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1

M. Hendijanizadeh1, S. M Sharkh1, S. J. Elliott2, M. Moshrefi-Torbati1

1Electro-Mechanical Engineering Research Group, Engineering Science,

Southampton, SO17 1BJ, UK

2ISVR, University of Southampton, Southampton, SO17 1BJ, UK

E-mail: m.hendijanizadeh@soton.ac.uk

Abstract. In some energy harvesting systems, the maximum displacement of the seismic mass

is limited due to the physical constraints of the device. This is especially the case where energy

is harvested from a vibration source with large oscillation amplitude (e.g., marine environment).

For the design of inertial systems, the maximum permissible displacement of the mass is a

limiting condition. In this paper the maximum output power and the corresponding efficiency of

linear and rotational electromagnetic energy harvesting systems with a constrained range of

motion are investigated. A unified form of output power and efficiency is presented to compare

the performance of constrained linear and rotational systems. It is found that rotational energy

harvesting systems have a greater capability in transferring energy to the load resistance than

linear directly coupled systems, due to the presence of an extra design variable viz. the ball

screw lead. Also, in this paper it is shown that for a defined environmental condition and a

given proof mass with constrained throw, the amount of power delivered to the electrical load

by a rotational system can be higher than the amount delivered by a linear system. The criterion

that guarantees this favorable design has been obtained.

Keywords- Constrained systems; Efficiency; Energy harvesting; Scaling effect; Vibration,

1. Introduction

Generating electricity from ambient vibration has been the subject of significant research and

development in the last decade. Piezoelectric [1, 2] ,electrostatic [3] and electromagnetic [2, 4-8]

transducers are the most commonly used methods that are utilized to convert ambient vibrations to the

electrical energy. However, due to the limitation of geometry and limited permissible deformation of

electrostatic and piezoelectric transducers, the electromagnetic induction method is the more appropriate

choice for large scale applications [9]. Movement of a backpack carried by a human during walking [10],

all-terrain-vehicle vibration [11], vertical movement of a sailing boat [12] are some examples of

relatively large scale vibration resource.

However, in addition to the configuration of energy conversion device, maximizing the output power and

efficiency are the main concerns in the process of design and optimization of vibration energy harvesters.

Efficiency is a fundamental parameter used to compare all kinds of energy harvesters with various sizes

and designs [13-15]. Usually the main goal of an energy harvesting system is to extract the maximum

power from the environment. In this paper, the efficiency of such systems when achieving maximum

power is studied. To achieve the maximum power condition, the parameters of the system need to be

selected and tuned carefully. Tuning the load resistance to its optimum value to ascertain impedance

matching in electromagnetic energy harvesters is reported in many research works to have improved

energy capture [16-18]. However, none of these works considered the maximum allowable displacement

of the oscillating mass as a constraint in the process of calculating the optimum load resistance. More

specifically, the optimum load resistance for harvesting maximum amount of energy is generally

calculated regardless of its effect on the relative displacement of the oscillating mass. However, it is

known that the load resistance can influence the overall system damping and hence the relative

displacement of the mass. In many transducers that are used in large size applications, due to size

limitations, the oscillating mass only moves within a specified range. Now, if the load resistance of the

transducer is selected without considering the maximum permissible through of the seismic mass, there is

a risk that the amplitude of the oscillating mass will exceed the physical dimensions of the transducer

Output Power and Efficiency of Electromagnetic Energy

Harvesting Systems with Constrained Range of Motion

2

thus affecting the performance of the device. Therefore, for these cases, at the design stage, the physical

parameters such as load resistance should be selected with regard to the constraints on the oscillating

mass.

In this paper, the maximum output power and the corresponding efficiency of two types of

electromagnetic energy harvesting systems (i.e. linear and rotational) for those with constraints on their

range of motion are studied. In linear electromagnetic energy harvesting systems (henceforth referred to

as linear system) such as those studied in [11, 19-21] a linear generator is employed. However, in a

rotational energy harvesting system (henceforth referred to as rotational system), an intermediate

mechanism, such as rack and pinion[7, 10, 22] or a ball screw[23-26], is utilized to convert linear motion

of the mass to rotational one to drive a rotary generator.

