Linear Time-invariant Elements Research Articles

Time-domain thermal noise simulation of switched capacitor circuits and delta-sigma modulators

This paper presents an accurate and efficient algorithm for simulating the effects of thermal noise in time-domain. The algorithm presented in this paper is based on Monte Carlo methods and is applicable to linear time invariant circuits. In addition to linear time invariant elements, the algorithm can easily be extended to include clock controlled switches and single or multibit quantizers. This expands the thermal noise analysis capability to switched capacitor circuits and oversampled delta-sigma modulators. Thermal noise generated by different elements is modeled, in the time-domain, by a random pulse waveform having a desired power spectral density. Typically the noise power level is much smaller than that of other desired signals in the circuit and accurate simulation is needed to obtain correct results. In addition, random noise waveforms require the computation of many time points in each transient analysis and efficient simulation methods are needed. In this paper, a new method for computing the transient response of linear time invariant circuits is used. This method accurately and efficiently computes the response to the random pulse waveforms. In addition to thermal noise, the method can also be used to simulate the effect of random dither signals in delta-sigma modulators. Examples of noise simulation are given and, when possible, comparison with measurements, previously published results, or analytical expressions is done.

Finite Time Stability Approach to Proportional Navigation Systems Analysis

The e nite time stability of proportional navigation guidance systems is considered. Assuming planar geometry and linear missile dynamics, a proportional navigation missile ‐target guidance model is e rst formulated. The model exhibitsa feedback cone guration consisting of a lineartime-invariant element and a time-varying gain. The dee nition of e nite timeglobal absolutestability isthen presented. Itisshown that by employing thecirclecriterion, the e nite time stability of the guidance dynamics can be analyzed. An analytic bound for the time of e ight up to which stability can be assured is established. The bound depends on the system parameters and the time of e ight. Less conservative results, as compared to previous works, are obtained. This approach enables not only analysis of the system behavior for given missile dynamics, but more importantly, enables generation of a tool for system design. Illustrative examples are presented showing the effect of the system parameters on the bound. In addition, some design implications, such as the relation to miss distance, are outlined.

Direct synthesis of constant resistance one-port networks

Two methods for constructing constant-resistance one-port networks of the self-dual type when the degrees of the two input terminals are arbitrarily specified are presented. The methods are direct in the sense that they deal with RLC elements from the very beginning. The ingenuity lies in the way these elements are connected together. General configurations are presented. All previously reported configurations are shown to be special cases of the present results. Since emphasis is on the topological aspect of the problem, only the use of passive linear time-invariant elements (i.e., RLC elements) is considered. >

Circuits, k-ports, hidden modes, and stability of interconnected k-ports

This paper exclusively considers lumped <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -ports and circuits which contain linear time-invariant elements, independent sources, and controlled sources. The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -ports are represented by their hybrid matrices. Tableau equations of circuits are used as a special case of polynomial matrix description. The hidden modes of a circuit are determined by inspection from its tableau equations in the Hermite row form. The same form is used also to determine the exponential stability of the circuit and that of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -port. Finally, necessary and sufficient conditions for exponential stability of interconnected <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -ports are given; the hidden modes of the interconnection are studied. The paper is self-contained.

Stability multipliers and multivariable circle criteria

This paper is concerned with the input-output stability of non-linear multivariable feedback systems. The class of feedback systems considered consists of a linear time invariant element in the feedforward path and a non-linear element in the feedback path ; although differing feedback configurations may be considered by use of suitable transformations. By expressing the linear time invariant part as the sum of a normal operator and the difference, graphical Nyquist-type conditions are shown to ensure the L2 stability of the non-linear multivariable feedback system. The frequency domain criteria present are of the circle type and may be viewed as the multivariable counterparts of the scalar multiplier results of Cho and Narendra (1968), Zames and Falb (1968).

Current Trends in Network Optimization

Some current trends in automated network design optimization which, it is believed, will play a significant role in the computer-aided design of lumped-distributed and microwave networks are reviewed and discussed. In particular, the adjoint network approach due to Director and Rohrer for evaluating the gradient vector of suitable objective functions related to network responses that are to be opttilzed is presented in a tutorial manner. The advantage of this method is the ease with which the required partial derivatives with respect to variable parameters, such as electrical quantities or geometrical dimensions, can be obtained using at most two network analyses. Least pth and minimax approximation in the frequency domain are considered. Networks consisting of linear time-invariant elements are treated, including the conventional lumped elements, transmission lines, RC limes, coaxial lines, rectangular waveguides, and coupled lines. To illustrate the application of the adjoint network method, an example is given concerning the optimization in the least pth sense using the Fletcher-Powell method of a transmission-line filter with frequency variable terminations previously considered by Carlin and Gupta.

