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.     

Robust gain-scheduled control for vibration suppression

Limited control authority is a key issue in the field of structural control and is a major research area since most of the practical control problems are dominated by constraints on the control signal. The paper presents a simple and practical gain-scheduled controller design procedure for active vibration suppression of a three-storey flexible structure. First, system identification experiments are performed and the plants uncertainty is derived. Next, robust controller design with constraint on the control signal is presented. For a better trade-off between control performance and control constraint a gain-scheduling approach is investigated. Stability analysis of the gain-scheduled controller is analysed using a parameter-independent Lyapunov function (quadratic stability) as well as a parameter-dependent Lyapunov function (biquadratic stability). Finally, the gain-scheduled controller is tested experimentally when the flexible structure is excited with a scaled historical earthquake record (1940 El Centro record). Successful experimental results show that the proposed robust gain-scheduled control approach offers good performance in the case of control authority limitation.Nomenclature M mass matrix - C damping matrix - K stiffness matrix - q relative displacement vector to base - m_a active mass - active mass acceleration - ground (base) acceleration - K_gap transfer gain of the displacement sensor - K_acc transfer gain of the acceleration sensor - K_AMD transfer gain of the active mass damper - _id frequency range of system identification experiments - r control reference - u control signal - d external disturbance - y control output - e control error - x state vector - p(t) time-varying parameter - N_p parameter boxs dimension - V(x) Lyapunov function - V(x,p) parameter-dependent Lyapunov function - largest parameter box where quadratic stability holds - S(s),T(s) sensitivity and complementary sensitivity transfer functions - G_n(s),G(s) transfer function of nominal and real plant - K_rate rate feedback gain - P_n(s) transfer function of nominal plant modified by rate feedback - P(s) transfer function of real plant modified by rate feedback - G_red transfer function of the reduced-order plant - _m(s) multiplicative uncertainty - W_S(s),W_T(s) performance and robustness weighting functions - G_c(s) controllers transfer function - G_c1(s),G_c2(s) transfer function of robust controller for vertex 1 and vertex 2 - G_cs(p,s) transfer function of the gain-scheduled controller - u_A amplitude of the control signal - K_min,K_max minimum and maximum controller gain - K(p) scheduled controller gain - J₁,J₂,J₃,J₄,J₅ performance evaluation parameters     

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.     

An efficient interval method for global analysis of non-linear resistive circuits

A new method for finding the set of all operating points of non-linear resistive circuits is suggested. It takes into account the fact that the circuit equations are in the so-called separable form where any equation f_i(x) is the sum of terms f_ij(x_j), each term depending on a single variable. The method is based on the introduction of an approximation of every real non-linear function f_ij(x_j) by an appropriate linear interval function, i.e. by a real linear function having an additive interval term. The parameters of the linear interval approximations are dynamically updated at each iteration of the computational process. Numerical experiments show that the computational efficiency of the new method is vastly superior to that of other known methods for global analysis, especially for circuits of large size. © 1998 John Wiley & Sons, Ltd.     

A quasi analytical procedure for determining the optimum commutation angles of PWM converters

Contents A method for calculating the optimum commutation times for PWM converters is proposed. The method is mainly analytical and leads to a simple, fast calculation procedure. The time behaviour of any variable, electrical or non electrical, which is considered to be the most characteristic of the behaviour of the system, can be optimized. The validity of the analytical approach is subject to some hypotheses which are often verified in practice. The computation power required by the method is so limited that, in the majority of cases, a minicomputer can be used.

---

Quasi analytische Bestimmung der optimalen Steuerwinkel für pulsbreitenmodulierte Frequenzumrichter

Übersicht Man beschreibt ein meist analytisches Verfahren für die Berechnung der optimalen Steuerwinkel für pulsbreiten-modulierte Frequenzumrichter. Durch dieses Verfahren wird es möglich, den Zeitverlauf der bedeutendsten elektrischen und nichtelektrischen Größen des Systems zu optimieren. Das Rechnungsverfahren beruht auf meistens erfüllten Annahmen; auf Grund seiner einfachen und schnellen Durchführung braucht man in den meisten Fällen nur Minicomputer zum Zweck.

