Derivation Of New Formulas Research Articles

Trapezoid-shaped resonators to design compact branch line coupler with harmonic suppression

This article proposes a new branch-line coupler design using two trapezoid-shaped resonators, applied in side and center. The standard 35 and 50 Ohms λ/4 transmission lines (TLs) are attained through the variations of the resonator’s TLs width. Since there are not explicit formulas for the suggested trapezoidal shape resonator, the design procedure will include the derivation of new formulas to configure the LC equivalent circuit. The proposed branch-line coupler, operating at 0.95 GHz, has achieved 79% of size reduction if compared to the conventional designs. Furthermore, the suggested design has successfully diminished the first six harmonics with a suppression level of (41, 29, 29, 28, 20, and 20) dB, respectively. The bandwidth is from 800 MHz to 1 GHz and The fractional bandwidth is 22.2%.

Local linear scale factors in map projections in the direction of coordinate axes

ABSTRACT This paper explains that the terms “horizontal and vertical scales” are not appropriate in map projections theory. Instead, the authors suggest using the term “scales in the direction of coordinate axes.” Since it is not possible to read a local linear scale factor in the direction of a coordinate axis immediately from the definition of a local linear scale factor, this paper considers the derivation of new formulae that enable local linear scale factors in the direction of coordinate x and y axes to be calculated. The formula for computing the local linear scale factor in any direction defined by dx and dy is also derived. Furthermore, the position and magnitude of the extreme values of the local linear scale factor are considered and new formulas derived.

Error-Rate Analysis of FHSS Networks Using Exact Envelope Characteristic Functions of Sums of Stochastic Signals

In this paper, we present a novel approach to analytically study and evaluate the error probability of frequency- hopping spread-spectrum (FHSS) systems with intentional or nonintentional interference over the additive white Gaussian noise and Rayleigh-fading channels. The new approach is based on the derivation of new formulas for the exact envelope characteristic function (EECF) of the general sum of n stochastic sinusoidal signals, with each of the n signals having different random amplitude and phase angle. The envelope probability density function (pdf) is obtained from the characteristic function (CF), which, in the important cases of interest, is shown to also give simpler formulas in terms of the Fourier transform (FT) of the Bessel functions. Previously, the Ricean envelope density had only been verified for the very special case where n = 1, and the phase is uniform and independent of amplitude. Here, a new formula for the exact density of the envelope of noisy stochastic sinusoids (EDENSS) is presented, which leads to the generalization of the Ricean envelope-density (GRED) formula under the most general conditions, namely, n ges 1 signals, dependent amplitudes, and phases having an arbitrary joint pdf. The EDENSS and GRED formulas are applied to compute the pdfs needed in noncoherent detection under noise. The derived formulas also lead to the exact formulas for the error probability of the FHSS networks using M-ary amplitude shift keying (MASK) without setting limits on the number of interferers or the symbol alphabet. The power of our EECF and FT methods is further demonstrated by their ability to give an alternative derivation of the exact general envelope-density (EGED) formula, which has previously been reported by Maghsoodi. The comparative numerical results also support the analytical findings.

Coefficients for studying one-step rational schemes for IVPs in ODEs: II

In [1], we derived variable order, variable stepsize, one-step nonlinear methods in which global formulation is facilitated by defining some certain coefficients. The analysis has a broader application and facilitates the derivation of new formulae. The methods are based on a variant self-adjusting rational interpolant of Luke et al. [2] and Otunta and Ikhile [3,4]. In this paper, we motivate an extension of this analysis to obtain a more generalized class of the rational methods in [1] for an arbitrarily desired order.

