# Development of An Analytical Method for Design of Electromagnetic Energy Harvesters with Planar Magnetic Arrays

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## Abstract

**:**

## 1. Introduction

^{−2}at the frequency of 4.10 Hz [9]. Green et al. (2013) studied the use of monostable and bistable nonlinear EMEHs for harvesting electrical energy from the vibration of bridges. Green found that such EHs are better suited to low-frequency excitations such as those caused by the vibration of bridges [10]. Pirirsi et al. (2013) proposed a speed-bump energy converter with a tubular permanent-magnet linear generator to harvest electrical energy from traffic. They showed that a proposed EH 1 m long, 3 m wide, and 10 cm in thickness can generate an electrical power of as much as 700 W for a typical passing car [11]. Gatti et al. (2016) developed an analytical model to determine the amount of energy that could be harvested from a passing train if the EH is installed on the sleeper vibrating in the vertical direction. They found that the optimum amount of energy harvested per unit mass of EH is proportional to the product of the square of the excitation acceleration amplitude and the square of the duration of excitation [12]. Takeya et al. (2016) proposed a tuned mass system in which a linear electromagnetic transducer has been used to harvest the unused reserve of energy in the damping system when attached to a bridge subjected to traffic loading [13].

## 2. Mathematical Modeling of EMEHs

_{bX}(t) along the X-axis. They are mounted inside two housings made of a non-magnetic material such as aluminum. The coil is attached to the base through a system of elastic springs and viscous dampers. The size of the vertical air gap between the coil and the left and right arrays along the Z-axis is denoted by Δ

_{gZ}. This is an important parameter that controls the strength of magnetic interactions between the arrays and the coil.

_{X}× n

_{Y}cuboidal PMs of the size a

_{m}× a

_{m}× a

_{m}which are separated from each other through air gaps of the size δ

_{gmX}and δ

_{gmY}along the X- and Y-axes, respectively. Therefore, the dimensions of the left and right arrays would be L

_{m}= n

_{X}a

_{m}+ (n

_{X}− 1)δ

_{gmX}, W

_{m}= n

_{Y}a

_{m}+ (n

_{Y}− 1)δ

_{gmY}, and H

_{m}= a

_{m}.

_{ci}(t) flowing in the copper wire of the coil whose direction varies with the motion of the coil. The direction of this electric current changes as per Lenz’s law in such a way that the induced magnetic field opposes the initial cause of change in the magnetic flux of the PMs, which is the motion of the coil. Eventually, this interaction causes the braking (damping) force F

_{c}that acts on the coil and its direction which is always opposite to the direction of the velocity of the coil, i.e., ${\stackrel{\text{\u22c5}}{\mathrm{u}}}_{\mathrm{sX}}\left(\mathrm{t}\right)$.

#### 2.1. Electromechanical Model

_{bX}(t) = u

_{bXmax}sin(ω

_{b}t) along the X-axis where u

_{bXmax}and ω

_{b}= 2πf

_{b}are the amplitude and the circular frequency of the excitation, and f

_{b}is the frequency of the excitation. The stiffness and mechanical damping coefficients of the elastic spring and viscous damper are denoted by k

_{s}and c

_{s}, respectively. The total mass of the SDOF system, consisting of the mass of the coil and the tip mass, is denoted by m

_{s}. The degree of freedom of the SDOF system is also denoted by u

_{sX}which represents the displacement of the mass center of the coil.

_{c}and R

_{l}, and an inductor L

_{c}connected in series to an altering-voltage source with the electromotive force V

_{emf}. R

_{c}and L

_{c}represent the resistance and induction of the coil, respectively, and R

_{l}is the resistance of the electrical load used to harvest the electrical power.

#### 2.2. Planar Arrangement of the PMs

_{X}= 5 and n

_{Y}= 5 in this study as n = 5 is the minimum number of PMs required to create a linear Halbach array. These arrays, therefore, consist of 25 identical cubic PMs with the side length a

_{m,}which are separated from each other by air gaps of the size δ

_{gmX}and δ

_{gmY}along the X- and Y-axes, respectively.