The paper is distinguished by three main contributions. First it investigates the optimum load resistance

for both constrained linear and rotational systems to address the maximum output power condition. It is

shown that for constrained systems the optimal load resistance is different from that of unconstrained

energy harvesting systems that is reported in literature [16]. Then the efficiency of both systems

corresponding to their maximum output power is obtained. For each system, an expression for the load

resistance corresponding to maximum efficiency is also derived. It is shown that for linear systems it is

not possible to achieve maximum efficiency when the maximum power is extracted from the transducer.

However, for rotational systems maximum efficiency occurs at maximum output power point.

The second contribution is the derivation of equations for power and corresponding efficiency of both

systems in unified forms so that proper comparison between them can be made. These unified forms are

developed based on the non-dimensional electromechanical coupling coefficient of systems introduced

by Elliott and Zilletti [27]. The comparison reveals that in the case of a linear system, the maximum

amount of power that can be transferred to the load is half the mechanical power transferred by the

harvester and the efficiency of system is always less than 50%. However, a rotational system can be

designed so as to have an efficiency greater than 50%. The criterion that guarantees the efficiency of a

rotational system is more than 50% is derived.

The third contribution is studying the effect of scaling the size of electromagnetic generator component

of the energy harvesting system on the output power and efficiency. It is shown that by increasing the

size of energy harvesting system the efficiency is increased for both constrained linear and rotational

systems.

2. Power generation from a constrained inertia system

2.1 Power Generation from a constrained linear energy harvesting transducer

A schematic diagram of a linear energy harvesting system using an electromagnetic generator is shown in

figure 1. In this diagram, m is the seismic mass, k is the spring stiffness,

m

c

represents the mechanical

viscous damping coefficient, and

e

c

is the electrical damping coefficient corresponding to the combined

power dissipated in the generator’s internal resistance and the power delivered to the load.

Figure 1. Schematic diagram of a linear energy harvesting system

3

The governing differential equation of motion for the system shown in figure 1, with respect to the

relative displacement of the seismic mass

z x y

, is

.

em

m z c c z k z m y

(1)

For a harmonic base excitation

sin y

y Y t

, when the driving motion is assumed to be independent

of the mechanical loading due to the harvester, the amplitude of the relative displacement Z, can be

shown to be

2

1/2

22

2.

em

Zm

Yk m c c

(2)

Figure 2. Equivalent circuit of an electromagnetic generator connected to a resistive load.

In many papers on generating energy from vibrations, the effect of the generator’s internal inductance is

ignored. Cammarano et al. [17] show that even in cases where the effect of internal inductance cannot be

ignored, due to a high oscillation frequency, the undesirable effect of the internal resistance can be

compensated by adding a capacitor in series with the circuit. The equivalent electrical circuit of the

energy harvesting device is shown in figure 2, in which a capacitor is added in series with the load

reactance to cancel the effect of the generator's inductance. Assuming that an electrical generator with an

emf constant

,

t

K

is directly coupled to the seismic mass, then the generated emf voltage is given by

.

emf t

V K z

(3)

Also, the electrical damping coefficient (

e

c

), corresponding to the power dissipated in the generator’s

internal resistance and transferred to the electrical load, is

where

and

where

ii

jL

and

1/

ll

jC

. In [17] it is shown that to deliver the maximum power to the load

l

R

, the effect of internal inductance should be compensated by tuning the capacitor such that

li

.

For a spring stiffness of k, the natural frequency of system is equal to the base excitation frequency

when

2

km

at which the corresponding relative displacement

r

Z

can be derived from (2) (for

n

).

2,

t

eli

K

cZZ

(4)

,

i i i

ZR

(5)

,

l l l

ZR

(6)

4

Then

sin( )

r n z

z Z t

and

cos( )

r n n z

z Z t

, by substituting the electrical damping coefficients

from (4) and considering

2

1/ n

li

CL

, the amplitude of the relative displacement is

2.

n

r

t

mil

m

Z

YK

cRR

(7)

The power delivered to the load resistance is

22

2

1 1 1 .

2 2 2

emf t

l out l l l

l i l i

VKz

P R i R R

R R R R

(8)

Substituting the maximum value for

z

, which is

rn

Z

in (8), the power supplied to the load is given by

2 2 2

2

1.

