Stabilization of carrier-frequency servomechanisms. III. Methods of obtaining required carrier phase-shift

In a servomechanism using a two-phase alternating current control motor, a 90° difference is required in the phases of the carrier-frequency voltages applied to the fixed and control windings. This part describes and compares various methods of obtaining the phase difference.The question of the possibility of a phase-shifting proportional-derivative parallel “T” is answered in the negative, by the result that in any parallel “T” transfer characteristic, if the quadratic factor in the numerator is of the proportional-derivative form at the correct resonant frequency, the amount of phase shift which may be obtained from the remaining portion of the transfer characteristic is less than are tan $\text{(}\text{2}\text{n}\text{)}$, where n is twice the carrier frequency divided by notch width. Thus for values of n high enough to have an appreciable stabilizing effect, the maximum obtainable intrinsic phase shift is negligible.In order to obtain a large phase shift it is necessary to add either a series input or a load impedance to the parallel “T,” or to use a phase-shifting network preceding or following the parallel “T.” Formulae and design charts are given for determination of the values of the components of phase lag networks.The method of calculation of tolerance requirements on the components, in terms of allowable deviation from the correct phase, is illustrated by an example of a phase lag network used in conjunction with a bridge “T” proportional-derivative network.     

Theory and application of antenna arrays : M.T. Ma. 413 pages, 171 figures, 6x9 in., John Wiley,New York, 1974. Price, $22.50.

The theory of transient operations of synchronous machines—the so called two-reaction theory—was developed during the years 1926–1938. Doherty, Nickle and Park made the first efforts to find a complete theory. The problem then was solved by Kron for a general rotating electrical machine. In this paper the two-axis-model-machine is described using a bond graph. An example is given in which state-space-equations and output-equations are derived from the bond graph. A power-conserving transformation between the electrical quantities of the armature windings of the model machine and those of the real three phase armature windings is developed. This transformation is shown as a displacement modulated transformer structure which is central to the bond graph model.     

An experimental investigation of forced vibrations

The solution of the differential equation y″ + 2Ry′ + n²y = E cos pt is written in a new form which clearly exhibits many important facts thus far overlooked by theoretical and experimental investigators. Writing s = n ? p, and Δn = n ? √n² ? R², it is found: (a) When s ≠ Δn, there are “beats,” and the first “beat” maximum is greater than any later maximum while the first “beat” minimum is less than any later “beat” minimum. The “beat” frequency is $\text{(s ?}\text{Δ}\text{n)}\text{2π}$. (b) When n² ? p² = R², there are no “beats,” and the resultant amplitude grows monotonically from zero to the amplitude of the forced vibration, (c) At resonance, when n = p, we still have maxima which occur with a frequency $\text{Δ}\text{n}\text{2π}$ in a damped system. (d) The absence of “beats” is neither a sufficient nor a necessary condition for resonance in a damped system.In the experimental investigation the upper extremity of a simple pendulum was moved in simple harmonic motion and photographic records obtained of the motion of the pendulum bob. Different degrees of damping were used, ranging from very small to critical.The experimental results are in excellent agreement with theory.     

Lotka's law: A testing procedure

Instead of the commonly accepted inverse square law, Lotka's original formulation was based on a more general inverse power law: xⁿ·y = c. The exponent and the constant must be estimated from the given set of author productivity data. A step-by-step outline is presented for testing the applicability of Lotka's law. Steps include the computation of the values of the exponent and the constant based on Lotka's method, and the test for significance of the observed frequency distribution against the estimated theoretical distribution derived from Lotka's formula.     

Lr-synchronization and adaptive synchronization of a class of chaotic Lurie systems under perturbations

The synchronous control of a class of disturbed chaotic Lurie systems is probed in. The conception of L_r-synchronization of drive-respond systems is presented. Via Lyapunov function analysis and comparison principle, L_r synchronous controller of the drive-respond systems under perturbation is given and its robustness is also discussed. Barbalat lemma is further used to derive the adaptively synchronous controller for the unknown disturbance situation and the globally asymptotical synchronization is realized. All designed controllers are verified by the simulations and the given controllers are linear, which are convenient and can produce rapid convergence speed of the error systems.     

