# Using a Lossy Electrical Transmission Line Model for Optimizing Straw Phonation Configurations
## 1 Introduction  
Over the past several decades, straw phonation has emerged as a novel method to improve vocal economy. Straw phonation involves the use of tube held between the lips through which the subject phonates and was originally developed in the 1960s using glass straws [1]. A wide range of studies have been conducted to measure the vocal quality before and after straw phonation for patients with muscle tension dysphonia [2], vocal fatigue [3], vocal fold paralysis [4], vocal fold nodules [5] and lesions [6] as well as singers with high vocal demand [7, 8]. Existing research on straw phonation has developed two main theories explaining increased ease of phonation: an acoustic-aerodynamic interaction and a mechano-acoustic interaction [9]. The former deals with the shape of the glottal flow pulse while the latter is focused on the material properties of the vocal tract wall, especially inertance on supraglottal pressure. However, both of these theories consider the importance of impedance, which creates a build-up of pressure after the glottis to facilitate oscillation of the vocal folds with decreased subglottal pressure [10]. Impedance is quantified as a complex number, whose real part is resistance and imaginary part is reactance. When displayed in vector form, the magnitude of impedance measures the ratio of pressure to flow velocity while the angle of the vector describes the phase difference between the pressure and velocity. Because impedance is useful for describing the effects of the vocal tract on the pressure and velocity of the sound wave as it travels from vocal folds to mouth, it is an important metric for measuring the effects of straw phonation and whether it improves the vocal quality of the subject. A robust, physically-based model for calculating impedance was developed by [PERSON] et al. in 1987 [11] and later refined and expounded upon by [PERSON] et al. in 2000 [12]. This model postulated the vocal tract as a series of cylindrical tubes, called \"tueblets\", each with a constitutive length and cross-sectional area. As sound travels through the vocal tract the characteristics are assumed to be constant within each tubelet, so each tubelet can be described by a \(2\times 2\) matrix which acts on a 2-dimensional column vector composed ofpressure and flow velocity. This iterated map outputs values for pressure and velocity at the end of each tubelet, taking into account the tubelet wall interactions with the sound wave, such as mass compliance, thermal conductivity, and viscosity. Since the tubelets are connected, the output of one is the input for the next tubelet in the line, until the sound wave reaches the mouth (or straw) opening and expands into the surrounding atmosphere. Given the successive multiplication of many matrices, this method of impedance calculation will be henceforth referred to as the \"chain matrix\" approach. However, as made apparent below, the equations for the 2 \(\times\) 2 matrix for each tubelet are complicated and not intuitive in their derivation. This presents challenges to fully understanding the computations involved and prevents expanding the model to other related vocal tract inquiries. Therefore, developing an alternative model and method to calculate impedance may be valuable for straw phonation research. The transmission line circuit analog of the vocal tract is one candidate for an alternative method to calculate impedance. Electrical transmission lines transmit a current from source to load based on the properties of the line. Acoustic-mechanical systems have many similarities to electrical circuits which make them convenient for analysis given the large amount of knowledge about electrical phenomenon in circuits. The oscillation of the vocal folds produces a periodic waveform similar to alternating current in a circuit. Table 1 provides a direct comparison between the properties of a vocal tract and electrical phenomenon in a circuit.