Acoustic techniques for studying high-voltage breakdown
The goal of this project was to understand the acoustic properties of heat-treated copper in order to relate the acoustic signatures of breakdown events to the underlying minor electromagnetic catastrophes taking place inside the accelerating structures. An optimistic outcome that proved unrealistic due to scattering of acoustic waves off grains in the copper would have been to develop a technique allowing derivation of an invertible acoustic Green's function for an individual copper structure. This Green's function would be used to predict the signals arriving at various sensors as functions of the acoustic excitation caused by a cavity breaking down. The inverse function, provided with data from a sufficiently large number of sensors, would yield information about the discharge. A more realistic outcome was to determine how well acoustic information can be used to localize breakdowns in NLC cavities.
During the first year of investigation we concentrated on building software tools and developing a small amount of laboratory infrastructure to begin learning about the problems we were confronting. Since these effects are well described by classical mechanics, the research proved ideal for participation by undergraduates. The students showed themselves to be remarkably productive and insightful.
Since NLC structures are held at high temperature when they are assembled by brazing, the copper's grain size grows so that sound waves must propagate through a crystalline medium with irregularly shaped grains a few millimeters in size which are oriented randomly. The speed of sound in copper is about five millimeters per microsecond, so acoustic waves with frequencies in the MHz range (whose wavelengths are comparable to the grain size) are disrupted by scattering as they propagate.
In our first year of efforts we worked with two sets of copper dowels on loan from the Fermilab NLC structure factory. The copper stock was from a shipment of material used to construct actual NLC test structures; one set of dowels was heat-annealed to bring up its grain size, while the other was not and, consequently, had microscopic grains.
We borrowed several 1.8MHz transducers (and associated signal conditioning electronics) from a colleague in Electrical and Computer Engineering as well as purchasing a pair of 500 kHz Panametrics transducers ourselves. A schematic diagram of our copper/transducer setup is shown in Figure 1. A variety of measurements for dowels of different lengths (including speed of sound, attenuation length, and beam spread) provide us with a nice set of experimental inputs with which to confront our acoustic models.
We developed a pair of models for the propagation and detection of acoustic waves in copper. Both describe copper as a (possibly irregular) grid of mass points connected by springs. We could vary the individual spring constants and the arrangement of interconnections to introduce irregularities representing grains into our simulated copper. The models may seem naïve, but they were able to support a variety of complex phenomena and it is a simple matter to tune various physical properties (such as the speeds of sound for compression- and shear-acoustic waves) through adjustments of the models' parameters.
Figure 1. Copper/transducer laboratory setup. We can listen for echoes returning to the transducer which fires pings into the copper, or listen to the signal received by a second transducer, or the sum of signals from the two transducers.
One of the models used MatLab as a computational engine, generating an analytic solution to the coupled equations describing the forces acting on each mass point. The other, written by two of the students, performed a fourth-order Runge-Kutta numerical integration to compute the response of mass points to acoustic perturbations. We found it invaluable to be able to compare the detailed predictions of the models in order to verify their accuracy: scattering off grains produced very complicated effects and it was important to confirm that our calculations wre accurate. Our numerical integration model was able to handle considerably larger systems than was possible with MatLab. However, when applied to smaller systems (with a few hundred mass points), both models agreed to an accuracy consistent with integration step size and machine precision.
Most of our model systems during our first year of investigation were two dimensional grids of 250 X 650 points. We would "drive" signals into them using a transducer model in which the piezoelectric device was described as a damped oscillator excited by shocks of short duration. Because of reflections at the ends of the cable used to drive the real transducers, the actual drive signal is complicated; we found we could model it adequately as a series of four closely-spaced impulses. Figure 2 shows a comparison of our simulation and measurements of the transducer signal for a pair of echoes in a copper dowel. We used the first echo to guide our selection of drive parameters; the shape of the second echo is well-reproduced.
Figure 2. Modeling the excitation of a piezoelectric transducer.
Propagation of a simulated acoustic wave in a homogeneous 250 X 650 point grid is shown in Figure 3a. The lateral spread of the pulse is a consequence of the relatively small number of mass points receiving the initial excitation.
Figure 3a. Acoustic pulse propagating in a homogeneous 250 X 650 point grid.
Click here to see the animation that corresponds to the stills of Figure 3a (4.9 MB AVI).
We could simulate the transducer signal as a function of time by summing the amplitudes at the "face" of a transducer as it experienced the effects of the acoustic pulse. Results, shown in comparison with a real oscilloscope record of transducer signal vs. time are shown in Figure 3b. They are promising but a considerable amount of refinement woild have been possible.
