Planning ofmobile complete set for a rural wind generator
Abstract
The aim of this thesis is to alleviate the chronic lack ofelectricity supply in the rural South African areas by designing a portablewind generator kit.
An extensive assessment on the rural village of Ga-Rampuru,in Limpopo Province, was conducted, to investigate the present needs, as wellas the availability of resources both human and material that would be neededto construct and assemble the system. From the inventory of recyclablematerials found during the investigation the author was more inclined tosuggest the design of a wind turbine that could be assembled and maintained bythe local artisans.
Atwo pole permanent magnet synchronous generator was designed using standardcommercial magnets, which were later replaced by recyclable loudspeaker magnetsthat were found in the village. This was done to compare the output of thegenerator in both cases. All the designs were modelled in FEMM, a software package,to estimate the induced voltage and flux of the generators.
Usingstandard commercial magnets the simulated voltage and flux levels were 9.4,5.1, 3.6V and 0.0489, 0.0186, 0.0175 Wb, respectively. Assuming a generatorcurrent rating of 1 amp this would yield 36 watts at the estimated average windspeed of 4 meters per second.
Thenwhen these were substituted with recycled speaker magnets the generator yieldeda voltage of 3.5V and a flux of 0.0171Wb. The estimated output power of therecycled generator was estimated to be 10.5W. This compared well with the poweroutput from the commercial magnets generators.
Fromthese preliminary results it is quite apparent that a viable generator can bedesigned from the recyclable magnetic components. The same design procedure asoutlined in this thesis can be used to design larger recycled generators withlarger outputs. The design of this wind turbine will obviously have a widerange of positive developmental benefits on the community of Ga-Rampuru.
The next stagewas practical construction to validate of the simulation results. This howevercould not be realised in time.
Chapter1. Introduction 1.1The subject of the report
Theaim of this thesis is to design a simple wind generator kit that can be easilyassembled and installed by rural artisans. The kit will use recyclablematerials that are found in the rural areas to ensure a cost effective andenvironmentally sustainable solution. 1.2 Background to researchand investigation of rural electrification
“Electricitybrings immeasurable benefits to human life. With electricity, comes lightingand the ability to extend the daylight hours, to study and to improveeducation. With electricity come cooling and heating and the ability to storefood and cooking. At its extended level, electricity facilitatescommunications, transportation and production and paves the way for theeradication of poverty, industrialisation and ultimately the growth of ourcountry’s economy”.[3]
Electricityis a basic necessity and access to it has a wide range of positive developmentalbenefits for communities [1], yet, in 2001 2.8 million South Africanhouseholds still had no access to electricity [2]. The majority ofthese households are poor and live in remote places which are located far fromthe central business districts and the country’s electricity grid. And thismakes it very expensive to connect them to the country’s electricity grid.
Asa national initiative to improve the quality of life in South Africa, NationalElectrification Programme (NEP) aims to provide universal access to all SouthAfricans by 2012 [4]. Hence, this has lead to the investigation ofother safe, cost effective and environmentally friendly alternative methods ofelectrifying rural areas in South Africa.
Renewableenergy resources such as wind and solar, are the fastest growing alternative meansof providing a reasonable amount of energy at the point of demand. TheGovernment of South Africa is also determined that renewable resources will bea major complement to the national mix [4]. 1.2.1Ga-Rampuru, a typical rural South African village
Ga-Rampuruis a small village located in Limpopo Province in South Africa. The village isin a fairly rural mountainous area, which is situated some 58 odd kilometresfrom Polokwane, the provincial capital city. The area has sparsely populated householdswith some trading stores and schools. Most of the people in the village areunemployed and rely on agriculture for their subsistence.
Peoplein the village have to travel long distances to collect wood or to purchasefuels like liquid petroleum gas (LPG) and kerosene to meet their cooking,lighting, refrigeration, infotainment and other needs. Figure 1 illustrates apicture of an LPG refrigerator in one of the trading store in Ga-Rampuru. Thispicture and others that will follow in this thesis we taken by the authorduring a visit at Ga-Rampuru last June vacation.
Thesupply of these fuels is both expensive and unpredictable. Additionally theproblems related to the use of fuels such as kerosene are incidences concerningburned houses and respiratory problem for children who use kerosene candles forreading is well documented world wide [6].
The authorpaid a visit to the Provincial ESKOM office to enquire about any plans toextend the grid to Ga-Rampuru village; and the Electrification Manager guaranteedthat ESKOM has plans to ultimately electrify the whole country by 2012. However,further discussions with people from Ga-Rampuru dismissed the ESKOM Manager’spromises as empty. They contended that they had heard similar promises but theystill lived in darkness.
Itwas the conclusion of the author that an alternative solution to the problem hadto be devised. Some means of generating electric power to meet loads such asthe refrigerator in figure 1, if only it could be an affordable design. Thebest design would clearly be one that uses local material and human resources.
1.2.2 Resource assessment
Theauthor spent the next three weeks exploring the resources available inGa-Rampuru that would support the design and sustainable construction ofelectricity generators.
Tobegin with Ga-Rampuru has two schools, namely Rampuru primary school andSeokeng secondary school, all which constitute a total population of roughly1400 pupils. On average 30% of school leavers will continue to tertiaryeducation, some will migrate to urban centres in search for jobs and asubstantial number will remain in the village.
Thisvillage is endowed with adequate human capacity with intermediate levels ofeducation. These would constitute a source of trainable technicians andpotential consumers of locally manufactured products. There are also localmechanics who fix cars and some electrical appliances. These people will beeasily trained as they have hands on experience.
