The Search for Clean Vehicles




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Conference session A8 Paper # 2128

Double-Layer Capacitors and Electric Vehicles


Andrew Wolfe (adw50@pitt.edu), Tom Pearce (tep14@pitt.edu)



Abstract An increase in awareness of the pollution caused by gasoline powered vehicles and the dependency of foreign oils has provided the challenge to make an electric powered automobile that could run conveniently for the average consumer. Consumers tend to stay away from electric cars because the batteries that power them have a poor lifespan, low power efficiency and they take a long time to charge [2]. One solution being researched is replacing the battery with an electric double-layer capacitor, also known as a super capacitor. In this paper, the efficiency and lifespan of these capacitors is explained in depth to provide an understanding of why the double layer capacitor is the best option for electric vehicles’ energy source. Electrical engineering is used to explain why this certain capacitor would allow an electric vehicle to be charged repeatedly without a decrease in lifespan. How the use of a double layer capacitor would provide consumers with an electric vehicle that has more power than a vehicle powered by a battery is also explained. The paper also discusses history of the supercapacitor, the types of super capacitors and the ethics that need to be considered when using these capacitors in electric automobiles. The ethics of using double layer capacitors in electric automobiles is taken into consideration as well as how this technology will provide consumers with convenient transportation.


Key Words—Super capacitor, Double-Layer Capacitor, Electric Vehicle, Electrochemical Battery, Electric Field

The Search for Clean Vehicles


For over a century, we have depended on gasoline in order to power our vehicles. Gradually, the negative effects of gasoline usage have made the scientific community shift their focus to new, cleaner sources of energy. One of these topics for the past fifty years has been electric vehicles.

Lately, electric vehicles have adopted the use of a lithium ion battery as its power source. These batteries are not capable of producing the same power as a gasoline engine. Also, the average electric car battery requires replacement every two to three years [1]. These disadvantages have kept the majority of consumers out of the electric vehicle market.

Recently, a device known as the supercapacitor has been discussed as a possible energy source for electric vehicles. The use of supercapacitors in electric vehicles will help remedy the aforementioned problems, but will also bring their own obstacles as well.

History of the Double-Layer Capacitor


The idea of the double-layer capacitor came about in the year 1853. A German physicist by the name of Hermann Von Helmholtz was the first person to give a description of how the double-layer capacitor would function. However, this idea took approximately a century to be put to use [3].

In the year 1957, the General Electric Company came up with the first patent of a capacitor using the idea of double-layer capacitance first presented by Helmholtz although no products were manufactured as a result of this patent. The capacitors made their debut in the market during that same year thanks to a company from Japan named Nippon Electric Company (NEP). The capacitors created by NEP were used to back up stored memory in the computers they produced. The technology that NEP used for its capacitors was originally the work of another company named Standard Oil Company, Cleveland, Ohio (SOHIO) before NEP licensed this technology. In 1970, SOHIO came up with its second capacitor, this time being in the shape of a disk. Once again NEP received the license for its technology in the year 1971 [3].

NEP was responsible for the production of the super capacitor, which became the first double layer capacitor to become popular in the market. The term supercapacitor has become one of the most widely used names for double-layer capacitors due to their success. During the 1980’s several other companies started producing their own double-layer capacitors using new innovations. Some of these include the gold capacitor produced by Matsushita Electric Industrial Company (Panasonic) and the ultra capacitor produced by the Pinnacle Research Institute (PRI). PRI is an organization that deals with military equipment. PRI used its ultra capacitors to enhance the military’s laser weapons and missile guidance systems [3]. As the double-layer capacitor continued to develop, numerous companies adopted their own versions to manufacture. Some companies that currently manufacture supercapacitors are listed in table I below.