Orbit consensus of quantum networks based on interaction design

For a general quantum network system with a non-zero Hamiltonian H composed of n identical m-level quantum subsystems, any symmetric consensus state in the interaction picture exactly corresponds to an orbit in the Schrödinger picture, which is called the H-orbit of the symmetric consensus state. By using the interaction picture transformation and the tool of the LaSalle invariance principle, this paper analyzes the orbit consensus of this quantum network and designs the corresponding swapping operators such that the system converges to the H-orbit of the target symmetric consensus state that exists in the interaction picture. In particular, we prove the convergence of the quantum network to the H-orbit when the quantum interaction graph is connected and the system Hamiltonian is permutation invariant. The orbit consensuses of a four-qubit network system and a quantum network of three identical three-level subsystems are achieved numerically, which verifies the correctness of our theoretical results and the effectiveness of the designed swapping operators.     

Temperature distribution in toroidal electrical coils of rectangular cross section

The essential content of a recent paper by the present writer comprises a comprehensive discussion of the physical bases underlying derivation of formulas for calculating the temperature distribution T, maximum temperature T_m and average temperature T_a in a toroidal electrical coil of rectangular cross section, internally generated heat and change of wire resistance with temperature being taken into account. Illustratively, the solution for the boundary value condition of constant surface temperature and uniform equivalent thermal conductivity was obtained.For the most part, however, problems that arise in practice are not encompassed in the comparatively simple boundary conditions of constant temperature. Experiment shows that in general the boundary condition is $\text{T ? T′ = ?}\text{K?}\text{?n}$; whereof n denotes the outward drawn normal to the coil surface, $\text{K = (}\text{k}{}_{\mathrm{n}}\text{h}\text{)}$ the ratio of the equivalent thermal conductivity in the direction of n to the emissivity of the boundary surface, and T and T′ are the corresponding temperatures in the coil surface and the immediately adjacent ambient medium. Again, it frequently ensues in practice that the thermal conductivity is substantially different in the directions of the two principal axes of the cross section.In the present paper formulas for T, T_m, and T_a are obtained for electrical coils of ratio of external to internal radius greater than (roughly) two whereof (i) the thermal conductivity is different in the directions of the two principal axes of the cross section, (ii) K is different on but constant over each of the four faces of the coil, and (iii) no restriction is made as to T′ except that over each face it be expressible in a generalized Fourier series. Determination of T is posed as a boundary-value problem in the mathematical theory of heat; the formal solution of T effected by expansions in orthogonal functions; and T_m and T_a then determined through use of their known relationships with T. The resulting formulas are in the form of rapidly-converging singly-infinite trigonometric-hyperbolic series. Illustrative of application of these general formulas, the maximum temperatures in a coil of given dimensions subject to two different sets of surface conditions are calculated and found to be in excellent agreement with the known measured values.The just-mentioned formulas encompass practically all cases encountered in practice except those coils which do not satisfy the restriction as to ratio of radii. For these latter formulas for T, T_m, and T_a are obtained pursuant to conditions of (i) equivalent thermal conductivity different in the directions of the two principal axes of the cross section, (ii) K, and likewise T′, different on but constant over each of the four faces of the coil. These formulas are in the form of rapidly-converging singly-infinite trigonometric- Bessel function (of zero order) series: Illustratively, the maximum temperature in a coil of given dimensions is calculated and found to be in excellent agreement with the known measured value.     

Modeling of synchronous electric machines for real-time simulation and automotive applications

Recent research in the field of vehicle electrification has indicated that synchronous machines, which include the permanent magnet synchronous machine (PMSM) and the externally excited synchronous machine (EESM), represent a viable solution for electric propulsion. A challenging problem for synchronous machines drives employed in automotive applications is to obtain accurate mathematical models which can deal with parametric variation and which are suitable for real-time simulations and synthesis of control laws. The goal of this paper is to provide a mathematical modeling framework for synchronous machines that can answer to this challenging problem. To this end, using the rotor reference frame, the mathematical models of PMSMs and EESMs are constructed taking into account also the parametric variation due to magnetic saturation and temperature variation. Then, a complex state-space bilinear model for both EESM and PMSM with parametric variation due to magnetic saturation and temperature are developed. Considering the parametric variation as a polytopic bounded disturbance, it is then shown how to split the bilinear complex model in two PWA variable parameter state-space models suitable for a cascade control structure. Based on the developed models, a dynamic unified simulator was constructed in Matlab^®/Simulink^®. Measurement data obtained in a real test-bench system were used to verify the accuracy of the simulator. The discrete-time simulator was then integrated in an industrial hardware-in-the-loop test bench for real-time evaluation of a current control scheme in EESM drives.     

