POWER (power optimization for wireless energy requirements) - A MATLAB based algorithm for design of hybrid energy systems

Journal of Power Sources 159 (2006) 758–780
POWER (power optimization for wireless energy requirements): A
MATLAB based algorithm for design of hybrid energy systems
K.A. Cooka, F. Albanob, P.E. Neviusc, A.M. Sastrya,b,c,∗
a Department of Biomedical Engineering, University of Michigan, Ann Arbor, MI 48105, USA
b Department of Material Science Engineering, University of Michigan, Ann Arbor, MI 48105, USA
c Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48105, USA
Received 12 July 2005; received in revised form 12 October 2005; accepted 17 October 2005
Available online 20 December 2005
Abstract
We have expanded and implemented an algorithm for selecting power supplies into a turnkey MATLAB code, “POWER” (power optimization
for wireless energy requirements). Our algorithm uses three approaches to system design, specifying either: (1) a single, aggregate power profile;
(2) a power system designed to satisfy several power ranges (micro-, milli- and Watt); or (3) a power system designed to be housed within specified
spaces within the system. POWER was verified by conducting two case studies on hearing prosthetics: the TICA (LZ 3001) (Baumann group at the
Tübingen University) and Amadeus cochlear implant (CI) (WIMS-ERC at the University of Michigan) based on a volume constraint of 2 cm3
. The
most suitable solution identified by POWER for the TICA device came from Approach 1, wherein one secondary cell provided 26,000 cycles of
16 h operation. POWER identified Approach 2 as the solution for the WIMS-ERC Amadeus CI, which consisted of 1 cell for the microWatt power
range and 1 cell for the milliWatt range (4.43 cm3
, ∼55% higher than the target volume), and provided 3280 cycles of 16 h operation (including
re-charge of the batteries). Future work will be focused on continuously improving our present tool.
© 2005 Elsevier B.V. All rights reserved.
Keywords: MEMS; Batteries; Hybrid; Algorithm; Cochlear; Implant
1. Introduction
Recently, we introduced an algorithm [1] to design hybrid
battery systems for multi-component, wireless microelectron-
ics. Proof of concept was established using the Wireless Inte-
grated Microsystems Engineering Research Center (WIMS-
ERC) Environmental Monitor Testbed (EMT) at the University
of Michigan. Use of our algorithm resulted in significant reduc-
tion in both mass and volume of power supplies, over trial-and-
error selection of batteries. For the WIMS-ERC EMT testbed,
we designed a power supply weighing 32 mg, comprised of thin-
film lithium-free [2] and prismatic lithium polymer secondary
cells; these were, respectively, the Ultralife UBC422030/PCM
and UBC641730/PCM [3].
Our methodology [1] constrained operating temperature,
energy/power density, and specific energy/power; we further
∗ Corresponding author.
E-mail address: amsastry@umich.edu (A.M. Sastry).
allowed requirements/constraints on rechargeability, mass, vol-
ume, and lifetime in selection of appropriate battery electro-
chemistries and configurations (i.e. parallel, series, or combi-
nations thereof). Our algorithm separately evaluated results of
threeapproachestosystemdesign,specifyingeither:(1)asingle,
aggregate power profile; (2) a power system designed to satisfy
several power ranges (micro-, milli- and Watt); or (3) a power
system designed to be housed within specified spaces within the
system, with device constraints on volume and surface area.
In this paper, we describe the expansion and implementation
of our algorithm into a turnkey MATLAB [4] code. We set out
the following objectives in this work, to expand our original
algorithm to its present realization:
(1) to implement simple models to account for capacity fade as
a function of discharge current and cycling, using our own,
and manufacturer-generated data on primary coin cells;
(2) to implement an algorithm for binning device voltage and
current requirements within the micro-, milli- and Watt
power ranges, along with expressions for calculating tar-
0378-7753/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.jpowsour.2005.10.062

K.A. Cook et al. / Journal of Power Sources 159 (2006) 758–780 759
Nomenclature
Alphabet
a number of cell configurations (integer number)
b voltage (V)
c cycle (integer number)
e energy (Wh)
I current (A)
L lifetime (cycles)
M mass (kg)
N number of cells (integer number)
P percent capacity fade (normalized number in the
interval [0,1])
p power (W)
t time (s)
V volume (L)
w weighted power (W)
X total capacity (Ah)
Greek symbols
χ capacity (Ah) at a given time increment
Superscripts and subscripts
c cycle
ctr counter
ctr com counter
i index
loc power site
p primary
r rth cell
s secondary
sys system
total summation
∼ specific property (kg−1)
ˆ density (L−1)
get micro-, milli- and Watt mass, volume and area target
values, based on user-defined battery numbers;
(3) to implement criteria in the algorithm to limit voltage and
current of power sites; and finally,
(4) to implement a discretization scheme for user-input current
profiles.
This new code, “POWER” (power optimization for wire-
less energy requirements), employs a graphical user interface
(GUI) to allow step-by-step input of system data by the user.
To verify our implementation, we conducted two case stud-
ies in power selection. The first was a re-examination of work
done at Tübingen University [5–8] in a fully implantable hear-
ing prosthesis designed to mechanically stimulate the tympanic
membrane, the Totally Implantable Communication Assistance
(TICA) [5–8]. The second case study comprised design of a
power system for a novel cochlear implant, the Amadeus, devel-
oped at the University of Michigan’s WIMS-ERC [9–11].
2. Background
2.1. Cell capacity
Theoretical cell capacity is determined as the ratio of the sum
of the electrochemical equivalent of the active materials, and
the total number of electrons involved in the reaction. Capacity
fade, i.e. loss of discharge capacity when the battery is inactive
(“calendar life” loss) or in use (“cycle life” loss), can sub-
stantially reduce performance [12]. This phenomenon has been
extensively studied in primary and secondary lithium-silver-
vanadium-oxide, lithium-manganese dioxide, lithium-thionyl,
zinc-silver oxide; and lithium, lithium-ion, lithium polymer,
and zinc silver nickel metal hydride cells, respectively, by the
biomedical device [13–15], defense [16], computer [17], hybrid
and electrical vehicle [18,19], and cellular phone [20] indus-
tries. It can be reversible, in which case it is commonly referred
to as self-discharge. Industrially, battery capacity lost in an open-
circuit, i.e. where no load is attached to the battery, is also called
local action [12,21–23].
Capacity fade is more pronounced at high rates of discharge
[24–27], and is further affected by depth of discharge (DOD)
[28,29], number of cycles [30–32], materials used (e.g. chemi-
cally co-precipitated calcium zincate as an active material in zinc
electrodes [33] and Si3-xFexN4 compound as a possible anode
for rechargeable lithium batteries [34]), and/or use of additives
(e.g. metallic bismuth in zinc electrodes [33], and amorphous
manganese oxides [35] and ketjen black dispersed in organic
solvents used in lithium-ion cells [36]). High operating temper-
atures (e.g. for lithium and lithium-ion cells [12,17,30,37,38])
and high storage temperatures (e.g. for lithium-ion batteries
[29,38,39]) can also exacerbate capacity fade. Restrictions on
operating and storage temperatures have limited use of lithium-
ion cells in self-heating portable electronics [17], under moder-
ate and high discharge currents.
2.2. Specific energy/power, power/energy density and rate
characterization
Throughout the rest of this paper, we classify ranges of spe-
cific power and energy for batteries as shown in Table 1, based
on common usage in the literature [40,41]. Table 2(a) (using
information from [42]) lists primary electrochemistries intrin-
sically high in specific energy. Table 2(b) (using information
from [16,41–43]) lists secondary electrochemistries intrinsi-
cally high in specific power. Batteries presently in the POWER
database were selected from the high specific energy/power
ranges defined in Table 2(a) and (b).
Table 1
Classification of specific power and energy ranges for primary and secondary
cells [16,42,58]
Specific power (W kg−1) Specific energy (Wh kg−1)
Low ˜p < 70 ˜p < 40
Medium 70 < ˜p < 300 40 < ˜p < 80
High ˜p ≤ 300 ˜p ≤ 80

