June 20, 2010
June 20, 2010
June 23, 2010
Electrical and Computer
15.1089.1 - 15.1089.7
Standing and Traveling Waves on Transmission Lines: Getting it Right Introduction
In the recent engineering education literature there have been numerous papers dealing with strategies for teaching electrical transmission lines. These papers approach the assistance in teaching from two points of view. The first is the use of simple low-cost experiments to demonstrate the concepts 1-3 and the second approach employs computer animation 4, 5. Another recent paper discusses the concept of teaching transmission lines early in the electromagnetics sequence 6.
Seeking clarification to the problem of sinusoidally driven, arbitrarily terminated, lossless transmission lines one of the authors asked the question, “How does the voltage on the line behave?” In order to answer this question some MATLABTM scripts were written to animate the voltage on the line. In examining these animations more questions arose about the concept of standing waves. In the examination of the animations the authors were confounded because for arbitrary termination impedance the resulting waveform did not “stand” as is the case for the commonly discussed short-circuit and open-circuit terminations. Consultation of a number of electromagnetics textbooks found that the term “standing wave” is often used to describe the interference pattern created by a combination of forward and backward traveling waves of the same wavelength (or frequency). As we shall show this definition is incomplete and leads to a semantic error in a number of contemporary textbooks.
Standing or Stationary Waves
After the search of electromagnetics textbooks it was decided to look on the web for information which is contained on physics websites. Standing waves are created by the addition of two oppositely traveling waves of equal wavelength and amplitude. There seems to be an agreement that the terms standing waves and stationary waves are synonymous and that such waves are characterized by nodes (points of zero response) and antinodes (points of maximum response) and that there is no energy transport in the direction of propagation 7.
A final resolution to the definition problem came with a search of Lord Rayleigh’s 1891 edition of The Theory of Sound 8 wherein the above stated definition was confirmed. The Rayleigh definition essentially states that sinusoidal waves are stationary (standing) if they are of the form w( z ,t ) = P cos( ωt - ε ) cos( βz - α ) (1) or after employing the appropriate trigonometric identity P w( z , t ) = [cos( ωt + βz - ε - α ) + cos( ωt - βz - ε + α )] (2) 2 We see that expression (2) is the sum of forward and backward traveling waves each of amplitude P/2. A similar result may be found in the handbook of Korn and Korn 9.
Jacquot, R., & Voltmer, D., & Steadman, J. (2010, June), Standing And Traveling Waves On Transmission Lines:Getting It Right Paper presented at 2010 Annual Conference & Exposition, Louisville, Kentucky. 10.18260/1-2--15792
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