Time-domain model and optimization for single-axis kinetic energy harvesters driven by arbitrary non-harmonic excitation

Consider figure 2(a), which shows the free-body diagram and electrical circuit of the microgenerator. The force experienced by the levitating magnet (due to the fixed magnet below it) is F_mag, the frictional force opposing motion is F_fric, the electrical damping force due to the opposing magnetic field (induced by the current flowing through the coil) is F_damp, and the force the magnet assembly experiences when it collides with the top of the microgenerator body is F_top. The external force acting on the tube and causing it to accelerate is F_acc. We can derive an expression for the mechanical component of the system model in terms of these forces. Unless otherwise indicated, all derivatives will be with respect to time.

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With reference to figure 1, recall that y_mag is the position of the center of the moving magnet (with mass m_mag) and y_tube is the position of the microgenerator tube (with mass m_tube). Now let:

$$\begin{eqnarray}&&{y}_{\mathrm{diff}}={y}_{\mathrm{mag}}-{y}_{\mathrm{tube}}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{y}_{\mathrm{diff}}^{{\prime} }={y}_{\mathrm{mag}}^{{\prime} }-{y}_{\mathrm{tube}}^{{\prime} },\end{eqnarray} \tag{ 2 }$$

then,

$$\begin{eqnarray}&&{y}_{\mathrm{tube}}^{{\prime\prime} }={F}_{\mathrm{acc}}/{m}_{\mathrm{tube}}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{y}_{\mathrm{mag}}^{{\prime\prime} }=({F}_{\mathrm{mag}}-{F}_{\mathrm{fric}}-{F}_{\mathrm{damp}}-{F}_{\mathrm{top}})/{m}_{\mathrm{mag}}.\end{eqnarray} \tag{ 4 }$$

Note that y_diff represents the displacement of the center of the lowermost magnet in the assembly, measured relative to the bottom of the tube, as shown in figure 1.

We can further expand equations (3) and (4) by adding an expression for the emf ${ \mathcal E }$ induced in the coil, which is itself a function of y_diff. Let ϕ be the flux linkage between the magnet assembly and the coil, as a function of the relative magnet position y_diff, then from Faraday's Law:

$$\begin{eqnarray}&&{ \mathcal E }=-\displaystyle \frac{{\rm{d}}\phi ({y}_{\mathrm{diff}})}{\mathrm{dt}}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&=-\displaystyle \frac{{\rm{d}}\phi ({y}_{\mathrm{diff}})}{{\rm{d}}{y}_{\mathrm{diff}}}\displaystyle \frac{{\rm{d}}{y}_{\mathrm{diff}}}{\mathrm{dt}}.\end{eqnarray} \tag{ 6 }$$

To describe the analytical electromechanical model, we thus need to find expressions for F_mag, F_fric, F_damp, F_top, and ϕ.

Consider the microgenerator design shown in figure 1, which shows the dimensions of the individual microgenerator components, and figure 3, which shows the position of the magnet assembly at rest and at its maximum height. If the microgenerator body is to allow the magnet assembly to move entirely through all the coils, we must define the height of the microgenerator body l_L as:

$$\begin{eqnarray}&&{l}_{{\rm{L}}}\geqslant {l}_{\mathrm{bth}}+{l}_{\mathrm{hover}}+2{l}_{{ \mathcal E }}+2{l}_{\mathrm{mcd}}(m-1)+2{{ml}}_{{\rm{m}}}+{l}_{\mathrm{ccd}}(c-1)+{l}_{{\rm{c}}}+{l}_{\mathrm{th}},\end{eqnarray} \tag{ 7 }$$

where l_bth is the thickness of the bottom of the tube, l_th is the thickness of the tube everywhere else, l_hover is the distance between the fixed magnet and the magnet assembly at rest, ${l}_{{ \mathcal E }}$ is the distance between the magnet assembly and the lowermost coil at rest. It is also the distance between the magnet assembly and the uppermost coil at the magnet assembly's maximum height. Parameters c and m are the number of coils and magnets, respectively, while l_ccd and l_mcd are the distances between the centers of adjacent coils and centers of adjacent magnets, respectively, and parameters l_c and l_m are the lengths of each individual coil and magnet, respectively.

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From equation (7), note the disproportionate effect of m, l_mcd, and l_m (which are all parameters related to the magnet assembly) on l_L: it is spatially expensive to have a large magnet assembly.

Due to the pole-matching, the levitating magnet experiences a repulsive force from the fixed magnet at the bottom of the microgenerator body, forming a magnetic spring. We approximate the force experienced by the levitating magnet due to the magnetic spring by a sum of three decaying exponentials, which can approximate the exponential decay inherent between two magnetic poles:

$$\begin{eqnarray}&&{F}_{\mathrm{mag}}({y}_{\mathrm{diff}})=\sum _{i=1}^{3}{s}_{i}{e}^{-{a}_{i}{y}_{\mathrm{diff}}},\end{eqnarray} \tag{ 8 }$$

where s_i is the size of each exponential and is an element of S = [s₁, s₂, s₃] and a_i is the rate of decay of each exponential and is an element of A = [a_i , a₂, a₃]. We find the elements of S and A using a curve-fitting procedure described in section 5.

We approximate the mechanical loss due to friction by modelling it as a constant damper that is proportional to the mass of the magnet assembly:

$$\begin{eqnarray}&&{F}_{\mathrm{fric}}={b}_{{\rm{m}}}{m}_{{\rm{m}}}y{{\prime} }_{\mathrm{diff}},\end{eqnarray} \tag{ 9 }$$

where b_m is the damping coefficient, which must be experimentally determined, and m_m is the mass of the magnet assembly.

We model the rebounding impact force experienced by the levitating magnet as it encounters the top of the microgenerator body as a joint spring-damper system that is not physically fixed to the levitating magnet. This can be adequately represented by the following relationship:

$$\begin{eqnarray}&&{F}_{\mathrm{top}}=H({y}_{\mathrm{diff}}-{y}_{\mathrm{diff},\ \max })\cdot (Q({y}_{\mathrm{diff}}-{y}_{\mathrm{diff},\ \max })+{b}_{{\rm{f}}}{y}_{\mathrm{diff}}^{{\prime} })\end{eqnarray} \tag{ 10 }$$

where Q is the strength of the spring, ${y}_{\mathrm{diff},\max }$ is the maximum magnet assembly displacement, relative to the tube, before it impacts the top cap of the tube. Note that we relate ${y}_{\mathrm{diff},\max }$ to the length of the microgenerator body with the following expression:

$$\begin{eqnarray}&&{y}_{\mathrm{diff},\max }={l}_{{\rm{L}}}-{l}_{\mathrm{th}}-{{ml}}_{m}-{l}_{\mathrm{mcd}}(m-1)+\displaystyle \frac{{l}_{{\rm{m}}}}{2}.\end{eqnarray} \tag{ 11 }$$

Also, in equation (10), b_f is the damping constant, which models the energy lost during the impact. We determine the value of b_f using a parameter search process, described in section 5. H is the Heaviside step function of the form:

$$\begin{eqnarray}H(x)=\left\{\begin{array}{ll}0, & x\lt 0,\\ 1, & x\geqslant 0,\end{array}\right.\end{eqnarray} \tag{ 12 }$$

which ensures that F_top acts on the levitating magnet only when it impacts the top of the microgenerator body. To model a rebounding impact with the top of the microgenerator body that has little 'compression', we select the spring strength Q to have a large value, Q = 1e7. The larger the spring strength, the less compression occurs, approximating a solid material. Note that a large value of Q does not influence the speed with which the magnet assembly rebounds after impact (this is, instead, determined by b_f), as the law of conservation of energy is not violated.

