ON THE COMPARISON, SCALING AND BENCHMARKING OF
ELECTROMAGNETIC VIBRATION ENERGY HARVESTERS
C. Cepnik* and U. Wallrabe
University of Freiburg – IMTEK, Department of Microsystems Engineering,
Laboratory for Microactuators, Freiburg, Germany
*Presenting Author: Clemens.Cepnik@imtek.de
Abstract: This paper introduces and compares different benchmark parameters for the performance of
electromagnetic energy harvesters, including the bottom-up (power according to the harvester model) as well as the
top-down view (maximum power that can be generated within a volume) and considering the dependency and
scaling of geometrical and physical parameters, such as the mechanical damping. The comparison is not only to
give assistance for a fair benchmarking of prototypes but moreover to better understand how high performance
harvesters should be designed. We summarize, which important data should be provided with a published harvester
to allow benchmarking, and conclude to a set of benchmark parameters that are of main practical importance.
Keywords: electromagnetic vibration energy harvesting, performance, benchmarking, scaling, damping
INTRODUCTION
The electromagnetic energy harvesters of different
shape and dimension that have been published for
more than a decade raise the question about what a
good design is like. The difficulty to answer the
question is, firstly, the complex impact of boundary
conditions, such as the volume and the excitation, on
the output power, secondly, the different boundary
conditions and fields of application harvesters have
been designed for, and, finally, the few data published
for most prototypes.
Different authors compared electromagnetic
harvesters and introduced benchmark parameters
(BPs). While Mitcheson et al. [1] applied the top-down
approach, i.e., referred to the maximum power that can
be generated within a certain volume, Arnold et al. [2]
came bottom-up and investigated a simplified linear
model as well as published harvesters to assess basic
scaling of the output power. Based on the same model
O’Donnel et al. [3] discussed further fundamentals of
harvester scaling.
We add a broader overview by a) accounting for
single physical parameters and b) investigating scaling
of physical as well as geometrical parameters and their
impact on the output power. In addition to discussions
of the electrical damping in [4] we will review the
impact of the mechanical damping on the output
power, i.e., damping by drag, Stokes and Couette
damping, squeeze film damping, anchor loss and
thermoelastic damping. We finally derive advanced
BPs, provide experimental examples and discuss the
parameters with respect to practical importance.
APPROACHES FOR NORMALIZATION
The maximum average power that can be
generated within a certain volume and was first
introduced by Mitcheson et al. [1]. If one assumes that:
The harvester volume is completely used for the linear
oscillator and the oscillator displacement, the
mechanical damping as well as the volume for coils,
casing, airgaps etc. are neglected and the oscillator is
only damped electrically, for the optimum oscillator
vs. harvester height of 0.5 the average power becomes
Pavg =
1
ωρm V ha
8π
(1)
with V the harvester volume, a the acceleration, ω the
angular frequency and ρm the mass density.
On the other hand, to describe the behavior of a
resonant harvester including the mechanical damping
dm the output power often is derived from the
normalized model [3] to
Pavg =
m2 a 2
m2 a 2
=
4 (dm + de )
8 dm
(2)
with m the oscillator mass and de the electrical
damping. Here, due to the normalization, the optimum
de equals dm, because the efficiency of the electrical
circuit is not considered. Nevertheless, equation (2)
mostly is a good approximation.
In contrast, we derived in [5] the output power
without normalization to better account for the
physical and geometrical parameters and found
Pavg =
V c kc B 2
m2 a 2
8dm (Vc kc B 2 + ρc dm )
(3)
where Vc is the coil volume, kc the copper filling
factor, ρc the resistivity and B the flux density
perpendicular to the wire and the direction of motion.
The difference compared to equation (2) becomes
obvious with the relation of de and dm [5].
Rl + Rc
dm
=
de
Rl − Rc
(4)
At high flux gradients and low dm the load resistance Rl
potentially is a magnitude larger than the coil resistance Rc, by which Rl and Rc become nearly equal [5].
SCALING
Based on eqs. (1)-(4) the scaling of the output
power with respect to the length scaling factor s can be
estimated. One needs to consider a) the environmental
boundary conditions: excitation frequency and
acceleration, b) the harvester parameters: aspect ratio
and total volume and c) the physical limitations:
mechanical and electrical damping. In contrast to a)
and b), the impact of the damping requires a more
detailed investigation.
