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Optimum Matching Of Two Unequal Impedances In Acoustical, Optical, Or Microwave Stepped Transmission Lines

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2001 Annual Conference


Albuquerque, New Mexico

Publication Date

June 24, 2001

Start Date

June 24, 2001

End Date

June 27, 2001



Page Count


Page Numbers

6.769.1 - 6.769.9

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Albert Biggs

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NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

Session 2793

Optimum Matching of Two Unequal Impedances in Acoustical, Optical, or Microwave Stepped Transmission-Lines

Albert W. Biggs University of Alabama at Huntsville


A simplified method is described for designing periodic structures with transformer sections connected in series to match impedances or provide transitions from slow wave to fast wave transmission lines in acoustical, optical, and microwave waveguides. For a given bandwidth, transformer section characteristic impedances are designed to provide a Chebyshev Polynomial, or equal ripple, reflection coefficient response, which reduces reflections to negligible levels. It extends Dolph-Chebyshev antenna theory to transmission line transformers. The earlier method was W.W. Hansen’s Binomial Coefficient design for several transformer sections. Over a bandwidth ratio of f2/f1 = 2.0, the Chebyshev Polynomial method has a VSWR of 1.02 to the binomial coefficient design with a VSWR of 1.13, for a line with five sections. Chebyshev polynomials are tedious to calculate, but an unwritten method, developed by Ross E. Graves at Stanford University, makes the calculations as simple as those in Pascal’s Triangle for Binomial Coefficients. My thesis advisors, Donald Reynolds and Myron Swarm at Stanford, were students under Professor Graves, and enjoyed my reference to Graves Pyramid for Chebyshev Polynomials.

1. Introduction

In microwave and optical waveguides, phase velocities are infinite at the cutoff frequency and are always greater than the velocity of light in the dielectric in the guide, where group velocity is zero at cutoff and is always less than the velocity of light in the dielectric in the guide. As the frequency increases far beyond cutoff, phase and group velocities both approach the velocity of light in the dielectric in metallic and dielectric guides. These phase velocities are described as fast waves. In traveling wave tubes, slow wave structures create phase velocities with velocities along the axis of the structure much less than the velocity of light. In slow wave structures such as traveling wave tubes, acoustical horns, and optical telephone circuits, it is necessary to couple these lines to fast wave structures for purposes of transmitting data or radiating electromagnetic and acoustical waves.

This paper describes designing transformer structures for matching acoustical and electromagnetic wave transmission lines with different characteristic impedances, and fast or slow wave Proceedings of the 2001 American Society for Engineering Education Annual Conference &Exposition Copyright O 2001, American Society for Engineering Education

Biggs, A. (2001, June), Optimum Matching Of Two Unequal Impedances In Acoustical, Optical, Or Microwave Stepped Transmission Lines Paper presented at 2001 Annual Conference, Albuquerque, New Mexico.

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