In order to calculate the electrical damping force F_damp and the emf, we need to model the magnetic flux linkage ϕ passing through the coil. The magnetic flux linkage is the amount of magnetic flux, produced by the magnetic field of the magnet assembly, that passes through the windings of the coil. We model the flux linkage as a function of the relative displacement between the magnet assembly and tube y_diff, the number of windings in the axial direction n_z, the number of windings in the radial direction n_w and our particular choice of magnet. We assume a single choice of magnet throughout our described process.

To develop our expression for the flux linkage, we begin by modelling it for a single-coil, single-magnet configuration. We then systematically extend this to a generalized model that we can apply to any number of coils and number of magnets.

3.6.1. Single-coil, single-magnet configuration

The simplest configuration of our microgenerator has one coil and one magnet. We have not been able to determine a closed form expression for the true value of ϕ as a function of coil parameters n_z and n_w, and believe this is not possible because the relationship is dependent on a number of hidden parameters, such as the magnetic field strength vector and magnetization of the magnet assembly, as well as the field's interaction with each winding in the coil. Hence, we will model the flux linkage as a weighted sum of k smooth kernels, with the kernel centers evenly distributed about ${y}_{\mathrm{diff}}\in [0,{y}_{\mathrm{diff},\ \max }]$, and sampled over evenly distributed points p ∈ y_diff. Modelling the flux linkage in this way provides two advantages. Firstly, such superpositions of smooth kernels have the ability to approximate a wide variety of smooth functions. Secondly, and more importantly, if we can find the relationship between n_z and n_w and the kernel approximation (for a particular choice of magnet), we can then predict the flux linkage as a function of n_z and n_w. We explore this particular concept further in section 6, where we construct a model that can predict the flux linkage from n_z and n_w for our particular choice of magnet.

Let us define each kernel as a function similar to that of a Gaussian radial basis function. If we define the center of a kernel as k_c and recall that we sample each kernel over evenly spaced points p, then we can express an individual kernel as:

$$\begin{eqnarray}&&{\bf{k}}({\bf{p}},{k}_{{\rm{c}}})={{De}}^{-\tfrac{{\left({\bf{p}}-{k}_{{\rm{c}}}\right)}^{2}}{2{s}^{2}}},\,\mathrm{with}\,{\bf{p}}\in {y}_{\mathrm{diff}}.\end{eqnarray} \tag{ 13 }$$

In equation (13) k_c controls the kernel location (which is set differently for each kernel so as to evenly distribute the kernels over ${y}_{\mathrm{diff}}\in [0,{y}_{\mathrm{diff},\ \max }]$), D controls the kernel magnitude and s controls the kernel width. We treat D and s as constants that we select, and thus omit D and s as parameters.

Next we combine the kernels into matrix K with dimensions (k, n_p ), where k is the number of kernels and n_p is the number of linearly spaced points in p. If we take the dot product of K with weight vector w_k , we can approximate the flux curve with the resulting vector Φ:

$$\begin{eqnarray}&&{\boldsymbol{\phi }}({p}_{{\rm{c}}})={K}^{T}\cdot {{\bf{w}}}_{k},\end{eqnarray} \tag{ 14 }$$

where we assume that our choice of w_k produces a weighted sum that approximates a smooth curve with a single peak at p_c, which corresponds to the center of the flux curve. We can easily achieve the desired p_c by linearly shifting the discrete sampling points p and kernel centers k_c of the kernels present in matrix K.

Thus, by carefully manipulating w_k , we can approximate different flux curve shapes. We show an example of this with p_c = 5 in figure 4.

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Additionally, when the levitating magnet's center aligns with the center of the coil, the total flux linkage reaches a peak, since at this position the flux entering and exiting the coil is equal, resulting in a net flux of zero. We can place the peak of ϕ by setting p_c equal to the location of the coil's center ϕ (p_c = c_c) for the single-coil, single-magnet (c = 1, m = 1) case. This allows us to model the position of the coil, relative to the position of the moving magnet.

We deliberately model the flux linkage as a function of the relative displacement y_diff between the magnet assembly and the tube, as this allows us to calculate the induced emf, it's derivative (velocity) and the flux linkage between the magnet and the coil given equation (6). Later, this is further utilized to couple the mechanical and electrical systems.

Note that equation (14) produces a vector of points, while we require a continuous function to later solve the set of equations that describe the electromechanical system. We therefore use a linear interpolation function, Γ, that accepts the resulting weight kernel sum ϕ and a single point p ∈ y_diff as arguments, and returns the interpolated flux value at this point p to define a new function ϕ₀:

$$\begin{eqnarray}&&{\phi }_{0}({y}_{\mathrm{diff}},{p}_{{\rm{c}}})={\rm{\Gamma }}(p={y}_{\mathrm{diff}},{\boldsymbol{\phi }}({p}_{{\rm{c}}})).\end{eqnarray} \tag{ 15 }$$

Now, given a kernel weight vector w_k , and a flux curve center p_c, we can use equation (15) to estimate the instantaneous flux linkage given the relative position y_diff between the magnet assembly and the coil for the case of c = 1 and m = 1.

3.6.2. Multiple-coil, single-magnet configuration

We now extend the flux model to allow for multiple-coils, but still assume a single magnet in the assembly.

As shown in figure 1, we wind adjacent coils in opposite directions and have their centers offset by the distance l_ccd. This ensures that, as the magnet assembly moves through the coils, the total flux linkage between the magnet and all coils adds constructively, resulting in an increased induced emf.

Using equation (14) and the knowledge that we space the coils uniformly, we can define an expression for the total flux linkage ϕ_c≥1,m=1 for the multi-coil, single-magnet configuration by adding the flux linkage contributions due to the individual coils for the spacing between the coil centers l_ccd:

$$\begin{eqnarray}&&{\phi }_{c\geqslant 1,m=1}({y}_{\mathrm{diff}})=\sum _{i=0}^{c-1}{\left(-1\right)}^{i}{\phi }_{0}({y}_{\mathrm{diff}},{p}_{{\rm{c}}}={c}_{{\rm{c}}}+{{il}}_{\mathrm{ccd}}),\end{eqnarray} \tag{ 16 }$$

where c_c references the position of the center of the lowermost coil, and c is the number of coils in the configuration.