TABLE I

Current Manufacturers of Supercapacitors for Utility-scale Applications [3]

Company name

Device name

Country

Voltage range (V)

Capaci-tance (F)

AVX

Bestcap

USA

3.5-12

0.022-0.56

Cap XX

Super Capacitor

Australia

2.25-4.5

0.09-2.8

Copper

Power stor

USA

2.5-5.0

0.47-50

ELNA

Dyna cap

USA

2.5-6.8

0.033-100

ESMA

Capacitor modules

Russia

12-52

100-8000

EPCOS

Ultra-capacitor

USA

2.3-2.5

5-5000

Evans

Capattery

USA

5.5,11

0.01-1.5

Kold Ban

Kapower

USA

12

1000

Maxwell

Bootstcap

USA

2.5

1.6-2600

NEC

Super capacitor

Japan

3.5-12

0.01-6.5

Nesscap

EDLC

South Korea

2.7

10-5000

Panasonic

Gold capacitor

Japan

2.3-5.5

0.1-2000



How Supercapacitors Work


The main idea of the double-layer capacitor which Helmholtz came up with is the use of an electric double-layer to store energy. This double-layer is formed by the use of an electrode and an electrolyte [4]. When the materials being used as the electrode and electrolyte are brought into contact with one another, the charges are distributed between the two with an extremely small separation, creating the electric double-layer [4]. When an external electric field is applied to the capacitor, this double layer is able to store energy. This energy can then in turn be harnessed by an external circuit.

The type of material that is used as the electrode in supercapacitors has a strong influence on the capabilities of the capacitor as a whole [3]. This is something that would have to be taken into serious consideration when developing a supercapacitor to power an electric automobile. The capacitance, or amount of energy a capacitor can store, is partially determined by what material is being used as the electrode [3]. In an electric automobile, the capacitance of the supercapacitor would directly correlate with the distance the automobile would be able to travel.

Some of the most common materials that are used as electrodes in capacitors include carbon, metal-oxides, conducting polymers, and hybrid polymers. The most commonly used of these is carbon due to its low cost and availability. The disk shaped capacitor created by SOHIO in 1970 contained carbon as its electrode. Many capacitors have contained carbon electrodes in the past, which also influences others to use it today [3].

Just as the electrode used in the supercapacitor affects the capacitor’s performance, the electrolyte being used also has a strong influence on its performance. The energy density of the capacitor is affected by the material used as its electrolyte [3]. Since the energy density is directly proportional to the power density of the capacitor, the power density is also influenced by the electrolyte being used. The power density of a supercapacitor used in an automobile would need to be as high as possible in order for the automobile to have more power than it would if it were to run on a battery.

There are two types of electrolytes that are used in supercapacitors: organic electrolytes and aqueous electrolytes. Organic electrolytes can achieve higher voltages than aqueous electrolytes, but they also have higher resistances, which limits the power output of the capacitor. Aqueous electrolytes cannot contain the high voltages that organic electrolytes can, but they have less resistance than organic electrolytes [3]. If supercapacitors are to be used in electric vehicles, the strengths and weaknesses that each electrolyte possesses will need to be taken into consideration.

Understanding how supercapacitors function helps provide knowledge of how well they would be able to power automobiles. Knowing how the materials that make up the capacitor affect its qualities is vital knowledge when producing a high powered and life sustaining automobile.

Types of Supercapacitors


Supercapacitors can be divided into three general categories: electrostatic, electrolytic, and electrochemical [3]. Each type varies in its performance, cost, and materials, which all will have an effect on supercapacitor use in vehicles.

Electrostatic


Electrostatic capacitors are the most basic type of supercapacitor. They generally do not have high capacitance compared to the other two types, but they are of simple design and are relatively easy to construct. As shown in figure I, they consist of two conducting parallel plates separated by a dielectric (non-conducting material). The dielectric is added to keep the two plates from contacting each other. Electrostatic capacitors store energy via storage of charge on the two plates. The voltage of this type of capacitor depends on the strength of the dielectric material [3].


FIGURE I

Simplified Parallel Capacitor [3]

Electrolytic


Electrolytic capacitors are similar to electrostatic capacitors, except for the addition of an electrolyte salt. The salt is added to the metal electrodes, which are coated with an insulating oxide layer. This addition gives added capacitance per volume, but its use poses safety risks as well. Dissolution of the oxide layer into the electrolyte can cause a short and even explosion in extreme cases [3].

Electrochemical


Electrochemical capacitors also use electrolyte solutions in their construction, but they have a porous structure that gives them a greater surface area which results in higher capacitance [5]. This category can be further divided into symmetric and asymmetric capacitors, which simply defines if the two electrodes are made of the same material (symmetric) or different materials (asymmetric). These types of double-layer capacitors are the most advanced and generally have the highest energy density, but they are also more expensive than the other types [3].

