Page 24 - Jan2013
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                    Fig. 1. Schematic of phased wavefronts approximating a plane wave.
simultaneously at the target site. With a little thought, it is easy to identify the circular wavefronts with their respective sources.
As the time-on-target site moves away from the array, the two types of time delays become increasingly indistinguish- able. In the implementation described later we use a time-on- target algorithm; the same code can be used to approximate a planar wavefield simply by making the distance to the tar- get large compared to the array dimensions.
Antenna patterns and design criteria
The antenna pattern of a phased array depends upon the radiation pattern of the individual transducer elements, the distribution of transducers in the array, the size of the array, and the frequency band of operation. There are extensive engineering criteria for selecting both the size of the individ- ual transducer elements as well as the number and distribu- tion of transducers to achieve various desired types of behav- ior. There are numerous useful criteria: some applications exploit tight collimation of the main lobe of the array’s response; some are dependent on minimizing the largest sidelobes while allowing greater spreading of the main lobe; some require other considerations.
The system that we built is very modestly engineered. It is a functional prototype constructed from off-the-shelf com- ponents with an eye to versatility and future experimenta- tion.
Simulating the array
We decided to invest the time to write a simple model- ing program for the phased array. The model is useful because it lets us study array performance in free space, a condition that is very hard to achieve experimentally, and because it is often faster to run a suite of modeling runs than it is to make observations on several experimental configurations. In the end, of course, it is the behavior of the physical system that matters.
A diagram of the transducer geometry we used is shown- in Fig. 3. Each transducer is a loudspeaker (tweeter) 0.03 m in radius. The transducers are mounted in a piece of rigid
Fig. 2. Simultaneous arrival of circular wavefronts at the target site.
Delrin 0.42 m by 0.22 m. The array is symmetric about the x and y axes but is not axisymmetric about z.
The system was modeled as an array of baffled circular pistons—each tweeter is a piston source with prescribed sinusoidal normal velocity mounted in an infinite rigid sheet. This model ignores interaction between transducers and also ignores acoustic coupling between the front and rear sides of the array: we assume the plastic mounting to be perfectly rigid and infinite in extent.
The field due to a single baffled circular piston is exactly described by a Green’s function integral over the face of the piston (Dowling, 1998) We implemented this integral numerically so that the results would be correct in both the near-field and the far-field. The net array response was com- puted by summing the signal from each transducer at each desired point in space. We specify a time-on-target location and the signal from any particular transducer is advanced or delayed in time by an appropriate amount.
Our calculations occur in a coordinate system the origin of which is the geometric center of the array. We take the orthogonal coordinate x as horizontal, y as vertical, and z increasing away from the array. To specify a calculation we
Fig. 3. Geometry of our transducer array. Axes are in meters.
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