**n**is the unit surface normal vector, ${M}_{\mathrm{rmIJ}}=\left({\mathsf{\gamma}}_{\mathrm{XIJ}}{e}_{\mathrm{X}}+{\mathsf{\gamma}}_{\mathrm{YIJ}}{e}_{\mathrm{Y}}+{\mathsf{\gamma}}_{\mathrm{ZIJ}}{e}_{\mathrm{Z}}\right){\mathrm{B}}_{\mathrm{rmIJ}}/{\mathsf{\mu}}_{0}$ is the magnetization vector of PM in which ${\mathsf{\gamma}}_{\mathrm{XIJ}}$, ${\mathsf{\gamma}}_{\mathrm{YIJ}}$, and ${\mathsf{\gamma}}_{\mathrm{ZIJ}}$ are constants showing the direction of vector, B

_{rmIJ}is the magnetic remanence, and

**r**and

**r**

_{0}are the position vectors of field and source points, respectively. Note that μ

_{0}= 4π × 10

^{−7}Tm/A is the magnetic permeability of the vacuum.

**B**

_{mla}(X,Y,Z) and right

**B**

_{mra}(X,Y,Z) arrays shown in Figure 2a can be calculated by summing up the magnetic flux density vectors of the PMs,

**B**

_{mIJ}is given by

**b**is a 2nd order tensor function in space defined as

^{2}+ Y

^{2}+ Z

^{2})

^{1/2}. Furthermore, in Equation (1c), X = X

_{cIJ1}, X = X

_{cIJ2}, Y = Y

_{cIJ1}, Y = Y

_{cIJ2}, Z = Z

_{cIJ1}, and Z = Z

_{cIJ2}are the coordinates of boundary surfaces surrounding the volume of IJ-th PM on the X-, Y-, and Z-axes, respectively.

#### 2.3. Thick Rectangular Coil

_{c}, the width is a

_{c}, the height is h

_{c}, and the winding depth is t

_{c}. In this figure, N

_{z}= h

_{c}/d

_{w}and N

_{t}= t

_{c}/d

_{w}denote the numbers of turns along the z-axis and the depth of winding, respectively, where d

_{w}is the diameter of the winding wire. Therefore, the total number of turns can be calculated as N

_{c}= N

_{z}× N

_{t}. It is assumed that when the electric current I

_{ci}(t) is counterclockwise in the xy-plane, I

_{ci}(t) is positive. This implies that the N- and S-poles are established at z = +h

_{c}/2 and z = −h

_{c}/2, respectively.

**K**′

_{bm}is modeled by an equivalent PM of the same dimensions as shown in Figure 4c. The magnetization vector

**M**′

_{rm}of the equivalent PM is related to

**K**′

_{bm}through

**K**′

_{bm}=

**M**′

_{rm}×

**n**, where

**n**is the unit surface normal vector of the equivalent PM [32]. The shell method [33] can be used to model a multi-layer turn coil referred to as the thick rectangular coil in this study. This method treats each single layer of turn as an equivalent cuboidal PM of the dimension a′

_{mq}× a′

_{mq}× h′

_{mq}where a′

_{mq}= a

_{c}−[2(N

_{t}− q) − 1]d

_{w}and h′

_{mq}= h

_{c}. The magnetic remanence of this PM is B′

_{rm}= μ

_{0}(N

_{z}I

_{ci}/h

_{c}) where B′

_{rm}= μ

_{0}M′

_{rm}and I

_{ci}= (h

_{c}K′

_{bm})/N

_{z}.

**B**

_{cq}is the magnetic flux density vector of the q-th turn of the coil that is calculated by Equation (1) for n

_{X}= n

_{Y}= 1 as per the shell method [33].

_{w}= 4N

_{z}N

_{t}(a

_{c}− t

_{c}), σ

_{c}= 58.58 MS/m, and ${\mathrm{A}}_{\mathrm{w}}={\mathsf{\pi}\mathrm{d}}_{\mathrm{w}}^{2}/4$ are the length, electrical conductivity, and cross-section area of the winding copper wire, respectively. The inductance of the coil is also given by [34],

_{cq}= a

_{c}− 2(N

_{t}− q)d

_{w}is the length of the sides of q-th turn.