2l

l out n t r

li

R

P K Z

RR

(9)

Equation (9) shows the relationship between the relative displacement, excitation frequency, load

resistance and the harvested power from a given generator.

Now, we design an optimum energy harvesting system to extract maximum energy from a given

vibration source with known amplitude and frequency of oscillation. This design will be accomplished

based on the parameters of a given generator that has given

t

K

and

i

R

values. It is also assumed that, due

to the transducer size limits, the maximum displacement of the oscillating mass is specified. Therefore,

the aim of design is the optimal selection of system parameters including k,

l

R

and m to harvest the

maximum power from the given generator within the specified range of motion. To this end, considering

0

r

Z

as the maximum allowable relative displacement of mass (i.e.,

0

rr

ZZ

is constant), the maximum

value of (9) is obtained when the load resistance is equal to the internal resistance of the generator in

which the output power is

0

2 2 2 /8

n t r i

k Z R

. The mass can then be selected from (7) to limit its maximum

displacement to

0

rr

ZZ

,

The natural frequency of system is equal to the excitation frequency when

2

km

, considering

n

in this condition the spring stiffness is given by

2.2 Power Generation from a constrained rotational energy harvesting transducer

A rotational energy harvesting system comprising a sprung mass coupled to an electrical generator

through a motion transmission system. The Ball screw is a conventional mechanism that converts linear

motion to the rotational one. A schematic diagram of this type of system is shown in figure 3. In this

device the base movement causes the mass to vibrate. The ball screw then converts the low frequency

linear motion of the seismic mass to high speed rotation. The governing differential equation of motion,

having an ideal ball screw, igure 3 is written as

22

22 .

e bg

m J z c c z k z my

ll

(12)

02.

2

rt

m

ni

ZK

mc

YR

(10)

02.

2

rt

nm

i

ZK

kc

YR

(11)

5

Figure 3. Schematic diagram of an energy harvesting system consisting of a sprung mass coupled to a

generator through a ball screw

Considering

l

as the ball screw lead, the equivalent reflected moment of inertia of the ball screw and the

generator is given by

2

2/Jl

, where J refers to the total moment of inertia of the system including the

moment inertia of the ball screw

b

J

and generator

g

J

and is defined as

Also

bg

c

includes the mechanical viscous damping of the combined ball screw connections

mb

c

and

generator

mg

c

, i.e.,

For a harmonic base excitation

sin y

y Y t

, the amplitude of the relative displacement is

The systems operates at its natural frequency when

n

where

n

is given by

In this condition, the relative displacement of the mass is given by

where

i

T

is the rotary generator emf-constant which is equivalent to

t

K

in linear systems. Then the

output power is therefore expressed as

.

gb

J J J

(13)

.

bg mb mg

c c c

(14)

2

1/2

22

22

2

.

22

e bg

Zm

Y

k m J c c

ll

(15)

1/2

2.

2

nk

mJ l

(16)

22,

2

n

ri

bg il

mY

ZT

c

l R R

(17)

6

Now, we assume that we wish to determine the parameters of a rotational system (i.e., l, m and Rl), based

on a specific generator and given mass (i.e., known

i

T

,

i

R

and m), such that the maximum energy can be

extracted from a specified vibration source (i.e., known Y and

).

Considering the maximum allowable displacement of mass

0

rr

ZZ

,

2

2/l

is then obtained from (17)

as

By replacing (19) in (18), the output power for the constrained system is obtained from the following

equation

From (20), the optimum load resistance to harvest the maximum power is obtained from

/ 0,

b out l

PR

which results in

the optimum ball screw lead can be derived as

and the spring stiffness can be calculated from (16).

3. Power and efficiency comparison between linear and rotational systems

3.1 Power and efficiency of an electromagnetic constrained transducer

Efficiency is a fundamental term that has been studied for different energy harvesting systems. Relying

solely on the assessment of the output power of energy harvesters does not reflect their quality of

performance and their capability to harvest the maximum amount of power. However, in the context of

vibration harvesting systems, the concept of efficiency has received less attention in the literature than

that of maximizing output power. Traditionally, efficiency is defined as the ratio of the electrical power

output to the mechanical input power; whilst, in a vibration-based energy harvester, the input

mechanical power itself is related to the device characteristics. Also, the efficiency cannot be defined in

terms of the potential mechanical power available from the source as, in some applications, the loading

by the harvester does not influence the dynamics of the source of vibration. Hence, the potential

mechanical power available from the source is effectively limitless [14]. To compare the power output

of various transducers, a dimensionless figure of merit, called effectiveness e, is introduced by Roundy

[28] which is defined as

2

2 2 2

2

12

.