Electrical damping
For the electrical damping
de =
l2 B 2
Rl + Rc
(6)
we find with the optimum load according to [5]
de = dm
V c kc B 2
Vc kc B 2 + 2ρc dm
(7)
B does physically not scale but can decrease at
miniaturization due to fabrication limitations and large
relative tolerances [4]. At high VckcB2 and low dm
(8)
de ∝ dm
while for low VckcB2 and large dm
(9)
de ∝ V c ∝ s 3
In case Rl = Rc, de in eq. (6) scales equally to (9).
However, the assumption Rl = Rc is only meaningful
when either the vibration amplitude needs to be
reduced (by increasing Lorentz force) or a power
management circuit with a similar input resistance is
used. However, in these cases the design should be
changed to allow a larger amplitude and respectively to
realiue a lower Rc. Thus, eqs. (8) and (9) apply.
Mechanical damping
Mechanical damping has been broadly investigated
for sensors and equally appears for energy harvesters.
Its scaling is more complex compared to the electrical
damping, as different models apply depending on the
geometry. A detailed comparison of different designs
with different aspect ratios requires extensive
measurements or simulations and is beyond the scope
of this paper. Our discussion is based on a review, how
dm scales according to the most relevant models.
a) Viscid drag damping. Drag damps a moving
body surrounded by a fluid in the laminar flow region.
The damping coefficient can be calculated to
dm = 6πµr ∝ µs
(10)
r is the representative dimension and µ is the viscosity.
b) Viscid lateral motion damping. Assume a small
gap g between a plate and a surface. The plate with the
surface A (dimensions much larger than g) moves
parallel to the surface. At a fully developed linear
time-independent flow one can model the damping of
the plate as Couette damping [6] and gets
dm = µA/g ∝ µs
(11)
In case the time dependency of the flow is considered
the damping is called Stokes damping [6]. The viscid
damping by the
fluid between plate and surface then is
!
1
√
(12)
ωρµ ∝ β ωµ s2
2
-1
with β = f(ωg), β ∝ 1 for large ωg and β ∝ s for small
dm = βA
ωg [6]. Thus, at very thin gaps and/or low frequencies
Stokes equals Couette damping. While the boundary
conditions in the Couette model neglect the force onto
the plate top surface, in the Stokes model this damping
component is considered by an infinite gap [6].
!
dm = A
1
√
ωρµ ∝ ωµ s2
2
(13)
For both, Couette and Stokes damping, edge and size
effects that appear at finite lateral dimensions only add
a factor but do not change scaling [7].
c) Viscid perpendicular motion damping. Squeeze
film damping occurs when a plate oscillates perpendicularly to a surface. For a small vibration amplitude
and a small characteristic squeeze film number, which
means that gas can flow out of the gap and is not
trapped within [8], dm is given by [9].
dm = µLw3 /g 3 ∝ µs
(14)
In case of large amplitudes Starr [10] found an
additional factor of (1-ε)-3/2 with ε the vibration
amplitude in relation to g. Again, edge and size effects
do not change scaling [11].
Viscid damping is more complex at high velocities
as higher order damping terms have to be considered.
According to [14] for turbulent drag follows
dm ∝ ρs2 ẋ
(15)
With the Reynolds number proportional to s, at large
volumes and excitations dm approaches s2 x˙ .
All viscid damping models depend on the ambient
pressure and can be reduced at rarefaction. According
to [11,12] different pressure regions
are distinguished
!
depending on the Knudsen number
Kn =
λ 0 P0
gP
(16)
with λ0 the mean free path at a known pressure P0 and
P the operating pressure. The first region, the slip flow
regime is defined at 0.01 < Kn < 0.1. Regions of lower
pressure are the transition (0.1 < Kn < 10) and the free
molecular regime (Kn > 10). Since a Kn of 0.1 in air
does already correspond to a representative physical
length (or airgap) of only 0.68 µm  P0/P, which by
now covers all harvester prototypes, we limit the
review to the slip flow regime. Here, the impact of P is
considered by the effective viscosity µeff, which was
found empirically [11,13] and slightly differs
depending on the publication. In each case dm is
reduced at lower pressures.
d) Other types of damping. Damping through
sound waves generated by the oscillation is called
acoustic damping [11] and is usually neglected.