Background to New Formulas for the Ultimate Limit State of Tubular Joints

Summary Tubular joints constitute one of the main high-cost and problem areas in design, construction, and maintenance of steel offshore structures and have been the subject of considerable research. Recently a study of various design documents was funded by the U.K. Dept. of Energy, with particular emphasis on the static strength of tubular joints. This study concluded that no two published design documents made identical recommendations and that major inconsistencies exist. Therefore, a complete review of available static strength test data was undertaken. This paper presents the derivation of new formulas describing the ultimate limit state of tubular joints. The ranges of application of the formulas are discussed in relation to typical joint configurations and loading, and areas for future research are identified. Introduction Most steel offshore structures consist of three-dimensional (3D) frames composed of cylindrical steel members with hollow sections, which provide the best compromise to the requirements of low drag coefficient, buoyancy, and high strength-to-weight ratio. The hollow sections also allow use of the inner space for transportation, fire protection, or additional strength. The principal design and fabrication problem for this form of structure principal design and fabrication problem for this form of structure is the welded joints between members. Typical simple tubular-joints and the notation system used in this paper are shown in Fig. 1. The word "simple" refers to joints without overlap of brace members and without the use of gussets, diaphragms, stiffeners, or grout. Design certification of such offshore structures has led to many codes of practice, rules, regulations, guidance notes, and published research data concerning tubular joints. Recently, a published research data concerning tubular joints. Recently, a study of the various design documents was carried out for the Underwater Engineering Group and funded by the U.K. Dept. of Energy. The study includes an exhaustive review of available static strength test data, since major inconsistencies exist in current design documents for the calculation of static strength, particularly in inherent levels of safety. particularly in inherent levels of safety. Database Careful study of all available test data revealed inconsistencies, particularly in documentation of material properties. In many particularly in documentation of material properties. In many cases, only the minimum specified material yield stresses are quoted, rather than actual values. The use of such values can lead to unconservative recommendations. Therefore, all such reported results were omitted from the data base. A number of test results were discounted where insufficient information was available or where inadequate testing procedures were adopted. The modes of failure for tubular joints are procedures were adopted. The modes of failure for tubular joints are discussed in Ref. 2. Test results where failure of the specimen was caused by a problem other than joint failure, such as brace yielding, were eliminated from the data base. The limit state design approach adopted in this paper differentiates between the two limit states of strength and serviceability. The former is taken as the maximum load achieved during the test and the second as a deflection or some other local damage criterion. Thus for the tension tests the maximum load achieved during the test has been used. The "first-crack" load is related to a serviceability limit state. The ultimate moment is taken as that value of moment at the intersection of the brace and chord rather than that at the centerline of the chord. The available test data for K-joints all relate to the case in which the net force perpendicular to the chord is zero, as is generally required by the design codes for the joint to be designed as a K-joint. Under such conditions, failure is always associated with the compression-loaded brace, and therefore the ultimate load values used for K-joints are those associated with this brace. After these criteria were applied, the data base presented in Ref. 1 was developed, giving the results for 207 tests (Table 1). General Form of Strength Equations In line with current trends, the format of the equations developed is based on brace member loads. This approach (1) avoids the need to introduce the concept of "punching shear," (2) is directly compatible with computer-based analysis methods that provide member loads, and (3) is also amenable to hand calculation. provide member loads, and (3) is also amenable to hand calculation. JPT P. 147

The Theory of Thermal Breakdown of Solid Dielectrics

The purpose of the paper is to correlate the work which has been done on thermal breakdown and to put it in a form in which it can be used by the electrical engineer in the calculation of breakdown voltage. Besides a treatment of the Fock theory, the paper includes the derivation of new formulas for breakdown of very thin and very thick samples and for internal temperature rise and current. A startling fact brought out by this analysis is that before breakdown the internal temperature never rises more than about 10 deg. cent. above ambient temperature. The Fock tables of breakdown voltage are extended to cover a wider range of temperature and thickness, and the results are put in several convenient forms for numerical calculation. Experimental data are also presented to verify the theory.

Thermal breakdown of solid dielectrics

The purpose of this paper is to correlate the work which has been done on thermal breakdown and to put it in a form in which it can be used by the electrical engineer in the calculation of breakdown voltage. Besides a treatment of the Fock theory, the paper includes the derivation of new formulas for breakdown of very thin and very thick samples and for internal temperature rise and current.