## 5 The Case Study
### The Governing Equations
#### 5.3.1 The Planar Micro-Coil Characterization  
The electrical DC resistance of the copper micro-coil was defined by authors as equal to 2 \(\Omega\), due to the field of application and the necessity to take into account skin and proximity effects; therefore, the actual parameterization value of the electrical resistance was identified as the equivalent resistance value for AC excitations. [PERSON] et al. in [16] proposed a detailed mathematical modeling of the Dowell's derived equations of an equivalent resistor to introduce in circuit simulation. The equivalent \(AC\) resistance expression \(R_{AC}\) was formulated as a combination of different hyperbolic functions \(\zeta\) and \(\zeta\). \[R_{AC}=R_{DC}\ F_{r}=R_{DC}\ \Delta\bigg{[}\zeta+\frac{2}{3}\bigg{(}p^{2}-1 \bigg{)}\zeta\bigg{]} \tag{1}\]  
The proximity \(\xi\) and skin \(\zeta\) effect [17] instead need to be numerically calculated according to Equation (2), through a combination of harmonic functions of the penetration ratio \(\Delta\). \[\xi=\frac{\sinh\Delta-\sin\Delta}{\cosh\Delta+\cos\Delta}\quad\zeta=\frac{\sinh (2\Delta)+\sin(2\Delta)}{\cosh(2\Delta)-\cos(2\Delta)} \tag{2}\]  
According to the set of Equations reported in Equation (3), the penetration depth \(\Delta\) is essentially a function of the physical and geometric parameters of the material. \[\Delta=\frac{\sqrt{\eta}d_{w}}{\delta}\quad\eta=\frac{t\sqrt{k}d_{w}}{h_{c}} \quad\delta=\sqrt{\frac{\rho}{\pi\mu f}} \tag{3}\]  
where  
* \(\eta\) is the porosity factor of the material. * \(d_{w}\) is the equivalent thickness of the material and for a round wire and is equal to \(\frac{\sqrt{\pi}d_{i}}{2}\), where \(d_{i}\) is the diameter of the wire. * \(\delta\) is the skin dept in the wire. Figure 5: Three-dimensional graphical representation of the MEMS electromagnetic actuator. \(t\) is the number of turns per layer. * \(k\) is the number of Litz wire, which is 1 for round wire. * \(h_{c}\) is the height of the core window. * \(\rho\) is the resistivity of the wire material. * \(\mu\) is the permeability constant equal to \(\mu_{r}\mu_{0}\). * \(f\) is the principal operating frequency. * \(p_{i}\) is the equivalent number of layers. A more in-depth description and identification of the parameters as well as the whole mathematical demonstration can be found in [16]. The other two main parameters needed for the characterization of the micro-coil were the stray capacitance and inductance. [PERSON] proposed in his work [18] a complete description of the parasitic capacitance behavior of a planar micro-coil. Starting from the geometry's properties, such as the number of turns \(n\), the internal \(R_{in}\), external radius \(R_{out}\), and the distance between two turns \(d=\left(R_{out}-R_{in}\right)/n\), the author proposed a set of equations for the analytical formulation of the equivalent stray capacitance. Modeling the whole planar coil as a series of \(i\)-th parasitic capacitances delimited by the polar coordinates \(a\) and \(b\) defined in Equation (4) with the potential and the electrical flux between two turns, it is thus possible to define the global \(X\) and \(Y\) constitutive Equations (5) and (6). \[a=R_{in}+\left(i-1\right)d;\quad b=a+i\ d;\quad for\ i=\{\ 1\leq i\leq\ n-1\} \tag{4}\]  
\[X=\frac{-b+\ln\bigl{(}b^{3}a\bigr{)}}{2a^{2}}+\frac{1}{4a^{2}}+\frac{1}{2b}- \frac{\ln\bigl{(}b^{2}\bigr{)}}{b^{2}}-\frac{1}{4b^{2}} \tag{5}\]  
\[Y=-3\ln\biggl{(}\frac{b}{a}\biggr{)}+b \tag{6}\]  
The effective height of the coil \(z\) is defined in Equation (7). \[z=8.25\times 10^{3}(n-10)+0.1401 \tag{7}\]  
Plugging the set of reported equations, it is possible to formulate the equivalent stray capacitance \(C\) as the sum of the \(i\)-th coupled capacitors \(C_{i}^{\prime}\). \[C^{\prime}=\frac{4z\pi\epsilon}{a^{2}}\frac{Y}{X} \tag{8}\]  
\[C=\sum_{i=1}^{n-1}C_{i}^{\prime} \tag{9}\]  
The inductive behavior instead is modeled through the mathematical formulation proposed by [PERSON] et al. in [19]. Herein are proposed different methods for the analysis of the planar coil.