#### 2.4. Magnetic Interaction of the Arrays with the Moving Coil

_{cXIJ}is the magnetic force applied to the coil due to the magnetic field of IJ-th PM. This force can be computed by the shell method as follows [33],

_{mXIJq}is the magnetic force applied to the q-th turn of the coil due to the magnetic field of the IJ-th PM and is given by [6,7],

_{mq}is the volume of the equivalent cuboidal PM enclosed by the q-th turn of the coil and fʹ

_{mXIJq}(.) is a dimensionless function in term of surface integrals carried over the equivalent cuboidal PM [7]. This function depends on the following parameters: (i) the aspect ratio of the turn α′

_{mq}= a′

_{mq}/h′

_{mq}in which a′

_{mq}=a

_{c}− (2q − 1)d

_{w}and h′

_{mq}= h

_{c}; (ii) the geometrical ratios ${\mathsf{\gamma}}_{\mathrm{mq}}^{\prime}={\mathrm{a}}_{\mathrm{m}}/{\mathrm{a}}_{\mathrm{mq}}^{\prime}$ and ${\mathsf{\beta}}_{\mathrm{mq}}^{\prime}={\mathrm{a}}_{\mathrm{m}}/{\mathrm{h}}_{\mathrm{mq}}^{\prime}$and (iii) the mass center eccentricity ratios ΔX

_{mIJq}/a′

_{mq}, ΔY

_{mIJq}/a′

_{mq}, and ΔZ

_{mIJq}/h′

_{mq}along the X-, Y-, and Z-axes, respectively, where Δ(.)′

_{mIJq}= (.)

_{mIJ}−(.)′

_{mq}.

## 3. Electromechanical Equation

**B**

_{mta}=

**B**

_{mla}(X, Y, Z) +

**B**

_{mra}(X, Y, Z) is described by the following two-degrees-of-freedom coupled electromechanical equation:

_{ctX}is the X-component of the total magnetic force applied to the coil, i.e.,

**F**

_{ct}=

**F**

_{cla}+

**F**

_{cra}in which

**F**

_{cla}and

**F**

_{cra}are calculated by Equation (5a–c). In Equation (6a), ${\ddot{\mathrm{u}}}_{\mathrm{bX}}\left(\mathrm{t}\right)={\ddot{\mathrm{u}}}_{\mathrm{bXmax}}\mathrm{sin}({\mathsf{\omega}}_{\mathrm{b}}\mathrm{t})$ in which ${\ddot{\mathrm{u}}}_{\mathrm{bXmax}}={\mathrm{u}}_{\mathrm{bXmax}}{\mathsf{\omega}}_{\mathrm{b}}^{2}$ is the maximum acceleration of the base.

#### 3.1. Decoupled Equation of Motion

_{ctX}and V

_{emf}as discussed below.

_{ctX}can be alternatively calculated by the Lorentz’s formula for a current-carrying wire in electromagnetism as follows,

_{ctX}= −c

_{e}${\stackrel{\text{\u22c5}}{\mathrm{u}}}_{\mathrm{sX}}$ where c

_{e}is the electrical damping caused by the magnetic interaction between the coil and the arrays. The correct sign of this force is implicit in Equation (7a). This force can be written in the following simple form,

_{f}is called electromechanical coupling coefficient or transformation factor [17] which is defined by

_{f}can be calculated by Equation (5) for I

_{ci}= −1 A because according to Equation (7b) as K

_{f}= F

_{ctX}for I

_{ci}= −1 A.

_{f}is constant and does not change with time. That is, they have assumed that K

_{f}= N

_{z}B

_{mtavg}(a

_{c}− t

_{c}) where B

_{mtavg}is the volume average of the magnetic flux density vector of the left and right arrays over the air gap between them. This is an oversimplified assumption that can result in error when estimating the harvested electrical power [35,36]. This error is lowest for thin copper coils placed in a narrow air gap.

_{emf}can be calculated by the following integral taken over the length of the coil,

_{ci}= V

_{emf}/(R

_{l}+ R

_{c}) and the velocity of the coil can be obtained as

_{ci}from Equation (6a) as follows:

_{s}is the critical mechanical damping ratio. The average electrical power harvested from the EMEH over the time interval [0,τ] is given by

_{l}is the instantaneous power consumed by the load which is given by

#### 3.2. Numerical Verification

_{c}= 0.25 in, wound by a copper wire of 18-AWG with d

_{w}≅ 1 mm and the ampacity current 16 A [38]. The two PMs are identical with the dimensions 1 in × 1 in × 0.5 in and the magnetic remanence B

_{rm}= 1.4 T (neodymium type N52). The size of the vertical gap between the coil and the PMs is ∆

_{gcZ}= 0.25 in.

