On the design of RC-active follow-the-leader. Feedback filters

For RC-active filters of the ‘follow-the-leader’ feedback type, a synthesis procedure is presented that is suitable for application also to the general case in which differently tuned biquadratic functional blocks are used. Their losses can either be concentrated in one of the blocks, or given by a prescribed distribution.     

Transformerless synthesis of transistor and thyristor switching RLC 1-ports

As an extension of the synthesis method developed in a previous paper, the transformerless synthesis of reciprocal switched 1-ports is presented in this paper. the transformerless structures of resistance switching n-ports described by impedance, admittance or hybrid matrices have been proposed. These structures enable the synthesis of dynamical switching RC, RL and RLC 1-ports. Using the state variables method, the realizability conditions and identification formulae for constant and sinusoidal input functions have been derived. Three numerical examples illustrate the developed methods. the present theory can be applied to create new thyristor and transistor arrangements, particularly compensators and auxiliary switching LC circuits facilitating the commutation of thyristors and transistors.     

On state-space synthesis of reciprocal n-ports

The paper considers the state-space synthesis of reciprocal n-ports with a minimum number of reactive elements. After formulating the synthesis problem new procedures are given for the lossless and for the passive cases which use a directly determinable reciprocal minimal realization as starting point. The methods are presented in terms of admittance matrices and then restated for impedance, hybrid and scattering matrix synthesis.     

Sufficient conditions for the unique solvability of linear networks containing memoryless 2-ports

Combinatorial necessary and sufficient conditions for the unique solvability of linear networks containing n-ports are well known for the ‘general’ case. They are only necessary if relations among n-port parameters are also taken into consideration. In the present paper combinatorial sufficient conditions are presented for linear networks containing RLC elements and memoryless 2-ports. The somewhat surprising result is proved that whether a 2-port can cause certain types of singularities can be predicted before the interconnection. A concept, similar to the normal tree (which intersects ideal transformers by one, gyrators by two or no edges) is introduced for arbitrary 2-ports. Its existence implies unique solvability. Relations to previous results and algorithmical aspects are also discussed.     

Simultaneous canonic synthesis of two dual impedances

It is proved that two dual passive impedances Z and R²/^Z (R real constant) each of degree m are simultaneously realizable by a lossless reciprocal 4-port of degree 2m closed on two constant resistances. Moreover, the lossless reciprocal 4-port is itself realizable as a cascade of such 4-ports of degrees 2 and 4. These are extensions of the Bott–Duffin and Fialkow-Gerst bridges but generally contain ideal transformers. As in Fialkow-Gerst synthesis, the only algebraic equation to be solved is Z(p) + Z(–p) = 0 but, in contrast with Fialkow-Gerst, the realization is minimal.     

Approximation method for decomposable filter pairs

The decomposable filter pair is presented, starting from the general form of the scattering matrix of a lossless reciprocal 3-port. The restrictions with respect to the general case are discussed. The two general categories of requirements imposed to a filter pair are presented and their incompatibility is established. An algorithm is described, allowing one to approximate simultaneously the two main transfer functions of a filter pair.     

An accurate rational 2-port model of a class of tapered lossless and RC lines

A rational 2-port model of a class of tapered LC and RC lines which includes the exponential, hyperbolic, trigonometric and square, or Bessel, lines is presented. The model for the exponential RC line (ERC ), considered as an example, accurately describes the actual frequency and transient response for various values of taper factor over wide ranges of frequency and time. In an ERC notch network configuration, application of this model predicts the results very closely for various values of taper factor.     

Cascade decomposition of a passive 2n-port network

This paper deals with a cascade decomposition of a passive 2n-port network. after a basic discussion relative to complex- and unit-normalized scattering matrices, necessary and sufficient conditions are presented under which given lossless 2n-ports D₁ and D₂ are simultaneously extractable from the front- and back-ends of the passive 2n-port N, respectively, to leave a passive 2n-port remainder Ne. Based on these conditions, some relations between the individual and simultaneous extractability of D₁ and D₂ from N for both degenerate and non-degenerate remainders N_e are discussed. the results are presented in terms of scattering as well as impedance matrices and can be considered as an extension of those given in Reference 1.     

