Existence condition of state equations of linear active networks over F(z)

If the physical parameter of each branch of a linear active network is considered to be an independently variable parameter, then it is called a network over the field F(z) of all rational functions in its physical parameters. This paper discusses the existence condition of state equation over F(z) for the active network. The condition depends only on the network structure, and is applied to two examples, which shows that the method is straightforward. Copyright © 2006 John Wiley & Sons, Ltd.     

Controllability and observability criteria of RLC networks over F(z)

RLC networks are a class of important linear systems. This paper discusses the problem of controllability and observability of RLC networks over the field F(z) of rational functions in independently variable parameters. Copyright © 2001 John Wiley & Sons, Ltd.     

Separability and reducibility criteria of RLC networks over Fz and their applications

The separability condition for the RLC network over the field F^z of rational functions in multi-parameters is derived. Some criteria and technique are proposed to check the reducibility of the characteristic polynomial of the network. The technique is straightforward since it is only necessary to observe its graph. © 1998 John Wiley & Sons, Ltd.     

Sensitivity of third- and higher-order filters

For any realization of a network function F(s) = N(s)/D(s), the sensitivities that can be most readily calculated are those of the coefficients in N(s) and D(s). A simple relationship is derived that enables one to calculate the root (pole and zero) sensitivities of F(s) in terms of the coefficient sensitivities. The root sensitivities, in turn, enable one to calculate the root pair Q and root frequency sensitivities, which can be used to characterize and compare different realizations of F(s). Application to 3rd- and 4th-order filters reveals formulations that are more elegant than those already known in the literature.     

Using residue sums to estimate high‐order Fourier harmonics of piecewise‐continuous transcendental functions: Application to Class A‐, B‐ and F‐type amplifier circuits

A discrete fast‐Fourier transform (DFFT) is the preferred method of choice for the rapid evaluation of a set of harmonics of a piecewise‐continuous and periodic transcendental form. For large sets of Fourier components specified to some stringent error criterion, the approach becomes increasingly unattractive owing to the presence of round‐off errors that result from the switching of one transcendental form to another. As an alternative, it might be wondered whether the high‐frequency components can be more efficiently estimated by employing a combination of residue sums and boundary integrals in the complex plane z = Re^iωt, where ω is the fundamental frequency and R = ∣z∣. The starting point is the construction of suitable contours that divide the complex plane into a number of sectors in accordance with the number of intervals of smooth behaviour of a periodic piecewise‐continuous real function along ∣z∣ = 1. Each sector encompasses the analytic extension of a real transcendental function on ∣z∣ = 1 to yield p(z)T(f(z)), where T(ζ) is meromorphic and p(z), f(z) are Laurent series. Fourier coefficients are subsequently expressed in terms of residue series and constant‐phase boundary integrals from each of the various sectors associated with a given p(z)T(f(z)). This approach is applied to the model for the drain current of a field effect transistor (FET), where in this case T(ζ) = tanh(ζ), which is subject to the modes of operation: ‘Class A’, ‘Class B’ and approximate ‘Class F’. In contrast to Classes A and B, the Fourier coefficients in the ‘Class F’ drain current decay slowly with frequency, suggesting that this mode might be more suitably analysed using a combined DFFT/residue procedure. Copyright © 2007 John Wiley & Sons, Ltd.     

A new method of inhomogeneous ladder network synthesis

A procedure to synthesize ladder networks in which each series arm impedance is f_iz(s) and each shunt arm admittance is g_iy(s) is presented. Given a specified network function T(s), the function G = z(s)y(s) is first established by a polynomial decomposition. Then the appropriate chain parameters associated with T(s) are constructed as polynomials in G. The constants f_i, g_i are determined from a straightforward continued fraction expansion of a related RL impedance function in G. Necessary and sufficient conditions for the steps in the procedure are established and examples are included.     