We cannot make the choice of coil center distance l_ccd arbitrarily. Consider figure 5, which shows the individual flux linkage due to each coil in a microgenerator with two coils and one magnet (c = 2, m = 1). Note how the flux linkage adds constructively when the gradient of the flux linkage for both coils has the same sign or zero for one of them (regions shown in blue). In contrast, destructive superposition occurs when the gradient of the flux linkage in the coils have opposite signs (regions shown in red). As our goal is to maximize the emf ${ \mathcal E }=\tfrac{{\rm{d}}\phi }{\mathrm{dt}}$, we must maximize the constructive superposition and minimize the destructive superposition by selecting an appropriate value of l_ccd. Figure 6 shows the RMS of the induced emf as a function of l_ccd for the same microgenerator configuration shown in figure 5, which we calculate by computing the RMS of the derivative of equation (16) (i.e. the emf) with respect to y_diff. We see that, when l_ccd is too small, RMS is reduced due to destructive superposition. Of particular interest is that an optimal l_ccd value exists that results in a maximum RMS. We therefore select this optimal value ${l}_{\mathrm{ccd}}^{* }$ to maximize the RMS of the emf:

$$\begin{eqnarray}&&{l}_{\mathrm{ccd}}^{* }=\mathop{\mathrm{argmax}}\limits_{{l}_{\mathrm{ccd}}}\,\mathrm{RMS}({{ \mathcal E }}_{c\geqslant 1,m=1}),\end{eqnarray} \tag{ 17 }$$

where ${{ \mathcal E }}_{c\geqslant 1,m=1}$ is the emf induced across all coils for the multiple-coil, single magnet microgenerator configuration:

$$\begin{eqnarray}&&{{ \mathcal E }}_{c\geqslant 1,m=1}=\displaystyle \frac{{\rm{d}}{\phi }_{c\geqslant 1,m=1}}{{\rm{d}}{y}_{\mathrm{diff}}}.\end{eqnarray} \tag{ 18 }$$

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3.6.3. Single-coil, multi-magnet configuration

Next, we extend our flux model to allow the microgenerator to have multi-magnets, but a single coil. This is similar to the multi-coil, single-magnet configuration described in the previous section.

As shown in figure 1, adjacent magnets in the magnet assembly have opposite polarity, and we separate their centers by a distance l_mcd. As for the single-magnet multi-coil microgenerator configuration, this ensures that the flux linkage of each magnet with the coil superimposes constructively, resulting in a larger induced emf. Recall from figure 1 and equation (1) that y_diff is the displacement of the center of the lowermost magnet in the assembly above the bottom of the tube.

We define an expression for the flux linkage between all m magnets and the single coil ϕ_c=1,m≥1 by adding the terms representing the contribution due to each magnet:

$$\begin{eqnarray}&&{\phi }_{c=1,m\geqslant 1}({y}_{\mathrm{diff}})=\sum _{i=0}^{m-1}{\left(-1\right)}^{i}\phi ({y}_{\mathrm{diff}},{p}_{{\rm{c}}}={c}_{{\rm{c}}}-{{il}}_{\mathrm{mcd}}),\end{eqnarray} \tag{ 19 }$$

where c_c is the location of the center of the coil, and the spacing between the magnet centers is l_mcd. Note here that we subtract the flux curve offset for additional magnets from the location of the coil, since y_diff describes the position of the lowermost magnet. This is in contrast to equation (16).

As with the multi-coil, single-magnet configuration, it is again important to select an optimal distance l_mcd between the magnet centers, so as to maximize the constructive superposition and minimize the destructive superposition of the flux linkages due to the individual contributions by the magnets. As for the case of multiple coils and a single magnet, selecting a value for l_mcd that is too small results in a reduced power output, due to the destructive superposition of the flux linkage. In this particular case, there is also a lower-bound on l_mcd due to the practical difficulty of assembling magnets in a pole-matched configuration. We denote this lower-bound as ${l}_{\mathrm{mcd},\min }$.

By inspecting equations (16) and (19) we see that there is no difference, from the perspective of the resulting flux curve, between the configurations (c ≥ 1, m = 1) and (c = 1, m ≥ 1). Therefore, as in equation (17), there must again exist an optimal value of ${l}_{\mathrm{mcd}}={l}_{\mathrm{mcd}}^{* }$ that maximizes the RMS of the emf induced for the single-coil, multi-magnet case:

$$\begin{eqnarray}&&{l}_{\mathrm{mcd}}^{* }=\mathop{\mathrm{argmax}}\limits_{{l}_{\mathrm{mcd}}}\,\mathrm{RMS}({{ \mathcal E }}_{c=1,m\geqslant 1}),\end{eqnarray} \tag{ 20 }$$

where ${{ \mathcal E }}_{c=1,m\geqslant 1}$ is the emf induced across the coil for the single coil, multi-magnet case:

$$\begin{eqnarray}&&{{ \mathcal E }}_{c=1,m\geqslant 1}=\displaystyle \frac{{\rm{d}}{\phi }_{c=1,m\geqslant 1}}{{\rm{d}}{y}_{\mathrm{diff}}}.\end{eqnarray} \tag{ 21 }$$

3.6.4. Multi-coil, multi-magnet configuration

Finally, we derive the flux model for the general case in which the microgenerator has multiple coils as well as multiple magnets.

As discussed in section 3.6.3, when a magnet assembly with multiple magnets moves through a single coil, it produces a flux curve described by the superposition of m curves as modelled by equation (19). When there are both multiple coils and multiple magnets, the produced flux curve will be a superposition of c curves, as described by equation (16), where each of these c curves is itself a superposition of m curves, as described by equation (19). Therefore, in this general case, we derive the microgenerator's flux curve from following double summation:

$$\begin{eqnarray}\displaystyle \begin{array}{rcl}{\phi }_{c,m}({y}_{\mathrm{diff}}) & = & \sum _{i=0}^{c-1}{\left(-1\right)}^{i}\sum _{j=0}^{m-1}{\left(-1\right)}^{j}\phi ({y}_{\mathrm{diff}},{p}_{{\rm{c}}}={c}_{{\rm{c}}}+{{il}}_{\mathrm{ccd}}-{{jl}}_{\mathrm{mcd}})\\ & = & \sum _{i=0}^{c-1}\sum _{j=0}^{m-1}{\left(-1\right)}^{i+j}\phi ({y}_{\mathrm{diff}},{p}_{{\rm{c}}}={c}_{{\rm{c}}}+{{il}}_{\mathrm{ccd}}-{{jl}}_{\mathrm{mcd}}).\end{array}\end{eqnarray} \tag{ 22 }$$

From equation (22) we see that both the coil-center distance l_ccd and the magnet-center distance l_mcd affect the flux curves in the same way. Therefore, their optimal values must be equal:

$$\begin{eqnarray}&&{l}_{\mathrm{ccd}}^{* }={l}_{\mathrm{mcd}}^{* }.\end{eqnarray} \tag{ 23 }$$

As an example of a multi-coil, multi-magnet flux curve, we show ϕ_c=2,m=2(y_diff), the case of two coils (c = 2) and two magnets (m = 2), in figure 7, together with the four individual flux curves superimposed by equation (22). Note the alternating polarity of each individual flux curve, due to the alternating polarity of adjacent magnets and the alternating direction in which we wind adjacent coils. In this particular example, the optimal coil center ${l}_{\mathrm{ccd}}^{* }$, and magnet center ${l}_{\mathrm{mcd}}^{* }$ distance are used.

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Note that, the values of l_ccd and l_mcd are independent of the number of coils c and number of magnets m. As a result, we only need to find either ${l}_{\mathrm{ccd}}^{* }$ or ${l}_{\mathrm{mcd}}^{* }$ for one of two cases: (c = 2, m = 1) or (c = 1, m = 2), respectively. We can then infer the other distance parameter using equation (23).

