Peter M. Osterberg

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.

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.