760 K.A. Cook et al. / Journal of Power Sources 159 (2006) 758–780
Table 2
Primary and secondary electrochemistries intrinsically high in specific energy
Anode Cathode Electrolyte Nominal
voltage (V)
Cell type Specific energy
(Wh kg−1)
Energy density
(Wh L−1)
Specific power
(W kg−1)
Operating
temperature (◦C)
(a) Primary cells
High specific energy and medium specific power
Li So2 Organic solvent 3.0 Cylindrical 260 415 90 −55–70
Li MnO2 Organic solvent 3.0 Button 230 545 65 −20–55
High specific energy and low specific power
Zn O2 (air) KOH (aqueous) 1.5 Prismatic 370 1300 8 0–50
Zn O2 (air) KOH (aqueous) 1.5 Cylindrical 300 800 8 0–50
Zn MnO2 KOH (aqueous) 1.5 Cylindrical 100 195 50 −60–85
Zn HgO KOH or NaOH
(aqueous)
1.35 Button 100 470 10.5 0–55
(b) Secondary cells
High specific power and low/medium specific energy
Pb PbO2 H2SO4 (aqueous) 2.0 SLI (starting lighting and
ignition) prismatic
35 70 1600 (10 s) to
800 kW
(0.1 s)5
−40–55
MH NiOOH KOH (aqueous) 1.2 Button, cylindrical, and
prismatic
75 240 2000–22002 −20–50
Zn NiOOH KOH (aqueous) 1.65 Cylindrical, prismatic
sealed and vented
50–60 80–120 300 −10–50
High specific power and high specific energy
Zn MnO2 KOH (aqueous) 1.5 Cylindrical 85 250 150 −20–40
C LiCoO2 Organic solvent 4.0 Cylindrical and prismatic 150 400 6503 −20–50
Zn AgO KOH (aqueous) 1.5 Prismatic 105 180 6004 −20–60
Data taken from [16,42,58].
2.3. Strategies employed previously, and present approach
Most power supplies for microelectronic devices are pre-
scribed after design is nearly complete. Power supplies are thus
frequently an afterthought: of the microelectronic devices listed
in Table 3 [44–49] only one was operated and tested with a
battery [45]. All others used external power supplies.
The devices in Table 3 require power in the milliWatt
range (0.3–25 mW) and voltages >3.3 V. Indeed, though not
evenly-spaced in terms of order-of-magnitude, the ranges of
micro-, milli- and Watt power arise commonly in wireless
electronics due to the intrinsic demands of their subcompo-
nents. Dynamic power switching, ubiquitous in wireless devices,
requires power in the milliWatt range [1], and is required
for device activation, volume fluctuation, wireless data trans-
mittal/reception, computation, heating/cooling, actuation, and
alarms (Tables 3 and 4). Innovations in the field have resulted
in reductions in supply voltage and increases switching fre-
quency [50–52], which in turn have resulted in reductions
in milli- and Watt power range consumption. In the milli-
Watt range, for example, improvements in adiabatic differential
switch logic and gate resizing for very large scale integrated
(VSLI) circuits have reduced power demands by 26% and
2.8–27.9%, respectively [50,53,54]. In the Watt range, improve-
ment of parallel Huffman decoders, and improvements in first
level filtering caches used for modem microprocessors have
reduced power demands by 50 and 58%, respectively [54,55].
It must be noted, however, that power reduction frequently
comes at the expense of speed of execution, bandwidth, clock
speed, and energy delay [1,55]. Thus, further reductions of
power in these established ranges will require examination of
tradeoffs.
Sample intrinsic specific power/energy, and energy/power
densities (which can presently supply power in these ranges
at needed rates of discharge) are listed in Appendix A. Most
electrochemistries provide nearly constant capacity values for
discharge rates within a 35% range, so that binning of power
according to power ranges of smaller steps (e.g. every 10 ␮W)
is excessively computationally intensive. Furthermore, power
consumption of complimentary metal oxide materials (CMOS)
devices, primarily a component of dynamic switching power, is
a function of the intrinsic material properties of CMOS materi-
als, namely capacitance due to charge/discharge switching [1].
Thus, the presently-used electrochemistries appear sufficiently
robust at this time to power the likely demands of microcircuits,
in the near term.
3. Methods
3.1. General methodology and definitions of terms
A flowchart for our algorithm is given in Fig. 1(a) and (b);
it is modified to reflect changes from our first work using this
approach [1]. The user provides target values for mass, volume,
and surface area, operational temperature, numbers of power
bundle locations, number of cycles, selection of primary or sec-
ondary cells, and mass or volume optimization. We have reduced
the number of user inputs in comparison to our past work [1],

Table 3
Typical discharge current requirements for microelectronics [44–49]
Microelectronic device Technology Size Power–current–voltage
requirements
Power source
Micro magnetic sensor Mineral insulated (Ml) sensor
constructed using CMOS IC
multivibrator circuit
Wire
diameter = 30 ␮m,
length = 2 mm
0.5–5 mW (pulse
current = 30 mA)
External power supply
Colpitts transmitter Five-turn dielectric suspended
inductor was fabricated using a
dissolved wafer process
Colpitts oscillator
transmitter
(5 mm × 5 mm area)
each coil is 25 ␮m
wide, 5 ␮m thick
100 ␮A with driving
voltage = 3.0 V
Operated with 3 V
battery
Si-based micro-machined
gas sensor
Sensor array was fabricated using a
post-process micro-machining
technique of standard CMOS process
Thickness = 1.2 ␮m,
active
area = 80 ␮m × 80 ␮m
9 mW of drive power
with 2.0 V drive
voltage
Amperometric potentiostat Potentiostat uses an ADC circuit that
allos the direct conversion of
electrode current in nanoampere
range to low-voltage CMOS levels
using four operational applifiers
Volume < 3 cm3 0.65 mW, 260 ␮A and
2.5 V
3 V lithium coin cell
suggested
Electrothermal actuator MEMS polysilicon surface
micromachined electroactuator uses
resistive Joule heating to generate
expansion and movement
462.5 ␮m × 15 ␮m
× 129.5 ␮m
∼7–25 mW External
programmable power
supply
Three-axial force sensor Si-based three-axial force sensor to
be used in a flexible smart interface
for biomechanical measurements
2.3 mm × 2.3 mm × 1.3 mm
sensors have
implanted
piezoresistors that are
6 ␮m × 30 ␮m
10–1 mW input
voltage = 3.3 V
wherein users were required to specify target values for the mass
andvolumeforeachpowerrange.Instead,thesevaluesarecalcu-
lated based on the maximum number of cells for each approach
specified by the user. Specifically, the target volume, Vi, and
mass, Mi, for each power range are computed from the expres-
sions
Vi =
Ni
Ntotal
Vsys i =
⎧
⎪⎨
⎪⎩
1 for microWatt power range
2 for milliWatt power range
3 for Watt power range
(1)
and
Mi =
Ni
Ntotal
Msys i =
⎧
⎪⎨
⎪⎩
1 for microWatt power range
2 for milliWatt power range
3 for Watt power range
(2)
where Ni (i = 1, 2, and 3) is the target number of cells for the
micro-, milli- and Watt power ranges, respectively, Ntotal is the
total number of cells, Vsys is the total volume and Msys is the
total mass of the desired power supply.
Table 4
Typical discharge current requirements for common commercial electronics [42]
Device Current drain (mA)
Cassette recorders 70–130 (low) 90–150 (medium) 100–200 (high)
Disk players 100–350
Calculators (LCD) <1
Cameras 800–1600 (photo flash) 200–300 (autowind) 500–1600 (digital cameras)
Cellular phones 300–800
Camcorders 700–1000
Computers 400–800 (palm held) 500–1500 (note book) 800–1000 (laptop)
Fluorescent lamp 500–1000
Flashlight 100–700
Memory 0.001
Remote control 10–60
Radios: 9 V battery 8–12 (low volume) 10–15 (medium volume) 15–45 (high volume)
Radios: cylindrical battery 10–20 (low volume) 20–30 (medium volume) 30–100 (high volume)
Walkman 200–300
Smoke detector 0.010–0.015 (background) 10–35 (alarm)
Motorized toys 600–1500
TV: portable 400–700

The user defines current and voltage in terms of time incre-
ments, prior to entry of device current and voltage values. A
duty cycle is the minimum time interval that can be repeated to
represent the lifetime usage profile of the device. For a cochlear
implant, for example, a typical duty cycle would be a single day,
and would include active usage that varied throughout waking
hours, with recharging occurring during sleep or off periods.
Device current requirements are rarely constant; for example,
the current versus time profile for a hearing aid [56] fluctuates
over a 60 s period (Fig. 2(a)).
Due to the impracticality of mapping small fluctuations, data
can be coarsened for input into POWER using two methods:
(1) consolidation of identical current values into the same time
interval, or (2) replacement of sufficiently similar current values
such that they produce nearly identical values of discharge rate,
either with the summed weighted averages of two current mag-
nitudes, or highest of the two current magnitudes; the approach
is shown schematically in Fig. 2(b). In the case of the hearing aid
current profile shown in Fig. 2(a), fluctuations in current reflect
variations in sound volume external to the user [56]. In the plot
shown in Fig. 2(b), common currents are combined, for data
entry into POWER.
Table 5 gives the relations used in computing of energy ei,
weighted power wi specific energy (energy per unit target mass)
˜ei, weighted specific power (power per unit target mass) ˜pi,
energy density (energy per unit target volume) êi, and weighted
power density (weighted power per unit volume) ˆpi. The nom-
inal voltage of the cell is the operating or rated voltage of the
cell specified by the manufacturer.
Devices are classified as having microWatt and milliWatt
power ranges, for powers requiring less than one milliWatt, and
less than 1 W, respectively. In our previous work [1], this logic
was applied iteratively: sub-devices contributing to the largest
power values within a particular power range were removed
and placed in a higher power range than their initial position,
as needed. Here, power ranges not meeting the power range
requirements are rearranged according to voltage value. Specif-
ically, devices within a power range are ranked in descending
order by operating voltage. Sub-devices contributing the largest
voltages within the microWatt or milliWatt power ranges are
Fig. 1. [2] Flowchart for logic implemented in POWER. [2] Flowchart for logic used in limiting mass, volume, surface area and number of cells prior to specific
energy, energy density and lifetime selection processes.

Fig. 1. (Continued).
Fig. 2. (a) Current vs. time data for ‘Digital Aid X’ hearing aid tested by Denis Carpenter of Rayovac [56]. (b) Data after data coarsening, for input into POWER.