From figure 3, we can see that it is possible to calculate the coil center distance, c_c, from the magnet length l_m, the magnet center distance l_mcd (in the case of multiple magnets), the magnet assembly hover height at rest l_hover, and the chosen distance l between the top edge of the magnet assembly and the lower edge of the first coil. This ensures that the magnet assembly has little coupling with the coil(s) at rest, thus increasing the possibility of realizing a large rate of change of flux, and thus emf, when the microgenerator experiences motion. From figure 1 and 3, we can express the coil center distance c_c as follows:

$$\begin{eqnarray}&&{c}_{{\rm{c}}}={l}_{\mathrm{bth}}+2{l}_{{\rm{m}}}+{l}_{\mathrm{hover}}+{l}_{\mathrm{mcd}}(m-1)+{l}_{\epsilon }+{l}_{{\rm{c}}}/2.\end{eqnarray} \tag{ 24 }$$

We can calculate the value of l_hover using the magnetic spring force from equation (8) and the weight of the magnet assembly:

$$\begin{eqnarray}&&{l}_{\mathrm{hover}}=\mathop{\mathrm{argmin}}\limits_{{y}_{\mathrm{diff}}}\,| {F}_{\mathrm{mag}}({y}_{\mathrm{diff}})-{m}_{\mathrm{weight}}| \end{eqnarray} \tag{ 25 }$$

We must know the coil resistance R_c in order to calculate the electrical damping force, F_damp, and therefore we must define a model for it. We first model the length of the copper wire used in the coil, l_coil. Given the assumed square-packing configuration of the windings and the parameters shown in figure 8, we can express the length of the wire as:

$$\begin{eqnarray}&&{l}_{\mathrm{coil}}={n}_{{\rm{z}}}\sum _{i=1}^{{n}_{{\rm{w}}}}2\pi \left({l}_{{th}}+{r}_{t}+{r}_{c}+2{{ir}}_{c}\right)\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&=\,2\pi {n}_{{\rm{z}}}\left[{n}_{w}{l}_{{th}}+{n}_{w}{r}_{t}+{n}_{w}{r}_{c}+2{r}_{c}\sum _{i=1}^{{n}_{w}}i\right]\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&=\,2\pi {n}_{{\rm{w}}}{n}_{{\rm{z}}}\left[{l}_{{th}}+{r}_{t}+{r}_{c}+2{r}_{c}\displaystyle \frac{({n}_{w}+1)}{2},\right]\end{eqnarray} \tag{ 28 }$$

where n_w is the number of windings in the radial direction, n_z is the number of windings in the axial direction, l_th is the tube wall thickness, r_t is the inner radius of the microgenerator tube, and r_c is the radius of the copper wire used for the winding.

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Once we have found l_coil, we can calculate the total resistance of the coil:

$$\begin{eqnarray}&&{R}_{{\rm{c}}}=2\pi \beta {n}_{{\rm{w}}}{n}_{{\rm{z}}}\left[{l}_{{th}}+{r}_{t}+{r}_{c}+2{r}_{c}\displaystyle \frac{({n}_{w}+1)}{2}\right],\end{eqnarray} \tag{ 29 }$$

where β is the wire gauge's resistance per unit length, which has a unit of Ω/mm.

As a magnet moves through a coil, it induces an emf. If there is a closed circuit, as shown in figure 2(b), Lenz's Law states that a current will flow in the direction that induces a magnetic field which opposes the magnetic field inducing the current (in this case the magnet assembly). This opposing magnetic field exerts a force on the magnet assembly that is proportional in magnitude to the current flowing through the coil. We therefore model the electrical damping force as directly proportional, with constant b_e, to the current i_cc flowing through the coil:

$$\begin{eqnarray}&&{F}_{\mathrm{damp}}={b}_{{\rm{e}}}{i}_{\mathrm{cc}}.\end{eqnarray} \tag{ 30 }$$

Given the electrical circuit in figure 2(b), we can calculate the closed-circuit current i_cc as:

$$\begin{eqnarray}&&{i}_{\mathrm{cc}}={ \mathcal E }\displaystyle \frac{{R}_{{\rm{l}}}}{{R}_{{\rm{l}}}+{R}_{{\rm{c}}}}\end{eqnarray} \tag{ 31 }$$

Substituting equations (6) and (31) into (30) gives the final expression for the electrical damping force:

$$\begin{eqnarray}&&{F}_{\mathrm{damp}}=-\displaystyle \frac{{b}_{{\rm{e}}}{R}_{{\rm{l}}}}{{R}_{{\rm{l}}}+{R}_{{\rm{c}}}}\left[\displaystyle \frac{{\rm{d}}\phi ({y}_{\mathrm{diff}})}{{\rm{d}}{y}_{\mathrm{diff}}}\displaystyle \frac{{\rm{d}}{y}_{\mathrm{diff}}}{\mathrm{dt}}\right].\end{eqnarray} \tag{ 32 }$$

By substituting equations (8) to (32), (22) and (10) into (3), (4) and (6), we obtain a set of equations that models the joint mechanical and electrical system:

$$\begin{eqnarray}&&{y}_{\mathrm{tube}}^{{\prime\prime} }={F}_{\mathrm{acc}}/{m}_{{\rm{t}}}\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}\displaystyle \begin{array}{rcl}{y}_{\mathrm{mag}}^{{\prime\prime} } & = & \left[\sum _{i=1}^{3}{s}_{i}{e}^{-{a}_{i}{y}_{\mathrm{diff}}}-{b}_{{\rm{m}}}{m}_{{\rm{m}}}{y}_{\mathrm{diff}}^{{\prime} }-{b}_{{\rm{e}}}\phi ^{\prime} \displaystyle \frac{{R}_{{\rm{l}}}}{{R}_{{\rm{l}}}+{R}_{{\rm{c}}}}\right.\\ & & \left.-H({y}_{\mathrm{diff}}-{y}_{\mathrm{diff},\ \max })\cdot \left(K({y}_{\mathrm{diff}}-{y}_{\mathrm{diff},\ \max })+{b}_{{\rm{f}}}{y}_{\mathrm{diff}}^{{\prime} }\right)\Space{0ex}{3.55ex}{0ex}\right]/{m}_{{\rm{m}}}\end{array}\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&\phi ^{\prime} ={ \mathcal E }=\displaystyle \frac{{\rm{d}}}{{\rm{d}}{y}_{\mathrm{diff}}}\sum _{i=0}^{c}\sum _{j=0}^{m}{\left(-1\right)}^{i+j}{y}_{\mathrm{diff}}^{{\prime} }{\phi }_{0}({y}_{\mathrm{diff}},{p}_{{\rm{c}}}={c}_{{\rm{c}}}+{{il}}_{\mathrm{ccd}}+{{jl}}_{\mathrm{mcd}}).\end{eqnarray} \tag{ 35 }$$

Using equations (33) to (35), we can numerically solve for y_t, y_m, and ϕ as functions of time, given initial conditions and input acceleration force F_acc. We can then use the solutions for y_t, y_m and ϕ to calculate the complete predicted electrical and mechanical behaviour of the microgenerator.

There are three categories of parameters that we require in order to solve equations (33) to (35): a set of parameters that are chosen by inspection, a set of parameters that are optimized to minimize the difference between model predictions and empirically-measured (or simulated) data, and a set of variable parameters that we can optimize to maximize the power that is generated.

