# Using a Lossy Electrical Transmission Line Model for Optimizing Straw Phonation Configurations
## 1 Introduction
Many previous acoustic and aerodynamic research papers have used the electrical analogy, but none so far have used it to study impedance [13; 14; 15]. One important note concerning transmission lines is the distinction between lossless and lossy. A lossless line does not dissipate the energy of the current as it travels along the line, and thus only contains values for inductance and capacitance. The vocal tract is not, however, a lossless line due to the interaction of the sound waves with the tube walls causing it to lose energy. Moreover, the vocal tract, once discretized into tubelets, represents a stepped transmission line by having different values of impedance for each line segment corresponding to each tubelet. In Figure 1, below, an abridged diagram of the electrical circuit model of the vocal tract is shown. Multiple of these circuit elements are connected to nodes at the ends of the elements. Since impedance from inductance and resistance is directly proportional of to the value of \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Variable** & **Electrical Interpretation** & **Physical Interpretation** \\ \hline I & Current & Flow of electrons per unit time & Velocity of fluid flow \\ E & Voltage & Difference in potential which drives the current & Pressure of air column \\ L & Inductance & Opposition to change in current & Inertia from mass of the air \\ C & Capacitance & Storage of electricity & Compressibility of air \\ R & Resistance & Non-conservative dissipation of electrical energy into heat & Viscous friction at vocal tract wall \\ G & Shunt Conductance & Leakage of current between conductors & Heat conduction at vocal tract wall \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Electrical equivalences to vocal tract properties. Figure 1: Diagram of electrical circuit model of the vocal tract with inductors, capacitors, resistors, and a shunt conductance terms. The arrow represents the break in the circuit where the rest of the tubelet circuit elements would be. An AC source represents the glottis and the resistor and inductor in parallel on the end represents the radiation impedance. the inductor or resistance, while impedance from capacitance and shunt conductance is inversely proportional, the inductors and resistors are shown in series while the capacitor and shunt conductor are in parallel to account for this difference. Although more advanced aerodynamic simulations and computational fluid dynamics (CFD) tools have been introduced to the field of human phonation studies in recent years, the proposed transmission line circuit analogy still possesses great value to this field. CFD often requires intensive calculations which take significant time and computing resources for each simulation, which makes the tool possibly problematic for analyzing multiple different vocal tract, straw, and frequency configurations. In contrast, a successful transmission line model provides an opportunity to significantly simplify the calculations by creating a Thevenin equivalent circuit for each phoneme. Additionally, while CFD can provide more detailed information about the aerodynamic interaction between the vocal tract wall and the air inside the vocal tract, this information such as vorticity, turbulence, and nasal interaction, is not necessarily relevant to the study of impedance, which is purely concerned with the ratio of pressure to volume velocity at the glottis. Therefore, the simplicity of the transmission line method enables a direct focus on the desired quantity of impedance. The objective of this study is to test the validity of the transmission line circuit analogy as a new method of calculating impedance, to use this method to find the optimal configuration of vocal tract shape and straw geometry for straw phonation, and to determine whether maximal impedance corresponds to maximal power transfer.