# Using a Lossy Electrical Transmission Line Model for Optimizing Straw Phonation Configurations
## 4 Discussion  
This study has demonstrated that the electrical circuit model of the vocal tract is not yet equipped fully to model the complex dynamics of human phonation. However, there is great value in developing a workable electrically based model, so future research should focus on the required alterations to Flanagan. Some suggestions are described in detail below. After the derivation of the electrical formulae for smooth, hard-walled tubes, Flanagan estimates the effect of energy lost due to cavity wall vibration of a soft vocal tract wall with  
Figure 4: Average power dissipated during phonation for the chain matrix (red) and transmission line (blue) impedance calculation methods for the example 17.5 cm long and 1 cm\({}^{2}\) cross-sectional area hard walled tube. non-infinite surface impedance. However, this discussion of the cavity wall admittance only changes the calculation of the imaginary portion of the propagation constant. As defined in Equation (4), the propagation constant is a complex value, with the real portion being the attenuation constant which describes how the amplitude of the wave is damped over time, and imaginary portion being the phase constant which describes change in phase per unit length of transmission line. Since cavity wall vibration provides another path for energy to be dissipated and stored, such vibration should have an effect on both parts of the attenuation constant. For a better understanding of interaction between phonation and vocal tract wall expansion, an analysis based on Hooke's Law may be fruitful. The Young's modulus of a material describes the amount of stress required to cause a certain amount of strain, or elongation, of that material. On a stress-strain curve, the Young's modulus represents the slope of the linear portion, where there is only elastic deformation. For vocal tract tissue, the Young's modulus, also called the modulus of elasticity, is on the order of \(10^{3}\)-\(10^{4}\) Pascals [23; 24]. Essentially, the vocal tract tissue acts a spring in the circumferential direction which expands and contracts based on the dynamics from the spring force provided by the tissue elasticity in opposition to the air pressure in the tract [25]. Additionally, this spring-like motion is damped due to the internal friction between the layers of tissue in the vocal tract wall. While this approach may yield some added realism to the existing LCRG transmission line model, an alternative electrical model could benefit from the use of transformers. A transformer is an electrical device which consists of two sets of wires wrapped around a shared metal core. The alternating current in one wire, called the primary, causes a changing magnetic flux in the core which, according to Faraday's Law, causes an induced current in the secondary loop. The ratio of the number of turns in each loop is equal to the ratio of voltages in each wire. Therefore, the transformer could help account for the discontinuous nature of the pressure and volume velocity along the vocal tract after it is discretized into tubelets. Beyond the changes required to account for the soft walled vocal tract, other changes could be valuable to extend the model beyond a healthy vocal tract for straw phonation. Three worthwhile cases are presented. First, continuous positive airway pressure (CPAP) has been studied before as an alternative to straw phonation, but recent results have been inconclusive [26; 27]. The introduced oppositional airflow requires greater subglottal pressure to accelerate the mass of air in the vocal tract, changing the relative inductance. Second, inflammation of the larynx is a common vocal disorder and simulating the effects of this change to the vocal tract wall may aid in providing treatment. Anatomical and biological changes to the vocal tract wall effect the viscous friction and heat conduction with the vocal tract wall, and thus resistance and shunt conductance, respectively.