# Analysis and Design of Helmholtz Protector to Improve High-Frequency Response of Insert Earphone
## 2 Analysis Method
### Acoustic Modeling  
To analyze the SPL performance of an earphone, all acoustic components need to be modeled in the analysis tool. The acoustic domain is modeled as follows. The acoustic components of a dynamic speaker unit are shown in Figure 4. There is a hole located in the center of the protector. As a result of the existence of a diaphragm, there is a front cavity and a back cavity in the unit. On the yoke, there are three holes. The cavity is modeled as the acoustical compliance shown in Figure 5.  
The related equation to define the acoustical compliance is as follows [15]. \[C_{a}=\frac{V}{\rho c^{2}} \tag{3}\]  
The small hole is modeled in Figure 6.  
Figure 4: Cavity and hole in dynamic speaker unit. Figure 5: Cavity model and its equivalent circuit. Figure 3: Equivalent circuit of mechanical domain. Figure 6: Hole model and its equivalent circuit. The mathematical expression for the small hole is described by the following equation:  
\[Z_{hole}=jwM_{a}+R_{a} \tag{4}\]  
where \(Z_{hole}\)\(M_{a}\) and \(R_{a}\) are the acoustical impedance, acoustical mass and acoustical resistance, respectively, which can be expressed as [16]:  
\[ \begin{array}{l}M_{a}=\frac{332-13x+7x^{2}+2x^{3}}{249-\alpha+4x^{2}+2x^{3}} \frac{4 pl}{\pi d^{2}}\\ R_{a}=8 fpl\frac{\sqrt{4096+45x^{3}+4x^{4}}}{d^{2}x^{2}}\end{array}  \tag{5}\]  
where \(x=d/2\beta\); \(\beta=\sqrt{(\eta/(\rho 2\pi f))}\) is the viscosity coefficient of the fluid (\(\eta=18.6\times 10^{-6}\) n-s/m\({}^{2}\) at 20 \({}^{\circ}\)C for air); \(\omega\) is the angular frequency; and \(l\) and \(d\) are the length and diameter of the hole, respectively. In addition to the dynamic speaker unit, a sound port and test jig are also present. Figure 7, demonstrates the acoustic structure. The structure consists of one back chamber, one conical tube, three cylinder tubes, and an IEC-60711 coupler. The tubes can be treated as transmission line models. Except for the cylinder tube, there is conical tube in front of the speaker unit. The tube modeling is shown in Figure 8.  
Figure 8: Tube models. Figure 7: Cross-section view of dynamic earphone experiment condition. The following equations are mathematic models of a cylinder tube and conical tube. \[\left( \begin{array}{c}p_{1}\\ q_{1}\end{array} \right)=\left[ \begin{array}{ccc}\cos(kl)&jZ_{w}\sin(kl)\\ (j/Z_{w})\sin(kl)&\cos(kl)\end{array} \right]\left( \begin{array}{c}p_{2}\\ q_{2}\end{array} \right) \tag{6}\] \[\left( \begin{array}{c}p_{1}\\ q_{1}\end{array} \right)=\left[ \begin{array}{ccc}(\frac{x_{1}}{x_{0}})\cos( kl)-(\frac{1}{x_{0}})\sin(kl)&(\frac{x_{0}}{x_{1}})jZ_{w}\sin(kl)\\ (\frac{j}{Z_{w}})[\frac{x_{1}}{x_{0}}+(\frac{1}{x_{0}})^{2}]\sin(kl)-(\frac{L} {x_{0}})(\frac{1}{x_{0}})\cos(kl)&(\frac{x_{1}}{x_{0}})[\cos(kl)+(\frac{1}{x_{ 0}})\sin(kl)]\end{array} \right]\left( \begin{array}{c}p_{2}\\ q_{2}\end{array} \right)\]  
The parameters are \(k=\omega/c\), where \(\omega=2\pi f\); \(l=\) tube length; and \(Z_{w}=\rho c\)/S = \(\rho c\)/\(\pi a^{2}\), where \(\rho\) is the air density and \(c\) is the speed of sound propagation. \(p_{1}\), \(q_{1}\), \(p_{2}\), and \(q_{2}\) are inward sound pressure, outward sound pressure, inward volume velocity and outward volume velocity. \(x_{1}\) and \(x_{0}\) are the lengths of the tube along the surface direction. If the diameter of the tube changes, there will be added length because of the existence of radiation impedance at the tube end. The tube end correction is shown in Figure 9.  
The tube end correction is as follows  
\[ \begin{array}{l}\Delta l_{1}=0.85d[1-1.47\frac{d}{d_{1}}+0.47(\frac{d}{d_{1 }})^{3}]\\ \Delta l_{2}=0.85d[1-1.47\frac{d}{d_{2}}+0.47(\frac{d}{d_{2}})^{3}]\end{array}  \tag{7}\]  
where \(d_{1}\), \(d\), \(d_{2}\), \(\Delta l_{1}\), \(l_{0}\) and \(\Delta l_{2}\) are the left side tube diameter, middle tube diameter, right side tube diameter, left side tube end correction length, middle tube length and right side tube end correction.