Application of direct synthesis techniques to customize filters with complex frequency response

Recently, direct synthesis techniques (DSTs) have been presented for filter synthesis. Unlike conventional synthesis techniques, DSTs derive the filtering polynomials of the filters to be synthesized directly in their own frequency domain. These filtering polynomials are real coefficient so that they might find applications in various fields. Furthermore, DSTs might be used to customize filters with a more complex frequency response, such as asymmetric frequency response or multi‐band frequency response. In this paper, DSTs are compared with some well‐known filter synthesis techniques. Then, the application of DSTs in the design of lumped‐element LC filters, distributed‐element filters, active RC filters, and infinite impulse response digital filters with complex frequency response is discussed. Some examples are presented for demonstration. Copyright © 2015 John Wiley & Sons, Ltd.     

Topologie des kopplungsfreien Reaktanz-2n-Poles mit geringstem Aufwand an Speichern

Übersicht In dieser Arbeit werden Kriterien abgeleitet, die cs erlauben, an einem vorgelegten kopplungsfreien Reaktanz-2n Pol die Zahl der überflüssigen Spulen und Kondensatoren zu ermitteln. Ausgangspunkt der Untersuchungen ist die im 2. Kapitel dargestellte Maschenimpedanzmatrix eines Reaktanz-2n-Poles und deren lineare Transformation in die Matrix einer äquivalenten Schaltung. Im 3. Kapitel wird ein Zusammenhang zwischen der geringsten Zahl von Spulen und Kondensatoren und topologischen Größen aufgezeigt. Im letzten Kapitel werden Erweiterungen der gewonnenen Sätze dargestellt.

---

Summary Criteria will be deduced, which permit to find out the number of superfluous coils and capacitors in a linear and losslessn-port without magnetic couplings. The investigation starts with the loop impedance matrix of a losslessn-port and its linear transformation in the matrix of an equivalent circuit. This is described in the second chapter. In the third chapter relations are to be presented between the minimum number of coils and capacitors and topological structures. Some extensions of the won theorems, especially on the invariance of the minimum number of coils and the minimum number of capacitors in equivalentn-ports, are given in the last chapter.

---

---

Mit 2 Textabbildungen     

Synthesis of low-sensitivity orthogonal digital filters

A novel method is proposed for the synthesis of low-sensitivity digital filters meeting any prescribed transfer function. The method is based on extracting an orthogonal matrix from the filter state matrix, resulting in structures that need n Givens rotations and at most n+1 multipliers. Thus the proposed realization is canonic in the sense that it has the same degrees of freedom as the original transfer function. It is also shown that when the filter transfer function is a reciprocal reactant bounded function, it can be decomposed into allpass functions that need only Givens rotations for their realization. As the basic module in the realization is the Givens rotation, the CORDIC computation algorithm can be applied directly. This means considerable savings in computation time and complexity. It also results in structures that are less sensitive to the effects of finite word length. Illustrative examples, including the design and synthesis of linear phase selective filters, are given to show the extremely low sensitivity with respect to finite word length of the resulting realizations when compared with other methods. © 1997 by John Wiley & Sons, Ltd.     

A theory of nonenergic N-ports

A circuit element is nonenergic if the instantaneous power flow into it is always zero. Well-known examples include the ideal diode, transformer, gyrator and circulator. Most of the interesting nonenergic elements are nonlinear N-ports with N ? 2, and many of their properties are quite counterintuitive. For example, there exists a surprisingly large class of nonenergic multiport capacitors and inductors, all of which, it turns out, are nonlinear and reciprocal. Nonenergic linear N-ports, on the other hand, are necessarily resistive and antireciprocal. In this paper, we present a rigorous fundamental theory of nonenergic N-ports that results in a general canonical representation. Special canonical forms are developed for nonenergic resistors, capacitors and inductors, and numerous examples are given.     

