A High Order, Lumped Parameter, Jet Dynamic Model for the Frequency Response of Laminar Proportional Amplifiers

Abstract

This paper presents a high-order, lumped parameter, jet-dynamic model for laminar proportional amplifiers (LPA’s). The governing equations for the lumped-parameter representation of the flow regimes found in the input of an LPA are derived in the Laplace domain, and an equivalent electrical circuit is obtained. The input governs the overall response of the LPA and may be modeled in its simplest form by five reactive components. The transmission of the signal from input to output is delayed by a transport time (determined by observation of flow visualization of a step response) equal to twice the average particle transit time. A pressure difference is then developed at the splitter that is proportional to the loading and the vent conditions. This signal is acoustically fed back to the control region of the jet, augmenting jet deflection when in phase. The vent inductance is found to have a significant influence on the low-frequency gain. Resonant regions determined by this model correspond closely to edgetone eigenfrequencies reported in the literature. Experimental data have shown good agreement with theory for the amplitude frequency response of LPA’s and excellent agreement for the phase shift. An engineering guide developed for the bandpass characteristics of LPA’s indicates that operating bandwidths of up to 14 kHz can be expected for amplifiers with a nozzle width of 0.25 mm, and ultrasonic operation appears feasible with devices having nozzle widths as large as 0.1 mm.