Thesis (MScEng (Electrical and Electronic Engineering))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Worldwide, more large antenna arrays are being deployed in areas of science previously dominated by other antenna geometries. Applications for large arrays include Radar, Satellite Communications and Radio Astronomy. Even though the use of large arrays solve some of the difficulties posed by more traditional antennas, new challenges are also faced. One of these challenges is the problem of noise coupling, and how the overall system performance is affected by it. The Focal Plane Array (FPA), which is a new example of a large antenna array, is currently being researched at a number of institutions worldwide for use in Radio Astronomy. As a result, FPA’s were used as an example element to demonstrate the practical importance of this research. In this study, the effect of mutual coupling on the noise performance of FPA’s was illustrated. This was done by calculating the mutual coupling between the elements of the array, and then calculating the noise power received by each element as a result of the mutual coupling. Next, the Active Noise Figure and Active Noise Temperature were calculated. These parameters were introduced to visualise the effect of mutual coupling on the overall noise performance of the array. Since FPA’s are by definition large, conventional brute-force analysis techniques are very resource intensive. Solving the coupling terms using these methods therefore requires the use of computer clusters even during the design phase of the antenna, which is very expensive. A method was therefore developed to calculate the coupling terms of a large array using Periodic Boundary Conditions. The method uses infinite array analysis, which resulted in an improvement in memory usage in orders of magnitude. This improvement comfortably places the memory requirements for the analysis of large arrays within the range of current personal computers. The results also displayed a reasonable amount of accuracy for use during the design phase of an array. The additional noise power on each element as a result of mutual coupling were also calculated. This was achieved by developing an equivalent circuit diagram that represents the system in terms of the noise and transmission parameters of the LNA of each receiver channel, and the coupling terms of the antenna array. Lastly, the active noise temperature and active noise figure are calculated. The theory was implemented by means of a script with a graphical user interface, to provide easy-to-use access to the theory. A quick reference table of estimated noise coupling penalty versus first term coupling and LNA noise temperature was also compiled. The results of an example calculation showed a significant amount of noise coupling in an 8×8 Vivaldi array. The noise coupling resulted in an increase in system noise temperature, Tsys, in the order of 9% of the LNA noise temperature, TLNA. According to the SKA Tsys budget, this results in an approximate Tsys increase of 1.3 Kelvin. In the context of Radio Astronomy, this additional source of noise cannot be ignored, as it can greatly affect the usebility of the telescope for certain areas of research. / AFRIKAANSE OPSOMMING: Groot antennaskikkings word deesdae al hoe meer ingespan in plek van ander tradisionele antennamodelle. Toepassings vir groot antennaskikkings sluit Radar, Satellietkommunikasie en Radioastronomie in. Alhoewel die gebruik van groot antennaskikkings baie van die probleme wat deur ander tradisionele antennamodelle veroorsaak word oplos, word nuwe uitdagings terselfdertyd geskep. Een van hierdie nuwe uitdagins is ruiskoppelling en hoe dit die ruisgedrag van die stelsel as ’n geheel affekteer. ’n Beeldvlakskikking (FPA), is ’n opwindende nuwe voorbeeld van ’n groot antennaskikking en die moontlikheid vir die gebruik daarvan in radioastronomie word tans wêreldwyd nagevors. Om hierdie rede is die FPA gekies as voorbeeldelement om die bruikbaarheid van hierdie navorsing in die praktyk te beklemtoon. In hierdie studie word die effek van wedersydse koppelling op die ruisgedrag van FPA’s geïllustreer. Dit word gedoen deur eers die wedersydse koppelling tussen die elemente van die antennaskikking te bereken en dan die ruisdrywing wat deur elke element ontvang word as gevolg van wedersydse koppelling. Daarna word die Aktiewe Ruistal en die Aktiewe Ruistemperatuur bereken. Hierdie nuwe parameters word bekendgestel om die gevolge van wedersydse koppelling op die ruisgedrag van die stelsel as ’n geheel te visualiseer. Omdat FPA’s per definisie groot is, vereis die analise daarvan deur middel van konvensionele metodes baie rekenaar hulpbronne. Hierdie metodes vereis dus die gebruik van rekenaarbondels of superrekenaars selfs gedurende die ontwerpfase van die antenna, wat baie duur en onprakties is. Daar is dus ’n metode ontwikkel wat gebruik maak van