This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).
The performance of more than 60 different electromagnetic energy harvesters described in more than 100 publications is benchmarked. The benchmarking is based on earlier published parameters from literature as well as on two novel parameters introduced in this paper. The former allow to compare different harvester conversion principles as well as harvesters of different electrodynamic design principles. The latter consider the impact of ambient and boundary conditions for the most important subgroup, namely the resonant electrodynamic harvesters. The special consideration of how the mechanical damping and the energy conversion effectiveness depend on these conditions enables a fairer benchmarking of this common harvester type. High performing prototypes are identified, and the key parameters are provided for explanation. Finally, beneficial design approaches and the main challenges to maximize the output power are pointed out.
In our very recently published review paper [
As the power density does not consider the excitation of a harvester, several authors reviewed and compared electrodynamic vibration harvesters based on different benchmarking parameters. Mitcheson [
In this paper we address the challenge to consider the complex impact of ambient and boundary conditions in a single benchmarking parameter. By considering the dependencies of key parameters such as the mechanical damping, an attempt to approach a fair benchmarking is provided. First, the benchmark parameters (BPs) from literature are briefly reviewed. Second, we introduce two novel parameters derived from the linear resonant harvester model. Third, based on the different BPs, the harvesters from our review paper [
To compare different harvesting mechanisms among each other or harvesters to energy sources such as batteries, the power density, subsequently named BP (a), is often used.
Mitcheson
Comparing only resonant electrodynamic harvesters allows to consider the impact of ambient and boundary conditions. A first benchmarking approach was derived from the linear harvester model by Arnold
In literature, the proportionality of
Whereas the coil filling factor
viscid drag
lateral laminar flow Couette damping
lateral laminar flow Stokes damping (Stokes damping here refers to the expression for small frequencies and air gaps)
perpendicular laminar flow squeeze film damping
As research in the field of sensors shows, viscid damping at small scales even dominates at very low pressures [
First, assume that laminar viscid damping dominates the system. With
(
The corresponding output power of a harvester can be found with the help of Equation
To relate the output power of a harvester to its particular volume, the length scaling factor can be replaced by
These three parameters have already been pointed out by Arnold [
Having derived a BP for the assumption that viscid damping dominates the system, one can similarly find a parameter for the case that material damping is dominant. According to Appendix A, material damping is independent of the excitation acceleration but proportional to the frequency and volume (compare Equations (
(
The two BPs (c) and (d) are quite different. Generally it is not clear under which conditions viscid damping or material damping is dominating, and the question of a fair benchmark parameter cannot be conclusively answered. On the one hand, one could claim that vacuum packaging of harvesters would allow to reduce viscid damping and generally apply BP (d). On the other hand, if harvesters are not used in vacuum and viscid damping is indeed dominating, the usage of BP (d) would treat harvesters with low
Before applying the BPs to published prototypes some limitations should be kept in mind. Firstly, BP (c) and (d) both assume that any dimension of the harvester scales proportionally to the length scaling factor. However, according to [
inversely proportional to
proportional to the acceleration at a given
proportional to
Secondly, the aspect ratio of the harvester defines the design freedom and might on the one hand have an impact on the mechanical damping. Additionally, it influences the ratio of the penetrated to the harvester volume. Since it is not clear how both effects can be mathematically considered, one has to take BP values of harvesters with different aspect ratio with care.
Thirdly, the “total” harvester volume reported for harvester prototypes does not always include the same components. In some cases an electronic circuit or a large, more robust housing is considered. An effective total volume that includes a minimal closed housing but excludes the electronic circuit could serve as common base for discussion.
Finally, additional aspects and limitations such as a required lifetime, temperature or shock resistance or a maximum flux density in the surroundings of a harvester can reduce the design freedom and consequently the harvester power.
Not least for the lack of data all these aspects can hardly be considered. If benchmarking results, however, are carefully interpreted, BP (c) and (d) can serve as an appropriate basis for the comparison of resonant electrodynamic harvesters.
The BP values in
As discussed in the previous sections, a low performance value must not necessarily be due to a disadvantageous harvester design but could also result from
measurements taken only at excitations where the BPs result in low values
an unfavorable aspect ratio, housing or included electronic circuit
restricting boundary conditions
a nonresonant harvester concept with different physics and, finally
the fact, that the harvester was not effectively optimized
Benchmark plots for electrodynamic harvesters. Plots show the maximum performance of each harvester according to the respective BP as well as available measurement data and with respect to the corresponding acceleration and frequency. Values are proportional to the area of circle and according to
In many publications the geometry is not completely given and the harvester is not fully characterized. The lack of data complicates the interpretation of the results, making it impossible to identify a single beneficial design approach.
In all benchmark plots one recognizes the trend that smaller harvesters have been designed for higher frequencies. The reason is probably the difficulty to design harvesters with fragile springs of low stiffness. The decreasing oscillator amplitude at higher frequencies as well as smaller volumes additionally helps to prevent fatigue and to reduce the mechanical damping. As the discussion of the other BP (a) and (b) refers to BP (c) and (d), the comparison of published harvesters starts with the latter parameters.
Harvester parameters corresponding to
#  Reference  

