The fuzzy two-load method for measurement of ducted one-port source characteristics

Knowledge of source characteristics is important for the calculation of sound pressure level and insertion loss of a silencer in sourced ducts. Measurement is usually the only feasible approach for the determination of source characteristics. The present paper is concerned with the measurement methods based on the Helmholtz-Thévenin equivalent of ducted one-port plane wave sources. Existing methods are classified as crisp and over-determined methods. In the crisp methods, the measured data are just sufficient for unique characterization of the source. This calls for two loads if their phase relative to the source is known, otherwise three loads are required. Over-determined methods use more number of loads than the crisp methods and are motivated for possible minimization of variations due to measurement errors. The point estimates for the source parameters are, however, still subject to some uncertainty, but estimation of confidence intervals is not feasible because the loads do not constitute a probability sample. The present paper proposes an approach which can yield the source characteristics in intervals from auto-power spectral density measurements with only two loads. The method is based on a novel Apollonian circle formulation. It is called the fuzzy two-load method, because uncertainty inherent to measurements is modelled by fuzzification of a characteristic parameter of the Apollonian circle of two loads. Fuzzy number transformations leading to the source pressure strength, sound pressure level and insertion loss interval predictions are discussed in depth. The paper includes an application showing the working features of the fuzzy two-load method.