The first category consists of parameters that we choose, by inspection, at the outset to conform to practical constraints. These include the specific magnet type and shape (controlled by l_m and w_m) used in the magnet assembly and magnetic spring, the inner radius r_t of the tube, the tube wall thickness l_th, coil wire radius r_c (and thus also the coil resistance per unit length β), the load resistance R_l, the magnet assembly to coil distance l and the microgenerator tube length l_L. We discuss the parameters belonging to this category in greater detail in section 4.

The second category of parameters consists of those that describe inherent physical properties of the microgenerator behaviour. These parameters are optimized to minimize the difference between the system model predictions and empirical measurements using an evolutionary process in conjunction with a number of physical reference devices. These include the magnet spring parameters (S, A), the mechanical friction coefficients b_m, the electrical damping constant b_e, and the impact force damping constant b_f. The process of discovering these parameters is discussed in section 5.

The third category of parameters consists of those that can be varied to alter the behaviour of the microgenerator, but without affecting the predictive accuracy of the model. We optimize these parameters to discover microgenerator designs that maximize the power delivered to the load. These include the number of coils c, number of magnets m, coil-center distance l_ccd, and magnet-center distance l_mcd. The base flux waveform ϕ₀ is a special case that also falls into this category, because it is dependent on the number of coil windings in the axial direction n_z and radial direction n_w (in addition to the choice of magnet), both of which we optimize to maximize load power. As a result, we give special attention to how we obtain ϕ₀ from n_z and n_w in section 6. Although not strictly an optimization parameter, we include the center height of the lowermost coil c_c in this parameter category also, since the coil center height c_c depends on parameters m and l_mcd (equation (24)) and we therefore need to update its value each time we change m and l_mcd during the optimization process. We discuss the optimization process for this third category of parameters in section 7.

The benefit of equations (33) to (35) is that, once we determine the first two categories of parameters (chosen parameters, and discovered parameters), we can predict the behaviour of the microgenerator for any input excitation F_acc and for any set of parameters that fall into the optimization category. This allows us to optimize the design of the microgenerator for any specific excitation.

It is critical to note that the parameters of the second and third categories (found parameters and optimized parameters) will be specific to the first category of parameters (those that we chose). As a result, if we modify any of the hand-chosen design parameters, we must repeat both the parameter discovery (including acquiring new measurement data) and parameter optimization processes for valid results. This is a direct result of the wide range of inter-dependencies between the parameters that are inherent to this microgenerator architecture, and contributes considerably to the difficulty of optimization.

We find the magnetic levitation force parameters of S = [s₁, s₂, s₃] and A = [a₁, a₂, a₃] by fitting equation (8) to data acquired by FEA simulation for the specific magnet shape and type that we selected in section 4.

We use the software package FEMM [36] to carry out the FEA force simulation. The axisymmetric test setup shown in figure 10. The simulation consists of an axially-magnetized magnet with a diameter of 10 mm and a height of 10 mm placed at a reference position. We specify the magnet material as N38 grade neodymium. The magnet dimensions and material match those of the magnets chosen for the physical microgenerator device. We position an identical magnet directly above the reference magnet, but rotated so that its bottom pole matches the top pole of the reference magnet. We translate top magnet upwards in 1 mm increments (starting at a distance of 0 mm and ending at 60mm) from the reference magnet, and calculate force experienced by the top magnet as a function of the distance between the two magnets.

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Using this FEA force data, we find the parameters S and A using a least squares optimization [37, 38]. For the particular magnet under consideration, we found that the parameter values that produce the best fit are S^* = [5.134, 26.291, 0.1256 × 10⁶], and A^* = [0.122, 0.352, 11.609]. A comparison between the FEA simulation and the curve-fit model of the magnetic levitation force in figure 11 shows strong agreement between the proposed model and the results of the FEA simulation.

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Next, we describe the parameter search process used to find values for the mechanical damper coefficient b_m, the electromechanical damper coupling factor b_e and the damping coefficient b_f of the impact force F_top. This process requires empirical, measured ground truth data of the relative magnet assembly position y_diff, the induced emf ${ \mathcal E }$ and the corresponding driving acceleration ${y}_{\mathrm{tube}}^{{\prime\prime} }$, as well as our results for the magnetic spring parameters S and A, discussed in section 5.1.

5.2.1. Reference devices and ground truth data collection

We require ground truth data in order to find parameters b_m, b_e and b_f. To achieve this, we physically construct and assemble n_devices = 4 possible reference microgenerator devices (A, B, C and D), each with different sets of chosen parameters as shown in table 2. It is important to have sufficiently many reference devices with differing designs to sufficiently explore and generalize over a wide design space. As a result, we configured the reference devices with substantial differences in their design parameters. For example, each device's coil has a different number of windings N, coil length l_c, coil width w_c, coil center c_c and coil resistance R_c. All other parameters match those chosen in section 4 and given in table 1.

Table 2. Microgenerator parameters for the four reference devices used to obtain ground truth data for the parameter search process.

Description	Param.	A	B	C	D
Number of coils	c	1	1	1	1
Number of magnets	m	1	1	1	2
Coil wire radius (mm)	rc	0.0715	0.0715	0.0715	0.0715
Coil center position (mm)	cc	72	74	76	90
Coil length (mm)	lc	8	12	14	25
Coil thickness (mm)	wc	0.75	0.67	0.48	1.43
Total coil windings	N	295	395	330	1760
Coil resistance (Ω)	Rc	12.5	23.5	17.8	80.1
Magnet length (mm)	lm	10	10	10	10
Magnet diameter (mm)	wm	10	10	10	10
Magnet center distance (mm)	lmcd	—	—	—	24
Tube inner radius (mm)	rt	5.5	5.5	5.5	5.5
Tube bottom thickness (mm)	lbth	3	3	3	3
Tube wall thickness (mm)	lth	1	1	1	1
Load resistance (Ω)	Rl	30	30	30	30
Microgen. body length (mm)	lL	120	120	120	150

Table 3. Parameters that are treated as optimizable. Each of these parameters are optimized using the search process presented in figure 20, in order to maximize the mean power ${\bar{P}}_{\mathrm{load}}$ delivered to the load.

Parameter	Description
c	Number of coils
m	Number of magnets in the magnet assembly
nz	Number of coil windings in the axial direction per coil
nw	Number of coil windings in the radial direction per coil
lmcd	Distance between the centers of each subsequent magnet
lccd	Distance between the centers of each subsequent coil

We 3D-printed the reference device bodies using PLA through an additive process, and wound the coils using a speed-controlled lathe.

We indicate the experimental setup in figure 12. We mounted a reference microgenerator to one end of a base plate and a camera, capable of high-framerate video, to the opposite end. We also mounted an accelerometer (ADXL345) to the base plate, directly behind the microgenerator. We connected the microgenerator coil to a Schottky diode full-wave bridge rectifier, whose rectified output powers a static resistive electrical load. This load voltage was attenuated using a voltage divider, and connected to the ADC input of a microcontroller (Teensy 3.5). We measured the voltage attenuation factor to be approximately 2.92: 1. Furthermore, each reference microgenerator body was manufactured with an axial groove down the sides, thereby exposing the magnet assembly to the camera. We programmed the microcontroller to record both the accelerometer output and the scaled load voltage, while the camera recorded video footage of the magnet assembly moving through the microgenerator body. This experimental setup allowed us to record both the mechanical system behaviour and the electrical system behaviour of the reference microgenerators simultaneously.