1  [ 
0.24  322  102  530  
2  [ 
0.3  6.2  51.6  0.83  45 
3  [ 
0.6  6  54  7.9  115 
4  [ 
1.2  3  143  1  12 
5  [ 
80  28  3000  
6  [ 
66  34.4  830  
7  [ 
10  11.8  545  
8  [ 
10.0  36.5  45.3  3  4200 
44.6  0.4  138  
9  [ 
55.9  2.0  2050  
2.5  3100  
10  [ 
27    57  0.98  
11  [ 
41.3    50  0.71  3000 
12  [ 
13.1  1.1  2000  
13  [ 
80.0    17  7  
3  
14  [ 
117  53  17.2  0.35  
15  [ 
130.7  55  50  13.9  
16  [ 
5.4  10  200, 000  
17  [ 
229  63  50  0.35  
18  [ 
500  100  6  0.71  15, 000 
Key parameters of selected harvesters in SI units
#  Design  Reference  





1  harvester (1)  [ 
2100  
2  harvester (1)  [ 
0.00075*  2000*  
3  harvester (27)  [ 
            
4  harvester (30)  [ 
7.7%  1.00  8.4×10^{‒3}  2900  
5  harvester (8)  [ 
            
6  harvester (14)  [ 
               
7  harvester (10)  [ 
3.7×10^{‒6}      
8  harvester (28)  [ 
0.38  7.7×10^{‒3}  0.18  0.96  3500  1.5%  
9  similar to (23)  [ 