2l

b out i r

li

R

P T Z

l

RR

(18)

0

2

2

2.

n

i

r bg il

mY

lT

Zc RR

(19)

0

0

2 2 2

22

1.

2ln

b out i n r

i

li r bg il

R mY

P T Z

T

RR Zc RR

(20)

max

1/2

2

2

,, ,

i

l rotational P i i bg

T

R R R c

(21)

0

1/2

2

1/2

2

2

2,

ri

bg

ni

i i i bg

ZT

lc

mY T

R R R c

(22)

22

0 max ,eQ

(23)

7

where, Q is the quality factor and is related to the damping ratio of the system,

is the coupling

coefficient of the transduction mechanism,

is the actual density of the device,

0

is a baseline density,

is the actual transmission coefficient and

max

is the maximum transmission coefficient. However, in

the “effectiveness” index,

Q

is related to the damping ratio of the system and does not have a

fundamental limit. Hence, this metric comparison does not reveal how well the device is optimized [8].

To investigate how close a device is to its optimum performance and distinguish between different

proof mass densities and geometries, Mitcheson et al. [14] introduce a “volume figure of merit”,

defined as

43

3

00

.

1

16

out

V

Au

P

FoM

YV

(24)

This dimensionless ratio compares the performance of the device with that of an ideal device. The device

has the same total package volume but with a proof mass equal to the density of gold (

Au

), occupying

half of this volume (

0

V

). The proof mass oscillates in the other half of this package. The power output

harvested by this hypothetical device is considered as the maximum possible output for the based

vibration with amplitude of

0

Y

at frequency of

. The power output of the transducer is compared with

the maximum possible output to evaluate the performance of a device as a function of its overall size.

Although the “volume figure of merit” facilitates the comparison of a harvesting device performance with

a reference ideal energy harvesting system, it does not enable the calculation of input power absorbed by

the system to produce a certain amount of output power.

Elliott and Zilletti [27] conducted research into scaling of linear electromagnetic transducers for power

harvesting and shunt damping. In this study the efficiency is defined as the ratio of output power to the

sum of the mechanical dissipated power, electrical power loss and electrical output power. This definition

is closer to the original definition of efficiency. This study shows that the efficiency of a linear

electromagnetic transducer depends on a non-dimensional electromechanical coupling coefficient which

will be discussed later in this paper. The coupling coefficient scales with the transducer’s size. However,

this research does not consider the constraint on the displacement of the proof mass. The mechanical

input power absorbed by the energy harvesting structure is given by

Here, we define the efficiency of a linear system,

l

E

, as the ratio of the electrical power harvested from

(9) to the supplied mechanical power from (25), which is

The load resistance corresponding to the maximum efficiency of the system, as opposed to the maximum

power output, can be obtained from

/0

ll

ER

, i.e,. differentiation of (26), which results in

By comparing the optimum load resistance for maximum output power (

max

,,l linear P i

RR

), and the load

resistance corresponding to the maximum achievable system efficiency derived in (27), it is realized that

the latter is always greater than the former. Therefore, in a practical linear system it is not possible to

achieve maximum efficiency at the maximum output power point. The mechanical input power absorbed

by the rotational system can be calculated as

222

1.

2t

l in m n r

li

K

P c Z

RR

(25)

2

22.

l out l t

ll in m l i t l i

P R K

EPc R R K R R

(26)

max

1/2

2

2

,, .

t

l linear E i i m

K

R R R c

(27)

2

222

12

.

2i

b in bg n r

il

T

P c Z

R R l

(28)

8

The harvesting efficiency,

b

E

, is defined as

Also, from (29), the load resistance corresponding to the maximum efficiency of the system can be

obtained from

/0

bl

ER

, which is

Comparison of (21) and (30) reveals that in the rotational system, the optimum load resistance to obtain

the maximum efficiency is the same as the load resistance corresponding to the maximum power. In the

other words, for a constrained rotational system the maximum efficiency occurs at the maximum output

power.