_{a}= 6 in. is used to model the air domain enclosing the PMs and coil. The center of this sphere is located at the center of air gap between the PMs where the origin of XYZ coordinate system is located. Table 1 shows the geometrical and material parameters of the FE model.

_{ci}, is not time-dependent. Therefore, the attractive magnetic interaction between the coil and the PMs can be described by the magnetostatic form of the Maxwell equations in the presence of an external current as follows [32],

**A**is the magnetic vector potential,

**J**

_{e}is the volume density vector of the electric current, and μ

_{m}is the magnetic permeability of the materials including the PMs (μ

_{m}≅ μ

_{0}) and air domain (μ

_{m}= μ

_{0}).

**A**= 0 (Coulomb gauge) must be satisfied. The Maxwell stress tensor is also used to calculate the magnetic force applied to the coil [6,7,39]. It is important to employ a very fine mesh along the edges of coil and PMs to make sure that the results are accurate, as illustrated in Figure 5a,b.

_{sX}= −0.216a

_{c}and I

_{ci}= −0.849 A. The magnetic flux density field of the AC is negligible.

_{bX}(t) = ü

_{bXmax}sin(ω

_{b}t) along the X-axis with ü

_{bXmax}= 0.05 g, f

_{b}= 3.5 Hz, and 0 ≤ t ≤ 2T

_{b}where T

_{b}= 1/f

_{b}= 0.286 s. Table 2 shows parameters of the SDOF model. The motion of this system is described by Equation (11). A numerical solver is used in MATLAB [40] to solve this equation.

_{sX}and I

_{ci,}we calculate the magnetic force by using the analytical model. Then it is compared to the corresponding value resulting from the FE model.

_{sX}= −0.199a

_{c}and +0.733a

_{c}and the currents I

_{ci}= −0.550 and +2.782 A, respectively, assuming a

_{c}=1 in. These values have been shown on Figure 6a,b with points 1 and 2, respectively. The results have been compared to those obtained from the analytical model as can be observed on Figure 6c. It is seen that there is a good agreement between the analytical and FE models. This validates the accuracy of the proposed analytical method in this paper.

_{f}is maximum at u

_{sX}= ±0.5a

_{c}and very low at large displacements; approximately beyond two times of the length of the air gap along the X-axis where the mechanical energy domain is totally decoupled from the electromagnetic energy domain.

## 4. Parametric Study

_{bX}(t) = ü

_{bXmax}sin(ω

_{b}t) along the X-axis for ü

_{bXmax}= 0.01 g, 0.05 g, and 0.10 g and f

_{b}= 3.5 Hz over the time interval 0 ≤ t ≤ 2 T

_{b}where T

_{b}= 1/f

_{b}= 0.286 s. The EMEH is tuned to undergo the resonant condition with the frequency f

_{s}= f

_{b}= 3.5 Hz. Table 3 shows the electromechanical parameters of the EMEH.

_{l}/R

_{c}for Array 1, 2, …, and 7. It is seen that the EMEH can deliver the highest electrical power when the PMs are arranged according to Array 2 in which the poles alternate along the X-axis. The average electrical power increases with the increase of the intensity of the base acceleration. It is seen that for ü

_{bXmax}= 0.1 g the maximum average electrical power is equal to P

_{lavgmax}= 513 mW that can be delivered by the load resistance R

_{lopt}= 0.25 R

_{c}= 186.6 mΩ. The amount of power is relatively very large and can be uses to power conventional sensors used for structural health monitoring of bridges. Larger amounts of electrical power can be delivered by adding a greater number of coils and having arrays installed in parallel.

_{lopt}< R

_{c}as shown in Figure 8b. However, this is not the case when the electrotechnical coefficient is assumed to be constant. The maximum average electrical power can be calculated by putting the derivative of P

_{lavg}with respect to R

_{l}equal to zero that results in the calculation of the optimal load resistance as follows [41],

## 5. Experimental Study

_{s}= 0.49 kg on the springs; a rectangular copper coil with a thickness of 0.25 in and the outside size 2.25 × 2.25 in and the inside size 1.75 × 1.75 in, two magnetic arrays consisting of 25 cubic neodymium PMs of the size 0.5 × 0.5 × 0.5 in with the magnetic remanence of B

_{rm}= 1.2 T (neodymium, type N42). The frequency of the system was estimated to be approximately about f

_{s}= 6.5 Hz.

