Using the Moebius transformation to predict the effect of source impedance on insertion loss

Knowledge of the strength and internal impedance of the sound source is essential to predict the overall acoustic performance of an exhaust or intake system. Source impedance is sometimes measured directly using a strong external source. At other times, a suitable external source is difficult to identify and an indirect multi-load method is used to determine source impedance and strength. Data can be processed using the commonly used circuit analogy or via a similar wave decomposition model. In this work, the two processing schemes are used to determine the source impedance and strength of a diesel engine. Results are compared to one another and to assumed source impedances and equations in the literature for source strength. The sound pressure in the exhaust is predicted from the measured source strength and impedance and compared with measurement. There is good agreement especially at the first few harmonics of the firing frequency. It is then observed that the equation relating the source impedance to the acoustic response is in the form of the Moebius transformation, which maps straight lines or circles in one complex domain into straight lines or circles in another complex domain. It is demonstrated that the range of source impedance, defined as an outline in the complex plane, can be transformed to a corresponding outline of the muffler insertion loss complex amplitude, and the minimum and maximum insertion loss can be computed at a given frequency. This permits the expected insertion loss range to be plotted as a function of frequency.