3.2 Comparison of output power and efficiency of systems

By replacing (7) in (9) for

0

rr

ZZ

the load power of a constrained linear energy harvesting system for

the load resistance corresponding to the maximum output power (

max

,,l linear P i

RR

), is

where

em

is a non-dimensional electromechanical coupling coefficient of an energy harvesting system

and is defined as [27]

for linear systems and

for rotational systems. By increasing this coefficient (i.e.,

em

) the maximum output power, given

by (31), approaches the following expression

This shows that the maximum theoretical power is determined by the environmental vibration

characteristics (

n

,

Y

) and also the system mass and the maximum allowable displacement. Note that

n

is a characteristic of the transducer, but here the system is designed such that the undamped natural

frequency of the device is equal to the frequency of excitation.

Considering (26), the efficiency of a constrained linear system for the load resistance corresponding to

the maximum output power (

max

,,l linear P i

RR

), can readily be shown to be [27]

For weak linear coupled systems, the efficiency is low. By increasing

em

the efficiency increases until

it reaches a maximum value of 50%, i.e.

2

22.

b out l i

bb in i l bg i i l

P RT

EPR R c T R R

(29)

max

1/2

2

2

,, .

ii

l rotational E i bg

RT

RR

c

(30)

max 0

3

,,

81 2

em

l out P n r

em

P m YZ

(31)

2,

t

em mi

K

cR

(32-a)

2,

i

em bg i

T

cR

(32-b)

max 0

3

,1

lim .

4

em l out P n r

P m YZ

(33)

max

,.

42

em

lP em

E

(34)

max

,1

lim .

2

em lP

E

(35)

9

However, considering the optimum load resistance for rotational systems from (21), the output power of

such systems from (20) can be written as

and for the case when

em

, the power is

Also, the efficiency of rotational systems corresponding to the maximum output power can be obtained

by replacing (30) in (28) and using (22), (32-a) and (32-b), which results in

Equation (38) indicates that in the case of a rotational system, it is possible to achieve an efficiency of

more than 50%. To achieve such favourable design, the condition below must be met

This condition is satisfied if

8

em

. Selecting the parameters according to this condition can lead to a

system with an efficiency above 50%. For the case when

em

, the efficiency of the rotational

system is

In the case that the a linear and a rotational system have same seismic mass, by replacing (7) in (25) and

(17) in (28), for

0

rr

ZZ

, it an be shown that the mechanical input power for both systems

is

0

3

1/ 2 nr

m YZ

, however, the linear system in the optimum condition can only transfer less than half of

this power to the load, while, the rotational system under certain condition, i.e.

8

em

, can harvest

more power.

3.3. Effect of the Scaling of constrained electromagnetic harvesters on the output power and

efficiency

It was shown earlier that by increasing

em

, the efficiency of a typical energy harvesting transducer is

improved. A question that arises here is “how do the output power and efficiency of a system change by

increasing the size of the generator?”.

Elliott and Zilletti [27] studied the relation between

em

and the characteristic length of a transducer [L].

In this study, assuming that

w

A

is the cross-sectional area of the wire used for the coil of the

electromagnetic transducer and

w

is its resistivity, the resistance of the coil is given by

Here h is the coil’s wire length, which is approximately given by

max 0

3

,2

1

1,

21 1 1 1

em em

b out P n r

em em em

P m YZ

(36)

max 0

3

,1

lim .

2

em b out P n r

P m Y Z

(37)

max

,2

1,

1 1 1 1

em em

bP

em em em

E

(38)

2 1 1 .

em em

(39)

max

,

lim 1.

em bP

E

(40)

.

iw

w

h

RA

(41)

,

C

w

V

hA

(42)

10

where

C

V

is the volume of the coil. For a well-designed transducer with saturated magnetic flux density

B, the emf-constant (

t

K

for linear systems and

i

T

for rotational systems) is proportional to the magnetic

flux density times the length of the wire in the coil (i.e.,

t

K

or

i

T

=Bh). Therefore, the electromechanical

coefficient of the transducer can be re-written as

The magnetic flux density (B) and wire resistivity (

w

) of the transducer depend on their material

properties, but not on the transducer dimensions. In general, the scale of the volume of the coil (

C

V

) is

[

3

L

], whereas the mechanical damping coefficient (

m

c

for linear systems and

bg

c

for rotational systems)

is related to the structure and the detailed mechanism of the transducer, but generally scales as [

L

][29].