The knowledge of the strengths and weaknesses of each type of supercapacitor is vital if they are to be used in electric vehicles. Each type will have to be analysed to find the best fit for the needs of an electric vehicle.

Supercapacitors Verses Electrochemical Batteries


Batteries provide a source of power to other devices when they are connected in a circuit with them. Because of their high energy density, lithium ion batteries are one of the best choices of batteries for electric vehicles [1]. Figure II shows the driving range of electric vehicles with different battery systems. If the most efficient type of supercapacitor is used and is made out of the right materials, the supercapacitor would prove to have a longer lifespan and higher power efficiency than the lithium ion battery.




FIGURE II

Driving Range For Battery Systems [8]

Lithium Ion Batteries


Batteries contain two electrodes, one of them being a cathode (positively charged) and the other an anode (negatively charged). When the two electrodes are connected by a conductor, an electric current flows from the cathode to the anode. The two electrodes are separated by an electrolyte, which conducts electricity. Once the electrodes are connected, a chemical reaction occurs between the anode and the electrolyte. Inside the battery, cations (positively charged ions) travel towards the cathode, while anions (negatively charged ions) flow in the direction of the anode. The resulting imbalance in charge causes electrons to flow into the cathode from the anode, thus providing a current which provides power for resistors connected to the circuit. [6].

Just as materials affect capacitor performance, the material used as the electrodes for batteries determines the capacitance and voltage of the batteries [6]. In a lithium ion battery the cation that is produced is a lithium ion. Lithium can be ionized easily because it has one electron in its outer shell that is easily removed. The electrolyte used most often consists of a mixture of a lithium salt and an organic solvent (containing carbon). The anode used is most often carbon. One common cathode used is lithium cobalt oxide. In a battery containing these two materials, the voltage output is approximately 3.6 volts [6]. Properties of other commonly used cathodes are included in table II.


TABLE II

Properties of Different Lithium Ion Cathode materials [7]

Cathode

Type

Energy Density (Wh/kg)

Relative Cycle Life

Voltage (volts)

LiCoO2

Energy/power

170 to 185

1

3.65

LiCo1/3Ni1/3Mn1/3O2

Energy

155 to 185

1

3.7

LiNi0.8Co0.15Al0.05O2

Energy/Power

145 to 165

3

3.65

LiFePO4

Energy

100 to 140

3

3.2

LiMn2O4

Energy/Power

90 to 120

>4

3.8

Lifespan of Supercapacitors Verses Lithium Ion Batteries


One downside of using batteries for electric vehicles is they require relatively frequent replacement [1]. Lithium ion batteries tend to die after 300 to 500 charge/discharge cycles, which usually occur within three to five years [17]. Part of the reason these batteries lose their capacity is exposure to heat [17]. The battery powering a vehicle would certainly be exposed to some amount of thermal stress. A battery’s life will also decrease if it is often kept at full charge [17]. Table III shows the capacity loss of batteries at different temperatures as well as whether they are kept at the recommend charge value or the typical user charge value.

In contrast to batteries, the capacity of a supercapacitor is only slightly affected by repetitive charging. After ten years its capacitance would only drop to eighty percent of its original value [16]. This translates to supercapacitors requiring replacement less frequently than batteries. If, however, voltages are applied that are too high for the supercapacitor to handle, its life will decrease faster [16]. Unlike lithium ion batteries, the life of a supercapacitor will not decrease when exposed to high temperatures [16]. They will function the same when exposed to any reasonable temperatures [16]. Table IV shows that the estimated service life of a supercapacitor powering an electric automobile is ten to fifteen years, while the service life for the lithium ion battery is only five to ten years.


TABLE III

Capacity Loss of Lithium Ion Batteries

Battery Temperature

Permanent capacity loss when
stored at 40% state-of-charge
(recommended storage charge level)

Permanent capacity loss when
stored at 100% state-of-charge
(typical user charge level)

0°C

2% loss in 1 year;

6% loss in 1 year

25°C

4% loss in 1 year;

20% loss in 1 year

40°C

15% loss in 1 year

35% loss in 1 year

60°C

25% loss in 1 year

40% loss in 3 months

Charge Time


One of the greatest advantages that supercapacitors have over lithium ion batteries is the time it takes to charge. A big turn off for consumers is that electric car batteries take a long time to fully recharge [2]. Since supercapacitors store energy via physical means and not chemical reactions like batteries, their internal reactions are easily reversed and as such they have a much shorter recharge time compared to batteries [4]. Table IV shows that a supercapacitor can fully charge within ten seconds while a lithium ion battery may take ten to sixty minutes. This significant difference in charge time would make owning an electric automobile more convenient.

