## 1 Introduction  
The nonlinear parameters of a miniature transducer play an important role in the sound quality of mobile devices. These parameters can be responsible for the output performance of a miniature transducer. Special techniques can be used to measure the harmonic and intermodulation distortion of signals excited by single-, dual-, and multi-tone sine waves/signals. In the last decade of research on transducers, multiple methods have been proposed to predict harmonic and intermodulation distortion, and these methods are useful in transducer design. The traditional method used to predict the sound pressure, diaphragm displacement, and electric current of a moving-coil transducer comprise the lumped parameter model (LPM) [1], which allows for the comprehensive analysis of a transducer's model parameters. The traditional LPM contains an important set of six parameters, namely, voice-coil inductance \(L_{e}\), electrical resistance \(R_{e}\), mechanical mass \(M_{m}\), mechanical resistance \(R_{m}\), mechanical stiffness \(K_{m}\), and force factor \(Bl\), which are used to represent the electrical and mechanical domains of the moving-coil transducer. The LPM helps to design a transducer quickly and accurately to estimate its performance, and it is currently the most common method used to assess the transducer quality. Electroacoustic measurement is characterized by multiple categories and methods [2]. Each measurement provider formulates different instruments and procedures for transducers with different dimensions. Different methods have been applied by scientists in the past, such as the added mass method [1; 3] and the optimization method [4], for transducer's linear parameters estimation. When the miniature transducer is driven in the large signal domain, abnormal vibrations have been observed due to large input signal and increased voltage imposed on voice coil for its displacement [5]. The moving diaphragm that was activated by pneumatic pressure [6] demonstrated that the transducer's model parameters cannot remain linear, but the performance of the transducer can be evaluated by position-dependent parameters [7] as nonlinear parameters. In 1990, [PERSON] [8] proposed a dynamic measurement system and used a high-precision laser vibrometry and noise signal to measure the linear and nonlinear parameters of a transducer LPM. The nonlinear parameters generate nonlinear effects, such as harmonic and intermodulation distortion; thus, the system that was proposed by [PERSON] is often used for transducer development, production, quality control, and diagnostics [5; 9]. [PERSON] et al. [10] proposed a nonlinear LPM to identify the coefficient of stiffness \(K_{m}(\upx)\) and the force factor (\(B\)I) and investigated the reduction of 2 nd-3 rd harmonic distortions by transducer parameter linearization. A simulation method [11] uses finite element analysis for the identification of the nonlinear parameters early, which is the meaningful way of characterizing the transducer in the product development process. In 2001, [PERSON] proposed [12] a parameter optimization method for minimizing the cost function between the predicted sinusoid type of displacements and the modeled response for obtaining nonlinear parameters. More proposed algorithms [13; 14] based on the system identification method are used to minimize the errors between the output response and the system output for obtaining optimal nonlinear parameters. However, the relative clay mass or sealed box were selected based on the transducer dimensions, thus the above methods increased the mechanical mass or acoustical stiffness of the transducer, which are not applicable to miniature and small transducers. Modern miniature transducers have been developed as part of the trend toward the miniaturization and they have been fabricated with metal membranes due to the low weights and dynamic responses.