Table 5
Relations used in POWER to calculate energy, weighted specific power, specific
energy, energy density and weighted power density [1]
Variable Units Expression
Power (W) pi(t) = ci(t) ×
vi(t), i = 1 :
N, no sum
Energy (Wh) ei = pi(t) t
Specific energy (for each
sub-device)
(Wh kg−1) ˜ei = pi(t) t
mx
Weighted specific power (for
each sub-device)
(W kg−1) ˜pi = ( t/tT )pi(t)
mx
Energy density (for each
sub-device)
(Wh L−1) ˜ei = tpi(t)
vx
Weighted power density (for
each sub-device)
(W L−1) ˜pi = ( t/tT )pi(t)
vx
Energy (for system) (Wh) Ex =
N
j=1
pj(t)t
Weighted power (for system) (W) Px(t) =
N
j=1
pj(t) t
tT
Energy provided by battery (Wh) ej =
tT
j=1
bjCj
t
tT
Energy factor [] xj = Ex
ej
Voltage factor [] yj = Vx
bj
Current factor [] zj = ix
ij
systematically removed from one power range and added to the
milliWatt or Watt duty cycle, respectively, until the power limit
is reached.
The total capacity required by a device for a duty cycle is
given by:
XE521 =
t=ttotal
t=1
χE521(I(t)), (3)
or simply the sum of capacity values, χ, for each time increment.
The number of cycles provided for a primary or secondary cell
without recharge, is:
Lp =
XV
Ek
, (4)
where X is the capacity of the cell, multiplied by the cell nominal
voltage, V, and Ek is the energy required; k refers to the system,
power range or site. Capacity losses were also considered, and
are discussed separately.
3.2. Selection of database batteries
Silver oxide cells (trivalent silver oxide, zinc/divalent silver
oxide and monovalent silver oxide) were included due to their
intrinsically high energy density (∼530 Wh L−1) in compari-
son to other primary aqueous electrolyte systems [24]. Because
of the inherent instability of trivalent and divalent silver oxide,
and the two-step discharge curve in the latter electrochemistry,
onlythezinc/monovalentsilveroxidesystemsareavailablecom-
mercially. We considered use of zinc-silver oxide primary cells
because of their high energy density (∼530 Wh L−1 [24]), high
power density [16] and commercial availability, which make
them good candidates for power sources for portable electronics
requiring low discharge currents (<1 mA). Though these cells
have demonstrated relatively high rate performance in appli-
cations where size and mass are key constraints [16], most
capacity data provided by manufactures is for very low discharge
rates/currents (∼0.02 to 0.24 mA [57,58]). Furthermore, many
portable electronics and implantable devices, such as defibril-
lators, require continuous discharge currents between 0.5 and
50 mA [13], which substantially exceed typical discharge cur-
rents used by manufacturers in testing, as shown in Table 4.
Lithium manganese and lithium thionyl chloride batteries
were also included in our database (e.g. batteries manufactured
by Maxell [57] and Renata [58], and Electrochem [59]). Lithium
thionyl chloride batteries were chosen because of their intrinsi-
cally high specific energies (∼275 to 715 Wh kg−1), their high
nominal voltage of 3.6 V and their flat discharge profile. These
batteries are manufactured in several sizes, ranging from small
button cells, to cylindrical and prismatic cells, with reported
capacities from 0.4 to 10,000 Ah [24]. Lithium thionyl cells,
whichuseSOCl2 asbothcathodeandelectrolytesolvent,contain
apassivationlayeroverthelithiumwhichinhibitsself-discharge.
This, in turn, results in long shelf life, but also results in some
voltage delay after storage. These cells operate over a wide tem-
perature range, −55 to 70 ◦C [60]. Lithium manganese dioxide
cells, which have a solid cathode, are nonpressurized (in contrast
with the soluble cathode lithium cell), and thus do not require
hermetic seals. They have lower discharge rates, however, than
soluble cathode batteries (including lithium thionyl) and infe-
rior low temperature performance (−20 to 55 ◦C) compared to
lithium thionyl batteries. Their specific energies range from 260
to 500 Wh kg−1 [24]. They also range in size, from button to
small cylindrical cells.
A detailed list of the batteries selected, along with their char-
acteristics, is found in Appendix A. Inherently, performance
tradeoffs must be considered with regard to duty cycle, size and
discharge current of the power supply. We specifically exam-
ined tradeoffs in capacity fade versus application of low-mass
batteries in pulse conditions, given the probable stringent size
constraints in implantable devices. For example, wristwatch bat-
teries of very low mass are available, but have not been widely
used in pulse applications.
3.3. Determination of voltage and current for each power
site location
In our previous work, a method for establishing maximum
current and voltage for each power site was not addressed; we
have added logic to do so the present version of POWER. Target
volumes and surface areas for each power site, are provided
by the user. Target voltage parameters supplied by the user are
sorted in descending order, and maximum voltages are assigned
to power site locations by rank. For example, for a system of
four devices, with voltages in Table 6(a) and (b), and allocation
for only two power sites, would result in assigned voltages for

Table 6
Sample system of four devices with varying voltages, used to demonstrate allocation of voltage values for power site locations; and the resulting assignment of
voltage values for two power site locations, based on the system defined
Device Voltage (V) Current (mA)
Device #1 15.0 0.001
Device #2 3.0 2.0
Device #3 5.2 1000
Device #4 6.0 0.25
Power bundle site Volume (cm3) Surface area (cm2) Voltage (V) Current (mA)
1 12.0 60.0 15 750
2 5.0 20.0 6.0 250
power sites 1 and 2, of 15 and 6.0 V, respectively. The energy
Ei required by each site is simply the volume fraction of the site
multiplied by the total energy of the system. The weighted power
required for each site, Pi, is similarly the area fraction of the site
multiplied by the total system power. The current for each power
site is obtained by multiplying area fraction of the site by the
maximum current at that site. Thus, the current for each power
site in Table 6 would be 0.75 and 0.25 A for sites having areas
of 60 and 20 cm2, respectively (Table 6). The surface area for
each cell in the database refers to the total surface of the cell,
and not one specific side or face.
3.4. Estimation of capacity fade, for primary and
secondary cells
Capacity fade as a function of both discharge current and
cycle number was estimated, where possible, using expressions
relating capacity fade as a function of cycle from online battery
manufacturer data [3,22,56–59,61]. Data used for the empirical
regression lines were inclusive of our experimental data and
values obtained from the manufacturer [3,22,56–59,61]. At least
fourdatapoints(e.g.capacityvalueasafunctionofcurrent)were
used in each plot.
For example, capacity for an Energizer 521 cell was deter-
mined via curve-fit of manufacturer-reported data [61] to be:
χE521 = −2.45 ln(I(t)) + 3.26, (5)
where I is the discharge current for time increment t. Similar
relations were generated for all cases using polynomials (lim-
ited to third order), logarithmic or power decay functions to
reflect the decay of capacity with increased discharge current
[24–27]. Correlation factors of >0.80 were deemed acceptable
for implementation. This method of computing capacity fade as
a function of discharge current was used for both primary and
secondary cells.
Capacity fade as a function of cycle was used only for sec-
ondary cells. Percent capacity fade as a function of cycle can be
expressed as the ratio of capacity provided by a cell at a certain
cycle by the maximum capacity the cell can provide, per
Pc =
X(ci)
X(c1)
. (6)
The total capacity a cell can provide, including all recharge
cycles, is thus:
XR =
c=total cycles
c=1
PcX(t) (7)
This capacity was used by our algorithm to determine the total
number of cycles a particular cell can provide for a specific duty
cycle, as:
LS =
XR
Ek
. (8)
The capacity value computed for non-rechargeable systems was
used for the energy factor calculation. Cycle time and recharg-
ing of cells is incorporated into POWER via Eqs. (6)–(8) for
accurate determination of battery solutions’ cycle life. Capac-
ity, X(t), is first computed as a function of discharge current over
time, per Eq. (3); total capacity as a function of cycle number is
then computed via Eq. (7). Pc drops monotonically with cycle
number; available capacity thus also drops monotonically with
increasing cycle number.
We also generated our own data on primary (i.e. non-
chargeable cells) silver oxide cells to estimate capacity fade.
Cells were discharged at currents one and two orders of magni-
tude above the manufacturer-recommended discharge currents,
for two reasons. First, many household appliances and electron-
ics (detailed in Table 4) require discharge currents that exceed
operational values provided by many manufacturers [57,58,61].
Second, our algorithm requires additional batteries to meet dis-
charge currents (current factor, xi) that exceed the maximum
discharge current allowed for each battery in the database. In
cases where manufacture data are provided for small nominal
discharge currents, additional batteries are suggested as a solu-
tion, to account for losses due to high rate operation.
Silver oxide primary cells (Table 7) were tested to inform
a simple model for the relationship between discharge current
and capacity. All cells were subjected to constant continuous
resistance discharges, wherein the initial open-circuit voltage
was approximately 1.55 V and then end voltage was less than
1.0 V. A schematic of the experimental setup is illustrated in
Fig. 3. Voltage per second was recorded for each cell, and the

Table 7
Characteristics of silver oxide cells tested
Manufacturer Part number Diameter (mm) Height (mm) Mass (g) Resistances tested (k )
Energizer 337 4.80 1.65 0.13 1.25, 1.50, 1.875
Duracell D379 5.79 2.15 0.23 1.25, 1.50, 1.875
Maxell SR516SW 5.80 1.65 0.20 1.25, 1.50, 1.875
Maxell SR616SW 6.80 1.65 0.30 1.25, 1.50, 1.875
Renata 337 4.80 1.65 0.12 100,6.8, 1.0,0.55
Renata 377 6.80 2.66 0.40 0.55, 1.0,2.5, 6.8, 100
Renata 364 6.80 2.15 0.32 0.55, 1.0,2.5
Renata 317 5.80 1.65 0.18 0.55, 1.0,2.5,6.8
Renata 319 5.80 2.70 0.29 0.55, 1.0,2.5, 6.8
Renata 321 6.80 1.65 0.25 0.55, 1.0,2.5,6.8
discharge current:
I(t) =
b(t)
R
(9)
was determined from the quotient of voltage per unit time, b(t)
and resistance, R. The average capacity for each cell was com-
puted as the product of the average current, Iavg and total time
of operation:
Cavg = Iavg × ttotal (10)
from an initial voltage of 1.55 V to a cutoff voltage of 1.2 V.
Cells were tested at various resistances, to allow curve-fit of a
plot of capacity versus discharge current.
3.5. Case studies: fully implantable hearing prosthesis
We selected two fully implantable hearing prostheses as case
studies. The first was a mechanical stimulator for the tym-
panic membrane, the TICA (LZ 3001) device [5–8], designed
by researchers at Tübingen University. Specifications on the
device’s power profile are listed in Table 8.
The second testbed was the WIMS-ERC Amadeus Cochlear
Implant [9–11,62,63], developed by researchers at the Univer-
sity of Michigan. Specifications on the device’s power profile
are listed in Table 9.
3.6. Conditionality statements
Conditionality statements were used to determine configura-
tion of the cells (series, parallel or a combination). Correcting
typographical errors in our original work [1], these values are
shown as Table 10(a) and (b). Cells can be placed in combina-
tions of series and/or parallel according to energy (x), voltage
(y) and current (z) factors (Table 10(a) and (b)). Factors (equa-
tions contained in our previous work [1]): x, y and z are ratios of
system requirements (energy, voltage and current, respectively)
to nominal cell values. Variables, n and s represent the system-
required total number of cells, and number of cells in series,
respectively. Cells can be placed in parallel to meet discharge
current and energy requirements, thus, w and u represent the
total numbers of cells placed in parallel, and required to meet
energy requirements, respectively.
Factors greater than 1 require additional cells to satisfy
energy, voltage and discharge system requirements. For exam-
ple, for a y of 2, two cells, in parallel, are required to meet the
system voltage requirement. Table 10(a) and (b) are circuit dia-
grams illustrating combinations of cells in series and/or parallel.
In some cases, additional cells necessary to meet energy require-
ments simultaneously result in satisfaction of discharge current
requirements, e.g. z = 5, y = 3 and x = 2 (Table 10(b)). Table 10(a)
and (b) also contain circuit diagrams illustrating cells in series
and/or parallel associated with various combinations of x, y and
z values.
After batteries were configured in series or parallel arrange-
ments according to the three approaches, mass, volume, surface
area, and number of cells in the configuration were exam-
ined. This portion of the algorithm is circled in Fig. 1(a), and
expanded with additional detail in Fig. 1(b). These iterative steps
(Fig. 1(b)) were implemented to enforce user-defined constraints
on maximum number of cells per configuration, surface area and
Fig. 3. Experimental setup for resistance testing of primary silver oxide cells.

