Assessing the effects of impedance modifications using the Moebius transformation

The Moebius transformation maps straight lines or circles in one complex domain into straight lines or circles in another. It has been observed that the equations relating acoustic or mechanical impedance modifications to responses under harmonic excitation are in the form of the Moebius transformation. Using the properties of the Moebius transformation, the impedance modification that will minimize the response at a particular frequency can be predicted provided that the modification is between two positions. To prove the utility of this method for acoustic and mechanical systems, it is demonstrated that the equations for calculation of transmission and insertion loss of mufflers, insertion loss of enclosures, and insertion loss of mounts are in the form of the Moebius transformation for impedance modifications. The method is demonstrated for enclosure insertion loss by adding a short duct in a partition introduced to an enclosure. In a similar manner, it is shown that the length or area of a bypass duct in a muffler can be tuned to maximize the transmission loss. In the final example, the insertion loss of an isolator system is improved at a particular frequency by adding mass to one side of the isolator.