Power


Supercapacitors are known to provide significantly more power than lithium ion batteries. Table II shows that the supercapacitor could provide up to ten times the amount of what a lithium ion battery can provide. This increase in power would allow the vehicle greater acceleration and speed on inclines, making it more attractive to consumers. Supercapacitors do however cost more than lithium ion batteries. A supercapacitor could cost consumers ten times the amount of money per Wh (Watt*hour) of energy it provides [16].

Potential Problems with Supercapacitors


Although there are many positive aspects of the supercapacitor, there are also disadvantages with their use. Properties such as high power density and quick charge time can have related converse effects. These problems will have to be overcome if supercapacitors are to be used widely in electric vehicles.

A major weakness of supercapacitors is the speed at which they lose their charge [10]. This downside actually comes from one of its prominent strengths, its high power output. Power is defined by energy used per unit of time [11]. This means that of two power sources of equal energy, one with a power twice as high as the other will use its energy twice as fast. This relates to supercapacitors and batteries as well.

Unlike batteries, which store energy in their entire mass, supercapacitors only store charge on their surface [9]. This leads to supercapacitors having less total energy than comparably sized chemical batteries. This would be a downside on electric cars where space is limited, and limits the effectiveness of supercapacitors until a more efficient way of storing energy can be discovered.

Also problematic for supercapacitors is maintaining a constant voltage. As a supercapacitor loses its charge, its voltage drops significantly when compared to a battery, and complex systems are required to keep a constant voltage across the supercapacitor [12]. This would add to the bulk already required by supercapacitors to compete with lithium battery energy storage and further limit its use in vehicles.

As with any technology, the weaknesses that come with its use must be taken into account as much as the strengths. Solutions to these problems will need to be found in order for the full potential of supercapacitors to be known.


TABLE IV

Performance of Supercapacitors and Lithium Ion Batteries [16]

Function

Supercapacitor

Lithium-ion (general)

Charge time

1–10 seconds

10–60 minutes

Cycle life

1 million or 30,000h

500 and higher

Cell voltage

2.3 to 2.75V

3.6 to 3.7V

Specific energy (Wh/kg)

5 (typical)

100–200

Specific power (W/kg)

Up to 10,000

1,000 to 3,000

Cost per Wh

$20 (typical)

$2 (typical)

Service life (in vehicle)

10 to 15 years

5 to 10 years

Batteries Paired with Supercapacitors


One solution being considered to solve the problems of capacitors and batteries on their own is coupling the two in electric vehicles to work together [3, 9]. The battery and supercapacitor can be used at alternate times to avoid the areas where either source lacks in performance.

In a car with a dual battery and supercapacitor, the supercapacitor would provide power-intensive functions, such as initial acceleration and climbing hills, while the battery would be used primarily to maintain cruising speeds [13]. This would alleviate much of the strain from the chemical battery and would allow the vehicle an estimated 10% increase in driving range [13]. This would also help eliminate the quick discharge time of the supercapacitor, as it would not be in use the entire time the vehicle was operating, and could recharge from the battery while idling or during periods of little power usage.

In addition, a vehicle using supercapacitors can benefit from regenerative braking, a method where energy is harnessed from the braking process and supplied to the capacitor to recharge. This is not practical for batteries because of the inability of the battery to charge quickly, but the supercapacitor does not share this drawback.

Supercapacitors In Electric Automobiles


The goal of attaining an electric vehicle that satisfies the needs and wants of consumers has become closer to reality due to significant advances in recent years. This reality would be a great deal closer to realization by incorporating supercapacitors as a power source for electric vehicles.

Just because supercapacitors can outperform electrochemical batteries in terms of power, lifespan and charging time does not mean that they are necessarily the best choice to power automobiles. Other factors such as endurance and ethics must also be taken into account.

Thermal and Mechanical Stress


If a supercapacitor were to be able to power an automobile it would have to be able to endure a certain amount of stress. The supercapacitor would have to be able to function in high temperatures and vibrations of high frequencies due to the terrain the vehicle will drive on.