periodiese randvoorwaardes om groot antennaskikkings te analiseer. Die metode benader ’n groot antennaskikking as ’n eindig-opgewekte oneindige skikking van antennas. As gevolg hiervan, word die geheueverbruik met ordegroottes verbeter. Hierdie verbetering plaas dus die analise van groot antennaskikkings binne die vermoëns van huidige persoonlike rekenaars. Die resultate wys ook ’n aanvaarbare graad van akkuraatheid vir gebruik gedurende die ontwerpfase van die skikking. Die bykomende ruisdrwying op elke element as gevolg van wedersydse koppelling is ook bereken. Om dit te vermag, is daar ’n ekwivalente stroombaandiagram ontwikkel wat die gekoppelde stelsel in terme van die ruis- en transmissieparameters van die laeruisversterker (LNA) aan elke ontvangerkanaal en die koppelterme van die antenna skikking voorstel. Laastens word die aktiewe ruistal en die aktiewe ruistermperatuur ook bereken. Die teorie is geïmplimenteer deur gebruik te maak van ’n grafiesegebruikerskoppelvlak (GUI). Die GUI verskaf aan die gebruiker maklike toegang tot die teorie wat onwikkel is in hierdie navorsing. Daar is ook ’n snelnaslaantabel geskep met benaderde waardes van ruiskoppelling vir ’n verskeidenheid waardes van LNA ruistemperature en eerste element koppelling. Die resultate van ’n 8×8 Vivaldiskikking voorbeeld, het ’n beduidende hoeveelheid ruiskoppelling getoon. Die ruiskoppelling het ’n maksimum toename in stelsel ruistemperatuur, Tsys, van ongeveer 9% van die LNA ruistemperatuur tot gevolg gehad. Volgens die huidige Tsys begroting van die SKA, kom dit neer op ’n Tsys toename van byna 1.3 Kelvin. In die konteks van die radioastronomie, kan hierdie toename in ruistemperatuur nie geïgnoreer word nie aangesien dit die bruikbaarheid van die teleskoop vir sekere velde van navorsing nadelig kan beïnvloed.

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In this dissertation, a finitely convergent disjunctive programming procedure, the Convex Hull Tree (CHT) algorithm, is proposed to obtain the convex hull of a general mixed–integer linear program with bounded integer variables. The CHT algorithm constructs a linear program that has the same optimal solution as the associated mixed-integer linear program. The standard notion of sequential cutting planes is then combined with ideasunderlying the CHT algorithm to help guide the choice of disjunctions to use within a new cutting plane method, the Cutting Plane Tree (CPT) algorithm. We show that the CPT algorithm converges to an integer optimal solution of the general mixed-integer linear program with bounded integer variables in finitely many steps. We also enhance the CPT algorithm with several techniques including a “round-of-cuts” approach and an iterative method for solving the cut generation linear program (CGLP). Two normalization constraints are discussed in detail for solving the CGLP. For moderately sized instances, our study shows that the CPT algorithm provides significant gap closures with a pure cutting plane method.

Liquid crystals are traditionally aligned via a rubbed substrate. The rubbing creates anisotropic defects and strain in the material which provide an energetically favorable orientation for the liquid crystal (LC). This is a well-developed technology that is used in virtually all liquid crystal displays (LCDs). However, it is only capable of uniform alignment on a large planar substrate. This work utilizes a new class of photoalignment materials (PM) that replace the traditional buffing technique. PMs allow for patterned LC alignment using polarized photo-lithography. Further, instead of using a nematic liquid crystal, a UV curable liquid crystal polymer (LCP) is coated on this patterned alignment material. This generates a cured, aligned, and patterned thin film that has retardance and diattenuation if a dichroic dye is incorporated in the LCP. Using these materials and methods, retarders, polarizers, and polarized fluorescent sources are fabricated with dimensions as small as a few microns. In addition to exploring the material and processing properties of the PM and LCP system, arrays of micropolarizer are fabricated for the construction of a prototype polarization camera, termed an imaging polarimeter. An imaging polarimeter is a device that measures not only the intensity but also the polarization state of a light field. My imaging polarimeter design incorporates a patterned LCP polarizer focal plane array (FPA) that is aligned and mounted to a charge-couple-device (CCD) image sensor. A polarizer FPA allows an individual pixel on a CCD sensor to detect a unique polarization state, such as a specific linear polarization orientation or right or left handed circular polarization. Neighboring pixels are designed to detect different states and each cluster can then estimate the incident polarization state. Results of a linear LCP polarimeter operating in the visible spectrum are presented.