Table 8
Input parameters for the Tübingen TICA (LZ3001) [5–8] tympanic membrane mechanical stimulator
Electronic components Input current (mA) Input voltage (V) Time interval (s)
Tübingen University—TICA implant—16 h operation
Microphone 0.05 1.25 60
Signal processor 0.4–0.6 1.25 60
Amplifiers 0.4 1.25 60
Memory (monitoring) 0.1a 1.25 60
Signal receiving circuit 0.1 1.25 60
Total time 16 h
Number of cycles 960 Number of power bundles 2
Surface area of each bundle site 1.0 cm2 Volume of each power bundle 1.0 cm3
Total area 2.0 cm2 Total volume 2.0 cm3
a Value corrected from original reference.
mass (mass prioritization) or volume (volume prioritization),
and also to compute the best solutions available, even if they did
not meet user requirements.
Table 1(b) schematically shows the methodology by which
battery solutions determined based on user-supplied mass or
volume prioritization. Specifically, if the number of battery solu-
tions in the database meeting the mass or volume requirements,
Nctr, specified by the user is greater than 10, then the number of
batteries meeting the minimum requirement for number of cells
in the battery solution is determined. So, battery solutions that do
not meet the mass or volume requirements are eliminated from
the pool of solutions that advance to the next step of analysis.
However, if insufficient solutions (Nctr = 10) meet the mass or
volume requirements, solutions that otherwise would have been
eliminated are allowed to advance to the next stages of analysis.
Specifically, the number of configurations within each
approach that satisfy the mass (mass prioritization) or vol-
ume (volume prioritization) target values are counted (Nctr,r for
Approach 1, Nctr,i where i = 1, 2 and 3 for micro-, milli- and Watt
power ranges; and Nctr,s, where s = 1:n loc). If Nctr,i is less than
10, a new target mass or volume is determined from the product
of minimum mass/volume of all battery configurations and 1.25.
For numbers of configurations that do not adhere to the maxi-
mum number of cells, nctr, less than 10, new target values for
the maximum number of cells are determined by multiplying the
minimum mass/volume of all configurations by 1.25. The code
iterates until at least 10 cells meet the mass/volume targets and
10 meet the number of cells per configuration requirements. The
number of cells that meet both requirements for mass/volume
and number of cells per configuration is determined, Nnctr com. If
Nnctr com is less than 5, both mass/volume and maximum number
of cells targets values are multiplied by 1.10 and iterated. The
number of cell configurations meeting the surface area, actr, is
checked and iterated in a similar manner, however, only two cell
configurations must meet the surface area requirement (Fig 1(a)
and (b)).
3.7. Cost analysis
Although not used as a constraint, we did examine the cost
of each power solution generated for the test cases. All specifi-
cations for batteries included in the database were readily found
online. In some cases, purchase of a large number of cells was
required to reduce cost per piece. Appendix A includes battery
cell characteristics, e.g. mass, volume, total surface area, elec-
trochemistry, shape and cost for purchases on a per piece basis.
4. Results
4.1. Experimental characterization of capacity fade
Primary silver oxide cells exhibited flat voltage discharge
curves and operated at a nominal voltage of 1.55 V, as expected.
An example of a discharge at a current of 0.8 mA is shown in
Fig. 4 (Maxell 516), with a corresponding plot of curve-fits for
capacity as a function of various discharge current shown in
Fig. 5. A number of silver oxide cells were subjected to contin-
uous constant resistance loads; in each case, voltage over time
was recorded. An expression for the line best fitting the capacity
as a function of discharge current was determined and included
Table 9
Input parameters for the WIMS-ERC Amadeus [9–11,62,63] cochlear implant
Electronic components Input current (mA) Input voltage (V) Time interval (s)
WIMS-ERC—Amadeus Cl—16 h operation
Electrodes 4.10 3.00 60
Microcircuits 0.08 3.00 60
Total time 16 h
Number of cycles 960 Number of power bundles 2
Surface area of each bundle site 1.0 cm2 Volume of each power bundle 1.0 cm3
Total area 2.0 cm2 Total volume 2.0 cm3

Table 10
Revised conditionality statements
(a) Condition Expression Examples and circuit diagram x y z nj sj Wj and Uj
z < x < y
nj = yz + |x − z|, sj = y,
Wj = z, uj = x
2 3 1 4 3 1 and 2
3 5 2 11 5 2 and 3
x = y > z 2 2 1 3 2 1 and 2
3 3 2 7 3 2 and 3
y < z < x and y = 1 4 2 3 7 2 3 and 4
z < y < x 3 2 1 4 2 1 and 3
5 3 2 9 3 2 and 5
y = z < x and y = 1 4 2 2 6 2 2 and 4
y = z < x and y = 1 nj = yz + |x − z|, sj = y,
Wj = 0, uj = x
3 1 1 3 1 0 and 3
y < z < x and y = 1 3 1 2 3 1 0 and 3
(b) Condition Expression Examples and circuit diagram x y z nj sj Wj
x = y = z
nj = yz, sj = y, Wj = z
1 1 1 1 0 0
2 2 2 4 2 2
4 4 4 16 4 4
x < y < z 1 2 3 6 2 3
2 3 5 15 3 5
y < x < z 2 1 3 3 1 3
3 2 5 10 2 5
x < z < y 1 3 2 6 3 2
1 5 3 15 5 3
x = y < z 1 1 2 2 1 2
2 2 3 6 2 3

Table 10 (Continued )
(b) Condition Expression Examples and circuit diagram x y z nj sj Wj
x = z < y 1 2 1 2 1 2
2 4 2 8 4 2
x = z > y 2 1 2 2 1 2
3 2 3 6 2 3
y = z > x 1 2 2 4 2 2
2 4 4 16 4 4
Fig. 4. Voltage vs. time curve obtained from constant resistance testing of a
Maxell 516SW silver oxide cell.
Fig. 5. Sample empirical ﬁt of capacity as a function of discharge current for
the Maxell 516SW silver oxide cell.
in our code. Table 11(a) and (b) provide the expression found
for each battery tested.
4.2. TICA (LZ 3001) device: 16-h duty cycle
Results for the 16-h operation of the TICA (LZ 3001) device
are shown in Table 12. The ﬁrst of the two tables show the best
secondary power solutions. Identical results were obtained for
the mass and volume prioritization. Application of Approach
1 resulted in a system comprised of a single cell, the Quallion
QL0170E, with a mass of 6.0 g and a volume of 2.62 cm3. The
lifetimes, in terms of cycle number, were calculated to be ∼28
and 25,800, for use of the cell as a primary and secondary source,
respectively.
Application of Approach 2 resulted in selection of two
Quallion-QL0170E cells (6.0 g and 2.62 cm3 per cell), one for
the micro power range and one for the milli power range, result-
ing in a total system size of 12 g and 5.24 cm3. The lifetimes, in
terms of cycle number, for both micro- and milliWatt power
ranges were 53,700 and 49,600, respectively, when recharge
cycles were included.
Using Approach 3, two Quallion-QL0170E cells were
selected (6.0 g and 2.62 cm3), one for each power site, resulting
in a total mass and volume of 12 g and 5.24 cm3. The lifetimes,
in terms of cycle number, were both 51,640 for each power
site, assuming recharge, i.e. use of the batteries as secondary
sources. When volume was selected as the priority, all the three
Approaches provided the same results as those determined for
the mass priority case.
For comparative purposes, we also used our algorithm to
determine the best systems for primary power supplies. One
Renata 380 cell was selected for Approach 1 and two Renata 377
cells were selected for Approach 3, one in each available power
site. Identical solutions were obtained for both mass and vol-
ume prioritization. For Approach 2, mass prioritization resulted
in selection of a lighter cell for the microWatt range (Duracell
D377, mass equal to 0.4 g); a Renata 380 (1.2 g) cell was selected
for volume prioritization. For the milliWatt power range, one
Renata 380 cell was selected for both mass and volume priori-