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To generate a ground truth sample, the camera and microcontroller recorded simultaneously at 120 FPS and 1600 Hz respectively. The base plate was moved in an approximately footstep-like motion for a single cycle. As illustrated in figure 13, the motion consists of an upwards acceleration across a distance of approximately 15 cm, a brief pause, and a subsequent downward acceleration followed by an impact. Figure 13(b) shows a typical such accelerometer measurement in the y-direction. We repeated this process n_input = 7 times for each device.

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After data collection, we labeled the position of the magnet assembly for each frame in the video recording. We also descale the ADC voltage measurements, using the voltage divider ratio, to calculate the true total load voltage. We use an Savitzky-Golay filter, which is a type of filter that is both simple to implement and easy to tune [39], which was empirically-tuned to reduce noise that was present on the output of the accelerometer.

5.2.2. Search process

The parameter search procedure is shown in figure 14. This process combines the ground truth measurements and our system of model equations to iteratively find the best set of parameter values for b_m, b_e and b_f.

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The search process begins by sampling new values for parameters b_m, b_e and b_f provided by the (1+1) evolutionary algorithm [40], as implemented in the Nevergrad computational library [41]. We deliberately chose to use an evolutionary algorithm to avoid the combinatorial explosion that occurs when using a exhaustive gridsearch with fine-grained parameter values. The sampled parameter values, along with magnetic spring parameters S and A and fixed parameters from table 1, are then substituted into the system of model equations (33) to (35), for each device. Next, we solve the model equations for each device using each of the n_acc = 7 corresponding measured acceleration as the input excitation using an ODE numerical solver of type Explicit Runge-Kutta method of order 5(4) [42], implemented in the SciPy library [38]. The solution provides estimates for the relative magnet position ${\hat{y}}_{\mathrm{diff}}$ and the load voltage $\hat{{V}_{{\rm{l}}}}$, the latter of which is then used to calculate a loss relative to the corresponding ground truth measurement of the load voltage V_l. We define our loss as the mean normalized load power difference across each reference device and every corresponding input excitation:

$$\begin{eqnarray}&&{ \mathcal L }=\displaystyle \frac{1}{{n}_{\mathrm{dev}}}\sum _{i}^{{n}_{\mathrm{dev}}}{{ \mathcal L }}_{i},\end{eqnarray} \tag{ 36 }$$

where ${{ \mathcal L }}_{i}$ is the mean normalized load power difference for the ith reference device:

$$\begin{eqnarray}&&{{ \mathcal L }}_{i}=\displaystyle \frac{1}{{n}_{\mathrm{acc}}}\sum _{j}^{{n}_{\mathrm{acc}}}\displaystyle \frac{{\hat{\bar{P}}}_{j}-{\bar{P}}_{j}}{{\bar{P}}_{j}}.\end{eqnarray} \tag{ 37 }$$

Here, ${\hat{\bar{P}}}_{{\rm{j}}}$ is the mean power dissipated in the load for the jth input excitation for the ith reference device, and ${\bar{P}}_{{\rm{j}}}$ is the corresponding ground truth measurement. The mean load power is calculated as follows:

$$\begin{eqnarray}&&\bar{P}=\displaystyle \frac{\mathrm{RMS}{\left({V}_{{\rm{l}}}\right)}^{2}}{{R}_{\mathrm{load}}}.\end{eqnarray} \tag{ 38 }$$

After calculation, we return the loss to the evolutionary algorithm, which updates the current best parameter set ${b}_{{\rm{m}}}^{* },\,{b}_{{\rm{e}}}^{* }$ and ${b}_{{\rm{f}}}^{* }$ if we find the loss is better than the current best. We repeated this process for n_evo = 2000 iterations, in an effort to converge to a parameter set that minimizes the loss function.

The search process found a parameter set of ${b}_{{\rm{m}}}^{* }=4.272,\,{b}_{{\rm{e}}}^{* }=5.096$ and ${b}_{{\rm{f}}}^{* }=3.108$ that minimized the loss function.

To evaluate these parameter values, we again executed the procedure shown in figure 14, this time using ${b}_{{\rm{m}}}^{* },\,{b}_{{\rm{e}}}^{* }$ and ${b}_{{\rm{f}}}^{* }$, and record the predicted relative magnet displacement ${\hat{y}}_{\mathrm{diff}}$ and predicted load voltage $\hat{{V}_{{\rm{l}}}}$. We then evaluate our model according to a number of criteria.

To evaluate temporal accuracy, we calculate the dynamic time warping (DTW) distance between the measured and the computed relative magnet distances and load voltages. This gives an indication of how well we are able to model the mechanical component and electrical component, respectively, of our reference microgenerators.

The DTW algorithm computes the degree to which two temporal signals can be made to align in time [43, 44], if we allow a certain degree of stretching and compression of the time axes. As a result, the smaller the DTW distance, the more similar two signals are in terms of their time behaviour. The literature reports DTW as a more robust measure of similarity than Euclidean (and related) distance metrics in multiple applications [45].

In addition to the DTW, we also calculate the normalized RMS difference between the measured and computed load voltages. This measure indicates how accurately the system model predicts the eventual overall power delivered to the load.

To ensure comparable results, we constrain the computed and the measured signals to the same length, and use interpolation to achieve the same fixed sampling rate for both. For DTW, we normalize the calculated distance by dividing it by the number of samples in the ground truth signals. This allows for ground truth measurements of different lengths to be compared. The normalized DTW distance between the computed and measured relative magnet distances is $\mathrm{DTW}({\hat{y}}_{\mathrm{diff}},{y}_{\mathrm{diff}})$ and the normalized DTW distance between the computed and measured load voltages is $\mathrm{DTW}({\hat{V}}_{\mathrm{load}},{V}_{\mathrm{load}})$. The normalized RMS-difference of the computed and measured load voltage is ${\mathrm{RMS}}_{\mathrm{diff}}({\hat{V}}_{{\rm{l}}},{V}_{{\rm{l}}})$. The RMS-difference is normalized by dividing by the RMS of the measured signal:

$$\begin{eqnarray}&&{\mathrm{RMS}}_{\mathrm{diff}}({\hat{V}}_{{\rm{l}}},{V}_{{\rm{l}}})=\displaystyle \frac{\mathrm{RMS}({\hat{V}}_{{\rm{l}}})-\mathrm{RMS}({V}_{{\rm{l}}})}{\mathrm{RMS}({V}_{{\rm{l}}})}.\end{eqnarray} \tag{ 39 }$$

Figure 15 shows $\mathrm{DTW}({\hat{y}}_{\mathrm{diff}},{y}_{\mathrm{diff}})$, $\mathrm{DTW}({\hat{V}}_{\mathrm{load}},{V}_{\mathrm{load}})$ and ${\mathrm{RMS}}_{\mathrm{diff}}({\hat{V}}_{{\rm{l}}},{V}_{{\rm{l}}})$ as a function of the evolutionary loss function ${ \mathcal L }$, defined in equation (36). Each point represents one of the possible parameter sets of b_m, b_e and b_f sampled by the evolutionary algorithm during for each of the n_evo = 2000 iterations during the parameter search process. The final chosen parameter set (consisting of ${b}_{{\rm{m}}}^{* },\,{b}_{{\rm{e}}}^{* }$ and ${b}_{{\rm{f}}}^{* }$) that minimizes ${ \mathcal L }$ is indicated in a different color and with arrows. We observe that every measure generally decreases as the loss decreases. This is significant, as it indicates that our parameter search process minimizes the loss (which is the difference between the calculated and measured load power as per equation (36)) by finding values of b_m, b_e and b_f that result in accurate predictions of the time-domain mechanical and electrical behaviour of the microgenerator devices. For example, it would be entirely possible for the parameter search process to find parameters that give accurate power predictions, but inaccurate time-domain behaviour predictions. Figure 15 indicates that this is not the case: our proposed search process (in conjunction with our proposed model) finds parameter values that lead to accurate time-domain behaviour prediction, which in-turn results in accurate load power estimations.