On the realizability of non-rational positive real functions

The paper deals with the analysis and synthesis of passive reciprocal one-ports composed of an infinite number of conventional elements (positive R, L. C and ideal transformers), considered as equivalent circuits of physical distributed one-ports. In the generalization from finite to infinite networks, several (generally overlooked) basic difficulties arise, which are discussed and partially clarified. Physically, a prescribed positive real function z(p) is only specified in Rep > 0, and a lossless infinite realization always exists. Since the value of the function in Re p < 0 is then deduced by z(p) + z(–p) = 0, the resistance r(α, ω) = Re z(α + jω) is such that r(0, ω) = 0, but the limit of r(α, ω) for α = +0 may be strictly positive, so that a lossless impedance may have a resistive behaviour in steady-state. The classical Foster and Cauer synthesis procedures may consequently all fail for lossless non-rational impedances, whereas the procedures of Darlington and Bott-Duffin (and sometimes Brune) succeed. Since every point is a transmission zero for an odd function, a cascade synthesis with all zeros at p = 1 always works, and explicit expressions for the element values are obtained. Many examples are treated in detail, and their sometimes pathological behaviour in Re p < 0 is discussed.     

Another FIR lattice structure

A new two multiplier FIR lattice structure is derived by using the digital two‐pair concept, which produces two transfer functions H_i(z) and H_i′(z) having the complementary relationship H_i′(z)=z^?iH_i(–z^?1), in contrast to the mirror image relationship, i.e. H_i′(z)=z^?iH_i(z^?1) satisfied in the conventional FIR lattice structure. The new structure should be useful in crossover networks as well as in multirate signal processing. Copyright © 2005 John Wiley & Sons, Ltd.     

Impedance and elektrodynamic force acting on a cylindrical conductor partly filling a semi-closed slot

Contents In this paper the inner impedance and the electrodynamic force acting on a cylindrical conductor partly filling a semi-closed slot are calculated. The investigations are made by using the separation of variables method.

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Übersicht Es werden die innere Impedanz und die elektrodynamische Kraft, die auf einen kreisförmigen Leiter wirkt, der eine halbgeschlossene Nut partiell ausfüllt, berechnet. Zur Berechnung wird die Methode der Trennung der Variablen angewendet. Die Ergebnisse sind in Kurvenform dargestellt.

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Symbole A z-component of the vector potential (complex r.m.s. value) - B _x, B_y components of the magnetic induction (complex r.m.s. value) - F electrodynamic forece - I current (r.m.s. value) - I _n(z) modified Bessel function of first kind - imaginary unit - J z-component of the current density (complex r.m.s. value) - L(o) inductance for direct current - R resistance - X reactance - Z impedance - z ^* conjugate complex number ofz - Rez, Imz, |z| real part, imaginary part and modulus of complex numberz - ₀ magnetic permeability - pulsation - Kronecker symbol - 1 atn=m o otherwise     

An optimal algorithm for rectilinear steiner trees for channels with obstacles

The channel rectilinear Steiner tree problem is to construct an optimal rectilinear Steiner tree interconnecting n terminals on the upper shore and the lower shore of a channel without crossing any obstacles inside the channel. However, intersecting boundaries of obstacles is allowed. We present an algorithm that computes an optimal channel rectilinear Steiner tree in O(F₁(k)n + F₂(k)) time, where k is the number of obstacles inside the channel and F₁ and F₂ are exponential functions of k. For any constant k the proposed algorithm runs in O(n) time.     

Messung des Dachverlaufs von Magnetfeldimpulsen

Übersicht Der DachverlaufH _z (t) eines trapezförmigen MagnetfeldimpulsesH _H (t) mit einer Länge von ca. 3 ms und einer maximalen AmplitudeH _Ho=35,8·10⁶ A/m (450 kOe) wird mit Spule und Integrator dadurch gemessen, daß man die von der Spule abgegebene Spannungu ₁(t) nur während der Dauer des Dachs integriert. Dagegen wird während der Flanken des Impulsesu ₁(t) unterdrückt.u ₁(t) erreicht während dieser Zeit Werte von der Größenordnung 1 kV. Diese werden mit einer mechanischen Torschaltung in Verbindung mit drei elektronischen gedämpft. Das Gerät erlaubt es deshalb,H _z (t) unabhängig vonH _Ho zu messen. Im vorliegenden Fall lassen sich Dachschwankungsbreiten bis herab zu 1, 2 A/m (15 mOe) mit weniger, als 1% Fehler nachweisen.