Figure 3b. Simulated transducer response to an acoustic pulse propagating in a homogeneous 250 X 650 point grid. The transducer is at the downstream edge of the array.
The effect of inhomogeneities on an acoustic wave is dramatic, as can be seen in Figure 4, below. Spring constants in the parallelogram-shaped region are half as large as those used elsewhere in the grid. The disruption suffered by the pulse dumps a significant amount of acoustic energy into the "bulk" of the copper. The version of the simulation shown in the figure does not include any damping. Even so, the echo returning to the transducer is badly disrupted. Notice the acoustic "glow" which washes over the transducer site due to scattering off the discontinuities in material properties evident in Figure 4. We see this sort of effect in the (real) heat-treated dowels when driving them with our 1.8 MHz transducers, as can be seen in Figure 5, below.
Figure 4. Propagation of an acoustic wave through an asymmetry in a two-dimensional grid. Spring constants inside the region indicated by the parallelogram are half as large as they are outside the parallelogram. Simulated signal in the transducer (which generates the initial pulse and then measures subsequent acoustic activity) is shown in the small graph below.
Figure 5. Acoustic glow at long times observed in a heat-treated (grainy) long copper dowel. Note that the scope trace is NOT showing noise: the fine structure is remarkably reproducible from shot to shot. Configuration uses a single transducer: ping, then listen to baseline signal as pulse travels into copper, pumping energy into acoustic baseline “glow.” Scope scales are 5 mV and 100 msec per division.
The amplitude of an acoustic pulse decreases because of attenuation as well as scattering of energy out of the pulse. Not surprisingly, a pulse bouncing back and forth in a heat-treated (grainy) dowel dies out more rapidly than does a pulse traveling in a dowel which hasn't been heat-treated. This is evident in Figure 6, which shows a comparison of the sizes of the first and second echoes for pulses traveling in short (2.5 cm length) copper dowels.
Figure 6. Pulses associated with the first and second echoes in short (2.5 cm) copper dowels. Note the relative sizes of the first and second echoes in each dowel; more energy is scattered out of the pulse traveling in the heat-annealed (grainy) copper. Horizontal and vertical scales are 2 msec and 500 mV per division respectively.
We refined our models, constructing software tools that allowed us to insert multiple grains into our grid and also worked on simulations of structures in three dimensions. Figure 7 shows the propagation of a mixed shear- and compression-wave in a homogeneous two-dimensional grid. Note that our model succesfully captures the different propagation speeds of the two classes of wave.
Figure 7. Propagation of a mixed amplitude wave that is 50% shear and 50% compression in a homogeneous (grain-free) two-dimensional grid.
Click here to see the animation that corresponds to Figure 7 (10.5 MB AVI).
Figure 8 shows the propagation of a mixed shear- and compression-wave in a two-dimensional grid with several hundred grains The shear and compression waves are badly disrupted by scattering off grains.
Figure 8. Propagation of a mixed amplitude wave in a grainy two-dimensional grid.
Click here to see the animation that corresponds to Figure 8 (13.6 MB AVI).
Shown in Figure 9 is an NLC structure model that we developed. We found that, in the absence of grains, we could reconstruct the location of a point impulse delivered to the copper as long as our simulated transducers provided us with directional information for acoustic waves striking their front surfaces in addition to a measure of the overall acoustic amplitude. "Vector" transducers that can do this are available, but are considerably more expensive than the simple devices we had purchased.
Figure 9. Simple three-dimensional model for NLC accelerating structure. Acoustic excitations are applied at the indicated point; the effects on mass points are shown for the plane containing the excitation.
Our reconstruction technique bore a certain resemblance to the autofocusing algorithms used by some cameras. We would play the signals received by the simulated transducers backwards, beaming them into the copper, and watching the evolution of the RMS deviation of mass points from their equilibrium locations. We chose the time corresponding to the maximum "roughness" of the displacements (the RMS deviation from equilibrium) as the time of origin of the initial acoustic impulse and identifed the largest point displacement amplitudes as the location of the impulse. Even in the presence of grains this worked reasonably well.
When the TESLA cryogenic technology was chosen for the ILC main linac we stopped pursuing this line of investigation.
- Figure 9 animation, without grains (6.1 MB AVI).
- Figure 9 animation, with grains (6.0 MB AVI).
- Driving transducer signals back into grain-free copper (4.6 MB AVI).
- Driving transducer signals back into grainy copper (5.3 MB AVI).