Brass instrument power efficiency and the relationship between input impedance and transfer function

Abstract

From observation of graphs of brass input impedance magnitude and transfer function vs. frequency, it is obvious that there is a strong relationship between the two. Both exibit a series of strong resonances extending from a low frequency limit to a cutoff frequency which is inversely proportional to the instrument's bell radius $f_c=c/(\pi a)$. However, the maxima of the impedance function correspond to the minima of the transfer function. As previously shown (Elliott et al., JASA, 1982), the relationship can be seen through a formula for $efficiency$ given by $power_out/power_in$. This formula leads to the squared transfer function being proportional to the efficiency times the real part of the reciprocal of the input impedance, divided by the real part of the radiation admittance. Curves for input impedance, transfer function, and efficiency have been measured, simulated, and compared for several brass instruments. For frequencies below cutoff, the efficiency has an approximate monotonically increasing relationship with frequency, where the log-log slope is dependent on internal losses.