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.     

Schwarz lemma for driving point impedance functions and its circuit applications

In this paper, a boundary version of the Schwarz lemma is investigated for driving point impedance functions and its circuit applications. It is known that driving point impedance function, Z(s) = 1 + c_p(s − 1)^p + c_{p + 1}(s − 1)^{p + 1} + ..., p > 1, is an analytic function defined on the right half of the s-plane. Two theorems are presented using the modulus of the derivative of driving point impedance function, |Z′(0)|, by assuming the Z(s) function is also analytic at the boundary point s = 0 on the imaginary axis with . In the obtained inequalities, the value of the function at s = 1 and the derivatives with different orders have been used. Finally, the sharpness of the inequalities obtained in the presented theorems is proved. Simple LC circuits are obtained using the obtained driving point impedance functions.     

Mathematisches Modell umrichtergespeister Gleich- und Wechselstrommotoren mit besonderem Augenmerk auf die Stromverdrängungsphänomene in den Ständer- und Läufernutenleitern

Übersicht In der vorliegenden Arbeit wird ein mathematisches Modell vorgestellt, welches es gestattet, Stromverdrängungseffekte in Gleichstrommaschinen und Asynchronmaschinen zu berücksichtigen. Der Vorteil des mathematischen Verfahrens besteht in der Beschreibung des Skineffektes bei eingeprägten Spannungen und nicht bei eingeprägten Strömen, wie sonst in der technischen Literatur üblich. Das Modell führt auf Integral-Differentialgleichungen vom Faltungstyp. Es wird eine Lösung angegeben, und die numerischen Ergebnisse zeigen den Einfluß der Wirbelströme in umrichtergespeisten Gleichstrom- und Asynchronmaschinen.

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Mathematical models of inverter-fed de and ac machines by taking into account skin effect

Contents The paper proposes a mathematical model which enables us to take into account the skin effect in de and ac machines. The advance of the mathematical procedure consists in describing the skin effect for impressed voltages and not for impressed current, as generally made in the technical literature. The model gives rise to integral-differential equations of delay-type. A solution is shown and numerical results are given to evidence eddy current influence in de and ac inverter-fed machines.

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Verwendete Symbole a Nutbreite - b Leiterbreite - d Zwischenraum zwischen zwei überlappenden Nutenleitern - l _t (t) Momentanwert der an den Bürsten induzierten Spannung - f Frequenz - f _c Chopper-Frequenz - g(x, t) Momentanwert der Nutstromdichte auf der Abszissex - h(x, t) Momentanwert des Magnetfelds der Nutstreuung auf der Abszissex - h _p Leiterhöhe - i(t) Momentanwert des Stroms - l _c Nutenleiterlänge - l _e Außennutenleiterlänge - l _m entsprechende magnetische Länge - m _s ,m _r Ständer- und Läuferphasenanzahl - p Polpaarzahl - q Nutenzahl pro Pol und Phase - s, r Kennzeichen für Ständer- und Läufergrößen - v(t) Momentanwert der Spannung - w Anzahl der Parallelzweige - z Zahl überlagerter Leiternuten pro Strang - z _r ,z _s Zahl überlagerter Leiternuten pro Nut - z _w Zahl nebeneinanderliegender Leiternuten pro Strang - B _M (t) Momentanwert der maximalen Luftspaltinduktion - D Bohrungsdurchmesser - L _m Selbstinduktivität pro Wicklungsphase - L _E Selbstinduktivität des Erregerkreises - M Drehmoment - M ^(r) Gegeninduktivität - N Spulenzahl pro Wicklungszweig - P Nennleistung - R _E Erregerkreiswiderstand - Re (A) Realteil der komplexen ZahlA - Im (A) Imaginärteil der komplexen ZahlA - S=b·h _p Leiterquerschnitt - V _EM Maximalwert der Erregerspannung - V _M Maximalwert der Ankerspannung - Zündwinkel des Ankerchoppers - Zündwinkel des Erregungschoppers - (t) Momentanwert des mit der Wicklung verketteten Flusses - ₀ Leerpermeabilität - Leitfähigkeit - Zonenfaktor - Kreisfrequenz - _r Läuferdrehgeschwindigkeit - * Faltungszeichen     

One-dimensional cylindrical diffusion of electromagnetic fields,part I

Contents We discuss magnetic fieldsB _z (r, t) andB (r, t) diffusing into homogeneous conducting circular cylinders of radiusr ₀ with boundary conditionsB _z (r ₀,t) orB (r ₀,t) proportional tot ⁿ. Laplace-transforms are used. The main difficulty is their inversion for larger values ofn. The procedures can be strongly simplified by the introduction of certain polynomials. They have very remarkable properties. They are also helpful for many applications. If one wants to calculate the dissipated Joule-heat for instance, one needs certain infinite sums related to the eigenvalues of the problems. These infinite sums can easily be evaluated with the help of the polynomials mentioned. The corresponding plane problems are also considered in order to show that these polynomials are the cylindrical analogues of Bernoulli- and Euler-polynomials. The relations between our polynomials and Fourier-Bessel-expansions are the same as those between Bernoulli- and Euler-polynomials and Fourier-expansions. Finally hollow cylinders are discussed, too. The results are similar but more complicated than for full cylinders.