---

List of symbols fundamental frequency - x(t) modulated wave (input wave) - y(t) variable to be optimized (output wave) - f ^*() transfer function - f ^*() f ^*() amplitude - ^*() f ^*() phase - f(n) f ^*() evaluated only in correspondence with () _f integer multipliers - a _n x(t) Fourier coefficients relative to the terms cos (n _f t) - b _n x(t) Fourier coefficients relative to the terms sin (n _f t) - a _n ,b _n asa _n ,b _n , but relative toy(t) - y _d (t) required output wave-form - a _d (n), b _d (n) y _d (t) Fourier coefficients - y _s (t) deviation function (defined asy(t)–y _d (t)) - a _s (n), b _s (n) y _s (t) Fourier coefficients - ^* rms value ofy _s - as ^* except for some constants - _i commutation angles - m ₁ commutations number in the first half of the period - m number of independent commutations in the period - _ij , _ij x _ij auxiliary variables dependent on the _i - V _k numerical values relative to the links imposed onx(t)'s,y(t)'s harmonics - peak values ofy(t) andx(t) harmonic of ordern - Y _n ,X _n rms values of ordern harmonics ofy(t) andx(t) - _j Lagrange multipliers - z number of constraints relative tox(t)'s harmonics Research supported by Italian Research Council (C.N.R.)     

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.     

Full flexible performance characterization of a feedback applied transistor with LNA applications

In this paper, the full flexible performance characterization of a transistor with series inductive/parallel capacitive feedback is carried out in terms of LNA applications. For this purpose, the input VSWR V_in–maximum available gain G_Tmax variations are constructed for a high technology low-noise transistor that is subject to the required noise figure F_req(f) ≥ F_min(f) along the device's operation band depending on the feedback. These V_in–G_Tmax variations result in the application of a design chart that indicates which value of feedback can be applied within which region of the operation band with the improvable trade-off between the V_in and output VSWR V_out for the Freq(f) ≥ Fmin(f). Following this, the optimum trade-off between V_in and V_out is made for the necessary operation frequency regions using the load impedance Z_L as an instrument with the predetermined source impedance Z_S. Finally, the LNA applications of a series inductive/parallel capacitive feedback applied transistor with the optimum V_in, V_out, and G_T subject to Freq(f) ≥ Fmin(f) ≥ are also presented as distributed across the entire bandwidth in the different operation bands. It can be concluded that this rigorous work will enable a designer to utilize the entire operation frequency band of transistor through using only a single series inductive/parallel capacitive feedback for the LNA designs of Freq(f) ≥ Fmin(f) with the optimum trade-offs among its performance measures.     

Effect of process parameter variation on f t in n-type junctionless FETs

In this paper we have studied the effect of fin width (W), fin height (H), gate oxide thickness (T _ox), gate length (L _g) and doping (N _d) values variations on unity gain cut-off frequency (f _t) in Junctionless FET by performing extensive Technology CAD (TCAD) simulations. The parasitic series resistance decreases as fin width, fin height and doping increases while the total input capacitance (C _gg) increases. Except the higher T _ox with elevated doping values, the device is in g _m dominated region resulting in increased f _t as fin width, fin height and doping values increase.     

Electrical properties of Pb0.98Eu0.02(ZryTi1−y)0.995O3 prepared by sol–gel processing

A Pb_0.98Eu_0.02(Zr_yTi_1−y )_0.995O₃ compound series, with y = 0.60, 0.53 and 0.45 was prepared. PZT samples were synthesized by sol–gel technique. The crystallization and quality of the compounds were analyzed by powder X-ray diffraction and electron microscopy. The shape of the ε′(ω) vs T curves can be considered typical of a ferro-paraelectric transition. The ferro-paraelectric transition temperature for each composition was 348, 328 and 307°C, for the y = 0.45, 0.53 and 0.60, respectively. σ′(ω) is strongly influenced by short range processes. For the logσ′(ω) curves as function of temperature, there is evidence of a non dispersive dc-conductivity component for the high temperature region. The associated dc-activation energies are larger than those calculated for the ac region (at lower temperatures).     