Table 11
Empirically-determined capacity vs. discharge current, for several silver oxide cells tested
Manufacturer Part number Resistance (k ) Capacity (mAh) Current (mA) Expression
(a)
Energizer 337
1.25 1.01 1.13 Capacity = 2870l2
− 8.9l + 0.008,
R2 = 0.99
1.50 1.99 0.95
1.88 2.17 0.78
Maxell SR516SW
1.25 5.00 1.18 Capacity = 4228l2
− 10.77l + 0.012,
R2 = 0.99
1.50 5.00 0.98
188 6.00 0.79
Maxell SR616SW
1.25 6.78 1.18 Capacity = 3854l2
− 11.84l + 0.015
R2 = 0.99
1.50 6.96 0.99
1.88 8.45 0.80
Duracell D379
1.25 0.13 1.03 Capacity = 14200l2
− 28.37l + 0.015,
R2 = 0.99
1.50 1.10 0.89
1.88 1.26 0.71
Renata 315
0.55 7.83 2.63 Capacity = 1359l2
− 8.28l + 0.02,
R2 = 0.96
1.00 9.72 1.47
2.50 16.9 0.60
Renata 317
0.55 1.58 2.46
Capacity = −0.002
ln(l) − 0.009,
R2 = 0.99
1.00 2.37 1.43
2.50 3.64 0.60
6.80 6.15 0.22
(b)
Renata
319 0.55 2.68 2.53 Capacity = −0.004 ln(l)
− 0.02, R2 = 0.991.00 4.48 1.44
Renata
321 0.55 1.18 2.53
Capacity = 0.0001l−05,
R2 = 0.97
1.00 1.28 1.43
2.50 3.22 0.60
Renata
337 0.55 1.89 2.52 Capacity = 1398l2
− 6l + 0.008,
R2 = 1.0
1.00 2.54 1.36
6.80 6.83 0.22
Renata
364 0.55 0.33 2.58
Capacity = 10−6l−09,
R2 = 0.97
1.00 0.49 1.45
2.50 0.62 0.60
Renata
377 0.55 1.78 2.60
Capacity = 0.02e−995,
R2 = 0.95
1.00 4.59 1.43
2.00 1.23 0.75
6.80 12.90 0.23
Renata
397 0.55 14.0 2.63
Capacity = 0.032e−328,
R2 = 0.99
1.00 18.50 1.48
2.50 26.90 0.61
tization. The cycle life resulting from application of Approach
1 was 5.08; each cycle was 16 h in length, resulting in a total
life of just over 3 days. The solution resulting from application
of Approach 2 for the microWatt range, provided 3110 cycles
of 16 h (∼5.66 years) for mass prioritization and 10,200 cycles
of 16 h (∼ 22 years) for volume prioritization. For the milliWatt
power range, a lifetime of 9.78 cycles (∼6.7 days) was computed
for both mass and volume prioritization. Approach 3 provided a
lifetime of approximately 4.4 cycles for both prioritizations.
4.3. WIMS-ERC Amadeus CI: 16-h operation
Results for a 16-h duty cycle for the Amadeus CI are given in
Table 13 (secondary cells). When mass was prioritized, applica-
tion of Approach 1 provided a solution consisting of a single cell,
the Quallion QL0170E, of size 6.0 g and 2.62 cm3. The num-
ber of cycles predicted was 3.51, without recharge and 3210,
with recharge. Application of Approach 2 resulted in selec-
tion of two cells, one Quallion-QL0100E cell (with a mass of
4.0 g and volume of 1.81 cm3) for the microWatt range, and one
Quallion-QL0170E cell (with a mass of 6.0 g and volume of
2.62 cm3) for the milliWatt range; the total mass and volume of
the system were 10 g and 4.43 cm3, respectively. The calculated
lifetime for the battery selected in the microWatt range was 105
cycles as a primary source, and 96,400 as a secondary source.
Application of Approach 3 resulted in selection of two Ultralife-
UBC641730 cells, one for each power site, resulting in a total
mass and volume of 9.0 g and 4.46 cm3. In this last case, we cal-

Table 12
Binning of devices
Range Device Power (mW) Voltage (V)
(a): Binning of devices into micro and milliWatt power ranges, before any
re-arrangement
Microwatt
1 0.006 1.5
2 0.950 2.0
3 0.750 7.0
Total 1.71
Milliwatt
4 1.5 15.0
5 2.5 16.0
Total 4.0
(b): Initial binning of devices within power ranges for a sample system
according to power value
Microwatt
1 0.006 1.5
3 0.750 7.0
Total 0.756
Milliwatt
4 1.50 15.0
5 2.50 16.0
2 0.95 2.0
Total 4.95
(c): Final binning of devices within power ranges, for a sample system,
according to voltage value
Microwatt
1 0.006 1.5
2 0.950 2.0
Total 0.956
Milliwatt
4 1.50 15.0
5 2.50 16.0
3 0.750 7.0
Total 4.75
culated a lifetime of 7.34 cycles without recharge, and 3200 with
recharge.
When primary cells were examined for both mass and volume
prioritization computations, the same batteries were selected
with application of Approaches 1 and 3. Three cells (Renata
380) were selected for Approach 1 and six cells (Renata 377)
were selected for Approach 3, i.e. three per power bundle. For
Approach 2 in the microWatt range, one Renata CR2032 (2.8 g)
cell was selected in the case of mass prioritization and a Renata
CN2450N (5.9 g) cell was selected for volume prioritization. For
the milliWatt power range, three Renata 380 cells were selected.
The cycle lifetime provided by Approach 1 was 1.9 cycles of
16 h each (∼1.5 days). The system designed by application of
Approach 2 for the microWatt range, provided 173,000 cycles
for mass prioritization and 712,000 cycles for volume priori-
tization. For the milliWatt power range, calculated lifetime as
1.9 cycles (∼1.5 day) for both mass and volume prioritization.
Approach 3 provided a cycle lifetime of 1.65 cycles (∼1 day)
for both prioritizations.
5. Discussion
We have implemented an algorithm into a turnkey battery
selection code, POWER, that can be used to design power supply
systems for a wide range of wireless devices. Our extension
of our original algorithm [] includes consideration of capacity
as a function of discharge current, capacity as a function of
cycle number, assembly of devices within power ranges based on
voltage rather than power, and battery number limitation based
on user input and rechargeability.
Table 13
Solutions generated by POWER for the TICA prosthesis implant (secondary batteries)
Manufacturer Part No. Total No. No. of cycles (no
battery re-charge)
No. of cycles (battery
re-charge)
Total mass (g) Total volume (cm3)
T¨ubingen TICA—mass priority—16 h of operation
Approach 1 Quallion QL0170E 1 28.10 25800 6.00 2.62
Approach 2
Micro Quallion QL0170E 1 58.60 53700 6.00 2.62
Milli Quallion QL0170E 1 54.10 49600 6.00 2.62
Totals 2 12.00 5.24
Approach 3
Site 1 Quallion QL0170E 1 56.30 51600 6.00 2.62
Totals 2 12.00 5.24
T¨ubingen TICA—volume priority—16 h of operation
Approach 2
Totals 2 12.00 5.24
Approach 3
Totals 2 12.00 5.24

5.1. Batteries selected and their efficiency in the cases
examined
The flat discharge curves of the zinc/monovalent systems
make them ideal for nearly constant voltage electronic appli-
cations such as watches, calculators, hearing aids and cameras;
typical capacities that range from 5 to 250 mAh [24]. These
cells also have demonstrated long storage life, retaining more
than 95% of their initial capacity after a one year at room tem-
perature. They also exhibit good low temperature performance,
and deliver approximately 70% of their capacity at 0 ◦C and
35% at −20 ◦C. Their optimal performance temperature range
is from 0 to 55 ◦C [24]. The open-circuit, nominal and cut-
off voltages of zinc-silver oxide cells are 1.5–1.6 V, 1.5 and
1.0 V, respectively [16]. The TICA and Amadeus have maxi-
mum discharge current and voltage values of 1.25 and 4.18 mA,
3.0 and 1.25 V, respectively. The discharge currents required
by these devices are smaller than majority of the devices listed
in Table 4. However, the desired battery cycle lifetimes for the
TICA and Amadeus are much longer than desired for majority of
the devices listed in Table 4. Thus, in comparison to many other
common electronics, our devices require batteries that are high
in energy density and specific energy and much less demanding
in regards to power density and specific power.
5.2. Key difference in power requirements for implanted
and explanted or other systems
Presently, biomedical implants such as neurostimulators,
drug pumps and implantable defibrillators require high pulse
power and long battery life, wherein steady current discharge
range could be 0.5–50 mA, and pulse discharge could be up to
several hundred mA [13]. The devices examined here, the TICA
and Amadeus, have maximum discharge current and voltage val-
ues of 1.25 and 4.18 mA, 3.0 and 1.25 V, respectively, with no
noted spikes in the current profile.
Approach 1, a homogeneous power supply system based on
the aggregate system profile, provided the best and, interestingly,
identical solutions for both the TICA [5–8] and Amadeus (6.0 g,
2.62 cm3, 1 cell [9–11]) implants in terms of smallest mass,
volume and number of cells amongst the three approaches—a
Quallion QL0170E, lithium polymer cell (6.0 g, 2.62 cm3, 1
cell). The optimal solution using the same criteria of mass,
volume and number of cells, found for the WIMS-ERC envi-
ronmental monitor testbed from our previous work [1], however,
was obtained from Approach 2, power selection based on divi-
sion of the power requirements based on power ranges of micro-,
milli- and Watt power. In this work, a hybrid solution consist-
ing of a thin-film lithium-free cell, 2 Ultralife UBC64130/PCM
lithium-ion cells and 5 Ultralife UBC422030/PCM lithium-ion
cells were selected. Approach 1 provides the best solution in
terms of mass and volume for the implantable system because
there are no current, voltage or power spikes/pulses in the power
profile, thus eliminating the gains associated with the use of high
power density and specific power materials for pulses and high
energy density and specific energy materials for the flat portions
of the power curve.
Both the WIMS-ERC cochlear and EMT call for use of
either lithium or lithium-ion electrochemistries because they fall
within the high specific power and high specific energy power
range for secondary batteries (Table 2(b)). However, complica-
tions associated with the cycling behavior of secondary cells
may make their application in implantable systems problem-
atic. Some workers (e.g. [8]) have identified several areas of
risk for the use of lithium-ion, lithium polymer, nickel cadmium
and nickel metal hydride; similar problems are associated with
lithium iodine cells used in cardiac pacemakers [8]:
1. Cellpackagingleakscanresultinlossofelectrolyte,resulting
in corrosion damage of electronics. All cell seals must adhere
to the standard MIL STD 883D.
2. Outgassing of oxygen and hydrogen at high rates of dis-
charge, cycling over an extended periods, or charge reversal
for certain arrangements of cells, can all lead to pressure
buildup and unavoidable deformation of cell housings in
these necessarily sealed systems.
3. High discharge rates and cycling for extended periods of time
can result in elevated temperatures that can lead to heating
of the external housing of the cell, implant and surrounding
tissue.
Capacity fade and cell swelling in lithium primary cells due
to chemical reaction of the electrodes with the electrolyte and the
passivation layer have led workers (e.g. [13]) to propose hybrid
primary battery systems of lithium iodine and lithium man-
ganese dioxide cells, to power implantable defibrillators. When
secondary cells were examined for our testbed cases, lithium-ion
cells were chosen for both the Amadeus and TICA devices and
Approach 1 provided the best results for mass (6.0 and 6.0 g) and
volume (2.62 and 2.62 cm3) for both cases, respectively. How-
ever, if lifetime is the foremost consideration in battery selection,
hybrid solutions clearly offer the best result for TICA device,
wherein battery cycle life for Approaches 2 and 3 were twice
the number of cycles (for both non-recharge and re-charge sce-
narios) calculated for the system resulting from application of
Approach 1.
This is not the case for the Amadeus device, which is operated
at a higher discharge current than the TICA device. Here, the
number of duty cycles calculated, when recharging is a factor,
is essentially the same for all approaches. The only exception
is for the microWatt range, wherein the discharge current is so
small (80 ␮A) that the number of cycles is an order of magnitude
higher than for the other cases. The impact of capacity fade as a
function of cycle is seen in the solution for the Amadeus, where
Approach 3 provides more duty cycles before requiring battery
recharge. However, the over number of duty cycles provided by
the configuration of two cells is nearly equal to those provided
by Approach 1.
We have considered the use of voltage regulators and oper-
ational amplifier to adjust for voltage in POWER. A prob-
lematic effect of these components is the generation of heat,
in implantable applications: in general, tissue can only dissi-
pate temperature gradients of less than 2 ◦C in the temperature
range of 37–41 ◦C [63]. Self-heating of voltage regulators and