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Of interest is that the minimum value of $\mathrm{DTW}({\hat{y}}_{\mathrm{diff}},{y}_{\mathrm{diff}})$ does not perfectly coincide with the minimum loss ${ \mathcal L }$. This indicates that, of all the possible parameter sets that were sampled, some lead to more accurate estimation of the mechanical time-domain behaviour, but worse estimation of the load power. This indicates the possible existence of some inaccuracies of the proposed model (possibly due to simplifying assumptions) or in the ground truth data collection process (possibly due to noise), or both. Notably, however, the difference between the parameter sets that lead to the best value of $\mathrm{DTW}({\hat{y}}_{\mathrm{diff}},{y}_{\mathrm{diff}})$ and the best value of ${ \mathcal L }$ is small, such that it does not significantly affect our ability to estimate y_diff.

We inspect the results for the best parameter set in ${b}_{{\rm{m}}}^{* },\,{b}_{{\rm{e}}}^{* }$ and ${b}_{{\rm{f}}}^{* }$ in figure 16, which shows distributions of the normalized Dynamic Time Warping (DTW) distance between the computed and measured values of the relative magnet assembly displacement $\mathrm{DTW}({\hat{y}}_{\mathrm{diff}},{y}_{\mathrm{diff}})$, the normalized DTW distance between the predicted and measured load voltage values $\mathrm{DTW}({\hat{V}}_{\mathrm{load}},{V}_{\mathrm{load}})$ and the normalized RMS-difference of the computed and measured load voltage ${\mathrm{RMS}}_{\mathrm{diff}}({\hat{V}}_{{\rm{l}}},{V}_{{\rm{l}}})$ for each reference device. We see generally good agreement between the predicted and measured results, indicating that the parameters of ${b}_{{\rm{m}}}^{* },\,{b}_{{\rm{e}}}^{* }$ and ${b}_{{\rm{f}}}^{* }$ found in the previous section are both accurate and able to generalize to varied microgenerator designs. In particular, note that the predicted load voltage RMS is almost always within 25% of the measured result. This is noteworthy given the free-form nature of the experimental setup, and suggests that we can be reasonably confident in the model's ability to accurately predict the behaviour and load power of the energy harvester with parameters ${b}_{{\rm{m}}}^{* },\,{b}_{{\rm{e}}}^{* }$ and ${b}_{{\rm{f}}}^{* }$.

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The relationship between coil parameters (n_z and n_w), magnet properties and the resulting base flux curve ϕ₀ is not easily determined. The calculation is typically performed using finite element analysis (FEA), which is computationally expensive and therefore not suitable to an iterative application. We propose an alternative method, which only uses FEA initially to generate a training dataset. This dataset is subsequently used to train a linear model for the computation of ϕ₀ that is inexpensive to apply for arbitrary values of n_z and n_w, including those not in the training set.

Recall the model given by equation (14), which approximates the flux curve ϕ by a weighted sum of k kernels across n_p points, defined as matrix K, by multiplying with a weight vector w_k . Since we fix K, we can consider the elements of weight vector w_k as trainable parameters.

If we attempt to make a prediction of the weight vector ${\hat{{\bf{w}}}}_{k}$ as follows:

$$\begin{eqnarray}&&\hat{{\boldsymbol{\phi }}}=K\cdot \hat{{{\bf{w}}}_{k}},\end{eqnarray} \tag{ 40 }$$

we can define a linear model that does so given some coil features x:

$$\begin{eqnarray}&&\hat{{{\bf{w}}}_{k}}={\left({\bf{x}}\cdot {W}_{w}\right)}^{T}.\end{eqnarray} \tag{ 41 }$$

This allows us to modify the model in equation (14) to predict the flux curve directly from the coil features:

$$\begin{eqnarray}&&\hat{{\boldsymbol{\phi }}}=K\cdot {\left({\bf{x}}\cdot {W}_{w}\right)}^{T},\end{eqnarray} \tag{ 42 }$$

where x is an f-dimensional row vector, where f is the number of features, and W_w is a matrix of feature weights with dimensions (f, k).

We can extend this to the general case for n sets of coil features X = [x₁, x₂,...,x_n ] to produce a corresponding set of n flux curve predictions $\hat{{\rm{\Phi }}}$:

$$\begin{eqnarray}&&\hat{{\rm{\Phi }}}={\left(K\cdot {\left(X\cdot {W}_{w}\right)}^{T}\right)}^{T},\end{eqnarray} \tag{ 43 }$$

where K still has dimensions (n_p, k), X has dimensions (n, f), and W_w still has dimensions (f, k). The resulting output $\hat{{\rm{\Phi }}}$ has dimensions (n, n_p). We treat the number of kernels k, the shape parameter s and the kernel magnitude D as model hyperparameters. Note that the only trainable component remains the matrix of feature weights W_w .

To find W_w , we first generate a flux curve dataset with FEA. Using ANSYS Maxwell, we simulate the n flux linkage curves Φ at n_p discrete points for a range of y_diff values for a single magnet passing through a single coil for all possible combinations of a set of n_z values {n_z1, n_z2,...} and set of n_w values {n_w1, n_w2,...}. We simulate the magnet moving at a constant velocity of 0.35 m/s, starting 80 mm above the center of the coil and stopping 80 mm below the center of the coil. All design parameters that are chosen, remain fixed at the values discussed and chosen in section 4, and shown in table 1. We select the range of windings in the axial direction to be n_z = {1, 6,...,200} and the range of windings in the radial direction to be n_w = {1, 6,...,200} for a total of n_p = 1681 combinations. It is important to note that both the FEA dataset and resulting trained weights W_w will be entirely specific to these same chosen parameters, and we must repeat the process if any of these parameters are chosen differently. We use least squares regression to find an optimal set of kernel weights ${W}_{k}^{* }$, that maps K to Φ with minimum error:

$$\begin{eqnarray}&&{W}_{k}^{* }=\mathop{\mathrm{argmin}}\limits_{{W}_{k}}\,\mathrm{SSE}({\rm{\Phi }},K\cdot {W}_{k}),\end{eqnarray} \tag{ 44 }$$

where W_k is the trainable feature weights matrix, and where SSE is the element-wise sum of the squared errors for two matrices:

$$\begin{eqnarray}&&\mathrm{SSE}(A,B)=\sum _{i,j}{\left({a}_{i,j}-{b}_{i,j}\right)}^{2}.\end{eqnarray} \tag{ 45 }$$