A‐map representations and asymptotically almost periodic responses

It was recently shown that for each member G of a large class of causal time‐invariant non‐linear input–output maps, with inputs and outputs defined on the non‐negative integers, there is a functional A on the input set such that (Gs) (k) has the representation A(F_ks) for all k and each input s, in which F_k is a simple linear map that does not depend on G. In this paper, we consider non‐linear maps G that have such ‘A‐map representations’. We observe that these Gs have extensions to a domain of inputs defined on the set of all integers. We show that these extensions possess some interesting properties including a certain important uniqueness property. As an application, we show that under the (very often satisfied) conditions of time invariance, causality, and approximately finite memory, and under typically mild boundedness and continuity conditions, the response of G to a discrete‐time asymptotically almost periodic input is an output that is always an asymptotically almost periodic function, and that the almost periodic part of the output is independent of the transient part of the input. We also give corresponding results for a continuous‐time case. Copyright © 2001 John Wiley & Sons, Ltd.     

An expanded dalton-cameron method: DC decay testing method at an arbitrary rotor position

The Dalton-Cameron method is a well-known method for determining direct and quadrature axis subtransient reactance (x′_d and x′_q) by standstill response testing. This method entails calculating x′_d and x′_q from the voltage and current measured when a rated-frequency single-phase voltage is applied to each armature winding (U-V, V-W, and W-U) in turn. The authors have developed a new method to calculate x′_d, x′_q and the impedance loci by applying a dc voltage instead of a single-phase voltage. This method was named the expanded Dalton-Cameron method. The method is a small-capacity standstill test, and is carried out by using the following three steps. The first is to short-circuit the U and V terminals while a dc current flows between these terminals, to measure the voltage and current (V_DC and I_DC) when the dc current flows between these terminals and to record the dc decay current (i(t)) after these terminals are short-circuited. This same procedure is also performed for the V-W and W-U terminals in turn. The second step is to draw the impedance loci from the measured V_dc, I_dc and i(t) by means of Fourier transformation and to divide it into the direct-axis and quadrature-axis impedance loci (Z_d(js), Z_q(js)). The third step is to calculated the values of x′_d and x′_q from Z_d(js) and Z_q(js) and the starting performance on the basis of the two-reaction theory. Experimental and calculated results on starting performance, as well as a comparison with calculated results of x′_d and x′_q by the Dalton-Cameron method, clearly show that this method is very useful. © 1998 Scripta Technica, Electr Eng Jpn, 123(2): 53–60, 1998     

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.     

INFLUENCE OF DEPOSITION TEMPERATURE AND N2 FLOW RATE ON HIGH QUALITY ZNO THIN FILM DEPOSITED ON SiO2/SI SUBSTRATE BY ULTRASONIC SPRAY PYROLYSIS

ABSTRACT

ZnO thin films were prepared on SiO₂/Si substrate by ultrasonic spray pyrolysis (USP) method using the aqueous solution of zinc acetate dehydrate. X-ray diffraction (XRD) and atomic force microscopy (AFM) were employed to analyze the crystalline and microscopic structure of the films. The properties of ZnO films were investigated with respect to deposition temperature (T_s) and N₂ flow rate (f). The results show that ZnO thin films exhibit hexagonal wurtzite structure and the highly preferential orientation along c-axis under T_s = 320°C and f = 5 L/min deposition condition.     