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Eindimensionale zylindrische Diffusion elektromagnetischer Felder, Teil I

Übersicht Wir behandeln magnetissche FelderB _z (r, t) undB (r, t), die in homogene leitfähige Kreiszylinder mit dem Radiusr ₀ diffundieren mit der Randbedingung, daß fürr=r ₀ die Felder proportionalt ⁿ sind. Dabei wird die Laplace-Transformation benutzt, wobei deren Inversion für größere Werten sehr umständlich ist. Durch die Einführung bestimmter Polynome kann das Vorgehen sehr erleichtert werden. Diese Polynome haben bemerkenswerte Eigenschaften und können für viele Anwendungen sehr nützlich sein. Will man z.B. die dissipierten Energien berechnen, so benötigt man dazu gewisse unendliche Summen der Eigenwerte der Probleme, die mit Hilfe der genannten Polynome leicht berechnet werden können. Die Behandlung der analogen ebenen Probleme zeigt, daß diese Polynome das zylindrische Analogon der Bernoulli-und Euler-Polynome sind. Die Beziehungen zwischen unseren Polynomen und Fourier-Bessel-Reihen sind dieselben wie die zwischen Bernoulli-und Euler-Polynomen und Fourier-Reihen. Abschließend werden auch Hohlzylinder behandelt. Die Ergebnisse sind ähnlich, jedoch erheblich komplizierter als für Vollzylinder.

     

A note on passivity and losslessness

This note gives new necessary and sufficient conditions for a linear system to be passive or lossless. It will be shown that, by using the theory of operators on an indefinite J(t)-Hilbert space, both the passivity and isometric conditions for an operator to be lossless can be combined into a single condition—namely an operator is lossless if and only if it is J(t)-contractive.     

Tangential nevanlinna-pick interpolation problem for bounded real matrices—A network approach

This paper deals with the so-called tangential Nevanlinna—Pick interpolation problem for bounded real matrices. This problem can be formulated as follows: given a set of n pairs {(p_i, K _i)}, where p_i are distinct complex numbers with Re p_i > 0 and K _i stands for 2m × l_i constant matrices, assuming that for every pair (p_i, K _i) with p_i complex there exists a complex conjugate pair (p _i, K _i) and that for every pair (p_i, K _i) in which p_i is real K _i is also real, find an m × m bounded real matrix S(p) such that [ S (p_i) 1_m] K _i = 0 for i= 1,…,n. The solution of this problem is obtained in an inductive way through the construction at each step of a real lossless multiport section that realizes two complex conjugate pairs or one real pair. After each step the number of pairs (p_i, K _i) is reduced by two (if p_i is complex) or by one (if p_i is real). the procedure is continued until all pairs have been considered. After the last step the final section may be terminated with any bounded real load. the scattering matrix S (p) of the resulting cascade multiport network is bounded real and satisfies the desired interpolation conditions. In this way the tangential interpolation problem is reduced to classical network cascade synthesis by the use of real lossless multiport sections.     

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.     

Piezoelectric properties of the Pb(Sc1/2Nb1/2) O3-Pb(Mg1/3Nb2/3) O3-PbTiO3 ternary ceramic materials near the morphotropic phase boundary

Abstract

The physical and electrical properties of xPb(Sc_1/2Nb_1/2)O₃-yPb(Mg_1/3Nb_2/3)O₃-zPbTiO₃(PSMNT 100x/100y/100z) ternary ceramic materials near the morphotropic phase boundary (MPB) were investigated. The MPB follows on almost linear region between the PSMNT 58/00/42 and PSMNT 00/68/32 MPB compositions of the binary systems. The maximum piezoelectric constant, d ₃₃ = 680 pC/N, were found at PSMNT 29/33/38, where ε^T ₃₃/ε₀ = 3,800, and electromechanical coupling factors, (k _p = 70%, k₃₁ = 43%, and k ₃₃ = 76%) and T _c = 207°C were obtained. The maximum electromechanical coupling factors, (k _p = 72%, k ₃₁ = 45%, and k ₃₃ = 77%) were found at PSMNT 29/34/37, where ε^T ₃₃/ε₀ = 3,000, d ₃₃ = 640 pC/N, and T _c = 205°C were obtained. These values are better than those of PZT.     