10  [ 
                
11  [ 
            2050    
12  harvester (1)  [ 
        820    
13  [ 
                
14  [ 
                
15  harvester (26)  [ 
               
16  harvester (5)  [ 
            177   
17  [ 
                
18  NOT equal to (7)  [ 
               
^{1}
For BP (c) and Equation (
As expected for viscidly damped harvesters, no correlation can be noted in the plot of BP (c) versus volume and the frequency of maximum power (which, for all resonant harvesters, is the eigenfrequency). The diagram versus volume and acceleration indicates that a higher performance can be realized at smaller acceleration. In fact, harvesters of low performance often seem to be deliberately measured at high accelerations to reach a considerable absolute power level. Additionally, since which harvester was limited by which damping type is not reported, a dominating impact of material damping could be a second reason.
As already stated, the best performing harvester is number 2 [
However, to interpret the high performance, two aspects have to be considered. Firstly, the harvester volume does not include a housing. Secondly and more important, this harvester is nonlinear due to a stiffening effect of the spring. Due to the stiffening, the output power increases when the harvester is measured at a frequency upsweep until it suddenly drops. The frequency of power break down is random and if the harvester is measured at a downsweep this point cannot be reached [
The comparison of publications [
In fact, even if the stiffening effect is not considered, the performance of number 2 is high, for which three reasons can be identified. Firstly, a low mechanical damping was realized due to a long clampedfree beryllium copper beam spring. Secondly, despite a small oscillator volume, the oscillator mass was increased by attaching a tungsten mass with a mass density of
A second well performing harvester with a volume more than an order of magnitude above is the cylindrical prototype 8 with design (28). Here, the maximum
However, compared with harvester 2 and the assumption
Other well performing harvesters are the commercial Perpetuum devices 14 and 15 [
A very similar performance was also realized with the harvester 4 according to design (30). It features the largest
All these harvesters contain a magnetic circuit with flux conducting components. In fact, two harvesters based on a simple magnet without flux guidance provided a good output power as well. The harvester 9 in [
The harvester 3 [
Finally, it is worth mentioning the harvester in [
Since all these well performing harvesters have not been comprehensively optimized, the question whether a certain design is beneficial cannot be satisfiably answered. Nevertheless, beneficial design aspects can be identified and will be summarized in
Benchmarking with BP (d) is based on a reference conversion effectiveness of
Compared with BP (c), the plot of BP (d) suggests increasing performance values for lower accelerations and (less significant) higher frequencies. This observation corresponds to the assumption that stress and material damping are reduced at the lower oscillator amplitudes.
As for BP (c) also at BP (d) harvester 2 is the best performing prototype. The reasons are the same, low
A very similar performance was reached by harvesters 11, 15, 8 and 14. Whereas the design of 11, 15 and 14 is not exactly known, the good performance of harvester 8 was due to maximum value of
Similar to what was explained for BP (d), as the harvesters have not been comprehensively optimized, the results only allow to identify beneficial design aspects, which are summarized in
The power density BP (a) versus volume and acceleration does not show a clear correlation. However, a comparison with BP (c) indicates the tendency that, despite a comparably unbeneficial harvester design, simply a large harvester volume or strong acceleration can result in a high power density.
For example, prototype 1 [
The nonresonant pendulum harvester 5 [
For comparison, the energy of lithium ion or alkaline batteries with an energy density of approximately
BP (b) requires the assumption of a mass density value. Since BP (b) is supposed to be a physical limit, the highest mass density of an element of the periodic table, the density of Osmium with
The maximum value of
A performance of
The performance of
The fourth best result of
The performance values realized according to BP (b) are small, because neither was the mass density of Osmium used, nor the power loss due to the mechanical damping and the coil resistance (which account for more than
The question which harvester design should be preferred is discussed based on the high performing harvesters according to BP (c) and (d). The best performing harvesters at BP (d) are also performing well at BP (c), and vice versa. This aspect shows that these designs stand out from the other published harvesters and feature beneficial design aspects.
A high
High values of
As long as the oscillator amplitude is small compared with the dimensions of the harvester, a low
Low damping was provided by a single fixedfree beam [
Another opportunity is membrane springs, because the closed boundary can reduce the impact of the clamping. Spring designs with long meandered or spiral beams can effectively distribute the stress that appears at the oscillator deflection. However, as our own measurements with harvester 8 suggested, membrane springs need to be designed carefully to prevent high stress from nonlinear geometrical effects. Additionally, for the spring design one has to consider the tradeoff between long beams providing low stress and a high stiffness ratio that ensures a defined oscillation at perpendicular forces.
Lowest damping was possibly realized with a single crystalline silicon membrane spring of comparably long beams [
Finally, as the example of the beam harvester 3 [
Therefore, to maximize the performance of electrodynamic harvesters, the harvesters should systematically be optimized under consideration of the mechanical damping phenomena.
The benchmarking parameters can serve as starting point to evaluate harvesters and to identify a good harvester design. Indeed, a completely fair benchmarking is practically not possible. First, the complex impact of all ambient and boundary conditions cannot be comprehensively considered. Second, the various different requirements on harvesters cannot be all quantified. Third, important data of prototypes, which are needed for benchmarking, are often not published. Due to the complex impact of the boundary conditions and different practical requirements that possibly limit the design freedom, multiple benchmarking parameters are necessary. Consequently, there cannot be
The novel benchmarking parameters BP (c) and (d), which were introduced in this paper, can nevertheless serve as a starting point for benchmarking. They allow to compare resonanttype electrodynamic harvesters based on the practical impact of the most important parameters: the excitation and the harvester volume. For the first time they consider the important dependencies of the mechanical damping and the conversion effectiveness.
The dependency of the material damping on the dimensions and excitation should be explained with the help of a single membrane spring, which can be modeled as a setup of multiple fixedguided Bernoulli beams assuming pure bending stress. For the given oscillator mass and eigenfrequency, the total spring stiffness
For a linear bending moment, the bending stress at the root of the fixedguided beams depends on the deflection
Now, whereas
The authors would like to thank the DFG for funding the graduate school 1322, Micro Energy Harvesting. Additionally, the first author thanks the Foundation of German Business (sdw) for his Ph.D. scholarship.
The authors declare no conflict of interest.