Aziz S. Inan

Impedance Between Adjacent Nodes of Infinite Uniform D -Dimensional Resistive Lattices

as the momentum density. This is the result obtained by Rowland and Pask, under the usual conditions pertaining to transverse waves on a string. Electronic mail: awalstad@pitt.edu D. R. Roland and C. Pask, ‘‘The Missing Wave Momentum Mystery,’’Am. J. Phys. 67, 378–388 ~1999!. W. C. Elmore and M. A. Heald, Physics of Waves ~McGraw–Hill, New York, 1969; Dover, New York, 1985!. We are speaking here of small longitudinal motions associated with a transverse wave. The wave equation itself is derived under the approximation of purely transverse motion and uniform tension. Due to curvature of the string, the tension forces at opposite ends of an infinitesimal segment do not cancel. The longitudinal component of the resulting net force is much smaller than the transverse component if we have u]h/]xu!1. The longitudinal motions may then be treated as a perturbation on the dominant transverse motions. Longitudinal motions are also produced by the variations in string tension associated with longitudinal waves; indeed, the physical impetus which establishes a transverse wave is likely to generate a longitudinal wave too, and longitudinal waves are essential to the conservation of momentum when a transverse wave encounters a density discontinuity. Interested readers should consult Rowland and Pask.

Calculating electromagnetic force and energy using singularity functions

The purpose of this article is to provide physical insight into a new analytical approach to calculate electromagnetic force and energy using singularity functions, specifically the unit impulse function and its integral, the unit step function. The electromagnetic force acting on a charged conductor is expressed in terms of a special singularity integral which is the sifting integral of the unit step function. The well-known parallel-plate capacitor problem in electrostatics is considered as an example to demonstrate the validity and use of this special integral. The authors believe that this technique can easily be extended to problems involving high-frequency microwave systems such as waveguides and resonant cavities.

Revisiting the sifting integral: an interesting special case

We provide a simple proof of the sifting integral as applied to a special-case discontinuous function of the form f(t)=g(t)+ku(t) and illustrate its application with an example involving a simple passive circuit. This special-case value of the sifting integral is either considered to be ambiguous and meaningless, or is completely avoided in the current educational literature. We hope that our discussion can serve the purpose of highlighting the importance of this property of the sifting integral and its applications in electrical engineering.

Calculating the per-unit-length circuit parameters of a coaxial transmission line using singularity functions

We apply a new analytical technique involving singularity functions to calculate the per-unit-length capacitance and inductance of a coaxial transmission line. With this technique, the force on each conductor of the coaxial line can be directly calculated by incorporating the sifting integral of the unit step function which is equal to 1/2. The per-unit-length parameters are extracted from the total stored energy in the coaxial line by varying the radius of one of the conductors and integrating the force expression on the conductor surface over the appropriate range of this radius.

What did Gustav Robert Kirchhoff stumble upon 150 years ago

Gustav Robert Kirchhoff is, without a doubt, one of the founding fathers of electrical engineering. The two circuit laws named after him, formulated in 1845 at age 21, constitute the foundation of electric circuit theory. Most electrical engineers assume that Kirchhoff is well known because of these two laws and are unaware of his other important contributions to science. About 150 years ago in 1859, while performing an experiment with a spectroscope, Kirchhoff unexpectedly made a major discovery that brought him and his colleague Robert Wilhelm Bunsen their greatest scientific fame: the spectrum analysis. The purpose of this article is to use this special 150th anniversary of his discovery as an opportunity to commemorate Kirchhoffs life and bring attention to his many accomplishments.

Special singularity integrals encountered in electric circuits [RLC circuit examples]

In this article, the authors first introduce four special singularity integral identities that involve the impulse, the doublet and the unit step functions, and provide a simple proof for each integral identity. Next, to demonstrate the use of these singularity integral identities, the authors consider the impulse response of two second-order RLC circuits and use these integrals to calculate the energy stored and dissipated in each circuit. The authors believe that these solutions are direct and easy to use in electrical circuit problems involving singularity functions. They hope that a wider coverage of these special integral identities and their applications will be offered in the educational literature.