A new simulation technique for RF oscillators

The study of phase-noise in oscillators and the design of new circuit topologies necessitates an efficient technique for the simulation of oscillators. While numerous approaches have been developed over the years e.g. [1-3], each has its own merits and demerits. In this contribution, an asymptotic numeric method developed in e.g. [4-5] is applied to the simulation of RF oscillators. The method is closely related to the stroboscopic and high-order averaging method in [6] and the Heterogeneous Multiscale Methods in [7]. The method is advantageous in that the same methodology can be applied for the simulation of general circuit problems involving highly oscillatory ordinary differential equations, partial differential equations and delay differential equations. Furthermore and counter-intuitively, its efficacy improves with increasing frequency, a feature that is very favourable in modern communications systems where operating frequencies are ever rising. Results for a CMOS oscillator will confirm the validity and efficiency of the proposed method

Nanogel formation of polymer solutions flowing through porous media

A gelation process was seen to occur when Boger fluids made from aqueous solutions of polyacrylamide (PAA) and NaCl flowed through porous media with certain characteristics. As these viscoelastic fluids flow through a porous medium, the pressure drop across the bed varies linearly with the flow rate, as also happens with Newtonian fluids. Above a critical flow rate, elastic effects set in and the pressure drop grows above the low-flow-rate linear regime. Increasing further the flow rate, a more dramatic increase in the slope of the pressure drop curve can be observed as a consequence of nanogel formation. In this work, we discuss the reasons for this gelation process based on our measurements using porous media of different sizes, porosity and chemical composition. Additionally, the rheological properties of the fluids were investigated for shear and extensional flows. The fluids were also tested as they flowed through different microfluidic analogues of the porous media. The results indicate that the nanogel inception occurs with the adsorption of PAA molecules on the surface of the porous media particles that contain silica on their surfaces. Subsequently, if the interparticle space is small enough a jamming process occurs leading to flow-induced gel formation

Identifying minimal and dominant solutions for Kummer recursions

We identify minimal and dominant solutions of three-term recurrence relations for the confluent hypergeometric functions $_1F_1(a+\epsilon_1 n;c+\epsilon_2 n;z)$ and $U(a+\epsilon_1 n,c+\epsilon_2 n,z)$, where $\epsilon_i=0,\pm 1$ (not both equal to 0). The results are obtained by applying Perron's theorem, together with uniform asymptotic estimates derived by T.M. Dunster for Whittaker functions with large parameter values. The approximations are valid for complex values of $a$, $c$ and $z$, with $|\arg\,z|<\pi$

Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials

Funder: Comunidad de Madrid; Id: http://dx.doi.org/10.13039/100012818Funder: Consejería de Educación e Investigación; Id: http://dx.doi.org/10.13039/501100010774Funder: Engineering and Physical Sciences Research Council; Id: http://dx.doi.org/10.13039/501100000266Funder: Cantab Capital Institute for the Mathematics of InformationFunder: Cambridge Centre for AnalysisAbstract: We study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex‐valued weight function, exp ( n s z ) , over the interval [ − 1 , 1 ] , where s ∈ C is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, s ∈ i R , due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as n → ∞ have recently been studied for s ∈ i R , and our main goal is to extend these results to all s in the complex plane. We first use the technique of continuation in parameter space, developed in the context of the theory of integrable systems, to extend previous results on the so‐called modified external field from the imaginary axis to the complex plane minus a set of critical curves, called breaking curves. We then apply the powerful method of nonlinear steepest descent for oscillatory Riemann–Hilbert problems developed by Deift and Zhou in the 1990s to obtain asymptotics of the recurrence coefficients of these polynomials when the parameter s is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter s approaches a breaking curve, by considering double scaling limits as s approaches these points. We see a qualitative difference in the behavior of the recurrence coefficients, depending on whether or not we are approaching the points s = ± 2 or some other points on the breaking curve

Asymptotic solvers for ordinary differential equations with multiple frequencies

We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms, focusing on the case of multiple, non-commensurate frequencies. We derive an asymptotic expansion in inverse powers of the oscillatory parameter and use its truncation as an exceedingly effective means to discretize the differential equation in question. Numerical examples illustrate the effectiveness of the method

Microfluidic systems for the analysis of the viscoelastic fluid flow phenomena in porous media

In this study, two microfluidic devices are proposed as simplified 1-D microfluidic analogues of a porous medium. The objectives are twofold: firstly to assess the usefulness of the microchannels to mimic the porous medium in a controlled and simplified manner, and secondly to obtain a better insight about the flow characteristics of viscoelastic fluids flowing through a packed bed. For these purposes, flow visualizations and pressure drop measurements are conducted with Newtonian and viscoelastic fluids. The 1-D microfluidic analogues of porous medium consisted of microchannels with a sequence of contractions/ expansions disposed in symmetric and asymmetric arrangements. The real porous medium is in reality, a complex combination of the two arrangements of particles simulated with the microchannels, which can be considered as limiting ideal configurations. The results show that both configurations are able to mimic well the pressure drop variation with flow rate for Newtonian fluids. However, due to the intrinsic differences in the deformation rate profiles associated with each microgeometry, the symmetric configuration is more suitable for studying the flow of viscoelastic fluids at low De values, while the asymmetric configuration provides better results at high De values. In this way, both microgeometries seem to be complementary and could be interesting tools to obtain a better insight about the flow of viscoelastic fluids through a porous medium. Such model systems could be very interesting to use in polymer-flood processes for enhanced oil recovery, for instance, as a tool for selecting the most suitable viscoelastic fluid to be used in a specific formation. The selection of the fluid properties of a detergent for cleaning oil contaminated soil, sand, and in general, any porous material, is another possible application

Construction and implementation of asymptotic expansions for Jacobi-type orthogonal polynomials

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to ∞. These are defined on the interval [−1, 1] with weight function: w(x)=(1−x)α(1+x)βh(x),α,β>−1 and h(x) a real, analytic and strictly positive function on [−1, 1]. This information is available in the work of Kuijlaars et al. (Adv. Math. 188, 337–398 2004), where the authors use the Riemann–Hilbert formulation and the Deift–Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in O(n) operations, rather than O(n2) based on the recurrence relation

Microdevices for extensional rheometry of low viscosity elastic liquids : a review

Extensional flows and the underlying stability/instability mechanisms are of extreme relevance to the efficient operation of inkjet printing, coating processes and drug delivery systems, as well as for the generation of micro droplets. The development of an extensional rheometer to characterize the extensional properties of low viscosity fluids has therefore stimulated great interest of researchers, particularly in the last decade. Microfluidics has proven to be an extraordinary working platform and different configurations of potential extensional microrheometers have been proposed. In this review, we present an overview of several successful designs, together with a critical assessment of their capabilities and limitations

Topological expansion in the complex cubic log-gas model. One-cut case

We prove the topological expansion for the cubic log-gas partition function, with a complex parameter and defined on an unbounded contour on the complex plane. The complex cubic log-gas model exhibits two phase regions on the complex t-plane, with one cut and two cuts, separated by analytic critical arcs of the two types of phase transition: split of a cut and birth of a cut. The common point of the critical arcs is a tricritical point of the Painleve I type. In the present paper we prove the topological expansion for the partition function in the one-cut phase region. The proof is based on the Riemann-Hilbert approach to semiclassical asymptotic expansions for the associated orthogonal polynomials and the theory of S-curves and quadratic differentials