Sin Wt Research Articles

Economic Cycle Math

In this short paper I would like to present a rigorous mathematical and yet heuristic description of modelling the economic cycles. The idea behind the cycles is that its dynamics could be decomposed into oscillation of two wave functions with low frequency (long-term cycle) and high frequency (short-term cycle). Finding the solution/s for these equations will be done with a use of a simple trigonometric functions such as sin⁡wt and cos⁡wt that lie at the heart of the wide amount of physical processes from Newtonian to quantum mechanics. Ultimately, we arrive at the energy equation – that is a driving force of this process.

Energy Considerations for Tracking in DC to DC Power Converters

A design of an adaptive controller applied to the boost and buck-boost converters to deal with the problem of tracking a sinusoidal voltage is proposed. The main contribution is to provide conditions on the design procedure in order to obtain a reduction in the DC voltage (offset of the sinusoidal signal) of the reference signal; in this way the AC energy is maximized. A nonlinear stable system is designed in order to produce the necessary inputs to exactly track the inductor reference current, which is a necessary condition to achieve the tracking behavior of the voltage reference signal ft=A+Bsin⁡(wt). A numerical example is provided to corroborate the result.

Trigonometrically fitted multi-step Runge-Kutta methods for solving oscillatory initial value problems

In this paper, trigonometrically fitted multi-step Runge-Kutta (TFMSRK) methods for the numerical integration of oscillatory initial value problems are proposed and studied. TFMSRK methods inherit the frame of multi-step Runge-Kutta (MSRK) methods and integrate exactly the problem whose solutions can be expressed as the linear combinations of functions from the set of $\{\exp (\mathrm {i}wt),\exp (-\mathrm {i}wt)\},$ or equivalently the set $\{\cos (wt),\sin (wt)\}$ , where w represents an approximation of the main frequency of the problem. The general order conditions are given and four new explicit TFMSRK methods with order three and four, respectively, are constructed. Stability of the new methods is examined and the corresponding regions of stability are depicted. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature.

Be Careful What You Assign: Constant Speed Parametrizations

SummaryWe explore one aspect of a multivariable calculus project: parametrizing an elliptical path for an airplane that travels at constant speed. We find that parametrizing a constant speed elliptical path is significantly more complicated than a circular path and involves taking the inverse of the elliptic integral of the second kind. We compare our constant-speed parametrization to the standard ellipse parametrization (x(t) = a cos wt, y(t) = b sin wt) and generalize to parametrizing other constant-speed curves.

Determination ofM2ocean tide parameters from satellite orbit perturbations

A sequence of 80 mean inclinations obtained from successive 2-day orbital arcs of a U.S. Navy navigation satellite (1967-92A) has been used to obtain M2 ocean tide parameters. By assuming solid tide parameters of k2 = 0.30 and δ2 = 0° the implied observation equation on terms of order 2 and degree 2, 4, 6, …, is (3.8″±0.60) × 10−2 sin [wt + (331°±10°)] = (0.99″/cm) × 10−2 C22+ sin (wt + e22+) + (0.94″/cm) × 10−2 C42+ sin (wt + e42+) + …, where w = 1 cycle/13.58 days and t is the time in mean solar days from 1900.0. This observation agrees well with published numerical solutions of Laplace's tidal equation.

A Transistorized Digital Analog Converter with Ac Amplification

A new type of analog-to-digital converter is described. This converter presents one analog input and several outputs with two possible states. Each time, only one output is activated. The converter is built up with the following elements: controlled oscillator, saturated tuned amplifiers, detectors, and inverters. The converter principle is: a voltage input E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> feeds a controlled oscillator whose output volttages are E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> sin wt and -E sin wt, with E constant. These voltages feed the two sides of a voltage divider made up of resistors of convenient values. As the two voltages mentioned above are in phase opposition, one point of the divider, for each E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , presents zero potential. The detection of this zero potential point furnishes an activated output, for each input E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> . A practical realization of the converter, all transistorized, is presented by the author.

A Compass Using a Thin Magnetic Film Component

The design and development of an allelectronic device which indicates bearing from local magnetic north is reported on. The device contains no moving parts and produces a d-c output voltage proportional to the bearing of a reference direction of the device from magnetic north. The sensing element of the compass consists of two thin magnetic films each of which contains a pumping coil and an output coil. When an a-c current is applied to the pumping coils an a-c voltage is developed in the output coil whose frequency is harmonically related to that of the pumping current and whose amplitude varies as H cos \theta, where H is the magnitude of an externally applied magnetic field and \theta is the angle between the external field and the axis of the output coil. By placing the two magnetic films in space quadrature and shifting the phase of the output voltages of the units so that the outputs are also in time quadrature, the output voltage of one unit will vary as H cos \theta sin wt while that of the other will vary as H sin \theta cos wt. The sum of these voltages will produce an a-c voltage whose phase shift depends upon \theta, according to the trigonometric identity: cos \theta sin wt + sin \theta cos wt = sin (wt + \theta). Detecting the phase shift of this signal will thus produce an output voltage proportional to the bearing of the magnetic field sensing element with respect to the external magnetic field. The method used in this device to detect this phase shift is to derive a series of pulses at the zero-crossings of the output voltage and apply these pulses to the set terminal of a bistable multivibrator. Another series of pulses derived from the zero-crossings of the pumping signal is applied to the reset terminal of the multivibratoro. The output of the multivibrator will then be a square wave whose duty cycle is proportional to the time interval between a reset pulse and the immediately preceding set pulse. Because these series of pulses are harmonically related, this time interval between the set pulse and the reset pulse will be proportional to \theta, and thus the d-c component of the output of the multivibrator will be proportional to \theta.