#### 5.1. Laboratory Testing

_{bXmax}= 0.2 g.

_{b}= 1.5 Hz to f

_{b}= 3.0 Hz. This range is quite short, and the frequencies in this range are less than f

_{s}= 6.5 Hz. This was due to the limitation of the actuator in compensating the gravity effects in the vertical direction (i.e., weight of the prototype EMEH). For this reason, exciting the prototype EMEH under a resonant condition when f

_{b}= f

_{s}= 6.5 Hz was not feasible in this experimental study.

_{l}≅ 0). These voltage signals were denoised by a third-order band-pass Butterworth filter with the cut-off frequencies 3 Hz and 21 Hz and a Nyquist frequency of 500 Hz defined by function [b,a] = butter(.) in MATLAB.

#### 5.2. Model Validation

_{b}= 2.3 Hz as shown in Figure 11c. The parameters of the SDOF system are m

_{s}= 0.49 kg, f

_{s}= 6.5 Hz, and ξ

_{s}. The mechanical damping ratio ξ

_{s}is given by

_{z}= 6, ${\mathrm{B}}_{\mathrm{mtavg}}\cong 0.3\mathrm{T}$ and ${\mathrm{a}}_{\mathrm{c}}-{\mathrm{t}}_{\mathrm{c}}=0.0508\mathrm{m}$ and assuming that ${\mathrm{R}}_{\mathrm{l}}\cong 0.01{\mathrm{R}}_{\mathrm{c}}$ (no load) and the air gap is very small. The total damping ratio ${\mathsf{\xi}}_{\mathrm{t}}$ can be estimated from the free vibration response of tip mass by using the logarithmic decrement method as illustrated in Figure 13. This figure shows the time history of acceleration of the SDOF system recorded during the lab test. The average total damping is estimated to be 4.3%, implying that the mechanical damping ratio should be equal to ξ

_{s}= 4.3−0.1 = 4.2%.

_{s}= 0.49 kg, ξ

_{s}= 4.2, and f

_{s}= 6.5 Hz. Table 4 shows the estimated value of natural frequency of the SDOF system for five successive peaks. The average value of the natural frequency is about 6.75 Hz, which is very close to the 6.5 Hz used in the analytical model.

#### 5.3. Field Testing

_{b}= 10 Hz to f

_{b}= 20 Hz. These frequencies are much larger than the frequency of the prototype which is about f

_{s}= 4 Hz. For this reason, it was difficult to put the prototype EMEH into a resonant condition with the vibration of the bridge. Figure 17 shows the time history of the voltage output from the prototype EMEH during the field test. It is seen that the magnitude of this voltage is less than 1.5 mV.

## 6. Conclusions

_{bXmax}= 0.1 g, the maximum average electrical power is equal to 513 mW that can be delivered by a harvesting circuit with an electrical resistance of 186.6 mΩ. Furthermore, a proof-of-the-concept prototype was fabricated for the laboratory and field testing in which Array 4 was selected to arrange the PMs poles. The laboratory testing showed that the RMSs of voltage outputted from the harvester when subjected to a base excitation with a frequency ranging from f

_{b}= 1.5 Hz to f

_{b}= 3.0 Hz (below the resonant frequency of the harvester) varies from 0.54 mV to 2.9 mV. The field testing of the harvester also showed that the harvester can generate a voltage as large as 1.5 mV for an excitation frequency in the range of 10 Hz to 20 Hz. The experimental study carried out in this paper was limited to one type of planar magnetic array (planar alternating) without considering the resonant condition. A more comprehensive experimental study of such electromagnetic harvesters with planar magnetic array will be the subject of future publication of the authors.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Configuration of the EMEH and its key components. (

**a**) Vertical cross section of the EMEH in the XZ-plane. (

**b**) Geometrical parameters of the arrays in details.

**Figure 2.**Electromechanical model of the EMEH consisting of a lumped SDOF dynamic system coupled to a lumped first-order RL circuit.