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.     

Rectifier circuits for optimal frequency conversion

It is proved that a circuit consisting of non-linear passive resistances and of any linear invariant passive elements cannot convert power from frequencies ω₁ and ω₂ into power at frequency mω₁ + nω₂ with an efficiency better than 1/(|m| + |n|)². Circuits attaining that efficiency are constructed for all m, n, so that the condition is both necessary and sufficient. For m = μt, n = vt, |μ| + |v| = 2^s (all literals are integers), the optimal circuit consists of a finite number of rectifiers and tuned circuits. For values of m, n that are not of the above form an infinite number of tuned circuits is necessary, but an efficiency better than 93 per cent of the optimum is attainable by simple finite circuits in all cases.     

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     

An optimal design of RC active filters using grounded capacitors

It is desirable, from the point of view of integration, to have all the capacitors grounded in RC active filters. Recently, such a network was reported with only a preliminary design procedure. An optimal design is presented in this paper for the above network. The new design leads to a simpler configuration, simultaneously minimizes the worst-case deviation in both the pole-Q(Q_p) and the pole frequency ω_p and also guarantees that the network is free from unstable modes of operation during activation. Although this design is based on Q_p and ω_p, it is also suitable for notch and allpass sections. It is shown that the gains of the operational amplifiers (o.a.s.), controlling the poles of the filter, can all be made equal to (4Q²_p-1)^1/3; this choice of the gains minimizes the effect of the finite bandwidths of the amplifiers on Q_p and ω_p. A simple scheme is proposed to compensate for this effect. The effect of compensation on the zeros, in the case of notch and allpass sections, is also studied. It is found that the effect on the notch frequency is negligible, while, for allpass filters, compensation for the poles automatically provides compensation for the zeros. A simple tuning scheme using only resistors is also presented, and the theoretical results are verified experimentally.     

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.     

Determining the periodic response of nonlinear systems by a gradient method

For systems of differential equations of the form ? = f(x) or x = f(x, t) , a periodic response may be identified by the requirement that x(kT) = x(0) , where k = 1, 2, … and T is the period, x(0) = x₀ being the initial-condition vector. We describe a gradient method for finding this x₀ vector by minimizing the square magnitude of the ‘discrepancy vector’ δ(x₀) = x(T)–x₀. The gradient of the scalar function P(x₀) = δ^t(x₀)δ(x₀) with respect to x₀ is calculated by one full-period forward integration of the original differential equation to obtain δ(x₀), and then one full-period backward integration of the adjoint variational equations, using δ(x₀) as the initial-condition vector. The gradient of P(x₀) is then twice the adjoint discrepancy vector. We use Fletcher's method of optimization to minimize P(x₀) .     

Morphotropic phase boundary and electrical properties of lead-free bismuth sodium lanthanum titanate—barium titanate ceramics

The important properties of lead-free piezoelectric ceramics have been investigated from Bismuth Sodium Lanthanum Titanate and Barium Titanate system: (1 − y)(Bi_0.5Na_0.5)_{(1 − 1.5x)}La _x TiO₃(BNLT)—yBaTiO₃(BT) where x = 0.017 and y = 0 − 0.2, respectively. The morphotropic phase boundary (MPB) was found to be around y = 0.1 by the x-ray diffraction and dielectric measurement at various amount of BT. The temperature dependence of dielectric constant (ε _r ) at various value of y showed the diffuse phase transition exhibiting the relaxor type ferroelectrics. The degree of diffuseness increased at a high doping content of about y = 0.15 where the second phase transition (T₂) of the ferroelectric to antiferroelectric phase disappeared. Moreover, this sample had the maximum piezoelectric coefficient (d ₃₃) of about 112 pC/N with relatively low dielectric constant. The optimum sintering temperatures and the microstructures of the dense BNLT-BT ceramics were also examined.     

Transfer functions interpolating an ideal amplitude with arbitrary phase and delay

A method is presented to generate polynomials with phase values and delay (first derivative) specified at given frequencies. Then a closed form solution is described for the scattering transfer function S₁₂(p) of a resistively terminated lossless reciprocal two-port network with ideal amplitude and arbitrary phase and delay.     

A sufficient condition of controllability of RLCM active networks over F(z)

This paper defines the separability of an RLCM active network and finds a sufficient condition that the active network is controllable and observable over F(z) if its passive network is controllable and observable over F(z). So the controllability (observability) criteria in (Proc. IEEE ISCAS, 2005) can be used to analyse and design active networks. Copyright © 2006 John Wiley & Sons, Ltd.     

