Zonal Harmonic Series Research Articles

AXISYMMETRIC ELECTRIC FIELD CALCULATION WITH ZONAL HARMONIC EXPANSION

The electric potential and fleld of an axially symmetric electric system can be computed by expansion of the central and remote zonal harmonics, using the Legendre polynomials. Garrett showed the usefulness of the zonal harmonic expansion for magnetic fleld calculations, and the similar radial series expansion has been widely used in electron optics. In this paper, we summarize our experience of using the zonal harmonic expansion for practically interesting axisymmetric electric fleld computations. This method provides very accurate potential and fleld values, and it is much faster than calculations with elliptic integrals. We present formulas for the central and remote expansions and for the coe-cients of the zonal harmonics (source constants) in the case of general axisymmetric electrodes and dielectrics. We also discuss the general convergence properties of the zonal harmonic series (proof, rate of convergence, and connection with complex series). Practical considerations about the computation method are given at the end. In our appendix, one can flnd many useful formulas about properties of the Legendre polynomials, various derivatives of the zonal harmonic functions, and a simple numerical integration algorithm.

High precision computation of solenoid magnetic fields by Garrett's methods

For magnetic field calculations of solenoids two methods, due to M.W. Garrett (1967), are developed. The first one consists of the expansion of the magnetic field in a zonal harmonic series in the central region of solenoid and in outer space. For both regions the expansion coefficients are given either by simple recurrence relations, or by explicit expressions. In particular, this method is suitable for high accuracy calculations of the field within windings. In the second method, the field is expressed in terms of incomplete elliptic integrals of the first and second kinds computed by use of raising Landen transformations. Given the high precision and high computation rate of this method, it has advantages in analyzing field problems for windings consisting of many radial sections carrying different currents, as occurs during the quench of superconducting multisectioned magnets. Also, the general formulas for mutual inductance and axial force between a pair of coaxial solenoids are given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Representation of harmonic functions in R4

Reciprocal integrals are constructed so that { H , h } constitutes a dual transform pair where H is a harmonic function in R 4 and h is the associated analytic function in C 3 . Necessary and sufficient conditions for the harmonic continuation to encounter singularities are linked to properties of the analytic continuation and conversely. Thus, Szego's theorem regarding zonal harmonic series and analytic functions in C 1 has a function theoretic extension to several variables.

Zonal harmonic series expansions of Legendre functions and associated Legendre functions

The Legendre functions pnu m(+or-cos theta ) of complex degree nu and integral order m may be expanded in terms of Legendre functions of integral degree and order, the latter being the zonal harmonic functions Pnm(cos theta ). Two methods for improving the convergence of a standard series expansion are discussed for the cases m=0 and 1. The first method involves repeated application of the relations between contiguous Legendre functions and the second employs integral relations between Legendre functions of different order. The formulae derived are suitable for computation and easily programmed.

Off-center spherical model for dosimetry calculations in chick brain tissue.

This paper presents calculations for the electric field and absorbed power density distribution in chick brain tissue inside a test tube, using an off-center spherical model. It is shown that the off-center spherical model overcomes many of the limitations of the concentric spherical model, and permits a more realistic modeling of the brain tissue as it sits in the bottom of the test tube surrounded by buffer solution. The effect of the unequal amount of buffer solution above the upper and below the lower surfaces of the brain is analyzed. The field distribution is obtained in terms of a rapidly converging series of zonal harmonics. A method that permits the expansion of spherical harmonics about an off-center origin in terms of spherical harmonics at the origin is developed to calculate in closed form the electric field distribution. Numerical results are presented for the absorbed power density distribution at a carrier frequency of 147 MHz. It is shown that the absorbed power density increases toward the bottom of the brain surface. Scaling relations are developed by keeping the electric field intensity in the brain tissue the same at two different frequencies. Scaling relations inside, as well as outside, the brain surface are given. The scaling relation distribution is calculated as a function of position, and compared to the scaling relations obtained in the concentric spherical model. It is shown that the off-center spherical model yields scaling ratios in the brain tissue that lie between the extreme values predicted by the concentric and isolated spherical models.

Accelerating the convergence of the zonal harmonic series representation in the Schumann resonance problem

A formula of Nickolaenko and Rabinowitz for the calculation of Legendre's function of the first kind of complex degree is corrected. A numerical test of the corrected formula is presented.

Speeding up the convergence of the zonal harmonic series representation in the Schumann resonance problem

Excitation of the isotropic homogeneous Earth-ionosphere cavity by a vertical electrical Hertzian dipole is considered. Solution of the problem is presented in the form of a zonal harmonic series representation (ZHSR). An algorithm for speeding up the ZHSR convergence is proposed and used. The high accuracy of the method enabled us to check the convergence of the conventional zonal harmonic series representation. The results developed can be used when computing the values of the Legendre functions of non-integer index.