Anchor loss or loss through imbalances is due to
tensions coupled into the clamping and can have a
significant impact at low pressures
[11,15,16].
However, it is neglected for pressures down to the slip
flow regime. Thermoelastic damping [15,17] is caused
by a temperature difference that appears when tensions
change in a material, e.g., in the harvester springs.
Also, similarly to anchor loss damping, it can be
neglected. Imperfections in the material of the springs
can additionally cause heat dissipation when chemical
or physical bonds are made or broken. This internal
friction is neglected at even lower pressures [15].
Table 1. Possible benchmark parameters with possible field of application and practical limitations.
(a)
(b)
(c)
(d)
(e)
(f)
Benchmark
P/V
P/(1/4 fρVha)
P/(V1/3a)
P/(V5/3 a2)
P/(V7/3a2)
P/(V2a2)
Possible field of application
volume comparison to batteries / other mechanisms
max. performance of different harvesting principles
el.-mag. harvesters in turbulent flow
el.-mag. harvesters in laminar flow, high ηmag
el.-mag. (micro) harvesters in laminar flow, low ηmag
el.-mag. harvesters in laminar flow (simplified)
10000
Practical limitations
no information about excitation
max. el. damping not considered
drag must be turbulent
drag must be laminar
drag must be laminar
drag must be laminar, simplified
[20] [19] [22] [21] [5]
[20] [19] [22] [21] [5]
[20] [19] [22] [21] [5]
Resoance frequency 1000
(*)
100
(*)
(*) commercial harvesters, no data
about design
published
10
[18]
0.001
[18]
0.1
10
[20] [19] [22]
0.1
1000
[5]
1000
[18]
10
1000
[20] [19] [22] [21] [5]
0.1
10
[20] [19] [22]
1000
Volume [5]
(*)
100
[21]
10
[21]
[18]
1
[18]
[18]
Figure 2. Benchmark plots for published electromagnetic energy harvesters. Value proportional to area of circle.
Thus, for benchmarking energy harvesters it is
appropriate to only consider viscid and electrical
damping as shown in figure 1. Due to the high
nonlinearity viscid damping can thereby only be
modeled for singular regions of volume, excitation and
pressure. The complexity and the lack of a
comprehensive theoretical model require to define a
set of BPs with respect to different boundary
conditions and matters of interest. For a lack of space
and harvester prototypes at reduced pressure we limit
the discussion to standard pressure conditions.
Damping [a.u.]
1E+04
1E+02
turbulent drag Stokes damping 1E+00
el.-mag. damping laminar drag, Couette &
squeeze film damping 1E-02
1E-04
0.01
0.1
1
10
100
Scaling factor s [a.u.]
Figure 1. Scaling of damping with respect to volume.
BENCHMARKING
Different benchmark parameters (BPs) as shown in
table 1 can be derived from practical aspects and
theory. An electronic circuit could require a certain
maximum voltage or power and certain average values.
However, we do not consider these values, because
they can easily be changed by volume scaling or
changing the boundary conditions. The power density
(BP (a)) is useful for a comparison with batteries and
other harvesting principles but does not consider the
practical dependency on the influencing variables.
According to equation (1), with ρm equal for all harvesters, the maximum power (b) that can be generated
in a certain volume on the one hand enables to
compare different harvesting principles. On the other
hand it does not consider mechanical damping and the
maximum possible electrical damping, which both are
the main reasons why the output power in practice is
much lower than one would expect from equation (1).
From eq. (2), which accounts for the mechanical
damping, we find the laminar scaling (d) in eq. (17)
assuming dm ∝ s (compare [2]).
Pavg ∝ s5 a2 ∝ V 5/3 a2
(17)
2
∝
In contrast, for dm s x˙ the turbulent scaling (c)
results when a nearly sinusoidal motion is assumed.
Pavg ∝ sa ∝ V 1/3 a
(18)
The difference!between eqs. (2) and (3) is the factor
ηmag =
V c kc B 2
Vc kc B 2 + ρc dm
(19)
which we call effectiveness of the magnetic circuit. For
benchmarking two scenarios are of general interest:
1.) For microharvesters low ηmag and laminar damping,
i.e., dm ∝ s, are realistic. 2.) For larger harvesters the
design freedom allows ηmag à 1, laminar as well as
turbulent damping should be considered. For the first
case BP (e) follows (compare [2]).