Therefore, the electromechanical coefficient shown in (32-a) and (32-b) is proportional to the square of

the characteristic length of the transducer [

2

L

]. Hence, an option in increasing the coupling coefficient of

a transducer is to increase its overall size. From (33) and (37) it is evident that, for both systems, by

increasing the size of device the electromagnetic coefficient and consequently the output power of the

system is increased.

In the case of a rotational system, considering the combined ball screw, mass, spring and the rotary

generator as the transducer assembly, the coupling coefficient related to the generator part of the

transducer can be defined as

where

mg

c

is the mechanical damping associated with the rotary generator. According to the discussion

presented above, it is expected that

emg

will scale with the square of the characteristic length of the

generator [L2]. This assumption will be examined in the next section by studying the specifications of a

set of commercial generators. For the rotational transducer assembly, the coupling coefficient defined in

(32-b) can be rewritten as

where

mb

c

is the mechanical damping due to the presence of other transducer’s mechanical components

such as ball screw, bearings and coupling shafts. Here, by increasing the size of the rotary generator, the

quantity

2/

ii

TR

scales as [L3], but

mg

c

scales as [L], while,

mb

c

does not scale up. Hence, it can be

understood that by increasing the generator size, the coupling coefficient of the overall transducer

assembly is increased but due to constant

mb

c

, the rate of scaling is higher than [L2]. For instance if two

rotational systems are designed based on two different rotary generators with electromechanical

coefficients

1

emg

and

2

emg

, the ratio of the non-dimensional electromechanical coefficient for these

generators scales as [L2], i.e.,

and from (45) the ratio of the overall electromechanical coefficient of the designed transducers is

22.

C

em wm

wm

w

Bh BV

hc

c

A

(43)

2,

i

emg mg i

T

cR

(44)

2,

i

em mg mb i

T

c c R

(45)

2

2 2 2

11

11

2

2

2,

i

emg mg i

emg i

mg i

T

cR L

T

cR

(46)

11

Therefore, if

1

/

mb mg

cc

is greater than

2

/

mb mg

cc

, then in comparison with

1

em

,

2

em

scales with a ratio

greater than [L2].

4. Numerical study

4.1. Linear system examples

This section investigates the relation between size and efficiency of energy harvesting devices under

constrained condition brought about by the employed commercial generator. It is assumed that a source

of vibration (for example a vertical movement of a boat) with a frequency of 0.5 Hz (

rad/sec) and

amplitude of 1 m (

1Y

m) is available. We are required to design an energy harvesting device such that

the maximum displacement of the seismic mass does not exceed 0.3 m.

First case is dedicated to the design of a linear energy harvesting structure based on figure 1. Table 1 lists

the parameters of a variety of linear electromagnetic actuators presented in [27] that are sorted in the

order of small to large scales. The last system represents a hypothetical case in which the size of the

actuator is much larger than model ASP400 (~8 times).

Table 1 Parameters of a number of linear electromagnetic inertial actuator models[27]

t

K

i

R

m

c

em

m

max

,l out P

P

Type

(N/A)

(Ω)

(Ns/m)

(kg)

(W)

Trust headphone actuator

0.74

8

0.38

0.18

0.03

0.007

Micromega(IA-01)

1.6

3.0

1.4

0.61

0.17

0.09

Aura

7

4.4

9

1.23

1.39

1.23

Motran (IFX 30-100)

10

1.6

44

1.42

7.18

6.93

Micromega (ADD-45N)

