. Jennings, Hongmei Jiang, Zenglu Li, Gail Mcclure, Rick Mcmullen, Jason H. Moore, Bindu Nanduri, Joan Peckham, Andy Perkins, Shawn W. Polson, Bhanu Rekepalli, Saeed Salem, Jennifer Specker, Donald C. Wunsch, Donghai Xiong, Shuzhong Zhang, Zhongming Zhao Jan 2013

No-Boundary Thinking In Bioinformatics Research, Xiuzhen Huang, Barry Bruce, Alison Buchan, Clare Bates Congdon, Carole L. Cramer, Steven F. Jennings, Hongmei Jiang, Zenglu Li, Gail Mcclure, Rick Mcmullen, Jason H. Moore, Bindu Nanduri, Joan Peckham, Andy Perkins, Shawn W. Polson, Bhanu Rekepalli, Saeed Salem, Jennifer Specker, Donald C. Wunsch, Donghai Xiong, Shuzhong Zhang, Zhongming Zhao

Electrical and Computer Engineering Faculty Research & Creative Works

Currently there are definitions from many agencies and research societies defining "bioinformatics" as deriving knowledge from computational analysis of large volumes of biological and biomedical data. Should this be the bioinformatics research focus? We will discuss this issue in this review article. We would like to promote the idea of supporting human-infrastructure (HI) with no-boundary thinking (NT) in bioinformatics (HINT).


Overview Of Large-Scale Computing: The Past, The Present, And The Future, Weng Cho Chew, Li (Lijun) Jun Jiang Jan 2013

Overview Of Large-Scale Computing: The Past, The Present, And The Future, Weng Cho Chew, Li (Lijun) Jun Jiang

Electrical and Computer Engineering Faculty Research & Creative Works

This is a brief review of the development of computational electromagnetics (CEM) to partially summarize its achievements, issues, and possibilities. © 1963-2012 IEEE.


Adaptive Multi-Vehicle Area Coverage Optimization System And Method, Ryan J. Meuth, John Lyle Vian, Emad W. Saad, Donald C. Wunsch Jan 2013

Adaptive Multi-Vehicle Area Coverage Optimization System And Method, Ryan J. Meuth, John Lyle Vian, Emad W. Saad, Donald C. Wunsch

Electrical and Computer Engineering Faculty Research & Creative Works

A mission planning system for determining an optimum use of a plurality of vehicles in searching a predefined geographic area (PGA). A discretizer subsystem may be used for sensing the capabilities of each vehicle to produce a point set defining a number of points within the PGA that the vehicles must traverse to completely search the PGA. A task allocator subsystem may determine an optimum division of the PGA into different subregions to be handled by specific ones of the vehicles, thus to minimize an overall time needed to search the PGA. A path optimizer subsystem may determine an optimum …