Tidal oscillation of water in a rotating rectangular basin of variable depth

The free oscillation of water in a rotating rectangular basin of variable depth is discussed. The depth is assumed to decrease from the center to the edges according to a paraboloidal law. The solution is obtained in terms of double series of zonal harmonics. Numerical solutions were worked out. Complete sets of modes will be evaluated by an electronic computer.

Electromagnetic Induction in a Conducting Sphere with a Concentric Shell

A quasi‐static derivation is outlined for the problem of a shell‐enclosed homogeneous solid sphere immersed in a slowly varying electromagnetic field. The secondary magnetic potential is expressed as a zonal harmonic series whose terms are proportional to a complex multipole response function xn of order n. The frequency dependence of the real and imaginary parts of xn is demonstrated to have diagnostic features related to the geometry and to the electric properties of the core and the shell.

Near fields and antipodal fields in the terrestrial Earth—ionosphere waveguide

Near fields of a vertical electric dipole are expressed as a series of zonal harmonics, or as a summation of waveguide modes. These representations are compared with dipole fields on a conducting ground plane, and with field calculations, using the method of images or ray theory. The series of zonal harmonics and the mode sum provide nearly identical near fields; their difference characterises the contribution of ‘branch-cut’ waves, and is negligible in the present computational model. In the near-field range, the method of images gives the simplest field representation for e.l.f.; for v.l.f., a useful approximation is provided by combining the ground wave and a single sky wave. At the upper waveguide boundary, the radial electric fields are finite, but the other field components vanish when approaching the source axis. The field components also exhibit similar variations when approaching the antipode of the source.

Thick Cylindrical Coil Systems for Strong Magnetic Fields with Field or Gradient Homogeneities of the 6th to 20th Order

For strong homogeneous fields the ideal loop and solenoid elements of a prototype system may be expanded into thick cylindrical coils whose current density is (a) uniform or (b) inversely proportional to the cylindrical radius. Error coefficients of the zonal harmonic series for field or gradient that are cancelled in any given prototype continue to vanish as the elements are expanded, even to fusion, if at each stage the coil boundaries are adjusted by an iterative method. This involves the use of (a) the harmonic source functions Un* or (b) the Legendre functions Pn; either set is best computed by a recursion formula. An alternative method based on Lyle's principle requires no iterations and only trivial calculations. Though it also is valid for systems of any order, it can only expand a set of loops into coils with square cross sections of limited area. Also, the true null coefficients are replaced by small residues proportional to 4th and higher powers of the section dimensions when the second method is used. The number of tabulated prototype systems of the 6th and 8th orders is increased to more than 200. Since multiply infinite sets of prototypes can be computed for every order higher than the 8th, univariate tables would be inadequate and only a few illustrative examples of such systems are given. Some 150 thick-walled systems of the 6th or 8th order are listed in several tables, but such a selection can only suggest the wide range of possibilities. These systems have more degrees of freedom than do the corresponding prototypes, and even the 6th order is multivariant. At the 8th order and beyond, the growing redundancy of variables permits the design of systems with a common current density in all coils and with all section dimensions proportional to a set of small integers. Systems of order 4n have paired coils with gaps for lateral access, an advantage that is lacking in the 6th-order designs. Efficient computer programs have been written to design systems of all the types so far enumerated, including multiple sets of both prototypes and fully adjusted systems of thick coils for orders from 8 to 20 in steps of 4. Since adequate tabulation of results is ruled out by their diversity and sheer bulk, in view of the wide range of potential applications the programs themselves must be considered the basic research tool. They will be described in detail elsewhere. As a measure of field inhomogeneity, including contributions from all orders and from both axial and radial field components, the concept of total vector error is developed. The tables include error limits for all systems, and some 40 systems or their error contours are shown in the figures.

Solid Angle Subtended by a Circular Aperture

Methods for calculating the solid angle subtended by a circle at an off-axis point are correlated and the rates of convergence of available series are compared. Averages over a plane surface or a cylindrical volume are derived from an analogy that permits the use of existing tables of mutual inductances. Rates of variation, for displacement or rotation of the source, and the technique of computing from the zonal harmonic series are briefly discussed.

A Formal Theorem on the Derivatives of a Series of Zonal Harmonics

A Formal Theorem on the Derivatives of a Series of Zonal Harmonics

Investigations on Series of Zonal Harmonics

Investigations on Series of Zonal Harmonics

On the application of harmonic analysis to the dynamical theory of the tides. Part I. On Laplace's 'oscillations of the first species,' and on the dynamics of ocean currents

1. By a transformation of the differential equations of Poincaré for the oscillations of a rotating mass of liquid, an equation for the tidal oscillations of the ocean is obtained in a form similar to that employed by Laplace. From this equation it is deduced that if the surface-value of the disturbing potential which gives rise to the oscillations is expressible as a series of zonal harmonics.