Dreidimensionale analytische Berechnung des Erregerfeldes eines Turbogenerators mit supraleitender Erregerwicklung

Übersicht Das Erregerfeld eines Turbogenerators mit supraleitender Erregerwicklung wird unter Berücksichtigung der genauen Wicklungsverteilung dreidimensional berechnet. Magnetische und elektrische Schirme werden in Form von idealen Berandungen berücksichtigt.

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Contents The magnetic field of a turbogenerator with a superconducting rotor is calculated in its three dimensions taking into account the exact geometric distribution of the winding. Magnetic and electric shields are considered in form of ideal screens.

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Übersicht der verwendeten Symbole A Strombelagshöchstwert - a Augenblickswert des Strombelags, örtlicher Wert des Strombelags - B Induktionshöchstwert - b Augenblickswert der Induktion, örtlicher Wert der Induktion - b Induktionsvektor (Augenblickswert) - I _n () modifizierte Besselfunktion 1. Art undn-ter Ordnung mit dem Argument - I _n () Ableitung vonI _n () nach dem Argument - I Gleichstrom - K _n () modifizierte Besselfunktion 2. Art undn-ter Ordnung mit dem Argument - K _n () Ableitung vonK _n () nach dem Argument - P Polpaarzahl - r radiale Koordinate - v Augenblickswert des Vektorpotentials - v Vektor des Vektorpotentials (Augenblickswert) - Z Leiter in Reihe geschaltet - z axiale Koordinate - Umfangskoordinate (räumlicher Umfangswinkel) - elektrische Leitfähigkeit - Ordnungszahl von Wellen, die sich in axialer Richtung räumlich und zeitlich sinusförmig ändern - ₀ magnetische Feldkonstante - _r Permeabilitätszahl - Ordnungszahl von Wellen, die sich in Umfangsrichtung räumlich und zeitlich sinusförmig ändern Indizes l Stator - (l) Grundwelle - 2 Rotor - const konstant - i Zählziffer - n Nut - r radial - z axial vom axialen Strombelag herrührend (zweiter Index hinterr oder ) - tangential in Umfangsrichtung vom tangentialen Strombelag herrührend (zweiter Index hinterr, oderz) - Welle mit der Ordnungszahl - Welle mit der Ordnungszahl Schreibweisen X(a, b, c) Funktion vona, b, c - X ₍₎ Fourierkoeffizient mit der Ordnungszahl - X _{(, )} Fourierkoeffizient mit den Ordnungszahlen und - X(x=x ₁) Funktionswert fürx=x ₁ - rs(i) Radius deri-ten Schicht - Laplacescher Operator     

One-dimensional cylindrical diffusion of electromagnetic fields, Part II

Contents This paper continues a previous one [1]. It discusses a magnetic fieldB _z (r,t) diffusing into a homogeneous conducting cylinder (of radiusr ₀). The difference between the two papers is that different boundary conditions are applied. The boundary condition now is an integrated one, the magnetic flux within a coaxial hollow cylinder (of radiusR ₀>r ₀) being proportional tot ⁿ (actually a more general problem is discussed). As in the previous paper the solution can be simplified by the introduction of certain polynomials, which are very useful and which have interesting properties. They are generalisations of the polynomials defined in [1]. The corresponding plane problem is discussed again, too. The polynomials defined in this case are related to generalisations of Bernoulli- and Euler-polynomials.

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Eindimensionale zylindrische Diffusion elektromagnetischer Felder, Teil II

Übersicht Die Arbeit stellt die Weiterführung einer vorhergehenden Arbeit [1] dar. In ihr wird die Diffusion eines Magnetfeldes in einen homogenen leitfähigen Zylinder (Radiusr ₀) diskutiert. Der Unterschied zwischen beiden Arbeiten liegt in den Randbedingungen. Hier ist die Randbedingung eine integrale. Der magnetische Fluß innerhalb eines koaxialen Hohlzylinders (RadiusR ₀>r ₀) ist proportional zut ⁿ (tatsächlich wird ein allgemeineres Problem behandelt). Wie in [1] kann die Lösung durch die Einführung bestimmter Polynome vereinfacht werden, die bemerkenswerte Eigenschaften haben und sehr nützlich sind. Sie stellen Verallgemeinerungen der in [1] eingeführten Polynome dar. Wie früher wird auch hier das analoge ebene Problem behandelt. Die dabei auftretenden Polynome hängen mit Verallgemeinerungen von Bernoulli-und Euler-Polynomen zusammen.