To test supercapacitors for electric automobiles, Fiorentino Conte and Franz Pirker (whom both have degrees in electrical engineering) conducted a study of supercapacitors under stressful conditions. For their experiment they used some of the most common supercapacitors available on the market. All of them contain electrodes based on carbon and organic electrolytes [14].

They fully charged each supercapacitor and placed them in a chamber with controlled temperature. They tested them at both twenty degrees Celsius and 43 degrees Celsius. They were tested at these temperatures while no mechanical stress was involved and with mechanical stress. To simulate the mechanical stress of an automobile, they placed the supercapacitors on a vibrating bench with vibration frequencies ranging from ten to sixty Hertz. The amplitude of the vibrations ranged from zero to twenty grams with an average of four to five grams [14].

The supercapacitors were charged and discharged with a current rate of forty milliamps per Farad during the mechanical and thermal stress to test the effects on their charge and discharge. On average, the discharge rate of the supercapacitors increased by approximately a factor of ten when the temperature was increased by twenty three degrees Celsius. This means that the supercapacitor would only be able to run for ninety percent of its normal lifespan when under thermal stress [14].

Also concluded by their results was that there was only a small decrease in the capacity of the supercapacitors when mechanical stress was applied. It was noted though that the ability of the supercapacitor to endure this stress varied significantly between the supercapacitors made by different manufacturers [14].

While under both the mechanical and thermal stress tests, one supercapacitor leaked fluid, which was most likely the electrolyte. The tests also showed that other supercapacitors similar to this one showed a large decrease in their discharge rate when under stress. The factors that resulted in this lower performance by these supercapacitors may have been due to the materials used by the manufacturer or the type of capacitor it was classified as [14].

The issues that Conte and Pirker highlighted in their experiment will require serious consideration from engineers that plan on implementing supercapacitors in electric vehicles. Systems that can regulate current output with temperature change, much like the systems used to regulate voltage, will need to be developed, which will in turn need to be improved to deal with issues of available space on vehicles. Due to the variation in stress endurance between capacitors, the types of supercapacitors used will again have to be taken into consideration regarding capacity under mechanical stress.

Ethics and Benefits to Society


Ethics play a vital role when dealing with automobiles due to the fact that they are an important part of many people's lives. The first rule of practice found in the National Society of Professional Engineers (NSPE) states that “Engineers shall hold paramount the safety, health, and welfare of the public” [15].

Pollution is an unfortunate byproduct of industrial progress. However, it is also controllable, and advances in engineering have already reduced pollution greatly. Further reducing the amount of pollution is an important part of improving the health of the public in the future. If the public were given an attractive alternative option to driving gas powered automobiles, the pollution due to gas emissions would decrease significantly. In order to keep up the general welfare, electric vehicles would need to meet the consumers’ desires. The electric vehicle must have enough power, be able to charge quickly, and be able to run for long distances. The use of the supercapacitor in electric automobiles has the potential to decrease pollution while providing consumers with satisfactory transportation.

Many people rely on automobiles for transportation, both public and privately owned. Because of this, it is important that a supercapacitor would not malfunction during vehicle operation, which could have bothersome or even devastating effects. During the experiment described in the previous section, a supercapacitor leaked fluid when put under thermal and mechanical stress. This would be unacceptable when designing a supercapacitor for an electric automobile. Engineers would need to make sure that they are using the right materials and the right type of supercapacitor when providing products for automobile companies to demonstrate ethical value.

The Best Power Source For Electric Automobiles


Supercapacitors are a relatively new technology and they have room for improvement. Research on them has recently become more popular since the idea of their use in automobiles. It is reasonable to assume that the performance of supercapacitors will increase as research progresses.

In order to produce a supercapacitor capable of powering an automobile, the right type of supercapacitor must be used. Whether it’s an electrostatic, electrolytic, or electrochemical supercapacitor will have an effect on the performance of the supercapacitor. Experiments also would need to be done to decide what materials should be used as the anode and cathode of the supercapacitor in order to maximize performance.

One of the most successful batteries in powering electric automobile is the lithium ion battery. When compared to this battery, the super capacitor is able to provide more power, run longer, and charge in less time. Even though it has its flaws, such as high cost, the supercapacitor proves to have better performance than the lithium ion battery in areas most important for powering automobiles.