operational amplifiers does not entirely prohibit their use in
implantable devices, but does merit further investigation on the
limits of their usage.
5.3. Lifetime of power designs for applications studied
There are a number of valid reasons to select primary, versus
secondary systems, for implantable applications, even if lifetime
is somewhat reduced. Chiefly, recharge of secondary systems
exposes the patient to potentially high currents, and introduces
other possible system failures. As described in Sections 4.2 and
4.3, the best primary systems had significantly reduced lifetime
over the best secondary systems examined here (i.e. 28/25,800
and 1.65/3210 battery cycles, for primary/secondary systems,
respectively, for the TICA (3001) and WIMS-ERC when sub-
jected to 16 h of operation). But the continuous development
of new primary power sources, along with diminishing power
demands in microcircuitry, may ultimately make primary sys-
tems more attractive.
For the longer lifetime, hybrid secondary systems, a weak-
link lifetime was reported, i.e. the lifetime of the shortest lived
power supply was reported as the system lifetime. This may
be rather overly conservative, since loss of low- or midrange
power might be reasonably compensated for by on-board cir-
cuitry shunting to the high power system. In any event, a logical
and necessary step in hybrid systems is to develop a protocol
for warning systems on essential and nonessential power, so
that continuous diagnostics can be run in these life-preserving
devices.
We also examined limitations on lifetime due to capac-
ity losses, which in turn are linked to operating conditions.
In batteries, the level of acceptable irreversible capacity loss
(ICL) greater than 20% over a 1–2 year period is generally
considered tolerable in portable electronic device batteries,
e.g. personal computers and cellular phones [12], but a satel-
lite battery must often retain 80% of its initial capacity for
18 years or more [12]. In the case of implantable systems,
the rate of battery capacity fade as a function of cycle has
not, to our knowledge, been previously examined. However,
implantable devices that prevent and/or limit life threaten-
ing physical malfunction require higher standards for battery
capacity fade than devices, such as the ones we have stud-
ied here, where failure of the devices is not necessarily life
threatening.
Low discharge currents allow for optimal capacity from high
energy density cells. Approaches 2 and 3 provided superior sys-
temsfortheimplantabledevices,intermsofcyclelife.Inthecase
of the TICA device, systems designed using Approaches 2 and
3 required more cells, two QL0170E cells, resulting in ∼50,000
duty cycles (including re-charge cycles). Approach 2 provides
the best solution for the Amadeus device in terms of battery
lifetime (∼96,400 cycles for microWatt and 3280 cycles for the
milliWatt power ranges, respectively). So, although Approach
2 does not provide the optimal solution in terms of the mass
and volume for the implantable systems, gains in battery cycle
life can be achieved with this technique. Since the power pro-
files for both implants were small in comparison (65–750 ␮W
[TICA] and 0.24–12.3 mW [Amadeus]) to the WIMS-ERC-
EMT (18 ␮W to 3.69 W), the key design factor for the fully
implantable system is battery cycle lifetime. Approaches 2 and
3 provide higher battery cycle lives because the power require-
ments are divided amongst power ranges (Approach 2) or power
sites (Approach 3). These implantable devices have discharge
current requirements that are small in comparison to many
electronic appliances, which generally require several hundred
milliWatts for operation (Table 4).
5.4. Effect of capacity loss profiles on selection of power
elements
Though generally, a nonlinear relationship between capac-
ity and discharge current is expected [64,42]. Some work has
been done to interrogate this relationship in specific systems;
for example, nonlinear degradation of capacity as a function
of discharge current in zinc-silver oxide cells appears to result
from reduced theoretical voltage and side reactions [65]. How-
ever, at present, there is insufficient support from a broad range
of electrochemical studies to support use of a single model.
Thus, in this present work, we considered polynomial, log-
arithmic and exponential fits to best fit experimental data,
obtained from our experiments and manufacturers’ published
data. The expressions are applicable within specific discharge
ranges noted in Table 11, and we state emphatically that these
relationships are not meant to be used to extrapolate behavior
outside of the bounds directly tested.
Consideration of capacity as a function of discharge current
allowed for inclusion of batteries that would have otherwise been
eliminated, if only high capacity values at very low discharge
rates provided by manufacturers were considered. For exam-
ple, Energizer suggests a nominal battery load of 100 k for
operation of cell 337 [61]; we demonstrated that these cells can
operate at loads up to several magnitudes lower, e.g. 1.25 k
(Table 11(a) and (b)). Thus, this battery can be considered for
applications where it would have otherwise either been elimi-
nated (from selection based on a 100 k requirement), or in a
case wherein a larger number of batteries was suggested, i.e. 100
cells, to meet a higher load.
Batteries were tested at lower discharge resistance values
than suggested by the manufacturer, to determine capacity ver-
sus discharge currents, at high currents. Cell fabrication and
use of additives [66,67] both play key roles in cell capacity,
as shown by the data in Table 11; cells having nearly identical
shape can exhibit very different capacities, e.g. Energizer ver-
sus Renata 337 cells. Other important factors affecting capacity
include storage time and temperature; as with any commercial
cell, these conditions cannot be fully known a priori, and thus
cannot presently be modeled.
Consideration of capacity fade as a function of both cycle
number and discharge current can provide a better estimate of
battery cycle life. POWER calculates the fraction of capacity
provided by a cell with each cycle. These values are used to com-
pute the number of battery cycles provided per recharge, where
the battery configuration identified by POWER is expected to
satisfy at least one duty cycle before recharge.

5.5. Power range device allocations
Currently, POWER transfers devices to higher power ranges
in order of descending voltage value, until the power range
requirements are satisfied. POWER, does not, however, go
through each combination of devices within a power range
to determine which configurations result in the minimal num-
ber, mass and volume of batteries; thus, though the solution is
improved over the original algorithm, it is not necessarily the
optimal one. In our prior work [1], power ranges were arranged
by first assigning portions of each device power profile into
appropriate power ranges, e.g. portions less than or equal to
1 mW were assigned to the microWatt range, those greater than
or equal to 1 mW and less than one Watt were allocated to the
milliWatt power range. Portions greater than one Watt were
assignedtotheWattpowerrange.Arrangementofdeviceswithin
power ranges according to voltage is effective because binning
devices with voltage requirements reduces the number of batter-
ies placed in series or the number of op-amps/voltage generators
needed.
5.6. Power site considerations
The current method of assignment based on descending
ranking of values led to some moderate system overdesign.
For example, suppose a system of five devices having voltage
requirements of 17, 16, 3, 1.5 and 1.2 V required two power site
locations (Table 12(a)–(c)). According to the current method of
voltage assignment, a 17 V would be assigned to site 1 and 16 V
would be assigned to site 2, which would require a minimum of
five lithium-ion cells for site 1, and five cells for site 2. How-
ever, the number of batteries placed in series to accommodate
the voltage requirement could be reduced by placing both the 17
and 16 V devices on one site, and the remaining three devices
on the other. Clearly, one site could be allocated to high voltage
applications and the other could be dedicated to lower voltage
application.
Also, the current assigned to each power site by POWER is
the product of the surface area ratio (surface area of individual
site to the sum of site areas) and maximum required current. If
the resulting current is less than current requirements of devices
surrounding the site, additional power programming is required
to combine current contributions from multiple sites. Obviously,
this eliminates the benefits of a ‘stand-alone’ system. In the cases
examined here, the solutions provided by Approach 3 were quite
close (in number of cells, mass and volume) to those recom-
mended by Approaches 1 and 2. However, this was not the case
for the WIMS-ERC-EMT system, where values of mass and vol-
ume were in close range of Approaches 1 and 2, but the number
of cells was 3.6 and 8.1 times those for Approaches 1 and 2.
5.7. Masses and volumes of power bundles
Since most manufacturers select power supplies post facto,
Approach 3 provides a means for designing to meet specific
surface area and volume constraints. The surface area used in
POWER, however, is quite conservative, in that the value of sur-
face area recorded in the POWER database is the entire surface
area of the battery. Specifically, if the cell is a rectangular pris-
matic cell, the surface area is the sum of the area of all six faces.
This could lead to elimination of some cells that may meet the
area constraints on one side.
Table 14
Solutions generated by POWER for the Amadeus cochlear implant (secondary batteries)
Manufacturer Part No. Total No. No. of cycles (no
battery re-charge)
No. of cycles
(battery re-charge)
Total mass (g) Total volume
(cm3)
WIMS—Amadeus (2005)—Cl—mass priority—16 h of operation
Approach 2
Totals 2 10.00 4.43
Approach 3
Site 1 Ultralife UBC641730/PCM/UMC005 1 7.34 3220 4.50 2.23
Totals 2 9.00 4.46
WIMS—Amadeus (2005)—Cl—volume priority—16 h of operation
Approach 2
Totals 2 10.00 4.43
Approach 3
Totals 2 9.00 4.46

