# What is an ultracapacitor?

Electric
double-layer capacitors, also known as supercapacitors, electrochemical double
layer capacitors (EDLCs) or ultracapacitors are electrochemical capacitors that
have an unusually high energy density when compared to common capacitors,
typically several orders of magnitude greater than a high-capacity electrolytic
capacitor.

The
electric double-layer capacitor effect was first noticed in 1957 by General
Electric engineers experimenting with devices using porous carbon electrode. It
was believed that the energy was stored in the carbon pores and it exhibited
"exceptionally high capacitance", although the mechanism was unknown
at that time.

General
Electric did not immediately follow up on this work, and the modern version of
the devices was eventually developed by researchers at Standard Oil of Ohio in
1966, after they accidentally re-discovered the effect while working on
experimental fuel cell designs. Their cell design used two layers of activated
charcoal separated by a thin porous insulator, and this basic mechanical design
remains the basis of most electric double-layer capacitors to this day. With
advances made on both materials and manufacturing process, today Tecate Group
PowerBurst® product show a superior advantage amongst all other ultracapacitors
in the market.

Generally,
capacitors are constructed with a dielectric placed between opposed electrodes,
functioning as capacitors by accumulating charges in the dielectric material.
In a conventional capacitor, energy is stored by the removal of charge
carriers, typically electrons from one metal plate and depositing them on
another. This charge separation creates a potential between the two plates,
which can be harnessed in an external circuit. The total energy stored in this
fashion is a combination of the number of charges stored and the potential
between the plates. The former is essentially a function of size and the
material properties of the plates, while the latter is limited by the dielectric
breakdown between the plates. Various materials can be inserted between the
plates to allow higher voltages to be stored, leading to higher energy
densities for any given size. For example aluminum electrolytic and tantalum
electrolytic capacitors, use an aluminum oxide film and a tantalum oxide film
as the dielectric, respectively. In contrast, Electric Double Layer Capacitors
do not have any dielectrics in general, but rather utilize the phenomena
typically referred to as the electric double layer. In the double layer, the
effective thickness of the “dielectric” is exceedingly thin, and because of the
porous nature of the carbon the surface area is extremely high, which
translates to a very high capacitance. Generally, when two different phases
come in contact with each other, positive and negative charges are set in array
at the boundary. At every interface an array of charged particles and induced
charges exist. This array is known as Electric Double Layer. The high
capacitance of an EDLC arises from the charge stored at the interface by
changing electric field between anode and cathodes.

**Figure
1: Ultracapacitor Charge Separation**

However,
the double layer capacitor can only withstand low voltages (typically less than
2.7V per cell), which means that electric double-layer capacitors rated for
higher voltages must be made of matched series-connected individual capacitors,
much like series-connected cells in higher-voltage batteries.

There
are 2 types of electrolytes used by EDLC manufacturers. One is water-soluble
and the other is non-water soluble. The non-water soluble electrolyte does
increase the withstand voltage per cell compared to that of a water soluble
electrolyte, hence producing a higher energy density. Tecate Group PowerBurst®
cells are made with non-water soluble electrolytes, and feature a small size
and light weight.

# What are Ultracapacitors advantages & challenges?

Each
application needs to be evaluated based on its requirements. Below are some of
the advantages and disadvantages when considering the use of EDLCs:

**Advantages:**

•
High energy storage. Compared to conventional capacitor technologies, EDLCs
possesses orders of magnitude higher energy density. This is a result of using
a porous activated carbon electrode to achieve a high surface area.

•
Low Equivalent Series Resistance (ESR). Compared to batteries, EDLCs have a low
internal resistance, hence providing high power density capability.

•
Low Temperature performance. Tecate Group PowerBurst® products, with their use
of patented technology, are capable of delivering energy down to -40°C with
minimal effect on efficiency.

•
Fast charge/discharge. Since EDLCs achieve charging and discharging through the
absorption and release of ions and coupled with its low ESR, high current
charging and discharging is achievable without any damage to the parts.

**Disadvantages:**

•
Low per cell voltage. EDLC cells have a typical voltage of 2.7V. Since, for
most applications a higher voltage is needed, the cells have to be connected in
series.

•
Cannot be used in AC and high frequency circuits. Because of their time
constant EDLCs are not suitable for use in AC or high frequency circuits.

The
specifics of ultracapacitor construction are dependent on the manufacturer, and
the intended application. The materials may also differ slightly between
manufacturers or due to specific application requirements. The commonality
among all ultracapacitors is that they consist of a positive electrode, a
negative electrode, a separator between these two electrodes, and an
electrolyte filling the porosities of the two electrodes and separators.

