Acoustic Analysis of a General Network of Multi-Port Elements - An Impedance Matrix Approach

Abstract

The present work deals with a generalized algorithm for analyzing a network of linear passive acoustic filters having multi-port elements which are interconnected to each other in an arbitrary manner through their respective ports or through general 2-port elements. A multi-port element is characterized by an impedance [Z] matrix, and the junctions through which these multi-port elements are connected are characterized by conditions of continuity of acoustic pressure and mass velocity. A connectivity matrix is written for the entire network, wherein the interconnections of the elements are taken care of by proper bookkeeping of the acoustic state variables. The acoustic pressures at the external nodes (at the network terminations) are related to mass velocity at the external ports by inversion of the connectivity matrix to obtain the global impedance matrix, characterizing the entire network. This characterization thereby offers a generalized formulation for dealing with a network of multi-port elements. Generalized expressions are obtained for determination of the acoustic performance parameters (Transmission loss (TL), Insertion loss (IL), and Level Difference (LD)) for a multi-port system in terms of the [Z] matrix and scattering [S] matrix. A simple method is proposed for evaluation of the [Z] matrices by means of the axial plane wave theory to characterize long chamber mufflers for a uniform area, conical and exponential duct having an arbitrary number of ports, whilst the ports can be located on the end faces as well as on the side surfaces. The [Z] matrices characterizing each of these multi-port elements are then used to analyze arbitrary networks of such multi-port elements, and the results are compared with those obtained by 3-D FEA and also against the existing literature.