We consider the Lighthill-Whitman-Richards traffic model on a network consisting of a highway with an off ramp, connected by a junction. We compare the known coupling conditions for the evolution of traffic at the junction and suggest a novel improvement to the existing conditions. That is, we resolve the spurious effects that arise in standard models, namely clogging of the main highway and vehicle destination changes. We achieve this by tracking vehicle density buildup in the form of a queue, which is modeled by an ODE. We define the solution to the Riemann problem at the junction using the supply and demand functions. The numerical approximation is carried out using a modified Godunov scheme, adjusted to take into account the effects of an emptying queue. Exact and numerical comparisons of the model with existing models verify that the number of vehicles who wish to exit are preserved and the nonphysical clogging of the main highway does not occur. / Mathematics

Reflectarray antennas uniquely combine the advantages of parabolic reflectors and phased array antennas. Comprised of planar structures similar to phased arrays and utilizing quasi-optical excitation similar to parabolic reflectors, reflectarray antennas provide beam steering without the need of complex and lossy feed networks. Chapter 1 discusses the basic theory of reflectarray and its design. A brief summary of previous work and current research status is also presented. The inherent advantages and drawbacks of the reflectarray are discussed. In chapter 2, a novel theoretical approach to extract the reflection coefficient of reflectarray unit cells is developed. The approach is applied to single-resonance unit cell elements under normal and waveguide incidences. The developed theory is also utilized to understand the difference between the TEM and TE10 mode of excitation. Using this theory, effects of different physical parameters on reflection properties of unit cells are studied without the need of full-wave simulations. Detailed analysis is performed for Ka-band reflectarray unit cells and verified by full-wave simulations. In addition, an approach to extract the Q factors using full-wave simulations is also presented. Lastly, a detailed study on the effects of inter-element spacing is discussed. Q factor theory discussed in chapter 2 is extended to account for the varying incidence angles and polarizations in chapter 3 utilizing Floquet modes. Emphasis is laid on elements located on planes where extremities in performance tend to occur. The antenna element properties are assessed in terms of maximum reflection loss and slope of the reflection phase. A thorough analysis is performed at Ka band and the results obtained are verified using full-wave simulations. Reflection coefficients over a 749-element reflectarray aperture for a broadside radiation pattern are presented for a couple of cases and the effects of coupling conditions in conjunction with incidence angles are demonstrated. The presented theory provides explicit physical intuition and guidelines for efficient and accurate reflectarray design. In chapter 4, tunable reflectarray elements capacitively loaded with Barium Strontium Titanate (BST) thin film are shown. The effects of substrate thickness, operating frequency and deposition pressure are shown utilizing coupling conditions and the performance is optimized. To ensure minimum affects from biasing, optimized biasing schemes are discussed. The proposed unit cells are fabricated and measured, demonstrating the reconfigurability by varying the applied E-field. To demonstrate the concept, a 45 element array is also designed and fabricated. Using anechoic chamber measurements, far-field patterns are obtained and a beam scan up to 25o is shown on the E-plane. Overall, novel theoretical approaches to analyze the reflection properties of the reflectarray elements using Q factors are developed. The proposed theoretical models provide valuable physical insight utilizing coupling conditions and aid in efficient reflectarray design. In addition, for the first time a continuously tunable reflectarray operating at Ka-band is presented using BST technology. Due to monolithic integration, the technique can be extended to higher frequencies such as V-band and above.