**Figure
4: Internal Cell Construction**

Today, in general, most manufacturers have adopted a cylindrical construction
method for their EDLCs. However, there are still products in the market that
use a prismatic design. Each method has its own advantages and disadvantages
which may or may not affect their use in specific applications. Tecate’s
PowerBurst® products use the round or cylindrical construction method. The cells
are constructed from activated carbon particles, mixed with a binder and then
deposited on aluminum foil. In this method, as shown in the following figure,
the electrodes are wound into a jellyroll configuration very similar to an
aluminum electrolytic capacitor. The electrodes have foil extensions that are
then welded to the terminals to enable a current path to the outside of the
capacitor.

**Figure
5: Cell Construction**

EDLCs
share the same equivalent circuit as conventional capacitors. The first order
model is represented by the circuit below. It is comprised of four ideal
components. The series resistance Rs which is also referred to as the
equivalent series resistance (ESR). This is the main contributor to power loss
during charging and discharging of the capacitor. It is also comprised of a
parallel

resistance
Rp which affects the self-discharge, a capacitance C and a series inductor Ls
that is normally very small as a result of the cell construction.

**Figure
6: First Order Equivalent Circuit**

Since Rp is always much larger than Rs it can be ignored. Also, because of the
porous material used on the electrode of EDLCs, they exhibit non-ideal behavior
which causes the capacitance and resistance to be distributed such that the
electrical response mimics transmission line behavior. Therefore, it would be
necessary to use a more general circuit, as shown in the figure 6, for
representing the real electrical response.

**Figure
7: Ladder Network**

However,
to simplify the circuit we can model the EDLC as an RC circuit. In this case
the charge stored is Q=CV. The energy stored in the capacitor in Joules
(watt-second) = 1/2CV2. Other useful formulas are discussed more in the sizing
section.

One
final note to consider in regards to EDLC, is the discharge characteristics of
the cells. Unlike batteries which can discharge a fairly constant voltage, the
EDLC cells act very similar to traditional capacitors and will drop their
voltage as they discharge their stored energy similar to what is shown in Figure
8.

**Figure
8: Ultracapacitor Discharge Curve**

# How do ultracapacitors differ from battery and traditional capacitors?

**Figure
2: Ragone Plot**

As
can be seen in Figure 2, the Ultracapacitors reside in between conventional
batteries and conventional
capacitors. They are typically used in applications where batteries have a
short fall when
it comes to high power and life, and conventional capacitors cannot be used
because of a lack
of energy. EDLCs offer a high power density along with adequate energy density
for most short
term high power applications. Many users compare EDLCs with other energy
storage devices
including batteries and conventional capacitor technology. Each product has its
own advantages
and disadvantages compared to other technologies as can be seen from the chart below:

**Figure
3: Ultracapacitors vs. Battery and Conventional Capacitors**

# What is the difference between power and energy?

Power
* Time = Energy

Power
is the rate of using energy.

Power Density vs Energy Density

# What are the key applications for ultracapacitors?

- Ultracapacitor
Functions
- Secure power
- Provides
reliable interim power, even if the primary source fails or fluctuates

- Energy storage
- Stores
energy from low power sources, enabling support for high power loads

- Pulse power
- Supplies
peak power to the load while drawing average power from the source

- User
Benefits
- Reduces the size
& weight of the battery / power source required
- Improves run-time
& battery life, particularly at cold temperatures
- Enables more
power-hungry features, being used more often
- Can remove the need
for a battery & harvest energy from clean sources
- Protects against
accidental power loss or fluctuations/interruptions
- Doesn’t need to be
replaced like batteries (unlimited discharge cycles)
- Environmentally
friendly & safe

# What is end of life and failure mode for an ultracapacitor?

In
general ultracapacitors do not have a hard end of life failure similar to
batteries. Their end of life is defined as when the capacitance and/or ESR has
degraded beyond the application needs.

Cap failure under typical use condition

Failure
under Abuse Conditions

**Over
voltage**

- Loss of capacitance
- Increase of ESR
- Bulging
- Possible venting

**Over
temperature**

- Loss of capacitance
- Increase in ESR
- Bulging
- Possible venting

**Mechanical
Stress**

- Deformation
- Broken lead
- Increase in ESR

# What is the self discharge or leakage current?

Self
Discharge: Is the voltage drop on a charged cell after a set period of time without a load.

