## 2 System and Its Mathematical Model  
The CAD diagram of the considered microgenerator, variations of potential energy, and electronic converter are depicted in Figure 1a-c.  
Figure 1: _Cont._  
The emf in armature coils 1 is induced as the result of the relative motion of magnets 3 along the \(y\) axis direction. Magnets 2 are stationary with respect to the armature and are used to develop an auxiliary magnetic force that brings nonlinear mechanical resonance into the system. The converter belongs to a class of nonlinear electromagnetic vibration energy harvesters whose insightful analysis including kinematic and dynamic effects can be found in [11] and [20,25]. Figure 1b shows variations of potential energy vs. relative displacement of permanent magnets (No. 3 in Figure 1a). The senses of magnetization of the magnets in Figure 1a determine the softening action of the magnetic force within the feasible range of displacement (between \(-5\) mm and \(+5\) mm), and two stable points around zero displacement (see Figure 1b). With the variations of potential energy due to magnets (magnetic energy) and that due to auxiliary cantilever spring (spring energy) there is one static stable point at zero displacement. The frequency range where the system produces electric power is approximately equal to 10 Hz. Due to lack of core element, the distribution of magnetic flux over the coil of the microgenerator shown in Figure 1a is too nonuniform to assume that the induced emf can be calculated considering the coil as a block. For this reason, the optimization routine assumes that the emf is estimated for each  
Figure 1: Considered system: (**a**) magnetic circuit of microgenerator—arrows denote magnetization senses of permanent magnets; (**b**) variation of potential energy vs. relative displacement of magnets 3; (**c**) circuit diagram of electronic boost converter [24] loaded by resistor \(R_{l}\). turn individually. This apparently simple computation causes, however, a burden that needs further organization. Consider a single racetrack-shaped coil in Figure 2.  
As shown in the figure, the turns over a cross section of a uniformly wound coil are distributed one after another and then layer after layer in order to fill the assumed area \(w\times h\) and are numbered repeatedly from left to right. Because the number of turns \(N_{t}\) has to be considered as a design variable, it should be noticed that it is not uniquely attributed to an electromagnetic quantity, such as the emf. It is only unique if it is attributed to turns ordered with respect to the value of the quantity. This attribution will be carried out as follows. For the system in Figure 1, the emf is induced only by the \(z\)-th component of the magnetic flux density, whose distribution due to the \(i\)-th magnet can be determined from:  
\[B_{zi}=\frac{\mu_{0}}{4\pi}\int\limits_{S_{i+}}\frac{3(z-z^{\prime})Q_{mi}}{ \left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}}ds-\frac{\mu_{0}}{4\pi}\int \limits_{S_{i-}}\frac{3(z-z^{\prime})Q_{mi}}{\left|\mathbf{r}-\mathbf{r}^{ \prime}\right|^{3}}ds, \tag{1}\]  
where \(\mu_{0}\) is the vacuum permeability, \(Q_{mi}=\mathbf{M}_{i}\cdot\mathbf{n}\) is the magnetic dipole density, \(\mathbf{M}_{i}\) and \(\mathbf{n}\) are the magnetization and surface-normal vector, respectively, \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) are the source and observation point, respectively, and \(S_{i+}\) and \(S_{i-}\) are the magnet surfaces parallel to the \(xy\) plane. The useful quantity is, however, a derivative of turn-linked magnetic flux not the flux itself, as the former is in proportion to the turn-induced emf. For the \(j\)-th turn, it can be conveniently calculated using [PERSON]'s theorem:  
\[\frac{\partial\lambda_{j}}{\partial y}=\frac{\partial}{\partial y}\iint \limits_{S_{ij}}\sum\limits_{l}B_{zi}dxdy=-\oint\limits_{L_{t}j}\sum\limits_ {i}B_{zi}dx, \tag{2}\]  
where \(S_{t}\) is the area enclosed by a single turn and \(L_{t}\) is the path along turn.