Foster's Theorem Research Articles

Beam Steering Using Active Artificial Magnetic Conductors: A 10-Degree Step Controlled Steering

An Active Artificial Magnetic Conductor (AAMC) is presented to steer the radiation pattern of a printed dipole working at 2 GHz. The elements that generates the phase shift are a set of Varactor Diodes, which are characterized using its spice model in order to obtain a phase shift - capacitance mapping. Overall beam steering of +/40° with a step size of 10° is achieved. A circuit model that describes any multilayer substrate AAMC unit cell, which uses fist form of Foster's theorem along with transmission line theory, is proposed. Our work is suitable to be used as low profile antenna; for example, street furniture antennas, which are located on the facades of houses or buildings, so that they can be visually mixed up with signs or advertisements. Simulations have been validated using a prototype consisting of an array of 22 × 14 AAMC elements; the overall structure measures 1.9λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> × 1.21λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . This reflector will generate a phase gradient in its columns, which will modify the reflection angle of an incident electromagnetic wave in the H-Plane. Beam switching control has been achieved using suitably amplified LPF PWM signals generated by two Arduino modules. A printed dipole with a Fractional Bandwidth of 17% is designed and manufactured to illuminate the structure at a distance λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> /4 above the surface. Far-field radiation patterns and reflection coefficients have been measured in an anechoic chamber using a spherical system. These compare favorably with simulations performed using the Time Domain solver in CST Microwave Studio.

Optimization of loosely coupled inductive data transfer systems by non-Foster impedance matching

Abstract. Wireless energy transfer is often used in industrial applications to power actors or sensors, for example in rotating applications as replacement for mechanical slip rings. In addition to the energy transfer, we have developed inductively coupled data transfer systems to expand the range of possible applications. The data transfer is accomplished by using loosely coupled coils on both sides of the power transfer system. In pure energy transfer systems, resonant coupling is used, meaning that the power transfer coils are both tuned to a common frequency to compensate the relatively small coupling factor between power transmitter and receiver and to achieve an impedance matching between both sides by compensating the inductive component of the transfer coils. In this case, capacitors can be connected in series or in parallel to the coils, leading to a sharp, narrow band resonance peak in the transfer function. In inductively coupled data transfer systems, this approach is often not useful because not just a pure sine wave has to be transferred but more likely a signal of a certain bandwidth. In one of our applications, a 100 Mbit s−1 Ethernet stream is transferred with an occupied bandwidth of 62.5 MHz. The coil structures used so far in our data transfer applications were intrinsically unmatched to the data transfer systems. Additionally, due to the small coupling factor between the data transfer coils, transfer losses in the range of up to 15 dB or worth had to be accepted. This is especially critical regarding the high noise level in vicinity of the energy transfer system and the cross coupling between the two transfer channels. For passive, lossless circuits, Foster's theorem states that the reactance increases monotonically with frequency. Subsequently, the inductive part of a circuit can just be exactly compensated with a capacitance for one single frequency. In contrast, active circuits like a negative impedance converter (NIC) can be used to achieve a non-Foster behaviour, for example a negative inductance can be realized. In theory, an inductance in series or parallel to a negative inductance of the same magnitude would be cancelled out for every frequency applied. For low power level applications like active receiving antennas, this approach has already been successfully used in the past to achieve improved matching of simple antenna structures over a comparably large bandwidth. We make use of non-Foster circuits, namely negative impedance converters, to compensate the inductive part of two loosely coupled inductors to achieve smaller transfer losses and better impedance matching, which should lead to a decreased transfer signal loss and higher signal to noise ratio. The results of this paper serve as a basis for this development. So far, we achieved almost complete cancellation of the reactive part introduced by the loosely coupled data transfer inductors. Unfortunately, the circuits active device used to form the negative impedance converter introduced a highly resistive element, greatly increasing the signal transfer losses. Nevertheless, the theory of loosely coupled inductors is shown in a compact form and a strategy to cancel the reactive part is presented. Simulations and measurements of a transfer system are carried out, both showing good agreement regarding the reactance cancellation. Based on this, optimised implementations will be developed in the future.

Simplicial networks and effective resistance

We introduce the notion of effective resistance for a simplicial network(X,R) where X is a simplicial complex and R is a set of resistances for the top simplices, and prove two formulas generalizing previous results concerning effective resistance for resistor networks. Our approach, based on combinatorial Hodge theory, is to assign a unique harmonic class to a current generatorσ, an extra top-dimensional simplex to be attached to X. We will show that the harmonic class gives rise to the currentIσ and the voltageVσ for X∪σ, satisfying Thomson's energy-minimizing principle and Ohm's law for simplicial networks.The effective resistance Rσ of a current generator σ shall be defined as a ratio of the σ-components of Vσ and Iσ. By introducing potential for voltage vectors, we present a formula for Rσ via the inverse of the weighted combinatorial Laplacian of X in codimension one. We also derive a formula for Rσ via weighted high-dimensional tree-numbers for X, providing a combinatorial interpretation for Rσ. As an application, we generalize Foster's Theorem, and discuss various high-dimensional examples.

Open Quantum Random Walks on the Half-Line: The Karlin–McGregor Formula, Path Counting and Foster’s Theorem

In this work we consider open quantum random walks on the non-negative integers. By considering orthogonal matrix polynomials we are able to describe transition probability expressions for classes of walks via a matrix version of the Karlin-McGregor formula. We focus on absorbing boundary conditions and, for simpler classes of examples, we consider path counting and the corresponding combinatorial tools. A non-commutative version of the gambler's ruin is studied by obtaining the probability of reaching a certain fortune and the mean time to reach a fortune or ruin in terms of generating functions. In the case of the Hadamard coin, a counting technique for boundary restricted paths in a lattice is also presented. We discuss an open quantum version of Foster's Theorem for the expected return time together with applications.