Solution pH change in non-uniform alternating current electric fields at frequencies above the electrode charging frequency

AC Faradaic reactions have been reported as a mechanism inducing non-ideal phenomena such as flow reversal and cell deformation in electrokinetic microfluidic systems. Prior published work described experiments in parallel electrode arrays below the electrode charging frequency (f_c), the frequency for electrical double layer charging at the electrode. However, 2D spatially non-uniform AC electric fields are required for applications such as in plane AC electroosmosis, AC electrothermal pumps, and dielectrophoresis. Many microscale experimental applications utilize AC frequencies around or above f_c. In this work, a pH sensitive fluorescein sodium salt dye was used to detect [H⁺] as an indicator of Faradaic reactions in aqueous solutions within non-uniform AC electric fields. Comparison experiments with (a) parallel (2D uniform fields) electrodes and (b) organic media were employed to deduce the electrode charging mechanism at 5 kHz (1.5f_c). Time dependency analysis illustrated that Faradaic reactions exist above the theoretically predicted electrode charging frequency. Spatial analysis showed [H⁺] varied spatially due to electric field non-uniformities and local pH changed at length scales greater than 50 μm away from the electrode surface. Thus, non-uniform AC fields yielded spatially varied pH gradients as a direct consequence of ion path length differences while uniform fields did not yield pH gradients; the latter is consistent with prior published data. Frequency dependence was examined from 5 kHz to 12 kHz at 5.5 V_pp potential, and voltage dependency was explored from 3.5 to 7.5 V_pp at 5 kHz. Results suggest that Faradaic reactions can still proceed within electrochemical systems in the absence of well-established electrical double layers. This work also illustrates that in microfluidic systems, spatial medium variations must be considered as a function of experiment time, initial medium conditions, electric signal potential, frequency, and spatial position.     

Adaptive stabilization of linear and nonlinear plants in the presence of large and arbitrarily fast variations of the parameters

The problem of adaptive stabilization of a class of continuous-time and time-varying nonlinear plants is treated in this paper. The control scheme guarantees that the state of the plant, with bounded time-varying parameters, asymptotically converges to zero. For the nonlinear case with n²+n unknown parameters (n time-varying and n² constant), when the control matrix B is unknown the controller has to adjust n²+1 parameters providing only local stability results. On the contrary, when the control matrix B is known only one parameter has to be adjusted and the proposed scheme provides global stability results. The general methodology is particularized for the linear case with 2n² unknown parameters (n² time-varying and n² constant), adjusting n²+1 parameters when the control matrix B is unknown and guarantees only local stability results, whereas in the case when the control matrix B is known only one parameter has to be adjusted and the proposed scheme provides global stability results.     

Stabilization of carrier-frequency servomechanisms. II. Design formulae for proportional-derivative networks

In an alternating current servomechanism, the error is proportional to the modulation envelope of a modulated-carrier error signal. It is shown in part I that for stability and fidelity of the servo, it is highly desirable that the effect of the controller includes a proportional-derivative action on the modulation envelope. This action may be obtained with various forms of RC networks, including the parallel “T,” bridge “T,” and Wien Bridge forms.This part contains detailed design procedures and tables of values for the various types of proportional-derivative networks. Several forms of parallel “T” networks arise from the fact that there are five independent time constants in the network, while in order to realize the desired transfer characteristic it is necessary to impose only four conditions. It is indicated how the remaining degree of freedom may be used to obtain the most suitable input and output impedances for the source and load impedances with which the parallel “T” is to be used. The derivations for the parallel “T” formulae are given in an Appendix.Tolerance requirements on the components of parallel “T” and bridge “T” networks are derived. If ±1 per cent components are used at 60 cycles, the resonant frequency will lie between 56.4 and 63.6 cycles, and the notch width (rejection band width) will be within ±0.99 cps. of the correct value. In order to guarantee that the phase shift at 60 cycles is within ±10°, the percentage deviation of each part must be less than ($\text{9.0}\text{T}{}_{\mathrm{d}}\text{ω}{}_0$), where ω₀ is the carrier angular frequency, T_d the derivative time constant.     

Stability criteria for certain classes of nonlinear difference equations

The general mth order difference equation X(n+m)+a₁X(n+m?1)+…+a_mX(n) = F[n,X(n),…,X(n+m?1)] is considered. The stability properties of its solutions are studied using the discrete form of Liapunov's direct method. A quadratic form is selected as a possible Liapunov function V(n,X) and a scheme is developed for determining appropriate conditions on this function to insure that its total difference ΔV(n,X) is negative semi-definite or negative definite with respect to the difference equation. The approach is applied to the fourth-order difference equation in full detail to illustrate the method for determining the conditions which imply either uniform stability or uniform asymptotic stability and specific results are obtained. Several comments on, and extensions of, the work done by Puri and Drake for the cases m = 2 and m = 3 are presented.The results of the present approach in the homogeneous case where F[n,X(n),…,X(n+m?1)] = 0 are compared with the usual Schur-Cohn criteria and are shown to be at least as good.     