Leakage
Current: Is the stable parasitic current expected when capacitor is held
indefinitely on charge at the rated voltage. This value is voltage and
temperature dependent.

# Series/Parallel combination of ultracapacitors?

The
voltage rating of PowerBurst® product is 2.7V per cell, which is mainly derived
from the electrochemical stability of the electrolyte and electrode materials.
The PowerBurst® family of products uses an organic electrolyte. The key
advantage of an organic electrolyte versus other (i.e. aqueous) electrolytes is
its higher voltage stability. In general, if cells are operated above their
rated voltage for a long period of time, the life is reduced. This is a result
of the electrolyte breakdown with exposure to high voltage. The amount of
damage varies based on the voltage and the amount of time the cell is exposed
to the over-voltage condition. Thus, occasional spikes above rated voltage will
not immediately affect the capacitor.

Since
in most applications the required voltage is above 2.7V multiple cells will
need to be placed in series. Depending on the required energy there could be a
need to then place multiple cells in parallel. When ultracapacitor cells are
placed in series or parallel they react very similar to conventional
capacitors. Below is a summary of key attributes when placing multiple cells in
series/parallel formation:

**Voltage**

*Series
connection*:
When placing cells in series the overall voltage is increased directly by the
number of cells in series.

Example:
4 cells (rated at 2.7V each) connected in series will have a maximum voltage of
10.8V.

*Parallel
connection*:
Placing cells in parallel will not affect the voltage.

Example:
4 cells (rated at 2.7V each) connected in parallel will have a maximum voltage
of 2.7V.

**Capacitance**

*Series
connection*:
When placing same value cells in series the system capacitance is reduced by
the number of cells placed in series based on the formula below:

Example:
4 x 10F cells (rated at 2.7V each) connected in series will have a capacitance
of 2.5F and a maximum voltage of 10.8V.

*Parallel
connection*:
Placing same value cells in parallel will increase the overall system
capacitance proportionally to the number of cells placed in parallel:

Example:
4 x 10F cells (rated at 2.7V each) connected in parallel will have a
capacitance of 40F and a maximum voltage of 2.7V.

**ESR**

*Series
connection*:
By placing same value cells in series the overall system ESR will increase
proportionally to the number of cells placed in series:

Example:
4 x 10F cells (DC ESR 75 mΩ each) connected in series will have a
total ESR
of 300 mΩ.

*Parallel
connection*:
Placing same value cells in parallel will decrease the overall system ESR
proportionally to the number of cells placed in parallel:

Example:
4 x 10F cells (DC ESR 75 mΩ each) connected in parallel will have a
total
ESR of 18.75 mΩ.

**Leakage
Current**

*Series
connection*:
Placing same value cells in series will not affect the leakage current. The
overall leakage current will be the same as the single cell**.

Example:
4 x 10F cells (Leakage current of 0.03mA) connected in series will have a total
leakage current of 0.03mA**.

*Parallel
connection*:
Placing cells in parallel will increase the overall leakage current
proportionally to the number of cells placed in parallel**.

Example:
4 x 10F cells (Leakage current of 0.03mA) connected in parallel will have a
total leakage current of 0.12mA**.

**It
should be noted that this does not take into account leakage current induced as
a result of cell balancing. In case of passive balancing the leakage current
will be dominated by the bypass resistor value. For additional information on
cell balancing refer to PowerBurst Product Guide under Design
Consideration/Interconnection section.

Product Guide

# Why do ultracapacitors require balancing? What are the balancing methods?

For
most applications a single cell at low voltage is not very useful and multiple
cells are required to be placed in series. Since there is a tolerance
difference between manufactured cells in capacitance, resistance and leakage
current there will be an imbalance in the cell voltages of a series stack. It
is important to ensure that the individual voltages of any single cell do not
exceed its maximum recommended working voltage as this could result in
electrolyte decomposition, gas generation, ESR increase and ultimately reduced
life.

This
imbalance is initially dominated by the capacitance difference between the
cells (i.e. a cell with a lower capacitance will charge to a higher voltage in
a series string). For example, if two cells of 10F each are connected in series
with one at +20% of nominal capacitance and the other at -10%, then the worst
case voltage across the capacitors can be calculated by:

Vcap1=Vsupply
x (Ccap1/(Ccap1 + Ccap2)

Assuming
Vsupply=5.4V

Vcap1=5.4
x (12/(12+9)) = 3.08V

As
can be seen, a proper cell balancing scheme needs to be placed within series
connected cells to ensure no cell sees higher than rated voltage.