Constant Resistance One-Ports Which Include Nonlinear Time-Varying Elements

Methods for generating constant resistance one-ports with nonlinear and time-varying elements are proposed. The first step is a general characterization of constant resistance one-ports with linear time-invariant elements. This characterization is then used to generate classes of one-ports whose elements may be nonlinear and time-varying. Examples are given of constant resistance one-ports that include one-ports which neither need to be currentcontrolled nor voltage-controlled. In one case, the necessary and sufficient conditions for constant resistance are given.

On the Properties of Nonlinear Integral Equations That Arise in the Theory of Dynamical Systems

This paper reports on some results concerning the properties of integral equations that govern the behavior of a large class of control systems or electrical networks containing linear time-invariant elements and an arbitrary finite number of nonlinear time-varying elements. In particular, for networks containing linear time-invariant elements and an arbitrary finite number of positive-slope nonlinear resistors, it is proved, under reasonable conditions, that the response to a periodic excitation applied at t = 0 is ultimately periodic with the same period as the excitation, regardless of the initial state of the network.

On the -Boundedness of Solutions of Nonlinear Functional Equations

Let e N denote the set of N-vector-valued functions of t defined on [0, ∞) such that for any real positive number y, the square of the modulus of each component of any element is integrable on [0, y], and let L 2N (0, ∞) denote the subset of e N with the property that the square of the modulus of each component of any element is integrable on [0, ∞). In the study of nonlinear physical systems, attention is frequently focused on the properties of one of the following two types of functional equations $\eqalignno{g &= f + KQf \cr g &= Kf + Qf}$ in which K and Q are causal operators, with K linear and Q nonlinear, g e e N , and f is a solution belonging to e N . Typically, f represents the system response and g takes into account both the independent energy sources and the initial conditions at t = 0. It is often important to determine conditions under which a physical system governed by one of the above equations is stable in the sense that the response to an arbitrary set of initial conditions approaches zero (i.e., the zero vector) as t → ∞. In a great many cases of this type, g belongs to L 2N (0, ∞) and approaches zero as t → ∞ for all initial conditions, and, in addition, it is possible to show that if f e L 2N (0, ∞), then f(t) → 0 as t → ∞. In this paper we attack the stability problem by deriving conditions under which g e L 2N (0, ∞) and f e e N imply that f e L 2N (0, ∞). From an engineering viewpoint, the assumption that f e L N is almost invariably a trivial restriction. As a specific application of the results, we consider a nonlinear integral equation that governs the behavior of a general control system containing linear time-invariant elements and an arbitrary finite number of time-varying nonlinear elements. Conditions are presented under which every solution of this equation belonging to e N in fact belongs to L 2N (0, ∞) and approaches zero as t → ∞.

On the Properties of Some Systems that Distort Signals - II

In this paper we study the recoverability of square-integrable bandlimited signals (with arbitrary frequency) bands) that are distorted by a freguency-selective time-variable nonlinear operator and subsequently are bandlimited to the original bands. The distortion operator characterizes a very general class of systems containing linear time-invariant elements and a single time-invariant nonlinear element. The subseguent bandlimiting of the system's output signals can be thought of as being due to transmission through a channel that performs filtering. Our principal result asserts that, under certain conditions that are satisfied by many realistic systems, it is possible to uniquely determine the bandlimited input to the system from a knowledge of the bandlimited version of the output, in spite of the intermediate distortion which generally produces signals thal are not bandlimited to the original frequency bands. We show that the input signal can be determined by a stable iteralion procedure in which the approximating functions converge to their limit at a rate that is at least geometric.

Signal Distortion in Nonlinear Feedback Systems

This paper reports on some properties of the solutions to the functional equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$s_{2}(t) = \varphi[{\bf C}s_{2}(t) + s_{1}(t)],$</tex> where ϕ is a nonlinear function, the operator C is a convolution, and s <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> is a known function belonging to a prescribed Banach space. The equation plays a central role in the theory of signal transmission through a general physical system containing linear time-invariant elements and a single time-variable nonlinear element. In particular we establish conditions under which s <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> (t) is the fixed point of a contraction mapping of the Banach space into itself and we discuss some consequences of this result. As a direct application, we consider the range of validity of two simple cascade flow graphs (i.e., flow graphs without feedback loops) for approximately determining the signal distortion in nonlinear feedback systems when the distortion is small. Our discussion is not restricted to specific types of nonlinear characteristics