We then perform a subsequent least squares regression to determine the feature weights ${W}_{W}^{* }$ that best map the coil features X to the optimal kernel weights ${W}_{k}^{* }$:

$$\begin{eqnarray}&&{W}_{W}^{* }=\mathop{\mathrm{argmin}}\limits_{{W}_{W}}\,\mathrm{SSE}({W}_{k}^{* },{\left(X\cdot {W}_{W}\right)}^{T}).\end{eqnarray} \tag{ 46 }$$

Substituting ${W}_{W}^{* }$ into equation (43) then produces a linear model that predicts a set of flux curves $\hat{{\rm{\Phi }}}$ from a set of coil features X:

$$\begin{eqnarray}&&\hat{{\rm{\Phi }}}={\left(K\cdot {\left(X\cdot {W}_{W}^{* }\right)}^{T}\right)}^{T}.\end{eqnarray} \tag{ 47 }$$

This approach to modelling the flux curves from the coil features eliminates the need for computationally-costly FEA simulation during the optimization process, and replaces it with the FEA synthesis of a training set that is only carried out once, before performing the microgenerator optimization. In this way it decouples the optimization process from the FEA simulations. The result is an end-to-end optimization process that requires measured input excitations and a trained flux curve model as input, and outputs an energy harvester design that has been directly optimized for the chosen design parameters in section 4 and the input excitation in question.

We trained the model on n = 840 randomly-selected samples (out of n_p = 1861 possible combinations), where each sample is a n_z and n_w value pair. We used the remainder of the dataset as a testing set. We randomly choose an additional development set of n = 100 samples from the training set, which we use to optimize the hyperparameters with the Nevergrad optimization library [41]. We transform every n_z and n_w value-pair into the coil features matrix X, where X is a fifth-degree polynomial feature matrix [46], that consists of all polynomial combinations of every n_z and n_w value-pair, resulting in f = 21 features. The hyperparameter optimization process found that the lowest average mean-absolute-error (MAE) across the development set was obtained for k = 53 kernels, a kernel magnitude of D = 1.883, and a shape parameter of s = 0.00485.

We show a plot of the MAE for every flux curve for small values of n_z and n_w in figure 17 and for larger values in figure 18. We show the plots separately for clarity. Every flux curve and predicted flux curve sets are normalized together prior to calculating the MAE to account for the large differences in magnitude in the resulting of the flux curves.

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The figures show that we observe small errors across almost the entire space, with the test set reporting an average MAE of 0.1821 × 10⁻⁶. To put this result into perspective, the average MAE is approximately six orders of magnitude smaller than the mean flux in the training set, which is 0.052 87 Wb. Poorer performance is only observed for small values of n_z and n_w, particularly when both n_z and n_w ≤ 6. This behaviour is also observed in figure 19, which shows a selection of ground truth FEA and predicted flux curves for different values of n_z and n_w. Here we show that for n_z and n_w ≤ 6 the flux curve model does not make useful predictions, apparently struggling when the magnitude of the flux curve is particularly small. However, for all other cases, the linear model provides an accurate approximation to the FEA curves. Importantly, the model's inaccuracy at particularly small values of n_z and n_w does not diminish its utility. Coils with n_z and n_w ≤ 6 correspond to small flux curves with small gradients that do not produce a large emf (and therefore power) as per Faraday's law. As a result, coils with n_z ∈ {1, 6} ∧ n_w ∈ {1, 6} can simply be excluded from the optimization without affecting the outcome of the optimization process.

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We executed the procedure shown in figure 20 for every possible combination of c = {1, 2}, m = {1, 2}, n_z = {6, 8, ⋯ ,200} and n_w = {6, 8, ⋯ ,200} for a total of n_psets = 38416 possible optimization parameter combinations. For the descriptions of these parameters and their effect on the resulting microgenerator design, recall figure 1.

We solve each possible parameter combination for n_acc = 15 different accelerometer measurements as input for the acceleration of the microgenerator tube ${y}_{\mathrm{tube}}^{{\prime\prime} }$, resulting in 576 240 solutions total.

We show the mean power delivered to the load ${\bar{P}}_{\mathrm{load}}$ for each optimization parameter combination in figure 21, separated into four 'bins', each representing a different {c, m} grouping. It is evident that values of n_z and n_w that are too small result in low power output, since the coil captures little flux. Similarly, for n_z and n_w values that are too large we also observe low power output, since in this case both the rate of change of flux is small (since a large portion of the magnet assembly's magnetic flux is captured by the coil(s) no matter the assembly's relative position) and the total coil resistance matches poorly the load poorly. Between these two extremes, however, there exists an optimum that provides a maximum power output. Furthermore, figure 21 indicates that this optimal point is also the global optimum for a particular {c, m} combination, and that the contour of the resulting load power 'surface' appears smooth. We also see that microgenerators with multiple coils are optimal when coils are shorter (smaller n_z) and narrower (smaller n_w) than microgenerator architectures with fewer coils. This is likely due to the increased resistance resulting of multiple coil configurations. The optimization also appears to prefer coils that are taller than they are wide (i.e. n_z > n_w). This is likely due to the particular choice of magnet and its corresponding magnetic field used in our microgenerator architecture.

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We show the optimal parameter combination, and corresponding power ${\bar{P}}_{\mathrm{load}}$ delivered to the load for each {c, m} combination in table 4. We find that the best configuration delivers a mean power ${\bar{P}}_{\mathrm{load}}=1.526\times {10}^{-3}{\rm{W}}$ to the 30Ωload across all input excitations (n = 15) with parameter combination c = 1, m = 2, n_z = 68, n_w = 20 and l_mcd = 17 mm. Furthermore, we find that the best design delivers slightly more power than for the best c = 2 and m = 2 configuration, suggesting that adding both additional coils and magnets does not always guarantee more power. Figure 21 shows that optimal parameters not only exist, but achieve vastly greater power output than most other designs that we considered in the exhaustive optimization set. As an example, let us consider the c = 1 , m = 1 microgenerator as a 'naive' reference design. Even with the best possible coil parameters for n_z and n_w, our optimum c = 1, m = 2 devices achieves ≈ 2.8x the load power output of this reference design. Here we note that the 'naive reference' is taken as the optimum c = 1, m = 1 device after optimization. Non-optimal choices of n_z, n_w leader to even lower output. This demonstrates the benefit of our proposed optimization process.

Table 4. Optimal combination of parameters n_z, n_w, l_ccd and l_mcd for each combination of {c, m} and the corresponding mean power ${\bar{P}}_{\mathrm{load}}$, over n_acc = 15 input excitations, that is delivered to the 30Ωload.

c	m	nz	nw	lccd (mm)	lmcd (mm)	${\bar{P}}_{\mathrm{load}}$ (W)
1	1	72	20	—	—	0.545 × 10−3
1	2	68	20	—	17	1.526 × 10−3
2	1	66	14	16	—	0.454 × 10−3
2	2	64	14	16	16	1.431 × 10−3

The complete set of design parameters for the optimal microgenerator design is shown in table 5.