Also,
when the cells are on charge for a period of time the leakage current will
dominate this difference (i.e. a cell with a higher leakage current will go to
a lower voltage distributing the voltage amongst other cells resulting in an
over-voltage). Proper cell balancing can eliminate this imbalance. There are
two balancing schemes to tackle this problem, and ensure a properly balanced
module. They are:

*Passive
Balancing*:
One technique to compensate for variations in parallel resistance is to place a
same valued bypass resistor in parallel with each cell, sized to dominate the
total cell leakage current. This effectively reduces the variation of
equivalent parallel resistance between the cells which is responsible for the
leakage current. For example, if the cells have an average leakage current of
10uA +/- 3uA, a 1% resistor which will bypass 100uA may be a good choice. By
using this resistor in parallel to each cell the average leakage current is now
110uA +/- 4uA. Introduction of this resistor has now decreased the variation in
leakage current from 30% to 3.6%.

By having the same value resistor in parallel with all cells, the cells with
higher voltages will discharge through
the parallel resistor at a higher rate than the cells with lower voltages. This
will help to distribute the
total stack voltage evenly across the entire series of capacitors.

Passive
voltage balancing is only recommended for applications that don’t regularly
charge and discharge the
ultracapacitors and that can tolerate the additional load current of the
balancing resistors. It is suggested
that the balancing resistors be selected to give additional current flow of at
least 10 times the worst-case
cell leakage current. Higher ratio can be used to balance the cells faster. A
typical tradeoff is based
on time to balance vs. leakage current. Once the system is balanced response
time to balance is less
of an issue unless a system it being severely cycled.

*Active
Balancing*:
For applications with a limited energy source or high level of cycling an
active voltage balancing
circuit is preferred since it typically draws much lower current in steady
state and only requires larger
currents when the cell voltage is out of balance. The active circuit forces the
voltage at the nodes of series
connected cells to stay below a fixed reference voltage.

In
addition to ensuring accurate voltage balancing, active circuits typically draw
much lower levels of current
in steady state, and only require larger currents when the capacitor voltage
goes out of balance. These
characteristics make active voltage balancing circuits ideal for applications
that charge and discharge
the cells frequently as well as those with a finite energy source.

# What are the temperature effects on an ultracapacitors?

One
of the main advantages of ultracapacitors is its wide temperature range. The
effect of temperature on ultracapacitor cells is two fold:

- Life: Operating at
high temperature extremes will reduce the life of the cells.
- Performance:
Operating at low temperature extremes will increase the internal resistance of
the cell.

# How to measure an ultracapacitor?

A
constant current discharge test may be useful for customer evaluation of the
product prior to application testing. All ultracapacitors are stored discharged
for safety. We recommend completely discharging any capacitors that will not be
installed into equipment.

Below
is a list of equipment required to perform a typical constant current discharge
test:

bi-directional
power supply (supply/load) OR

separate
power supply and programmable load (constant current capable)

voltage
vs. time measurement and recording device (digital scope, or other data
acquisition)

current
vs. time measurement and recording device (optional if you can trust the power
supply and load settings)

Before
testing, connect data acquisition equipment to the device terminals, and set
recording speeds as fast as reasonably possible (<<100msec preferred, the
faster, the more accurate the calculations).

**Setup**

Set
the power supply to the appropriate voltage and current limits, and turn the
supply output OFF.

The
current limit can be anything at or less than the maximum rated current for the
cell. When performing repetitive high current testing, cooling air should be
provided.

The
voltage limit is the maximum cell voltage, times the number of cells in series.
A single cell should be limited to 2.7 volts. Six cells in series (for example)
can be operated at any voltage up to 16.2 volts (6 x 2.7V = 16.2V).

Connect
the ultracapacitor to the power supply (having pre-set the current and voltage
limits).

Cooling
air may be required to keep the ultracapacitor within operating temperature
limits, depending on the test current and duration.

Connect
the voltage and current measuring/recording devices.

**Charge**

With the power supply pre-set, and the ultracapacitor connected, turn the
supply output ON.

Charge
the ultracapacitor at the appropriate current to the appropriate voltage.

**Discharge**

Note:
If using a separate programmable load instead of an integrated bi-directional
power supply, disconnect the charging power supply prior to discharging. (Don’t
simply turn it off or change its set points, as many supplies will sink current
when not regulating.)

Set
the load to the appropriate constant current, and discharge to 0.1V, or as low
as the load can be controlled.

IMMEDIATELY
remove the load once the minimum voltage is reached, allowing the device's
voltage to "bounce" back.