Table 15
Commercial biomedical devices [68–74]
Implantable device Medical condition Description of device Location of device Battery type Battery
lifetime
Device volume (cc)
and mass (g)
Cardiac pacemaker
[64]
Conduction disorders
(bradycardia); heart
failure
Three parts: pulse
generator, one or two
pacing leads and a
programmer
Pacemaker: implanted under
the skin in upper chest,
attached to one or two leads,
which are placed next to or in
the heart muscle
Lithium iodine
(primary)
2–10 years Pulse generator:
8–16.6 cc, 18–37 g,
leads: 46–58 cm
Cardiac defibrillator
[42]
Ventricular and atrial
tachyarrhythmi a and
fibrillation
Three parts:
defibrillator, one or
two pacing leads and
a programmer
Defibrillator: implanted under
the skin in the upper chest
and is attached to one or two
leads, which are placed next
to or in the heart muscle
Lithium iodine
(primary)
5 years Defibrillator:
34–65 cc, 70–118 g,
leads: 65–110 cm
Muscle stimulators
[65]
Urinary and faecal
incontinence;
gastroparesis
Five parts:
neurostimulator,
programmer, an
extension, a lead, and
control magnets
Neurostimulator: implanted
subcutaneously in the
abdomen; lead placed
adjacent to sacral nerve and
attached to neurostimulator
with extension
Lithium iodine
(primary)
6–9 years Stimulator: 34 cc/42 g
Neurological
stimulators [66]
Tremor (e.g. due to
Parkinson’s disease);
pain management
(lower leg and back)
Fully implanted
system:
neurostimulator, lead,
extension,
programmer, patient
programmer, control
magnet
Battery: implanted or worn
externally; neurostimulator:
placed under skin in abdomen
or chest cavity for
Parkinson’s; lead: placed near
spine for pain and in brain for
Parkinson’s, extension
connects lead and the
stimulator. If external system
is used, antenna must be
placed on skin with adhesive
patch to receive stimulation.
External
system: 9 V,
internal:
lithium iodine
(primary)
4–6 weeks
(9 years)
Pulse generator:
8–16.6 cc, 18–37 g
leads: 46–58 cm
Cochlear implants Hearing disorders Consist internal and
external components
Internal components: implant
package implanted in
temporal bone behind the ear
and electrode array is
introduced into inner ear
(cochlear and labyrinth);
external components:
microphone, speech
processor, and external cable
[67]
AA batteries
or specialized
lithium-ion
batteries
3–5 days Depends on
manufacturer
Monitoring devices Syncope; seizures Consist of electrodes
on the surface that
sense the hearts
electrical activity [68]
Recorder: placed in upper
chest cavity; activator placed
over heart after seizure to
save response information
Primary 1 year 8.8 cc
Drug pumps Pain caused by:
cancer and its
treatments, injuries,
diabetes;
(external/internal
pumps), - spasticity
(intrathecal baclofen
pumps)
Drug delivery system
to treat pain:
implantable pump,
intrathecal catheter,
external programmer
[69]
Pump: placed in abdominal
subcutaneous pocket;
catheter: inserted into
intrathecal space of spine,
and tunneled under skin and
connected to the pump
Primary 3 years 10–80 cc
Left ventricular assist
devices
Heart failure; bridge
to transport or
recovery
Three components:
pump, tube and power
pack
Pump device is implanted
into the upper part of the
abdominal wall; tube from
the pump fits into the left
ventricle, and another tube
extends outside of the body
and is attached to a small
battery pack worn on a
shoulder holster [70]
AC outlet or
two 12 V
secondary
batteries
5–6 h 119.025 cc, 280.66 g

5.8. Extension of mass, volume and area target values
The code currently examines a minimum of 10 cells, e.g.
if only one power solution meets the mass target set by the
user, an additional nine power configurations are examined to
assure that the configuration also best meet the cell number, areal
and specific energy requirements. The advantage of finding the
minimum values of solution to meet target values of mass and
volume (10 cells), minimum number of cells (5 cells) and sur-
face area requirements (2 cells) so that the algorithm does not
converge to a solution in one iteration. Thus, some battery con-
figurations that meet the immediate mass or volume target do
not necessarily provide the best specific energy or energy den-
sity requirements. As the number of batteries in the database
increases, the need for increasing the target values in order
to have several available solutions should diminish. Selecting
from among the 189 primary and 60 secondary cells in the bat-
tery database (Appendix A), only 1 cell configuration met the
volume constraint and number of cells constraint (2 cm3 and 1
cell) for the Amadeus, the Ultralife UBC322030. However, the
solution actually provided by POWER, the Quallion QL0170,
though slightly higher in volume (2.6 cm3) provides a higher
energy density, of 268 Wh L−1, than does the Ultralife cell,
223 Wh L−1.
Use of the total surface area of the cell does appear to
eliminate batteries that may be feasible solutions if assem-
bled on a certain face or side. For example, the solution
provided by POWER for the Amadeus, was the Quallion,
QL0170 lithium polymer cell, with a total surface area of
12.41 cm2. The target area was multiplied by 1.25 until a
minimum of three cells met the new target surface area,
since none of the battery configurations met the original tar-
get area constraint (1.0 cm2). This resulted in identification
of three that met the volume, number of cells, energy den-
sity and new area constraints: the Quallion QL0110V (1 cell,
0.0026 L, 153.62 Wh L−1 and 12.41 cm2), Quallion QL0100E
1 cell, 0.0018 L, 223.07 Wh L−1 and 9.34 cm2) and Qual-
lion QL0170E (1 cell, 0.0026 L, 268.38 and 0.0026 L). How-
ever, a cell that was smaller in volume that did not meet the
area constraint was the Ultralife UBC641730 (1 cell, 0.0022 L,
330.41 Wh L−1 and 15.08 cm3). Because the surface area of
largest face of the QL0110, QL0100, QL0170 and UBC641730
are 3.28, 1.248, 3.28 and 5.58 cm2, respectively, none met the
surface area target, but all were closer to the target values than
the total surface area of the entire cell.
5.9. Use of secondary versus primary cells
Among the primary cells, the most common electrochemistry
that our algorithm selected was the zinc-silver oxide; lithium
cells were selected only for the microWatt power range.
Secondary cells selected by POWER for the cochlear implant
(16-h operation) weigh less (<5 g, per Tables 13 and 14) than
some power systems currently used by commercial cochlear
implants (Table 15 [68–74]), such as a 23 g alkaline cylindri-
cal cell (Energizer 391-AA [61]). As expected, Approach 2,
through at a penalty of slight increases in mass and volume,
provided a higher number of cycles than Approach 1, with and
without recharge. It can be seen in Tables 13 and 14 that in
all cases, there obviously significant increase in the number of
cycleswhenrechargeabilityisincluded,butalsoatlowdischarge
current, for the microWatt power range.
Because the CI operates at a higher voltage than the TICA
device (3.0 V versus 1.25 V), the number of cells required for the
former case, for all Approaches. Although Approach 3 presents
the smallest mass and volume for all approaches, it requires the
highest number of cells (six cells in two bundles); its inherently
greater complexity makes it somewhat less appealing than the
other approaches. The lifetime for all primary solutions was
limited to two cycles.
5.10. Cost analysis
From Appendix A, we see that on average, primary cells
meeting the design constraints of the testbed are less expensive
than secondary cells. Further, most primary cells listed in the
database could be purchased readily online, while the secondary
cells were often sold by whole sellers, who required purchase
of several hundred cells.
6. Conclusions
Based on the volume constraints (2 cm3) specified by the
workers at Tübingen university in Baumann group [5–8] for
the TICA (LZ 3001) device, the most suitable power solution
would be the one identified by POWER for Approach 1, sec-
ondary cells. Consisting of just 1 cell type Quallion QL0170E
(2.62 cm3), this solution had a volume ∼24% higher than the tar-
get value, 2 cm3. As far as the lifetime is concerned, this solution
can provide power for 28 cycles of 16 h each, without need to
recharge (448 h, i.e. 18.6 days). Our algorithm also accounts for
rechargeability and capacity fade as cells are recharged; there-
fore, the actual lifetime of 26,000 cycles of 16 h, i.e. 416,000 h
or ∼48 years of continuous use. This solution provides a lifetime
10 times longer than the Ni–Cd battery pack that was designed
in 1998 [6,8] for the TICA device.
For the WIMS-ERC Amadeus CI [9–11], the best solution
among the power sources our code identified was the one of
Approach 2, secondary cells. Specifically, a cell type Qual-
lion QL0100E was selected to fulfill the power requirements
of the microWatt range sub-devices (microcircuits and micro-
processors) and a Quallion QL0170E cell for the milliWatt range
(electrode array). The calculated lifetime of this system would
be 3280 cycles, corresponding to ∼6.7 years of continuous use.
Accounting for system shutdown during 8 of 24 h of usage
(sleep), the actual lifetime becomes ∼10 years.
The primary power solutions presented in the current study
allowed only a few days’ operation. Even so, primary cells
deserve further investigation as they present some advantages
over secondary power sources. Specifically, primary cells do
not rely on patient compliance to operate the implant [75]. Fur-
ther, primary cells exhibit less outgassing than secondary cells,
and thus pose fewer safety concerns in that area [17,18].