Single frame ultrasound B-mode image quality is largely governed by the ability to focus the ultrasound beam over a range in depths both in transmission and reception. By developing a comprehensive understanding of acoustic wave propagation two signal processing methods were identified for solving the transmission problem. We made use of both the impulse response using the classical point spread function (PSF) and the spatial sensitivity function (SSF) which describes the spatial distribution at a particular time. Using the angular spectrum method, an accurate analytical model was developed for the field distribution arising from a finite geometry, apodized and focused, plane piston transducer. While there is a thorough understanding of the radiated field arising from uniformly excited plane piston transducers, the focused equivalent (i.e., one that allows a continuous change in phase over the plane piston surface) is incomplete and assumes the Fresnel approximation. Our model addresses the effects of diffraction and evanescent waves without the use of the Fresnel approximation and is applicable at all near- and far-field locations in a lossless medium. The model was analyzed to identify new insights into wave propagation and compared with the Fresnel approximation and the spherically-focused, concave transducer. The piston transducer model was then extended to an attenuating and dispersive medium. After analysing existing models of power-law frequency dependent attenuation, a causal, spherical wave Green’s function was derived from the Navier-Stokes equation for a classical viscous medium. Modifications to the angular spectrum method were presented and used to analyze the radiated field of a focused, planar piston transducer. Finally, after presenting our signal processing strategy for improving imaging spatial resolution through minimization of the SSF, two signal processing methods were derived and analysed in simulation: a deconvolution technique to remove the effects of the ultrasound excitation wave and suppress additive noise from the received ultrasound signal, and a retrospective transmit focusing method that changed the response from a predefined transmit focus to an arbitrary transmit focal depth. Proof-of-concept simulations were presented using a variable number of scatterers and compared with the traditional matched filtering and envelope detection technique.

Magnetic deflection techniques have been employed to measure proton angular distributions from some (d,p) and (t,p) reactions. In many cases the distributions exhibit typical stripping patterns and their analysis in terms of plane wave and distorted wave theories of stripping has enabled spin and parity assignments to be made for a number of excited nuclear levels. These theories are briefly outlined in Chapter 1 of this thesis and the experimental procedures are described in Chapter 2. The (d,p) reaction has been studied at an incident energy of 3 MeV with target nuclei B¹⁰, B¹¹, C¹², C¹⁴ and O¹⁶ and an account of this investigation is given in Chapter 3. Although the plane wave theory gives a good account of the angular distributions corresponding to the low-Q transitions (say Q ≤ 2 MeV) it is not a good approximation for the high-Q transitions. This is in accord with Wilkinson's suggestion that distortion effects should be quite small even at low deuteron bombarding energies providing also that the reaction Q-value is low. Agreement with the high-Q ground state distribution for B¹¹ could only be obtained with distorted wave theory if a cut-off radius were used. This may indicate the need for taking into account finite range effects in this theory. The Se⁷⁶(d,p)Se⁷⁷ reaction has been studied at 7.8 MeV bombarding energy and eleven angular distributions corresponding to the ground and ten excited states of Se⁷⁷ have been analysed using distorted wave theory. This investigation is described in Chapter 4. Deuteron and proton elastic scattering measurements have also been made from Se⁷⁶ and Se⁷⁷, respectively. The optical model potentials required to describe the stripping distributions are found to be entirely consistent with those derived from the elastic scattering data. In Chapter 5 an account is given of a systematic study of the (t,p) reaction for target nuclei B¹⁰, B¹¹, C¹², C¹⁴, O¹⁸, Si²⁸, Si²⁹ and Ca⁴⁰. This investigation was carried out at triton energies between 8 and 13 MeV. In the majority of cases the angular distributions are observed to be strongly forward peaked and these have been analysed in terms of Newn's plane wave theory of double stripping. With the exception of the B¹⁰(t,p)B¹² and C¹²(t,p)C¹⁴ reactions the agreement in general is found to be very satisfactory. In addition to the ground state, nine excited states of B¹³ were observed and information on the