(The
discharge can actually be stopped at any voltage. Depending on equipment, some
units can be discharged to 0.1V, and others discharged to ½ of the initial
voltage. Values of capacitance will be slightly higher when discharged to ½
initial voltage rather than 0.1V.)

Measure
the following parameters: (reference figure 1)

Vw
= initial working voltage Vmin = minimum voltage under load

Id
= discharge current Vf = voltage 5 seconds after removal of load.

td
= time to discharge from initial voltage to minimum voltage

Capacitance
calculation:

Capacitance
= (Id * td)/(Vw – Vf) = (Id * td)/Vd

(This
change in voltage (Vw – Vf) is used because it eliminates the voltage drop due
to the equivalent series resistance)

Equivalent
Series Resistance (at “DC”) calculation:

ESR
= (Vf – Vmin)/Id

(An
LCR meter or bridge can be used to measure ESR at higher frequencies. The ESR
at frequencies up to 100Hz will typically be 50-60% of the “DC” ESR. The
capacitance will be much lower, due to the structure of the electrode.)

(Note
that calculations for Capacitance and Resistance can also be done on the
charge)

Figure
1: Representative measurement points for constant current test

**Safety
Considerations**

As
in all electrical testing, you as the investigator should take appropriate
cautions in the design and execution of the test. Proper precautions for the
appropriate voltage should be observed. Any interconnections should be sized
for the maximum anticipated current, and insulated for the appropriate voltage.
If repeated testing will be performed, cooling air may be required to keep the
test unit within its operating temperature range.

# How to size an ultracapacitor for your application?

Tecate
Group offers a large selection of ultracapacitor cells and modules for various
applications. In order to size the appropriate ultracapacitor cell for any
application, we will need to determine the system variables needed. Using this
information we can calculate the appropriate size and number of cells needed.

In
order to get a complete solution, the following parameters will need to be
defined:

- Maximum
Charged Voltage (Vmax), if different from Working Voltage then also (Vw)
- Minimum
Voltage (Vmin)
- Required
Power (W) or Current (I)
- Duration
of Discharge (td)
- Duty
Cycle
- Required
Life
- Average
Operating Temperature

The last three parameters are used to determine the life degradation factor to
use. This is not discussed here but is a consideration to
be taken by user. In order to know the appropriate size and also the number of
cells required one needs to perform some simple sizing exercise. Most
applications can be categorized into two categories: constant current
applications or constant power applications. We will examine each one
separately.

During the discharge cycle of an ultracapacitor there are two parameters to
consider. The drop in voltage due to internal resistance, and the drop in
voltage due to capacitance, as shown in Figure 1.

**Figure
1: Discharge Curve**

As
can be seen above during a discharge cycle the initial drop in voltage is due
to the Equivalent Series Resistance of the part (ESR). The amount of drop is a
function of the ESR and discharge current as indicated by the equation below:

**Equation
1**** **

dVESR=
I * ESR

After
the initial instantaneous drop due to ESR, the capacitor will discharge
according to its capacitance and discharge current as indicated in equation 2:

**Equation
2**** **

dVcap=
I * td/C

By
placing these two equations together the total voltage drop can be calculated
per equation 3:

**Equation
3**

dVTotal
= I *td/C + I * ESR

A
brief overview of the variables in above equation:

dVTotal=
The drop in voltage when the capacitor is discharged. This is the difference
between the Vw and Vmin as

indicated
on Figure 1. As can be seen in equation 3 this is the sum of the resistance and
capacitance drop.

*Note:*

*Allowing
a larger dV will reduce the capacitance size used. Typically by allowing the
capacitor to drop to ½ Vw,*

*75%
of the capacitor energy is discharged.*

I=
Current in Amps used to discharge the capacitors. For equation 3 we assume this
to be a constant current

discharge.

td=
Duration in Seconds to discharge the capacitor between Vw and Vmin.

C=
the total capacitance of the ultracapacitor. If a single cell is used, then it
is the cell capacitance. If multiple cells are used the equivalent capacitance
is based on the number of capacitors in series or parallel. For capacitors in
series the capacitance is additive at 1/C. For capacitors in parallel the
capacitance is additive.

**Equation
4**

CTotal
= Ccell * (# parallel/# series)

ESR=
the total resistance of the ultracapacitor. If single cell is used then it is
the cell resistance. If multiple cells are used the equivalent resistance is
based on the number of capacitors in series or parallel. For capacitors in
series the resistance is additive. For capacitors in parallel the resistance is
additive at 1/resistance.