Understanding and Tackling the Root Causes of Instability in Wireless Mesh Networks

We investigate, both theoretically and experimentally, the stability of CSMA-based wireless mesh networks, where a network is said to be stable if and only if the queue of each relay node remains (almost surely) finite. We identify two key factors that impact stability: the network size and the so-called effect, a consequence of the hidden-node problem and nonzero transmission delays. We consider the case of a greedy source and prove, by using Foster's theorem, that three-hop networks are stable, but only if the stealing effect is accounted for. We also prove that four-hop networks are, on the contrary, always unstable (even with the stealing effect) and show by simulations that instability extends to more complex linear and nonlinear topologies. To tackle this instability problem, we propose and evaluate a novel, distributed flow-control mechanism called EZ-flow. EZ-flow is fully compatible with the IEEE 802.11 standard (i.e., it does not modify headers in packets), can be implemented using off-the-shelf hardware, and does not entail any communication overhead. EZ-flow operates by adapting the minimum congestion window parameter at each relay node, based on an estimation of the buffer occupancy at its successor node in the mesh. We show how such an estimation can be conducted passively by taking advantage of the broadcast nature of the wireless channel. Real experiments, run on a nine-node test-bed deployed over four different buildings, show that EZ-flow effectively smooths traffic and improves delay, throughput, and fairness performance.

Numerically Efficient Computation of a Complete Set of Eigenfunctions in Complex Cavities

Abstract It is demonstrated, that both the divergence-free resonance modes and the irrotational electric and magnetic eigenfunctions are necessary for a rigorous modal expansion in complex cavities. A numerically efficient computation of the irrotational eigenfunctions based on the generalized scattering matrix technique is presented. A systematic searching strategy for the cavity eigenfunctions which is based on Foster's theorem is discussed. Finally two improvements of the modal expansion are suggested which significantly increase the accuracy and the numerical efficiency of this method, namely, the removal of the non-uniform convergence of some field series at the coupling apertures and the estimation of the asymptotic values of some slowly converging series related to the modal expansion.

Generalization of the Wittrick-Williams formula for counting modes of flexible structures

A recent (1971) result due to Wittrick-Williams relating the number of modes of a flexible structure to the zeros of a matrix function is generalized and an independent proof given. It is applied to Timoshenko models of a class of interconnected beamlike lattice structures and an interpretation of the zeros given as pinned modes of the structure. N determining modes of continuum models of flexible structures, particularly when the mode density is high, it is often necessary to make sure that no modes have been missed. Here, a result due to Wittrick and Williams1 (which is at once an algorithm and often referred to as such) may be used to determine the total number of modes in any given interval of frequency. In the words of the authors, it is an algorithm in the sense that it enables one to calculate how many natural frequencies lie below any chosen frequency, without determining them and hence to converge to any required natural frequency to any specified accuracy. References 2-5 are illustrative of the actual use of this algorithm for determining modes of beamlike structures. Our concern in this paper is in the formula itself rather than its use as a computational tool for finding modes of structures. Indeed, it can be regarded as a generalization of Foster's theorem for onedimensional loss-less impedances, familiar in classical electrical circuit theory; see Ref. 6. Also, the proof given by the authors in Ref. 6 leans heavily on a classical theorem of Lord Rayleigh7 and many arguments peculiar to mechanics of structures. Here we formulate the problem in the generality of an abstract wave equation in a Hilbert space and at the same time present an entirely independent mathematical proof. We show, in addition, how it can be applied to Timoshenko models of interconnected beamlike lattice structures.

An Extension of Foster's Network Theorem

Consider an electrical network onnnodes with resistorsrijbetween nodesiandj. LetRijdenote theeffective resistancebetween the nodes. Then Foster's Theorem [5] asserts thatwherei∼jdenotesiandjare connected by a finiterij. In [10] this theorem is proved by making use of random walks. The classical connection between electrical networks and reversible random walks implies a corresponding statement for reversible Markov chains. In this paper we prove an elementary identity for ergodic Markov chains, and show that this yields Foster's theorem when the chain is time-reversible.We also prove a generalization of aresistive inverseidentity. This identity was known for resistive networks, but we prove a more general identity for ergodic Markov chains. We show that time-reversibility, once again, yields the known identity. Among other results, this identity also yields an alternative characterization of reversibility of Markov chains (see Remarks 1 and 2 below). This characterization, when interpreted in terms of electrical currents, implies thereciprocity theoremin single-source resistive networks, thus allowing us to establish the equivalence ofreversibilityin Markov chains andreciprocityin electrical networks.

Energy in lossless and low-loss networks, and Foster's reactance theorem

Energy stored in a lossless network driven by a voltage (current) source is related to a frequency derivative of the network susceptance (reactance). This relation is proven to be valid also for low-loss networks at frequencies far from the network resonance frequencies. Close to the resonance frequency, the stored energy is expressed in terms of the network susceptance (reactance) and the bandwidth of the resonance. A proof of Foster's reactance theorem follows from this consideration. It is also proven that Foster's theorem is a direct consequence of causality. >

Some remarks on dynamic receptance, stiffness, and impedance of mechanical systems

In this paper, several general properties are proved for the dynamic receptance, stiffness, and impedance matrices of linear mechanical systems. First, undamped systems are considered and Foster's theorem, which is well known in network theory, is generalized; its new, generalized form seems to be unknown also in network theory. Two additional theorems are also proved for damped systems, the second one for a special case of modal damping. The results are illustrated by several examples.

Moments for stationary and quasi-stationary distributions of markov chains

A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.

Moments for stationary and quasi-stationary distributions of markov chains

A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.