**Figure 3.**Planar arrangement of the PMs poles in seven different multipole arrays with their corresponding parameters: (

**a**) uniform, (

**b**) X-linear alternating, (

**c**) Y-linear alternating, (

**d**) planar alternating on the left and right sides; and the top faces of (

**e**) X-linear Halbach, (

**f**) Y-linear Halbach, (

**g**) planar Halbach arrays on the right sides, and the top faces of (

**h**) X-linear Halbach, (

**i**) Y-linear Halbach, and (

**j**) planar Halbach arrays on the left side.

**Figure 4.**Geometrical parameters of the coil in details; (

**a**) cross-section in the xy-plane; (

**b**) cross-section in the xz-plane; and (

**c**) the equivalent PM of p-turn.

**Figure 5.**FE model used for validation of the force-current model developed in this study: (

**a**) longitudinal cross section on the XZ-Plane, (

**b**) XY-Plane, (

**c**) FE model enclosed by the air domain; and (

**d**)

**B**

_{mt}-field on the XZ-plane at Y = 0 for u

_{sX}= −0.216a

_{c}and I

_{ci}= −0.849 A.

**Figure 6.**Comparison between the analytical model and the FE model to calculate the magnetic force; (

**a**) displacement of the coil, (

**b**) electric current induced in the coil, and (

**c**) magnetic force acting on the coil.

**Figure 7.**Variation of the electromechanical coupling coefficient with (

**a**) time and (

**b**) displacement of the coil.

**Figure 8.**Average electrical power harvested from the EMEH versus the ratio of the load resistance to the coil resistance (R

_{l}/R

_{c}) for (

**a**) Array 1, (

**b**) Array 2, (

**c**) Array 3, (

**d**) Array 4, (

**e**) Array 5, (

**f**) Array 6, and (

**g**) Array 7.

**Figure 9.**Proof-of-concept prototype of the EMEH; (

**a**) 3D view of the fabricated device in laboratory with details of the geometry of (

**b**,

**c**) the arrays and (

**d**) the copper coil.

**Figure 10.**Experimental setup established to test the proof-of-concept prototype of EMEH in a laboratory environment under harmonic excitations.

**Figure 11.**Power spectral density of the acceleration of base recorded by an accelerometer of the model Data Logger X2-5.

**Figure 12.**Voltage output from the prototype EMEH subjected to five different base excitations with the frequencies from f

_{b}= 1.5 Hz to f

_{b}= 3.0 Hz; RMSs of these voltage signals are (

**a**) 0.54 mV, (

**b**) 0.93 mV, (

**c**) 2.9 mV, (

**d**) 1.6 mV, and (

**e**) 1.3 mV.

**Figure 13.**Time history responses of the SDOF system recorded during the lab testing; (

**a**) base acceleration f

_{b}= 2.3 Hz, (

**b**) tip mass acceleration (proper), and (

**c**) tip mass acceleration under a free vibration.

**Figure 14.**Voltage output from the SDOF model subjected to base excitations with the frequency from f

_{b}= 2.3 Hz.

**Figure 15.**Field testing of the proof-of-concept prototype of EMEH (

**a**) placement of the device on the middle of a horizontal lateral bracing of the deck, and (

**b**) testing setup.

**Figure 16.**Acceleration of the base excitation recorded by an accelerometer of the model Data Logger X2-5 during the field testing: (

**a**) 5 min time-history of the acceleration and (

**b**) power spectral density of the acceleration.

**Table 1.**Geometrical and material parameters of the coil and PMs in the FE model developed in COMSOL Multiphysics software.

Parameter | Value | Unit | Description |
---|---|---|---|

a_{m} | 1 | in | Length of the sides of the PMs |

a_{c} | 1 | in | Length of the sides of the coil |

h_{c} | 0.5 | in | Height of the coil (N_{z} = 13) |

t_{c} | 0.25 | in | Winding depth (N_{t} = 6) |

d_{w} | 1 | mm | Diameter of the copper wire (18-AWG) |

Δ_{gcZ} | 1/16 | in | Size of the vertical gap between the coil and the PMs |

B_{rm} | 1.4 | T | Magnetic remanence of the PMs (Neodymium, type N52) |

σ_{c} | 58.58 | MS/m | Electrical conductivity of copper wire |

Parameter | Value | Unit | Description |
---|---|---|---|

fb | 3.5 | Hz | Frequency of the base excitation |

ü_{bXmax} | 0.05 g | m/s^{2} | Maximum acceleration of the base excitation (u_{bXmax} = ü_{bXmax}/ω_{b}^{2} = 15.2 cm) |