**Equation
5**

ESRTotal=ESRcell
* (#series/#parallel)

**Constant
Current Application**

In
the example below we will look at an application where the load requirement is
a constant current.

**Example
1:**

Let’s
assume we have a requirement as follows:

Vmax=
15V

Vmin=
9V

I=
4 A

td=
5 sec

Using
the information above we can use two methods to try and find the appropriate
capacitor size.

**Method
1:**

We
can ignore the ESR effect on the voltage drop to get an estimate on the
capacitance and then resolve Equation 3 to see if the size picked is
appropriate with the ESR effect, and if not, to increase to the next cell size.

So
by taking out the ESR portion of equation 3 and solving for C we get:

**Equation
6**

C
= I*td/dV Equation 6

By
substituting the above parameters we get:

C
= 4*5/ (15-9) = 3.33F

Note
that this is the total capacitance at 15V. Since each ultracapacitor cell is
typically rated at 2.7V, then we divide Vmax by 2.7 and round up:

15/2.7
= 5.5, so 6 cells in series.

Using
equation 4 and assuming one parallel set then we estimate the cell needed will
be in the range of:

Ccell=
CTotal * # in series = 3.3 * 6 = 19.8F

Looking
at the product offering we see the closest size is a 22F cell with an ESR value
of 45 mohm. Taking the capacitance and ESR value of this cell we can plug it
into equation 3 to see if the total voltage drop is within the application
limit of 6V (note that we need to calculate the system C and ESR for 6 cells in
series):

dVTotal=
(4*5/ (22/6)) + (4* (.045*6)) = 5.46 + 1.08 = 6.54V

As
can be seen the above voltage drop is more than the 6V allowed. Therefore we
need to move to the next size up cell and redo the calculation. Also please
note the values used are the initial specifications. To take into consideration
end of life degradation one needs to apply the degradation factors to the
capacitance and resistance numbers.

**Method
2:**

For
the same parameters as above we can solve for the cell value using the RC time
constant. The RC time constant of an ultracapacitor is the product of its
capacitance value and resistance value. For the Tecate product TPL series we
can use 0.8 seconds as an average.

Since
R*C=0.8 seconds, then R=0.8/C

By
substituting the above in equation 3 we get the following:

dVTotal=
I*td/C + I*0.8/C

Solving
for C, we get:

C
= I/dVTotal*(td+0.8)

By
substituting the value give (dV=6, I=4 and td=5) we solve the above equation
for C:

C=
3.86F

Please
note this is the stack capacitance and we need to solve for the cell
capacitance. Based on the Vmax of 15V we determined we need 6 cells in series.
So using equation 4 we solve for the cell capacitance as follows:

Ccell=CTotal*
# series

Ccell=
3.86 * 6 = 23.16F

We
will then use the data sheet to round this up to the closest cell available,
which in this case is 25F.

**Constant
Power Application**

In
the example below we look at an application where the load is on a constant
power discharge.

**Example
2:**

Let’s
assume we have a requirement as follow:

Vmax
= 15V

Vmin
= 9V

Power
(P) = 60W

td
= 5 sec

Using
the information above we can use two methods to try and find the appropriate
capacitor size.

**Method
1**:

A
simple method will be to calculate an average current based on the above
parameters and then apply the constant current sizing method to calculate the
most appropriate cell:

Imax
= Power/Vmin = 60/9 = 6.67 Amps

Imin
= Power/Vmax = 60/15 = 4 Amps

Iavg
= (6.67 + 4)/2 = 5.34 Amps

Using
the method 1 sizing of the constant current application mentioned earlier we
get a cell value of 26.7F. Rounding it up to the next cell value we find out
the 30F cell is the appropriate cell for this application. Please note we
typically recommend oversizing the cell by 20-30% to accommodate any
degradation over time.

**Method
2:**

We
can also calculate the appropriate cell size based on the energy needed. This
method works well for low power application where the loss due to ESR is
minimal.

Energy
Needed = 60W * 5 sec = 300 Joules

The
equation calculating the energy stored in a capacitor is:

**Equation
8**** **

E (joules) = 0.5 * C (V^{2}max – V^{2}min)

Substituting
the values in equation 8 and solving for C we get:

300
= 0.5 * C (225 – 81)

C
= 4.17 F

Since
we have 6 cells in series the cell capacitance needed is:

4.17
* 6 = 25.02 F

Note
that the above method did not take into account any losses due to ESR, thus
resulting in a slightly smaller cell value.