Waveform Design for Maximum Pass-Band Energy

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Doctor, Steven
Gibbs, Alan
Gribble, R
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Review of Progress in Quantitative Nondestructive Evaluation
Center for Nondestructive Evaluation

Begun in 1973, the Review of Progress in Quantitative Nondestructive Evaluation (QNDE) is the premier international NDE meeting designed to provide an interface between research and early engineering through the presentation of current ideas and results focused on facilitating a rapid transfer to engineering development.

This site provides free, public access to papers presented at the annual QNDE conference between 1983 and 1999, and abstracts for papers presented at the conference since 2001.


One way to maximize the sensitivity of an ultrasonic inspection is by establishing the pulser output voltage waveform to provide the maximum possible fraction of its energy in the pass-band of the piezoelectric transducer. An analytical study is reported that is backed up with experimental verification. Two pulser constraints are analyzed in this study. The first constraint is to study the common and easily generated waveform shapes for which each waveform has unit energy and compare to the optimum waveform shape with unit energy that is determined analytically. The second constraint is to repeat the first analysis with waveforms having unit amplitude rather than unit energy. The analysis for the first constraint shows that the numerically intractable problem of summing a very large number of Fourier coefficients can be replaced by a mathematically equivalent evaluation of the pass-band energy which requires only the integration of smooth functions. This alternative formulation also leads to the result that the optimized waveform is the eigenfunction of a particular integral operator corresponding to the largest eigenvalue. The eigenvalue itself gives the maximum attainable passband energy. The optimized waveform is compared with sine waves, rectangular waves, trapezoidal waves, triangle waves and exponential spikes for 1/2, 1 and 3/2 cycle durations. The analysis for the second constraint shows that the unit amplitude is in the form of an inequality which is outside the realm of the classical calculus of variations. An exact characterization of the optimized waveform was not found but numerical integration techniques were employed to determine the pass-band energies for the waveforms considered under the first constraint. Finally, a breadboard pulser model is constructed and extensive comparisons of the various waveshapes, sensitivity studies, spectral distributions and experimental verification are made for each constraint.