f_{s} | 3.5 | Hz | Frequency of the SDOF system |

ξ_{s} | 5 | % | Critical mechanical damping ratio of the SDOF system |

m_{s} | 41.8 | gr | Mass of the SDOF system (m_{s} = m_{w} = 41.8 gr) |

R_{c} | 129 | mΩ | Resistance of the coil |

R_{l} | 129 | mΩ | Resistance of the electrical load (R_{l}/R_{c} = 1) |

Parameter | Value | Unit | Description |
---|---|---|---|

a_{m} | 0.5 | in | Length of the sides of the PMs |

δ_{gmX} | 1 | mm | Size of the gap between the PMs along the X-axis |

δ_{gmY} | 1 | mm | Size of the gap between the PMs along the Y-axis |

n_{X} | 5 | Number of the PMs along the X-axis | |

n_{Y} | 5 | Number of the PMs along the Y-axis | |

a_{c} | 2.5 | in | Length of the sides of the coil |

h_{c} | 0.5 | in | Height of the AC (N_{z} = 13) |

t_{c} | 0.5 | in | Winding depth (N_{t} = 13) |

d_{w} | 1 | mm | Diameter of the copper wire (18-AWG) |

Δ_{gcZ} | 1/16 | in | Size of the vertical gap between the coil and the PMs |

B_{rm} | 1.4 | T | Magnetic remanence of the PMs |

σ_{c} | 58.58 | MS/m | Electrical conductivity of copper wire |

f_{b} | 3.5 | Hz | Frequency of the base excitation |

ü_{bXmax} | Var. | m/s^{2} | Maximum acceleration of the base excitation |

f_{s} | 3.5 | Hz | Frequency of the SDOF system (f_{s} = f_{b}) |

ξ_{s} | 5 | % | Critical mechanical damping ratio of the SDOF system |

m_{s} | 241.7 | gr | Mass of the SDOF system (m_{s} = m_{w}) |

R_{c} | 746.4 | mΩ | Resistance of the coil |

R_{l} | Var. | mΩ | Resistance of the electrical load |

**Table 4.**Estimating the total damping ratio and frequency of SDOF system from free vibration results.

i | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

${\left[{\ddot{\mathrm{u}}}_{\mathrm{sX}}+{\ddot{\mathrm{u}}}_{\mathrm{bX}}\right]}_{\mathrm{i}}\left(\mathrm{g}\right)$ | 0.7090 | 0.5610 | 0.4160 | 0.3410 | 0.2410 |

${\mathrm{t}}_{\mathrm{i}}$ (sec) | 22.952 | 22.248 | 22.400 | 22.544 | 22.688 |

Estimated Parameters | |||||

${\mathsf{\xi}}_{\mathrm{t}}$ (%) | 3.7 | 4.8 | 3.1 | 5.5 | |

${\mathrm{f}}_{\mathrm{s}}$ (Hz) | 6.6 | 6.9 | 6.6 | 6.9 |

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## Share and Cite

**MDPI and ACS Style**

Amjadian, M.; Agrawal, A.K.; Nassif, H.H. Development of An Analytical Method for Design of Electromagnetic Energy Harvesters with Planar Magnetic Arrays. *Energies* **2022**, *15*, 3540.
https://doi.org/10.3390/en15103540

**AMA Style**

Amjadian M, Agrawal AK, Nassif HH. Development of An Analytical Method for Design of Electromagnetic Energy Harvesters with Planar Magnetic Arrays. *Energies*. 2022; 15(10):3540.
https://doi.org/10.3390/en15103540

**Chicago/Turabian Style**

Amjadian, Mohsen, Anil. K. Agrawal, and Hani H. Nassif. 2022. "Development of An Analytical Method for Design of Electromagnetic Energy Harvesters with Planar Magnetic Arrays" *Energies* 15, no. 10: 3540.
https://doi.org/10.3390/en15103540