Employing a larger volume battery may be a tradeoff that
would allow higher reliability and safety. For a volume of
∼6 cm3 (corresponding to 200 Renata 337 cells), a lifetime of
more than 2 years can be achieved (∼500 cycles of 16 h). How-
ever, incorporation of 200 cells would certainly increase the
probability of failure, which should be weighed in selection of
the final design.
7. Future Work
7.1. Evolution of POWER
Currently the POWER battery database of consists of 189
primary and 60 secondary cells. Additional batteries and other
types of power supplies should certainly be included, to contin-
uously take advantage of design innovations.
POWER currently calculates recharge cycles by assuming
that the cells are only recharged after at least one duty cycle,
at 100% depth of discharge. However, batteries often provide
better cycle life when they are recharged at higher levels of
DOD. Thus, consideration of depth of discharge would poten-
tially allow for less overdesign, and also allow for inclusion
of power scavenging, wherein batteries could be charge during
periods of low operation or sleep mode, increasing the number
of cycles provided by the system.
7.2. New applications
Several workers have proposed the use of hybrid implantable
power systems for neurostimulators, drug pumps and defibrilla-
tors (all of which generally have power requirements in excess of
those required for pace makers) to combat problems generally
associated with implantable batteries: lifetime, swelling (vol-
ume change), self-heating and capacity fade [13]. Defibrillators
uselithium-silveroxovanadiumandlithium-manganese-dioxide
cells for power, which are operable at relatively high rates of
discharge [13]. Lithium iodine cells are commonly used in pace-
makers [13,14].
Most pacemakers consist of a pulse generator, pacing leads,
and a controller. The pulse generator and controller have inter-
mittent power profiles, which allow for longer battery lifetimes
than continuously-discharged devices. However, the solid elec-
trolytes used in lithium technologies may prevent their use in
cochlearimplants,duetorequiredhighdischargecurrentsneces-
sitated by the high internal resistance in such cells.
These devices, along with more recent devices employing
telemetry for physiological monitoring, often outside the clini-
cal setting, have created a need for increased discharge current,
although not necessarily greater energy capacity [14]. A num-
ber of potential power sources have been examined for such
applications, including biogalvanic cells [14]. Nuclear batteries
such as those using plutonium 238 as a fuel [14] have also been
proposed. However, the extreme toxicity of these materials [14]
may preclude their use, even under seal.
Other new elements to consider in novel power supplies
include containment of potentially harmful outgas by-products,
containment of toxic active materials, implementation of
specialized power management software, development of
circuitry to monitor charge and tight control of discharge to
prevent overheating, overcharge and charge reversal in cells.
Operationally, change in temperature and volume during opera-
tion, and heat generation, must also be considered. Future work
will include these, and other considerations, in continuously
improving our present tool.
A systematic approach to selection and design of power
systems for microelectronics has not, to our knowledge, been
previously reported. The novelty of our procedure is that it takes
into account mass and volume design constraints set by the user,
and user specific energy/power and energy and power density, to
provide concrete solutions. POWER is useful because it incor-
porates all of the steps in power selection based on mass and
volume, and provides a rational means for comparison of power
systems.
Appendix A
[22,56–59,61]
Manufacturer Part No. Capacity (mAh); Xi(I)
Renata
CR1927 XCR927 = 7.92l2 − 10.97l + 34.4, R2 = 0.95
CR1025 XCR1025 = −281.77l2 − 22.46l + 31.8, R2 = 1.0
CR1216 XCR1216 = 68.86l2 − 39.5l + 26.6, R2 = 0.91
CR1220 XCR122 = −69.75l2 − 1.93l + 38.2, R2 = 0.97
CR1225 XCR1225 = 4.17l2 − 8.94l + 48.9, R2 = 0.97
CR1616 XCR1616 = −7.12l2 − 2.33l + 50.2, R2 = 0.86
CR1620 XCR1620 = 6.51l2 − 14.7l + 69.1, R2 = 0.93
CR1632 XCR1632 = −1114.6l3 + 489.4l2 − 69.5l + 128.3, R2 = 1.0
CR2016 XCR2016 = −41.97l2 − 0.40l + 82.2, R2 = 0.99
CR2025 XCR2025 = −1632.7l3 + 765.5l2 − 101.0l + 173.9, R2 = 0.99
CR2032 XCR2032 = −814.9l3 + 468.4l2 − 85.1l+240.1, R2 = 0.99
CR2320 XCR2320 = 8.05l2 − 12.0l + 152.3, R2 = 0.98
CR2325 XCR2325 = −685.68l3 + 320.2l2 − 46.0l + 192.7, R2 = 0.96
CR2430 XCR2430 = −2.61l2 − 0.17l + 285.6, R2 = 1.0
CR2440N XCR2440N = −9.95l3 + 14.5l2 − 7.9l + 542.2, R2 = 1.0
CR2477N XCR2477N = −5.01l2 − 0.62l + 956.0, R2 = 0.99

Manufacturer Part No. Capacity (mAh); Xi(I)
Electrochem
4301 X4301 = −0.045l2 + 0.32l + 5280.7, R2 = 1.0
44230 X44230 = 1.28l2 − 40.3l + 1775.6, R2 = 1.0
3B960 X3B960 = −1.14l2+1.08l + 792.0, R2 = 1.0
3B880 X3B880 = 9.28l2 - 62.7l + 1006.2, R2 = 1.0
3B940 X3B940 = −0.156l2 − 1.09l + 1900.1, R2 = 1.0
4006 X4006 = 60.61l2 − 56.4l + 63.0, R2 = 1.0
4030 X4030 = −26.0 ln(l) + 534.2, R2 = 0.89
4161 X4161 = 0.185l2 − 4.3l + 824.1, R2 = 1.0
4260 X4260 = 0.128l2 − 13.2l + 5619, R2 = 1.0
4204 X4204 = 0.014l2 − 1.4l + 1622.8, R2 = 1.0
Capacity [Ah]
Energizer
521 X521 = −2.45 ln(l) + 3.3, R2 = 0.96
528 X528 = 3.67l2 − 11.3l + 9.0, R2 = 1.0
539 X539 = 11.02l2 − 3.48l + 0.29, R2 = 0.98
E91 XE91 = 0.42e−0.47·l, R2 = 0.92
E92 XE92 = −0.17 ln(l) + 0.012, R2 = 0.94
Manufacturer Capacity (Ah) Xi and capacity ratio [] Pc,j
Panasonic XCGR17500 = −0.05l + 0.84; R2 = 1
Pc,CGR17500 = 4 × 10−7c2 − 4 × 10−4c + 0.98; R2 = 0.98
Panasonic XCGR18650HG = +0008l2 − 0.86l + 1.84; R2 = 1
Pc,CGR18650HG = 4 × 10−7c2 − 4 × 10−4c + 0.98; R2 = 0.98
Panasonic XCGR18650A = +0.001l2 − 0.02l + 1.98; R2 = 1
Pc,CGR18650A = 4 × 10−7c2 − 4 × 10−4c + 0.98; R2 = 0.98
Panasonic XCGR18650C = −0004l2 − 0.012l + 2.17; R2 = 1
Pc,CGR18650C = 4 × 10−7c2 − 4 × 10−4c + 0.98; R2 = 0.98
Panasonic XCGA523436 = −0.18l2 + 0.14l + 0.7; R2 = 1
Pc,CGR18650C = 4 × 10−7c2 − 5 × 10−4c + 0.98; R2 = 0.97
Panasonic XCGA523450A = 0.1l2 + 0.09l + 0.93; R2 = 1
Pc,CGR523450A = −5 × 10−9c2 − 2 × 10−4c + 0.99; R2 = 0.98
Panasonic XCGA633450A = −0.0084l2 − 0.015l + 1.053; R2 = 1
Pc,CGA633450A = 6 × 10−8c2 − 4 × 10−4c + 0.98; R2 = 0.98
Panasonic XCGA103450A = −0.013l2 − 0.01l + 1.94; R2 = 1
Pc,CGA633450A = 6 × 10−8c2 − 4 × 10−4c + 0.98; R2 = 0.98
Quallion Pc,i = 5 × 10−6c2 − 0.0134c + 100; R2 = 0.99
i = QL0003l, QL0700l, QL0110V, QL0900V, QL0100E, QL0170E, QL0320E, QL010KA, QL015KA
Ultralife XUBC422030 = −333.34l2 − 35l + 149.25; R2 = 1
Pc,UBC422030 = 96.72e−00004·l; R2 = 0.98
Ultralife XUBC641730 = −250l2 − 35l + 199; R2 = 1
Pc,UBC641730 = 96.78e−00004·l; R2 = 0.98
Ultralife XUBC383450 = 11.77l2 − 33.7l + 604; R2 = 0.99
Pc,UBC36106102 = 0.057l + 96.63; R2 = 0.99
Battery type Part number Approximate cost q = quantity
Lithium polymer rechargeable UBC641730/PCM/UMC005 q = 1, $12.07
q = 12, $11.110
q = 24, $10.41
q = 48, $9.720
Lithium polymer rechargeable UBC433475/PCM/UBC001 q = 1, $17.390
q = 12, $16.01
q = 24, $15.010
q = 48, $14.000
q = 12, $11.380
q = 24, $10.66
q = 48, $9.950

Battery type Part number Approximate cost q = quantity
q = 12, $10.06
q = 24, $9.430
q = 48, $8.80
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