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Contents The pulse top shapeH _z (t) of a trapezoidal magnetic field pulseH _H(t) with about 3 ms duration and a maximum amplitude ofH _Ho=35.8·10⁶ A/m (450 kOe) is measured by pick-up coil and integrator as follows: The voltageu ₁(t) induced in the pick-up coil is integrated only within the duration of the pulse top. It is suppressed during the pulse edges. At that timeu ₁(t) can have values of the order of 1 kV. They are damped by a mechanical gate in connection with three electronical ones. The apparatus is able to measureH _z (t) without an influence ofH _Ho. In the case in point its sensivity is limited to 1.2 A/m (15 mOe) with an error of less than 1%.

     

A simplified design-oriented analysis of switched-capacitor active filters

The method presented in this paper provides a simplified approach to obtain the z-domain equivalent circuit of an SC active filter by one-for-one substitution of each basic SC element with its z-domain equivalent admittance, if a specific terminal condition is met. Then, traditional circuit analysis techniques can be used to derive the z-domain transfer function of the filter. This method makes the analysis and the design of the SC active filter simple and similar to the S-domain method. Two application examples are given.     

Electrodynamic forces acting on a conductor with elliptic cross section in the slot of an electric machine

Contents In this paper, radial electrodynamic forces acting on a conductor entirely filling a semi-closed slot with elliptic cross section are investigated. The calculations have been made by using the Bubnow-Galerkin method. The results are compared with published data.

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Elektrodynamische Kräfte auf einen Leiter mit Ellipsen-form in der Nut elektrischer Maschinen

Übersicht In diesem Artikel wurden die Kräfte untersucht, die auf einen Leiter mit Ellipsenform in halbgeschlossener Nut wirken. Die Berechnungen wurden mit der Bubnow-Galerkin Methode vorgenommen. Die Berechnungsergebnisse wurden mit Literaturangaben verglichen.

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List of Symbols A z-component of vector potential (complex r.m.s. value) - B ,B normal and tangential components of the magnetic induction in the elliptic-cylindrical system of coordinates (r.m.s. value) - F electrodynamic force - F _m , F _a arythmetical mean and alternating components of the total force - J z-component of current density (complex r.m.s. value) - imaginary unit - 2a, 2b major and minor axis of ellipse - c distance of focus from center of cllipse - l conductor lenght - I=I e^j complex value of current, |I|=r.m.s. value - z,z ^* modulus of complex numberz and complex conjugate ofz - Rez, Im z real and imaginary part of complex numberz - ₁ angle of slot opening - conductivity - magnetic permeability - pulsation - cross section area     

A quasi-steady-state approach to the generation of non-sinusoidal orthogonal waveforms at microwave frequencies

By the application of a quasi-steady-state time-domain analysis approach some advance is made in the generation of non-sinusoidal orthogonal waveforms at microwave frequencies. The main study concerns the time-domain response of short-circuited cascades of commensurate transmission lines. It is found that with simple prescribed pole location for the driving-point impedance. Z₁¹(z), on the unit circle in the z-plane, a quasi-steady-state response is obtained, having a known periodicity. Furthermore, this response is automatically a member of an orthogonal set of waveforms which is uniquely associated with every choice of periodicity.     

Approximate formulae for numerical inversion of Laplace transforms

Most methods for the numerical calculation of inverse Laplace transformations f(t) = L⁻¹[F(s)] have serious limitations concerning the class of functions F(s) that can be inverted or the achievable accuracy. The procedures described in the paper can be used to invert rational as well as irrational or transcendental functions of the complex variable s. The required accuracy of the results can be enhanced without changing the algorithm, only at the cost of a longer computation time. The described methods were verified with many examples including transients in lumped/distributed systems with sections of lossy multiconductor transmission lines or with distributed RC elements. © 1998 John Wiley & Sons, Ltd.     

Analysis of work of liquid metal induction pump

Contents In the paper the finite element method is presented so as to determine electromagnetic field distribution in a cylindrical liquid metal induction pump. The effect of exciting current frequency, the channel width and the channel material conductivity on dynamic parameters of the pump has been analysed. — The obtained calculation results have been shown in the graphs.

