Error analysis for the four-load method used to measure the source impedance in ducts

Abstract

The impedance of a sound source can be measured using direct approaches such as the standing wave and transfer function methods or indirect approaches such as two-load, three-load, and four-load methods. Unlike the direct and two-load methods, the three-load and four-load methods do not require the measurement of a complex quantity such as the acoustic transfer function. Therefore, a single-channel signal analyzer can be used to measure the autospectra needed in the case of the three-load and four-load methods. The formulation for the source impedance gives two and three second-order, nonlinear algebraic equations for the three-load and four-load methods. The unknowns in these equations are the real and imaginary parts of the source impedance. The two nonlinear equations in the three-load method can be solved graphically for each frequency of interest or by using well-known mathematical methods. However, in the four-load method, there are three nonlinear equations in terms of two unknowns which means one of the equations is redundant. The three nonlinear equations could also be reduced to two linear equations and the two equations could be solved for the real and imaginary parts of the source impedance. The accuracy of the solutions depends on how accurately the original (nonlinear) equations are satisfied when the real and imaginary parts are substituted into these equations. An error analysis of the formulation in the four-load method of source impedance measurement was done. It was found that the error due to apparent reduction of the nonlinear equations to linear equations was very large. The source impedance values computed using the four-load method were compared with those obtained from the direct (transfer function) method. Although the formulation of the four-load method satisfies the mathematical procedure, the error due to computation could be large, and further work is needed in the application of the three-load and the four-load methods.