Jonathan H. B. Deane

Modeling of chaotic DC-DC converters by iterated nonlinear mappings

In parameter ranges where conventional methods break down, DC-DC converters may be described by iterated mappings, a nonlinear discrete modeling technique. The underlying principles are explained and are applied to the example of a PWM-controlled buck converter. Stable behavior and bifurcations to chaos are predicted by numerical evaluation of the governing mapping and are confirmed by experiment. >

Instability, subharmonics and chaos in power electronic systems

The concept of chaos is applied to a variety of nonlinear power electronic circuits. With the onset of instability, the phenomena of subharmonics, quasi-periodicity, and chaos are predicted and observed. The following examples are dealt with: diodes with charge storage (with application to resonant converters); a ferroresonant circuit; a controlled thyristor rectifier circuit; and a Buck converter controlled by pulse-width modulation (PWM). The examples were chosen to model aspects of real power electronic systems. Analytical and experimental results demonstrate the complex nonlinear phenomena such as subharmonics, quasi-periodicity and chaos can arise in even the simplest systems. >

Chaos in a current-mode controlled boost DC-DC converter

A mapping is derived in closed form, without approximations, for an idealized current-mode controlled boost converter. This circuit is known experimentally to behave chaotically for certain values of the reference current, and to produce subharmonics of the clock frequency at others. Numerical iteration of the mapping indicates chaotic operation and the presence of subharmonics. Two mechanisms of bifurcation are explained. >

Reformulation of the standard theory of Fowler-Nordheim tunnelling and cold field electron emission

This paper presents a major reformulation of the standard theory of Fowler–Nordheim (FN) tunnelling and cold field electron emission (CFE). Mathematical analysis and physical interpretation become easier if the principal field emission elliptic function v is expressed as a function v(l′) of the mathematical variable l′≡y2, where y is the Nordheim parameter. For the Schottky–Nordheim (SN) barrier used in standard CFE theory, l′ is equal to the ‘scaled barrier field’ f, which is the ratio of the electric field that defines a tunnelling barrier to the critical field needed to reduce barrier height to zero. The tunnelling exponent correction factor ν=v(f). This paper separates mathematical and physical descriptions of standard CFE theory, reformulates derivations to be in terms of l′ and f, rather than y, and gives a fuller account of SN barrier mathematics. v(l′) is found to satisfy the ordinary differential equation l′(1−l′)d2v/dl′2=(3/16)v; an exact series solution, defined by recurrence formulae, is reported. Numerical approximation formulae, with absolute error |ε|<8×10−10, are given for v and dv/dl′. The previously reported formula v≈1−l′+(1/6)l′ ln l′ is a good low-order approximation, with |ε|<0.0025. With l′=f, this has been used to create good approximate formulae for the other special CFE elliptic functions, and to investigate a more universal, ‘scaled’, form of FN plot. This yields additional insights and a clearer answer to the question: ‘what does linearity of an experimental FN plot mean?’ FN plot curvature is predicted by a new function w. The new formulation is designed so that it can easily be generalized; thus, our treatment of the SN barrier is a paradigm for other barrier shapes. We urge widespread consideration of this approach.

Analysis, simulation and experimental study of chaos in the buck converter

A buck DC-DC power converter, whose input voltage is controlled by naturally sampled, constant-frequency PWM (pulse width modulation), is operated in the continuous conduction mode. Two versions are treated, a first-order and second-order circuit. Their behavior is modeled analytically and numerically. For certain values of the circuit parameters instability occurs. Strange phenomena of multiple pulsing, shipped cycles, subharmonics, and chaos are predicted theoretically and observed experimentally, including a period-doubling route to chaos. There is good agreement between theory and experiment.<<ETX>>

Modeling the dynamics of nonlinear inductor circuits

The Jiles-Atherton (J-A) model is applied to the problem of describing the dynamics of a nonlinear circuit driven by a square wave voltage source and comprising a linear resistor and capacitor in series with a nonlinear inductor, whose core displays saturation and hysteresis. The presence of hysteresis is shown to increase the order of the circuit by one. Period-multiplication and chaos are observed and excellent agreement is obtained between experiment and simulation. >

Calculation of the periodic spectral components in a chaotic DC-DC converter

A simple mapping is derived, which describes the behavior of a peak current-mode controlled boost converter operating chaotically. The invariant density of this mapping is calculated iteratively and, from this, the power density spectrum of the input current at the clock frequency and its harmonics are deduced. The calculation is presented, along with experimental verification. The possibility of a novel application of chaos-amelioration of power supply interference-is discussed,.

Transmission coefficients for the exact triangular barrier: an exact general analytical theory that can replace Fowler & Nordheim's 1928 theory

In field electron emission theory, evaluating the transmission coefficient DET for an exact triangular (ET) potential energy barrier is a paradigm problem. This paper derives a compact, exact, general analytical expression for DET, by means of an Airy function approach that uses a reflected barrier and puts the origin of coordinates at the electrons outer classical turning point. This approach has simpler mathematics than previous treatments. The expression derived applies to both tunnelling and ‘flyover’ (wave-mechanical transmission over the barrier), and is easily evaluated by computer algebra. The outcome is a unified theory of transmission across the ET barrier. In different ranges of relevant physical parameters, the expression yields different approximate formulae. For some ranges, no simple physical dependences exist. Ranges of validity for the most relevant formulae (including the Fowler–Nordheim 1928 formula for DET) are explored, and a regime diagram constructed. Previous treatments are assessed and some discrepancies noted. Further approximations involved in deriving the Fowler–Nordheim 1928 equation for current density are stated. To assist testing of numerical procedures, benchmark values of DET are stated to six significant figures. This work may be helpful background for research into transmission across barriers for which no exact analytical theory yet exists.

Piecewise contractions are asymptotically periodic

We show that, given a finite partition of the plane C such that the map G acts as a linear contraction on each part, for almost every choice of parameters every orbit of G is (asymptotically) periodic.

Free carrier lifetime modification for silicon waveguide based devices

We investigate the effect of silicon ion irradiation on free carrier lifetime in silicon waveguides, and thus its ability to reduce the density of two-photon-absorption (TPA) generated free carriers. Our experimental results show that free carrier lifetime can be reduced significantly by silicon ion implantation. Associated excess optical absorption from the implanted ions can be reduced to an acceptable level if irradiation energy and dose are correctly chosen. Simulations of Raman scattering suggest that net gain can be achieved in certain cases without the need for an integrated diode in reverse bias to remove the photo-generated free carriers.