spins and parities of six of them have been obtained. The excitation energies of only four excited states were previously known. C¹⁶ had not previously been observed and the present investigation has shown this to be stable by 4.25 MeV against neutron emission, in good agreement with the predicitons of Zel'dovich. The ground state was confirmed to be O⁺ and the first excited state at 1.753 MeV excitation is probably 2⁺. The delayed neutron emission for C¹⁶ has also been studied and its measured half-life found to be 0.74 ± 0.03 seconds. Angular distributions were measured for the ground and nine excited states of O¹⁸ and ground and four excited states of O²⁰. Only one state, at 4.45 MeV excitation in O¹⁸, could not be interpreted by a double stripping process. Spin-parity assignments from the reactions Si²⁸(t,p)Si³⁰, Si²⁹(t,p)Si³¹ and Ca⁴⁰(t,p)Ca⁴² are in good agreement with earlier measurements. The Be¹¹ nucleus has been studied using the Be⁹(t,p)Be¹¹ reaction at 6 and 10 MeV triton energies. This investigation is described in Chapter 6. At the higher bombarding energy six energy levels of Be¹¹ were observed and three of these were found to have natural widths in excess of 10 keV. Proton distributions were measured at both energies for the ground and first excited states. Their interpretation in terms of a double-stripping mechanism is complicated by the presence of large backward peaks but the distributions are not inconsistent with the spins of 1/2 ^- and 1/2 ⁺, respectively, predicted by Talmi and Unna. Angular distributions from the C¹²(t,α)B¹¹ reaction at 10 MeV triton energy were also studied in an attempt to obtain information on the spins and parities of some of the states in B¹¹ which are involved in the beta-decay of Be¹¹. In Chapter 7 an account is given of triton elastic scattering measurements made at incident energies 6.4, 6.8 and 7.2 MeV from C¹², O¹⁶, O¹⁸, F¹⁹ and Ca⁴⁰. Only the scattering from F¹⁹ and Ca⁴⁰ can be described by the optical model, although the optical parameters are ambiguous. The scattering distributions from O¹⁶ at all three energies exhibit large backward peaks suggestive of compound resonance scattering. Optical model parameters derived from the triton scattering data have been uesd by Rook and Mitra to analyse the proton distributions from Ca⁴⁰(t,p)Ca⁴², using distorted wave theory. A brief account of the results is given in Appendix C.

This work is focused on thermodynamic modelling of isobaric vapour-liquid-liquid equilibrium (VLLE) (homogeneous) and (heterogeneous) for binary, ternary and quaternary systems. This work uses data for organic/aqueous systems; historically these mixtures were used in the production of penicillin and were required to be separated by continuous fractional distillation. Modelling of the separation required phase equilibrium data to be available so that predictions could be made for equilibrium stage temperatures, vapour compositions, liquid compositions and any phase splitting occurring in the liquid phase. Relevant data became available in the literature and work has been carried out to use relevant theories in correlating and predicting as was originally required in the distillation equilibrium stage modelling. All the modelling carried out was at atmospheric pressure. The modelling has been done using an Equation of State, specifically Peng Robinson Styrjek Vera (PRSV), combined with the activity coefficient model UNIversal QUAsi Chemical (UNIQUAC) through Wong Sandler mixing rules (WSMR). The success of all correlations and predictions was justified by minimizing the value of the Absolute Average Deviation (AAD) as defined within the thesis. Initially the integral Area Method and a method called Tangent Plane Intersection (TPI) were used in the prediction of liquid-liquid equilibrium (LLE) binary systems. This work used a modified 2-point search, suggested a 3-point search and has successfully applied both of these methods to predict VLLE for binary systems. It was discovered through the application of the TPI on ternary VLLE systems that the method was strongly sensitive to initial values. This work suggested and tested a Systematic Initial Generator (SIG) to provide the TPI method with realistic initial values close to the real solution and has demonstrated the viability of the SIG on improving the accuracy of the TPI results for the ternary systems investigated. In parallel with the TPI another method the Tangent Plane Distance Function (TPDF) was also investigated. This method is based on the minimisation of Gibbs free energy function related to the Gibbs energy surface. This method consistently showed it was capable of predicting VLLE for both ternary and quaternary systems as demonstrated throughout this work. The TPDF method was found to be computationally faster and less