Peculiarities of Variation of the Dielectric Permittivity and the Pyroelectric Coefficient with Temperature in PZT-Based Ferrosoft Ceramics Modified with Different Additives of PbGeO3

ABSTRACT

The temperature dependencies of the dielectric permittivity ?( T) the dynamic pyroelectric coefficient γ_d( T) and the quasi-static pyroelectric coefficient γ_st( T) of prepoled samples of the low-coercive materials belonging to the PZT-based multicomponent system modified with 0.5%; 1% and 2% PbGeO₃ and having the Curie temperatures T _c = 207, 215 and 206°C, respectively, were studied. Polarization of the samples was performed under three different conditions. It was been established that in the ?( T) curve one can observe a step-like variation of ?( T) or a slope variation at temperatures below the temperature maximum by 40–50°C. The additional low- and high-temperature maxima are observed in the γ_d( T) and γ_st( T) curves.     

On the steady-state analysis of the inverter-fed induction motor

Contents The analysis of the induction motor based on the signal flow graphs is carried out. It is shown that the state transition matrix can be determined in a systematic and straightforward way using Mason's gain formula. This method is found computationally very much superior to other methods available in the literature. There is a considerable reduction in the CPU time. The presence of an input signal also does not pose any problems, as inherently the transition signal flow graph includes its effect also.

---

Das stationäre Verhalten der stromrichter-gespeisten Asynchronmaschine

Übersicht Es wird in der vorliegenden Arbeit das stationäre Verhalten der stromrichter-gespeisten Asynchronmaschine auf Grund des Signalflußdiagramms untersucht. Die Anwendung der Masonschen Regel liefert unmittelbar die, State Transition Matrix' mit einem CPU-Zeitaufwand, der geringer ist, als der der in der Literatur bekannten Methoden. Beliebige Eingangssignale lassen sich in einfacher Weise behandeln, da diese im Signalflußdiagramm repräsentiert werden können.

---

List of Symbols i _ds,i _qs direct and quadrature axis currents of stator - i _dr,i _qr direct and quadrature axis currents of rotor - i _dr(0),i _qs(0) initial values of d-q currents of stator - i _dr(0),i _qr(0) initial values of d-q currents of rotor - I _ds(s),I _qs(s) Laplace transforms of d-q currents of stator - I _dr(s),I _qr(s) Laplace transforms of d-q currents of rotor - L _s,L _r Self inductances of stator and rotor - M Mutual inductance between stator and rotor - R _s,R _r Resistance per phase of stator and rotor windings - R ₁,R ₂,R ₃,R ₄ Roots of characteristic equation - V _ds V _qs Direct and quadrature components of stator voltage - _ds, _qs Direct and quadrature components of stator flux linkages - _dr, _qr Direct and quadrature components of rotor flux linkages - _s Synchronous angular velocity - _r Rotor angular velocity - T _d Torque developed     

Simple solution curves of non-linear resistive networks

A generalized sufficient condition for the existence of a (unique) simple solution curve defined by f_i(x₁, x₂,…, x_n) = 0 for i = 1, 2. … n ? 1 is developed. the condition is deduced based on a parametric representation of a solution curve. Modified networks having identical (n ? 1) network equations are generated and topological criteria associated with the sufficient condition are obtained from the examination of modified networks.     

Characterization of BaTiO3—BaZrO3 Solid Solution Thin Films Prepared by MOCVD

Thin films of the solid solution system Ba(Ti_{1 ? y} Zr _y )O₃ (y = 0, 0.37, 1) have been grown on ?111? Pt-coated silicon substrates by means of metal-organic chemical vapor deposition (MOCVD). Commercially available liquid precursors typical for the fabrication of perovskite thin films were used: Ba(thd)₂(diethylene-triamine), Ti(OⁱPr)₂(thd)₂, and Zr(OⁱPr)₂(thd)₂ (thd = C₁₁H₁₉O₂) dissolved in diglyme (C₆H₁₄O₃). Growth took place in a horizontal type reactor (AIX-200). The vaporizer technology based on a TriJet? liquid delivery system provided by AIXTRON. It allows to feed in four metal-organic species in a pulsed injection mode. We combined process-related studies (growth rate, rate-limiting factors) with material-related properties (crystallinity and morphology) using various analytical techniques (X-ray analysis, scanning electron microscopy). Furthermore we characterized the samples electrically with respect to the voltage dependence of permittivity and dielectric losses.     

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.     

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.

---

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.

---

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.