Pavg ∝ s7 a2 ∝ V 7/3 a2
(20)
Scaling in the second case equals eq. (17) and (18).
Finally, the difference between BP (d) and (e) is small.
In case it is difficult to decide which of both is more
appropriate, we suggest to use the average BP (f)
Pavg ∝ s6 a2 ∝ V 2 a2
(21)
which interestingly results from eq.(2) with dm = const.
Figure 2 contains the BPs for published el.-mag.
harvesters with respect to their volume and resonant
frequency. If measurements at different excitations are
available for one prototype we use the best result in
each diagram. The plots show that only a few
harvesters are high performing according to all BPs.
Available and estimated key parameters of best
performing prototypes are collected in table 2.
Considering the design of these harvesters shows that
different approaches are target-aiming. A low
performance mostly is due to a small effectiveness of
the magnetic circuit per total volume. In case a
prototype is only high performing according to one BP
reasons could be:
- the harvester has a small effectiveness of the
magnetic circuit but profits from a certain BP
- measurements are only available for a certain
excitation, at lower / larger excitations the
performance can increase according to (d) / (c)
- a low aspect ratio increases (a) but decreases the
other parameters and vice versa
- the concept is different to the standard oscillator
(e.g., a pendulum or frequency up-conversion)
For our cylindrical prototype figure 3 shows the BPs
and mechanical damping vs. the acceleration. This plot
indicates that BP (d)-(f) are appropriate for laminar
drag and (c) for turbulent drag.
Table 2. Key parameters of high performing harvesters
in SI units computed or cited / estimated / simulated.
Vc/V
[18]
[19]
[20]
[21]
[22]
[5]
0.013
0.020
0.020
0.050
0.077
0.053
kc
Bavg
dm
ρm
xmax/h
m/V
0.15
0.50
1.00
0.31
0.02
0.29
0.36
0.41
0.33
0.64
0.0130
0.0046
0.2-0.4
~7600
~7800
~16000
~7000
~7800
~7800
0.13
< 0.04
0.40
< 0.02
1100
2100
3500
1000
2900
3650
1E05
1.0
1E04
[a.u.]
Benchmarks [a.u.]
*[18]: frequency up conversion, [21]: circulating pendulum
1E03
(d), (e), (f)
1E02
1E01
(a)
0.2
(b), (c)
4
2
1
Acceleration 0.2
laminar
damping
0.1
0.2
2
1
4
Acceleration Figure 3. BPs for the measured power and damping
(open circuit) of our harvester [5] vs. acceleration.
For small accelerations laminar damping dominates,
for accelerations > 1ms-2 damping increases.
CONCLUSION
Since it is impossible to find a benchmark parameter
(BP) that accounts for all boundary conditions at
different magnitudes and since different energy
sources scale differently, we can conclude that
depending on the focus of research or the potential
application different BPs need to be used:
1. (a), (b) to compare different harvesting principles
2. (c) to benchmark el.-mag. harvesters with turbulent flow, i.e., large volume or strong excitation
3. (d) to benchmark el.-mag. harvesters with laminar
flow, i.e., small excitation and very small volume
4. (e) to benchmark el.-mag. harvesters with laminar
flow, i.e., small excitation but larger volumes
5. (f) to benchmark both, (d) and (e).
To enable a better harvester characterization and
design evaluation measurements with different
boundary conditions should be provided. This
especially includes measurements of the
1. output power,
2. output voltage and
3. vibration amplitudes (open circuit and with Rl)
at different acceleration amplitudes. To compare the
effectiveness of the magnetic circuit geometrical and
material parameters of the design and the prototype
should be provided. We encourage the reader to
provide as many data as possible to contribute to the
challenge of finding most beneficial harvester designs.
Finally, from the application point of view, additional
parameters are of broader interest: robustness against
temperature, tolerated peak acceleration and lifetime.
ACKKNOWLEDGEMENTS
The first author thanks P. Mitcheson and E.M.
Yeatman for the fruitful discussions about [1].
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