A short account of the theory of lubrication. III. dimensional theory,with applications

In Part I properties of the scale coördinate, of the form: B(n + θ) are discussed. n is shown to be associated with the operation of counting scale marks, θ with the operation of estimating between them, and B, with the operational and configurational aspects of that part of apparatus which lies adjunct to the scale system.In Part II three types of measurement codification are discussed: (a) the differential interval; (b) the finite amorphous interval; (c) the scale interval; a relationship among them is postulated.In Part III the finite differences in scale coördinates are defined and simple theorems are used to illustrate these definitions. Simple difference equations in scale coördinates are solved to illustrate macroscopic “selection principles” arising partly out of the methodology of codifying a coincidence in scale coördinates.In Part IV an example of causally related dimensional systems is described by use of the scale coördinate. This example is taken from the perfect gas law and Van der Waals' gas law.     

Spatial selective manipulation of microbubbles by tunable surface acoustic waves

A microfluidic device based on a pair of slant-finger interdigital transducers (SFITs) is developed to achieve a selective and flexible manipulation of microbubbles (MBs) by surface acoustic waves (SAWs). The resonance frequency of SAWs generated by the SFITs depends on the location of its parallel pathway; the particles at different locations of the SAWs'' pathway can be controlled selectively by choosing the frequency of the excitation signal applied on the SFITs. By adjusting the input signal continuously, MBs can be transported along the acoustic aperture precisely. The displacement of MBs has a linear relationship with the frequency shift. The resolution of transportation is 15.19 ± 2.65 μm when the shift of input signal frequency is at a step of 10 kHz. In addition, the MBs can be controlled in a two-dimensional plane by combining variations of the frequency and the relative phase of the excitation signal applied on the SFITs simultaneously. This technology may open up the possibility of selectively and flexibly manipulating MBs using a simple one-dimensional device.     

Transmission Line Circuit Models for Dielectric Waveguide

A general procedure is outlined for obtaining single or coupled transmission line models to represent the propagation of surface wave modes in conductively unshielded dielectric waveguides. The procedure uses a homogeneous electrically or magnetically walled waveguide having the same dimensions as the dielectric of the surface waveguide, to produce a set of orthogonal eigenfunctions. These eigenfunctions are projected upon Maxwell's equations resulting in a system of transmission lines coupled together through a wave immittance, which represents the ratio of a longitudinal and a transverse field component at the dielectric-air interface. Examples are given for various modes of the dielectric slab and the dielectric rod, in particular the HE_1n modes for the latter. The transmission line models derived for these examples consist of a single trasmission line found directly by projection or reduced from a coupled transmission line model by port elimination, or of two transmission lines coupled together. All circuit models derived preserve the basic properties of surface waves (e.g. no solution below cut-off), and any of the single line models can be solved to give explicit approximate algebraic formulae for the propagation constant as a function of frequency. Numerical results show that the dispersion curves calculated from the models versus exact values are generally excellent over the entire frequency spectrum.     

On the Algebra of Multiple Bilinear Transformations

In this paper a closed-form relationship between the original coefficients of a 2-D analog transfer function and the coefficients of the transformed function after application of the double bilinear transformation in an alternative manner is given. The extension to the n-dimensional case, as well as the 1-dimensional case are presented. Examples are given to illustrate the usefulness of the derived relationship.     

Non-iterative pole placement technique: A step further

This paper comes back to the hard problem of pole placement by static output feedback: let a triplet of matrices {A;B;C}∈R^n×n×R^n×m×R^p×n be given, find a matrix K∈R^m×p such that the spectrum of A+BKC equals a specified set. More precisely, this article focuses on the derivation of non-iterative techniques, based upon the notion of eigenstructure assignment, to solve the problem, especially when Kimura's condition does not hold. Some solutions can sometimes be found.     

An application of the theory of the complex variable to the mapping of some potential functions.

Using the theory of the logarithmic potential and the methods of intersecting circles and radii, plots are obtained for the lines of flow and equipotentials for eight cases of two sources, two sinks, or a source and a sink of varying strengths. A detailed analysis of the equilibrium equipotentials is included. The problem of three unit sources placed at the vertices of an equilateral triangle is investigated; and the general expression of the equilibrium equipotential for n similar unit sources (or sinks) placed at the vertices of a regular n-sided polygon is developed.