Darlington's theorem and complex normalization

Two novel proofs are presented to establish the equivalence between complex normalization and Darlington representation of the loads. Schwarz reflection of analytic functions and Hermitian metrics serve in these proofs as natural concepts for the analysis of lossless multiports. By use of these tools, the algebraic structure that underlies the physical notion of losslessness is analysed, and a class of matrix representations of lossless 2n-ports is derived. the fundamental transformations between these matrices are established to facilitate easy conversion of results related to different choices of co-ordinates (voltages/currents, waves, impedances, reflectances). By virtue of this theory, complex normalization is tacitly extended to include not only impedances but also reflectances as references. Explicit expressions for a 2n-port normalized to complex loads are derived. Invariance of the complex normalized scattering matrix in a lossless cascade circuit is shown to be the most comprehensive generalization of the well known invariance of scalar reflection magnitudes. the relation between this viewpoint and the general theory of invariants of linear fractional maps in terms of cross-ratios is pointed out explicitly. the factorization problem, common to complex normalization as well as Darlington representations, is also reviewed, and it is shown that the concept of minimal factorization (well known in linear prediction theory) leads to more satisfactory results than those of established factorization recipes in complex normalization.     

The existence and the determination of terminal equations for hybrid connected N-ports

A new procedure for the existence and the determination of terminal equations for the hybrid interconnection of two n-port networks is given. It is shown that this procedure generalizes the previous works of Duffin, Trapp and Mitra for the hybrid sum of terminal impedance matrices which are either Hermitian positive semi-definite (HSD) or arbitrary matrices with complex entries. For the application of the new procedure it is shown that each n-port need not be terminal solvable or well defined. In fact, even in the case of active RLC n-ports considered in sinusoidal steady-state, the previous results of Duffin, Mitra and Trapp established for HSD matrices or arbitrary complex matrices can still be utilized with the use of modified terminal equations, to obtain the terminal equations of the resulting n-port networks. It is also noted that the method introduced in this paper is independent of the choice of generalized inverses.     

Necessary and sufficient conditions for absolute stability of linear n-ports

A set of necessary and sufficient conditions is stated and proved for the absolute stability (under any passive terminations), in the ‘bibo’ sense, of a linear n-port characterized by its open-circuit impedance matrix. A more explicit set of such conditions is derived for the special case n = 3. In the process of deriving this set of conditions, some results and theorems concerning finding the zero sets of two and three parameter functions, are stated and proved. Also derived is a set of necessary and sufficient conditions for the absolute stability of two-ports characterized by the scattering parameters. Such sets have already been derived previously, but the present set is considerably simpler.     

Switched capacitor filters with a row dominant nodal admittance matrix

Stray capacitance insensitive switched capacitor filters are studied, whose nodal admittance matrix is row-dominant for arbitrary capacitor values. Such filters have low passband sensitivities, because their transfer function is intrinsically bounded by 1, as is the case for LC-filters. Furthermore, row-dominant filters have the additional advantage that internal voltages are bounded by one, and reach this bound at each attenuation zero, which ensures optimal dynamic range. However, we show that the attenuation zeros of row-dominant filters are located at the frequencies z² with (z²)^m = 1, where m is smaller than or equal to the number of operational amplifiers. Therefore, such filters are not suitable for conventional filter applications. Two examples of such filters are given.     

An efficient numerical derivation of group delay characteristic from prescribed characteristic function

This paper proposes an efficient method of deriving the group delay characteristic only from the prescribed characteristic function of a resistively terminated LC 2-port network with the aid of a special numerical integration technique. As a by-product of the method, the transfer function of the network can also be derived. the idea of the method is based on the Hilbert transform relation between the attenuation and phase of the minimum-phase polynomial. A simplified Hilbert transform relation is presented which avoids the improper integral of its original form. the analysis of the error due to the numerical integration applied in the method and the comparison with a conventional method concerning execution time and accuracy are included.     

Some new results on the controllability and observability of linear systems and networks

This paper introduces the concept of ‘hybrid-dual systems’ and presents some new results on the controllability and observability of these systems. Theorem 1 states that for hybrid-dual systems, which include dual systems as a special case, controllability of one implies observability of the other. Theorem 2 gives the topological conditions for RLC n-ports such that controllability and observability imply each other. Theorem 2 also shows that for any RLC n-port, observability implies controllability, but the converse is not true.     

Representation of noise in a particular type of linear n-port network

A noisy linear N-port network that can be represented by a noise-free linear N-port network with voltage and current noise sources at the ports is considered. If the linear N-port network has a particular type of admittance matrix, it is possible to show that as many as N/2 when N is even and (N – 1)/2 when N is odd of the ports may be cleared of noise sources. The total number of unique representations of such a network is given for both the general case and for the case when a particular port or ports must be free of noise sources.