Graphical Aids in Interpreting the Performance of Crystal Transducers

Abstract

When the mechanical damping of a transducer is large, as by acoustic radiation from one or both faces into a liquid or solid, the circular diagram that represents its characteristics requires special treatment. As a background for this treatment, the uses and limitations of the conventional circle for a resonator with small losses is first reviewed. The problem of the transducer with large losses is then considered with special reference to the equations and graphs for a thickness-type transducer with unsymmetrical loading. For plane-wave transducers the expressions are exact for all loads and at all frequencies, including harmonics. Either the voltage or the current may be constant. From the admittance or impedance diagrams the magnitude and phase of current, voltage, particle velocity, and vibrational amplitude at any frequency can be obtained immediately. Similar results would be found with plates in lengthwise vibration. A new type of diagram is developed for representing vibrational amplitudes. As an illustration, the case of a quartz plate radiating into three liquids of widely different acoustic properties is treated. When the load is unsymmetrical, there is no true node anywhere in the crystal except when the load is zero or infinity. There is, however, a plane of minimal vibration, the amplitude and location of which are derived. The equations indicate certain peculiar effects when the specific acoustic resistance of the medium is just twice that of the crystal.