20

4

35

2.86

8.11

11.10

ASP 400

21

1.6

30

9.19

16.02

30.60

Hypothetical case

42

0.8

60

36.75

111.01

224.8

For each presented linear actuator type, the proof mass is calculated such that the oscillation at excitation

frequency (

rad/sec) occurs within the given constraint (i.e.,

00.3

r

Z

m). For each inertial

generator

em

and the seismic mass are calculated from (32-a) and (10), respectively. Then, at optimum

load resistance (

max

,,l linear P i

RR

), the output power is obtained from (9). As table 1 shows, by increasing

the transducer dimensions,

em

is increased and that is in agreement with the result presented in section

3. Also, by increasing the size of the linear actuator, the overall damping of the system gets larger, thus,

requiring a bigger mass to reach the same displacement (i.e.,

00.3

r

Z

). In addition, it is seen that by

increasing the size of the linear actuator, the output power increases. However, as in this case, mass is the

design variable (and for hence the absorbed mechanical power is different for each design), system

efficiency would therefore be a more appropriate criterion to be used in order to compare the different

harvesters. Figure 4 shows the efficiency of the designed system corresponding to their maximum output

power calculated from (26). It is seen that by increasing

em

due to the increase of the transducer size, the

2

22

2 2 1

11

1

2

11

2

2

1

.

1

imb

mg mb i

em emg mg

mb

em emg

i

mg

mg mb i

Tc

c c R c

c

Tc

c c R

(47)

12

efficiency of the energy conversion system is improved. However, even in the case of a hypothetical

system where the size has been increased dramatically, the efficiency of the system does not exceed 50%

which is in agreement with the result obtained from (35).

Figure 4. Efficiency of linear electromagnetic energy harvesting systems versus

em

for the linear

actuator shown in table 1

4.2. Rotational system examples

Table 2 presents the size and specifications of a number of commercial PM (permanent magnet)

generators where h and r, respectively, are the length and the radius of the rotary generator coupled to the

ball screw as presented in figure 3. Here, for each generator,

emg

is calculated from (44), see table 2.

Figure 5 shows the variation of the coupling coefficients of the generators in comparison with the size of

the reference generator (Model a). A reasonable fit to

emg

shows that it is linearly proportional

to

2

3

1

/

i

VV

, where

1

V

is the volume of generator model a, and

i

V

is the volume of the selected generator.

This result validates the statement made in section 3 that the electromechanical coupling coefficient of a

generator scales up with the square of the characteristic length of the device [L2]. Also in each case

em

which represents the electromechanical coefficient of the transducer assembly is calculated from

(45). Note that

mb

c

is not a function of the generator size and is assumed to be 3.0E-3 (mN.m.s.rad-1) for

all the designed transducers. A comparison of

em

and

emg

reveals that the

em

scales with a ratio

higher than that of

emg

. This agrees with the discussion presented in section 3.

13

Table 2. The parameters of PM motors from Faulhaber [30]

h

r

i

T

i

R

mg

c

emg

em

l

max

,b out P

P

Type

(mm)

(mm)

(mNm/A)

(Ω)

( mN.m.s.rad-1)

(mm)

(W)

a

6

20

1.13

9.1

6E-5

2.33

0.05

1.2

0.4

b

12

26

2.77

2.3

4.2E-4

6.78

0.95

1.5

6.2

c

16

28

3.86

4.3

4.8E-4

7.22

0.99

1.6

6.4

d

20

36

6.34

3.4

1.3E-3

9.20

2.75

2

12.0

e

30

56

12.74

1.6

6E-3

16.20

10.80

3.8

20.6

f

35

64

14.52

0.6

1.4E-2

24.40

20.20

6.1

24.2

g

44

90

23.83

0.23

6E-2

39.94

38.4

13.5

27.3

Figure 5. The coupling coefficient of rotary generators presented in table 2 versus ratio of their sizes to the

reference generator in power of two over 3.

Now, it is assumed that the environmental vibration condition and the constraint on the maximum

allowable displacement of the seismic mass are the same as the values considered in the first case

(

01Y

m,

). In this case, based on each of the PM generators presented in table 2, a rotational

harvesting system is designed. It is assumed that the energy harvester has a mass of 8.1 kg, and the

design variables are

l

and

l

R

. The optimum load resistance for each case is obtained from (21), and then

the optimum lead size for the ball screw is calculated from (22). Table 2 presents the ball screw lead

values and the generated power of each system corresponding to the relevant selected PM generator in

each case. It is seen that by increasing the size of the generator,

em

and consequently the output power

of the system is increased which is in agreement with (37).