When put under the stress similar to what they would experience in an automobile, supercapacitors hold up fairly well; however, there is plenty of room for development. To ensure consumers with a satisfactory vehicle, the durability of the supercapacitor would need to be improved so their performance is not greatly affected by outside factors, such as temperature change and mechanical stress.

The high power and decrease in charge time of the supercapacitor would provide owners with the convenience that a combustion engine provides without the constant maintenance they require. Hopefully, as development increases, the cost per Watt-hour for supercapacitors will drop to the range of lithium ion batteries, and their use will become widespread.

References


[1] “The High power Lithium-ion.” Battery University. [Online] Available: http://batteryuniversity.com/learn/article/the_high_power_lithium_ion

[2] “Electric Car Range” GreanStream. [Online]. Available: http://www.owningelectriccar.com/electric-car-range.html

[3] T.S. Bhatti, Pawan Sharma. (December 2010) “A review on electrochemical double-layer capacitors.” Energy Conversion and Management, Volume 51, Issue 12 [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0196890410002438 p.

2901-2912

[4] “Principles Behind Electric Double-layer Capacitors, and Their Features.” ELNA America Inc. [Online]. Available: http://www.mouser.com/pdfdocs/ELNAPrinElectricDoubleLayerCaps.pdf

[5] Laura Howes. (May 2011). “New carbon material boosts supercapacitors” Royal Society of Chemistry [Online] Available: http://www.rsc.org/chemistryworld/News/2011/May/13051102.asp

[6] “Lithium-ion batteries”. American Physical Society [Online]. Available: http://physicscentral.org/explore/action/lithium-1.cfm

[7] Antti Vayrynen, Justin Salminen. (21 September 2011) “Lithium ion battery production.” The Journal of Chemical Thermodynamics, Volume 46. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0021961411003090 p. 80-85

[8] Ralph Brodd, Martin Winter. (2004) “What Are Batteries, Fuel Cells, and Supercapacitors?” Chemical Review Issue 104. [Online] Available: http://pubs.acs.org/doi/pdf/10.1021/cr020730k p. 4252-4259

[9] Joel Schindall. (November 2007) “The Charge of the Ultra – Capacitors” IEEE Spectrum.[Online] Available: http://spectrum.ieee.org/green-tech/advanced-cars/the-charge-of-the-ultra-capacitors

[10] Josie Garthwaite. (July 2011) “How ultracapacitors work.” GigaOM [Online] Available: http://gigaom.com/cleantech/how-ultracapacitors-work-and-why-they-fall-short/

[11] Jearl Walker.(2011) “Fundamentals of Physics.” John Wiley & Sons, Inc.. 8th Edition. P. 184

[12] C. Park, K. No, P. H. Chou. (Sept. 2011) “TurboCap: A Batteryless, Supercapacitor-based Power Supply for Mini-FDPM.” University of California.

[13] G. Guidi. “Efficient Use of Electric Double Layer Capacitor as Energy Source on Board of Electric Vehicles.” Yokohama National University. [Online]. Available: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5637740&tag=1

[14] F.V. Conte, F. Pirker. (2005). “Electrical Performances of High Power Electric Double Layer Capacitors Under Thermal and Mechanical Stress.” NESSCAP. [Online]. Available: http://www.nesscap.com/support/papers_list.jsp?pg=2&schFld=whole&schWd=

[15] “NSPE Code of Ethics for Engineers”. NSPE. [Online]. Available: http://www.nspe.org/Ethics/CodeofEthics/index.html

[16] “Supercapacitor”. Battery University. [Online] Abailable: http://batteryuniversity.com/learn/article/whats_the_role_of_the_supercapacitor

[17] “How to Prolong Lithium-based Batteries”. Battery University. [Online]. Available: http://batteryuniversity.com/learn/article/how_to_prolong_lithium_based_batteries

Additional Resources


Marshall Brain. (24 May 2005) “How Electric Cars Work.” [Online]. Available: http://srinivasarao.webs.com/electric-cars.pdf

Acknowledgements


We would like to thank the authors of the sources which we used to research supercapacitors. We also thank everyone from the writing center that came to our Engineering Computing lecture to help us further understand the assignment. We also received helpful advice from Bert Cos.



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