sensitive to the initial values. Some of the methods investigated in this work were also found to be applicable as phase predictors and it was discovered that the TPDF and the SIG methods were successful in predicting the phase regions; however the TPI method failed in identifying the 2 phase region. Applying the techniques described to newly available quaternary data has identified the strengths and weaknesses of the methods. This work has expanded the existing knowledge and developed a reliable model for design, operation and optimisation of the phase equilibria required for prediction in many separation processes. Currently available modelling simulation packages are variable in their predictions and sometimes yield unsatisfactory predictions. Many of the current uses of VLLE models are particularly focused on Hydrocarbon/Water systems at high pressure. The work described in this thesis has demonstrated that an EOS with suitable mixing rules can model and predict data for polar organic liquids at atmospheric and below atmospheric pressure and offers the advantage of using the same modelling equations for both phases.

The main results of this dissertation show evidence supporting the Successive Surface Scaffolding Conjecture. This is a new conjecture that, if true, guarantees the existence of all the wye-delta-order minimal obstructions of a surface S as subgraphs of the irreducible triangulations of the surface S with a crosscap added. A new data structure, i.e. an augmented rotation system, is presented and used to create an exponential-time algorithm for embedding graphs in any surface with a constant-time check of the change in genus when inserting an edge. A depiction is a new formal definition for representing an embedding graphically, and it is shown that more than one depiction can be given for nonplanar embeddings, and that sometimes two depictions for the same embedding can be drastically different from each other. An algorithm for finding the essential cycles of an embedding is given, and is used to confirm for the projective-plane obstructions, a theorem that shows any embedding of an obstruction must have every edge in an essential cycle. Obstructions of a general surface S that are minor-minimal and not double-wye-delta-minimal are shown to each have an embedding on the surface S with a crosscap added. Finally, open questions for further research are presented. / Graduate

The asymptotic behaviour of an elastically supported infinite string and an elastic isotropic half plane (in frames of specific asymptotic model) under a moving point load are studied. The main results of this work are uniform asymptotic formulae and the asymptotic profile for the string and the exact solution and uniform asymptotic formulae for a half plane. The crucial assumption for both structures is that the acceleration is sufficiently small. In order to describe asymptotically the oscillations of an infinite string auxiliary canonical functions are introduced, asymptotically analyzed and tabulated. Using these functions uniform asymptotic formulae for the string under constant accelerating and decelerating point loads are obtained. Approximate formulae for the displacement in the vicinity of the point load and the singularity area behind the shock wave using the steady speed asymptotic expansion with additional contributions from stationary points where appropriate are derived. It is shown how to generalise uniform asymptotic results to the arbitrary acceleration case. As an example these results are applied for the case of sinusoidal load speed. It is shown that the canonical functions can successfully be used in the arbitrary acceleration case as well. The graphical comparative analysis of numerical solu- tion and approximations is provided for different moving load speed intervals and values of the parameters. Vibrations of an elastic half plane are studied within the framework of the asymp- totic model suggested by J. Kaplunov et al. in 2006. Boundary conditions for the main problem are obtained as a solution for the problem of a string on the surface of a half plane subject to uniformly accelerated moving load. The exact solution over the interior of the half plane is derived with respect to boundary conditions. Steady speed and Rayleigh wave speed asymptotic expansions are obtained. In the neighborhood of the Rayleigh speed the uniform asymptotic formulae are derived. Some of their interesting properties are discovered and briefly studied. The graphical comparative analysis of the exact solution and approximations is provided for different moving load speed intervals and values of the parameters.

In this paper we consider a radially symmetric nonlinear Dirichlet problem in a ball, where the nonlinearity is "superlinear" and "superlinear with jumping."