Figure 6 shows the efficiency of the designed rotational systems versus

em

. It is seen that by increasing

the size of PM generators, the efficiency of the system increases. Here, in contrast with linear systems, an

efficiency above 50% is achievable. This occurs for those systems whose

em

meet the criterion

presented in (39), i.e., systems designed based on generators e, f and g. However, if

em

does not satisfy

the condition presented in (39), i.e.

8

em

, designing a rotational energy harvesting system may result

in a sub-optimum energy harvesting device in comparison with the linear system. For instance

comparison of the designed systems based on the generators a, b and c with the linear system designed

based on Micromega (ADD-45N), reveals that although the rotational systems utilize the same mass, they

produce less power compared with the linear system. Therefore, for constrained applications, in the

14

design process of the energy harvesting systems, a rotary generator should be selected carefully to allow

the designer to take advantage of the superiority of the rotational systems over the linear systems.

Figure 6. Efficiency of rotational electromagnetic energy harvesting systems versus

em

for the rotary

generators shown in table 2

5. Discussion and conclusion

In some energy harvesting systems, the maximum displacement of the oscillating mass will be limited

due to the physical constraints of the device. In systems where this limitation does not exist, choosing the

optimum load resistance with the goal of maximizing the energy harvested from the environment is a

process that takes place after the machine design. This is why, in these cases, the phrase “tuning” is used

to refer to the selection of the resistance load. However, in systems where the maximum displacement of

the mass is limited (constrained systems), choosing the optimum load resistance is part of the actual

design process and cannot be done independently of choosing other parameters.

In this article, the maximum power condition and the corresponding efficiency for constrained vibration

based linear and rotational energy harvesting devices were presented. For convenience, and for enabling

the comparison of different systems, the definition for the coupling coefficient of an energy harvesting

device given by (32) is employed [27].

In a linear system, electromechanical coupling coefficient (

em

) is shown to increase with the size of the

transducer according to its characteristic length squared. However, in the case of a rotational system,

although

emg

of the rotational generator, itself, increases as [L2], the value of

em

for the whole

transducer assembly (including the ball screw) scales by a ratio greater than [L2].

It is shown that in a system with linear motion and constrained throw, even with the assumption of

negligible mechanical losses, the maximum harvestable power (at optimum condition, i.e.,

max

,,l linear P i

RR

) is half of the mechanical power that can be absorbed by the transducer.

Also, it is shown that the output power and efficiency of linear systems increase by increasing the size of

the structure. However, the maximum efficiency for such devices cannot be more than 50%.

In contrast, rotational systems with a constrained throw show greater capability in transferring energy to

the resistance load. In these systems, the ratio of the optimum load resistance and the internal resistance

of the generator can be written according to equation (21) and (32) as follows:

Therefore, by increasing

em

, this could be achieved by the enlargement of the rotary generator size, the

ratio of the generator internal resistance to the load resistance increases.

max

,, 1.

l rotational P em

i

R

R

(48)

15

Figure 7 shows the logarithmic plot of

em

against the generator volumes over reference volume to the

power of two over three for both linear and rotational transducers, respectively presented in tables I and

II. The generators volume and the related reference volume for the linear transducers have been obtained

from [27]. It is seen that

em

for rotational systems scales with a greater ratio in comparison with the

linear systems. Hence, scaling the generator part in a rotational system, can be more beneficial in terms

of improvement the system efficiency and output power.

Figure 7. Log-log plot of

em

versus volume over the reference volume to the power of two over three for

linear and rotational systems presented in tables I and II.

It is demonstrated that these transducers can be designed to operate with efficiencies above 50%. The

criterion that guarantees this superior efficiency was derived in (39) which can be used in the design

process. This superiority of rotational systems over linear systems is due to the presence of an

intermediate mechanism viz ball screw that can provide an extra design variable, thus enabling us to

optimize the power output of the system subject to displacement constraint more desirably.

For a defined environmental condition and a given proof mass with constrained maximum allowable

displacement, the amount of power delivered to the electrical load by a rotational system can be as high

as twice the amount delivered by a linear system.

Acknowledgements

The authors are very grateful to Mr Mike Russell for providing financial support to the project.

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