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Non-reciprocal network synthesis using the generalized Richards transformation
Abstract
A generalization of the Richards transformation is presented and its properties are derived. the new transformation is appropriate for the canonic synthesis of non-reciprocal network sections based on the transformation of an impedance, and it is shown that under certain constraints a transformerless realization is possible. the generalized Richards transformation can also be applied to the synthesis of network sections without degree reduction of the impedance to be transformed. This kind of application is important for the realization of transfer functions. A dual transformation for an admittance is established. Finally, it is shown how to use the generalized Richards transformation for the synthesis of transfer functions by doubly terminated lossless two-ports.
This paper investigates the synthesis problem of one‐port mechanical networks consisting of one damper, one inerter, and a finite number of springs, where the damping coefficient and inertance are some values in given intervals. Due to the construction limitation, the element values of dampers and inerters are usually required to be constrained in certain ranges instead of any positive values. A necessary and sufficient condition is derived for any positive‐real admittance to be realizable as such class of networks, which is in terms of the coefficients of the admittance and the upper and lower bounds of element value constraints. Moreover, the corresponding synthesis procedure is presented based on the derivation process and realizations of three‐port spring networks. Finally, numerical examples are given for illustration. The results of this paper can be utilized in the design of inerter‐based mechanical control systems. This paper solves the realization problem of any positive‐real admittance as a one‐port mechanical network containing one damper, one inerter, and a finite number of springs, where the damping coefficient and inertance are constrained to be some values in given intervals. A necessary and sufficient condition for the realizability is presented in Theorem 2, and the corresponding realization procedure is presented in Procedure 1. In addition, numerical examples are presented to illustrate the results.
This paper is concerned with the realization problem for a class of high-pass as well as low-pass transfer functions as a two-port RC ladder network with a specified gain, which requires that the actual gain of the realization configuration is equal to the gain of the given transfer function. From a cascade realization approach, it is shown that any transfer function belonging to the class under investigation can be realized as a two-port RC ladder network with any specified gain in a continuous interval. Finally, a numerical example is presented to illustrate the result. Copyright
This paper is concerned with a generalized theorem of Reichert for biquadratic minimum functions, which states that any biquadratic minimum function realizable as the impedance of networks with n reactive elements and an arbitrary number of resistors can be realized with n reactive elements and two resistors. First, a series of constraints on networks realizing minimum functions are presented. Furthermore, by discussing the possible resistor edges incident with vertices of the reactive-element graphs, it is proved that any minimum function realizable as the impedance of networks with three reactive elements and an arbitrary number of resistors can be realized with three reactive elements and two resistors, from which the validity of the case of n = 3 follows. Similarly, the validity of the case of n = 4 is proved. Together with the Bott–Duffin realizations, the generalized theorem of Reichert for biquadratic minimum functions is finally proved. The results of this paper are motivated by passive mechanical control. Copyright
This paper is concerned with the realizability problem of n-port resistive networks that contain 2n terminals. A necessary and sufficient condition for any real symmetric matrix to be realizable as the admittance of an n-port resistive network containing 2n terminals is obtained. This condition is based on the existence of a parameter matrix. Furthermore, the values of the elements are expressed in terms of the entries of the admittance matrix and the parameter matrix. Finally, a numerical example is used to illustrate the results. Copyright © 2013 John Wiley & Sons, Ltd.
We consider on-line network synthesis problems. Let N = {1,··· ,n} be a set of n sites. Traffic flow requirements between pairs of sites are revealed one by one. When ever a new request rij = rji (i < j) between sites i and j is revealed, an on-line algorithm must install the additional necessary capacity without decreasing the existing network capacity such that all the tr affic requirements are met. The objective is to minimize the total capacity installed by the algorithm. The perf ormance of an on-line algorithm is measured by the competitive ratio, defined to be the worst-case ratio between the total capacity by the on-line algorithm and the total optimal (off-line) capacity assuming we have priori information on all the requirements initially. We distinguish between two on-line versions of the problem depending on whether the entire set of sites is known a prior or not. For the first version where the entire set of site is unknown, we present a best possible algorithm along with a matching lower bound. For the second version where the entire set of sites is known a priori, we present a best possible algorithm for n · 6.
Necessary and sufficient conditions are derived for a function to be the driving-point impedance of a physically realizable network consisting (essentially) of lumped resistors and lossless transmission lines. The circuits so developed are thoroughly practical for pure reactances and in many other special cases, but, in general, ideal transformers are sometimes required. A rigorous correspondence between lumped-constant circuits and line-resistor circuits is established. This correspondence immediately extends the usefulness of a wealth of theorems and techniques.
Necessary and sufficient conditions for the synthesis of nonreciprocal lossless n-ports are shown also to be necessary and sufficient for the realization of a nonreciprocal lossless 2-port as a cascade of reciprocal and nonreciprocal lossless 2-ports. Nonreciprocal transmission zeros are realized by four canonic nonreciprocal 2-ports, which are analogous to the Darlington A, B, C, D networks that are required for realization of reciprocal transmission zeros.
A synthesis procedure is presented for a lossless positive real impedance matrix Z(s) . The procedure, based on a choice of a suitable state-space representation for Z(s) , uses the minimum possible number of reactive elements and, simultaneously, the minimum possible number of gyrators.
The well-known Richards' Theorem of the continuous-time filter theory is reformulated in the digital domain in a convenient manner, leading to a simple derivation of cascaded lattice digital filter structures, realizing lossless bounded transfer functions. The theorem is also extended to the matrix case, leading to a derivation of m -input p -output cascaded lattice filter structures with lossless building blocks, that realize an arbitrary p times m digital Lossless Bounded Real (LBR) transfer matrix. Extensions to the synthesis of arbitrary, stable p times m transfer matrices in the form of such cascaded lattices is also outlined. The derivation also places in evidence a means of testing the stability of an arbitrary p times m transfer matrix of a discrete-time linear system.
New method for cascade synthesis of 1‐port passive networks
- Kao C.
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Minimum‐gyrator cascade synthesis of losslessn‐ports
- Belevitch V.