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Analyse der Pumpe für ein flüssiges Metall

Übersicht Im Beitrag wurde die Methode der finiten Elemente zur Bestimmung der Verteilung des elektromagnetischen Feldes in einer zylindrischen Pumpe für ein flüssiges Metall verwendet. Man hat den Einfluß des Erregerstroms, der Kanalbreite und der Leitfähigkeit des zum Aufbau des Kanals benutzten Stoffes — auf die dynamischen Parameter dieser Pumpe untersucht. Die erhaltenen Ergebnisse hat man in Form von Diagrammen dargestellt.

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List of symbols A vector potential - A vector potential (complex r.m.s. value) - A z-component of vector potential (complex r.m.s. value) - A _i ,A _j ,A _k vector potential values in nodal pointsi, j, k (complex r.m.s. values) - B magnetic induction - B magnetic induction (complex r.m.s. value) - B _x ,B _y components of magnetic induction (complex r.m.s. values) - F _t electrodynamic force - F mean force - F _a alternating force - F _x ,F _y components of the mean force - conductivity of the liquid metal - J current density (complex r.m.s. value) - J _a current density - J _w exciting current linear density (complex r.m.s. value) - l length of the channel - magnetic permeability of the liquid metal - M torque acting upon the liquid metal - current pulsation - p pressure of transported metal - Q pump efficiency - v _x ,v _y components of the liquid metal's velocity in the 0X and 0Y direction - z ^* conjugate complex number ofz     

Berechnung der magnetischen Felder in den Stirnraumwänden elektrischer Maschinen

Übersicht Ausgehend von der Beschreibung des magnetischen Feldes im Stirnraum elektrischer Maschinen wird die Induktion in den nichtleitend und hochpermeabel angenommenen Stirnraumwänden berechnet. Ferner wird versucht, die wirklichen Materialbeiwerte nachträglich zu berücksichtigen.

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Contents The magnetic field in non-conductive and highly permeable walls of the end-region of electrical machines is calculated by means of the field in the air-part of the end-zone. In a second step the properties of real materials are considered.

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Im Text verwendete Symbole a Vektorpotential - A , A, A_z Komponenten des Vektorpotentials in der zyl. Maschine - A _y, A_z Komponenten d. Vektorpotentials im abgewickelten Modell - a radiale Bauhöhe des Stirnraums im abgewickelten Modell - a , a_z; a_y, a_z dimensionslose Koeffizienten der - b , b_z; b_y, b_z Reihenwicklung des Strombleags - B , B, B_z Komponenten der Induktion in der zylindrischen Maschine - B _y, B_z Komponenten der Induktion im abgewickelten Modell - c axiale Abmessung des Stirnraumes - c _I–c _VI Konstanten der homogenen Lösungen der Wandflüsse - d _I–d _VI (die Indices kennzeichnen einzelne Wandzonen entsprechend Bild (B 2)) - d Eindringmaß - magnetische Feldstärke - i , i, i_z Ströme - F Strombelag - J , J, J_z Komponenten des Strombelags - j , j_z Strombelagsmaximum für ein Wicklungselement - Drehoperator - k, n Separationsparameter in der zyl. Maschine - l ₀, m, n Separationsparameter im abgewickelten Modell - l komplexer Separationsparameter - p Polpaarzahl (=Separationsparameter i. d. zyl. Maschine) - R Reduktionsfaktor - |R| Betrag des Reduktionsfaktors - d Wegelement - u, v, w natürliche Zahlen - flußdurchsetzte Zone in den idealisierten Stirnraumwänden - elektrische Leitfähigkeit - Permeabilität - ₀ Permeabilität des Vakuums - Grundwellenpolteilung im abgewickelten Modell - magnetischer Fluß - Kreisfrequenz Funktionen I _p(k ) Besselfunktionen erster und zweiter Art - N _p(k ) Besselfunktionen erster und zweiter Art - I _p(n ) modifizierte Besselfunktionen erster und zweiter Art - K _p(n ) modifizierte Besselfunktionen erster und zweiter Art - S _{u, p}(k ) Hilfsfunktionen nach Lommel (L₃) Koordinaten , ,z Zylinderkoordinaten - x, y, z cartesische Koordinaten - z ₁,z ₂,z ₃ Einheitsvektoren für Zylinderkoordinaten - ₁, ₂; ₁, ₂;z ₁ Koordinaten des Wicklungselementes mitj -undj -Strombelagskomponenten - ₁; ₁, ₂;z ₁,z ₂ Koordinaten eines Wicklungselementes mitj -undj _z-Strombelagskomponenten - ₀ Wellenradius - ₃ Außenwandradius hochgestellte Indices (i) ideell - (h) homogen - (p) partikular     

