Analysis and Measurement of Anti-reciprocal Systems By

Loudspeakers, mastoid bone-drivers, hearing-aid receivers, hybrid cars, and more – these “anti-reciprocal” systems are commonly found in our daily lives. However, the depth of understanding about the systems has not been well addressed since McMillan in 1946. The goal of this study is to provide an intuitive and clear understanding of the systems, beginning from modeling one of the most popular hearing-aid receivers, a balanced armature receiver (BAR). Models for acoustic transducers are critical in many acoustic applications. This study analyzes a widely used commercial hearing-aid receiver, manufactured by Knowles Electronics, Inc (ED27045). Electromagnetic transducer modeling must consider two key elements: a semi-inductor and a gyrator. The semi-inductor accounts for electromagnetic eddy currents, the “skin effect” of a conductor, while the gyrator accounts for the anti-reciprocity characteristic of Lenz’s law. Aside from the work of Hunt, to our knowledge no publications have included the gyrator element in their electromagnetic transducer models. The most prevalent method of transducer modeling evokes the mobility method, an ideal transformer alternative to a gyrator followed by the dual of the mechanical circuit. The mobility approach greatly complicates the analysis. The present study proposes a novel, simplified, and rigorous receiver model. Hunt’s two-port parameters as well as the electrical impedance Ze(s), acoustic impedance Za(s), and electroacoustic transduction coefficient Ta(s) are calculated using transmission and impedance matrix methods. The model has been verified with electrical input impedance, diaphragm velocity in vacuo, and output pressure measurements. This receiver model is suitable for designing most electromagnetic transducers, and it can ultimately improve the design of hearing-aid devices by providing a simplified yet accurate, physically motivated analysis. As a utilization of this model, we study the motional impedance (Zmot) that was introduced by Kennelly and Pierce in 1912 and highlighted by many researchers early in the 20th century. Our goal for this part of the study is to search for the theoretical explanation of the negative real part (resistance) observed in Zmot in an electromechanical system, as it breaks the positive-real (PR) property of Brune’s impedance, as well as the conservation of