Analytical and numerical modelling of a stationary temperature field in a three dimensional electric heating system

Contents In the paper, a stationary temperature field in a three dimensional system of a direct floor heater was modelled. This installation was described by an elliptic boundary problem. On this basis two independent simulations, analytical and numerical, were carried out. In the former, the cable core was modelled with thin axes and the triple Fourier series was used. In the numerical simulation the heat sources were right octagonal prisms inscribed into the cable core. In this case the finite element method was applied. The results of both simulations are approximate with good accuracy. The results obtained are presented graphically.

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Analytische und numerische Modellierung des stationären Temperaturfeldes einer dreidimensionalen Fußbodenheizung

Übersicht In der vorliegenden Arbeit wurde das stationäre Temperaturfeld im direkten dreidimensionalen System der Fußbodenheizung modelliert. Die Anlage wurde in der Form eines elliptischen Randproblems beschrieben. Hierauf wurden zwei unabhängige Simulationen durchgeführt: analytische und numerische Simulation. In der ersten Simulation wurde die Kabelstränge mit Hilfe von dünnen Achsen modelliert, und es wurde eine dreifache Fourierische Reihe genutzt, in der numerischen Simulation Wärmequellenals achteckige, regelmäßige Prismen, die in die Kabelstränge einbeschrieben wurden. Im letzten Beispiel wurde die Finite-Elemente Methode benutzt. Die Ergebnisse der beiden Simulationen sind sich in hinreichender Genauigkeit ähnlich. Die gewonnenen Ergebnisse wurden in graphischer Form dargestellt.

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List of symbols A dimensionless constant, determined by (9b) - (2a, 2b, l) dimensions of the floor panel Fig. 1a - B_mni coefficients of the series (8) - {G} heat source vector - G(z) gate function determining the position and length of the heating cable along the axis OZ (Fig. 1b) - g _k (x, y, z) volumetric power density of the system with thek-th cable section - K number of cable sections - k index of thek-th cable section (k=1, 2, 3,...,K) - P _k active power of thek-th cable section - q _k linear power density of thek-th section of the heating cable core - R cable radius - r radius of cable core - {T} node temperature vector - T(x, y, z) total temperature field in the floor panel - T _m mean temperature - T _k (x, y, z) temperature field component from thek-th cable section (with the others switched off), - T ₀ air temperature (far from the floor surface), - u dimensionless filling factor of the lengthl by the cable (Fig. 1b, u 0, 1) - (x, y, z) coordinates of a point in the floor panel - (x _k,y _k,z) coordinates of the position of thek-th section of the cable core (forzul, (1–u)l) - averaging coefficient of heat transfer to air (sum of the radiation and convection coefficients) - _n successive positive roots of equation (9a) - (x–x _k), (y–y _k) Dirac's deltas shifted tox _k andy _k respectively - convergence index of series (10b) and (10c)-ratio of the module of the sum of the last ten terms to the module of the total sum (table 1) - heat conductivity matrix - averaging heat conductivity of micro-rein-forced concrete - _c cable core heat conductivity - v _k(x,y,z) k-th component of an increase in the thermal field of over the value T₀ caused by thek-th section of the cable (with the others switched off) - V _k ⁽¹⁾ two dimensional component of an increaseV _k (x, y, z), determined by eq. (10b) - V _k ⁽²⁾ three dimensional component of an increaseV _k (x, y, z), determined by eq (10c) - l(z) unit step function The work (Code No W/WE/3/96) was carried out in Biaystok Technical University under the financial support of State Committee for Scientific Research, Poland.