Someof the people who left the village for jobs in the cities come back to settledown in the village and build big houses like the one indicated in Fig 2. Thisclearly indicates that this people can afford the electricity tariffs if theywere to be supplied with power.
Movingfurther around the village there was evidence of old windmills used for pumpingwater. Figure 3 shows one of the windmills. These windmills operate satisfactorilyproviding enough water to the villagers. The presence of these windmills inthis area is evidence that there is some wind resource in the area.
Furtherinvestigations took the author to various waste-dump sites and a range ofdisused old gadgets that could potentially be re-used, as shown in appendix A,were discovered. These included cables from an old car, loudspeaker magnets,drums and old machines that were used for grinding grain.
Theother natural resource in the area (of course) is the sun but from theinventory of recyclable materials found during the investigation it is moreinclined to suggest the design of a wind turbine.
1.3 Objectives of the report
Inlight of the above background, the main objective of this thesis is to design asmall wind generator for Ga-Rampuru village using recyclable materials found inthis village. The idea is to build an easily assembled and manufactured machinethat can be build by the rural artisans. This wind generator must of course becost effective.
Theresource assessment of Ga-Rampuru village is conducted in order to investigatethe present needs, as well as the availability of resources both human andmaterial that would be needed to construct and assemble the wind turbine usingrecyclable materials. Furthermore, the resource assessment analyses lead to anappropriate wind generator design specifically for Ga-Rampuru village. 1.4 Method of investigation
Theinvestigations were conducted in July 2006 at Ga-Ramrupu village in Limpopoprovince. The author collected information regarding this village in thefollowing manner:
1. The author grew up in Ga-Rampuru village and therefore knows theproblems and challenges that the villagers face on a day-to-day basis livingwithout electricity. This was an advantage in terms of moving around thevillage doing the resource assessment analysis.
2. One of the store owners in the village, Mr Morifi was interviewedregarding the issues he faces in supplying power to his store, especially tothe refrigerator he has in store. The store owner mentioned that he has torefill the petroleum gas (LPG) in his store every two weeks. He also added thatthis is very expensive as there are also transport costs involved.
3. Face to face interviews were conducted with some of the villagers wheremany concerns and challenges were raised. Most of the villagers said that ithas been several years since they have been promised to be electrified andnothing has been done to date.
4. The author paid a visit to the Provincial ESKOM office in Pretoriato enquire about any plans to extend the grid to Ga-Rampuru village. The ESKOMElectrification Manager, Jack Bandile was interviewed in this regard. 1.5 Plan of development
Thereport begins with a brief background of the thesis and introduction of the ruralarea for which the wind generator will be designed for. Then, the remainingproject researches are outlined as follows:
· Chapter 2 reviews the design of a small wind generator and afterthat a wind generator suitable for Ga-Rampuru village is designed usingrecyclable materials that where found in this village.
· Chapter 3 details the procedure undertaken to design a permanentmagnet synchronous generator for Ga-Rampuru village wind turbine.
· Chapter 4, the generator geometry discussed in chapter 3 ismodelled in FEMN using recyclable and commercial magnets to analyse andestimate both machine designs.
· Chapter 5 discusses the results found in chapter 4.
· Chapter 6 details all the steps that were taken in an attempt toassemble a prototype of the wind generator.
· Chapter 7 & 8 concludes the discussion based on the analysesand finally presents recommendations.
Chapter 2. Design of the wind turbine prototype 2.1Background on wind energy
Windpowered systems have been widely used since the tenth century for waterpumping, grinding grain and other low power applications [9]. Sincethen, this has lead to an investigation and attempt to build large wind energysystems to generate electricity.
Windenergy has proven to be cost effective and reliable in the past years. The maindevelopment of this technology has been with large wind turbines in theindustrialized world, but there is scope to deliver decentralized energyservice in the rural areas of developing countries [6].
Furthermore,wind energy is an attractive option to generate electricity since it does notconsume fossil fuels nor emit greenhouse gases. The land on which the windgenerators are build may also be used for agricultural purposes such as ploughingthe land or domestic animal gazing.
Duringits transition from the earlier day’s wind ‘mills’ to the modern electricgenerators, the wind energy conversion systems (WECS) have transformed tovarious sizes, shapes and designs, to suit the applications for which they areintended for [5]. In this chapter, the main components of a simplesmall wind generator will be investigated and a wind generator suitable forGa-Rampuru village will be designed using recyclable materials found in thearea.
Theavailable wind resource is governed by the climatology of the region concernedand has a large variability from one location to the other and also from seasonto season at any fixed location [9]. Hence, it is important that thewind generator is designed for a specific area; this will ensure that the wind energyin that specific area is exploited to generate maximum power from the wind.
2.2 Wind turbine basic principles
Thewind generators are specially designed and build to extract power from turningblades with the maximum efficiency and minimum complexity [6]. Themagnet rotor disk rotates as a result of the force of the wind on the turbine’sblades.
Atypical small wind generator consisting of blades, tower, PM generator and thecabling is illustrated in figure 2.1. The main components, which are common tomost wind generators, will be discussed below.
/>
Fig2.1 Basic features of a typical small wind generator [6] 2.2.1The blades
Modernwind turbine rotors usually have two or three wooden blades. A larger number ofblades would create more turning force (torque), but would not be capable ofdriving the PM generator fast enough to generate the required voltage, becausethe rotor would turn more slowly [6]. The rotor blades are designedin such a way that they extract the maximum power from the wind.
Powersupplied by the blades to the generator is [7]:
/> (Eq 2.1)
where/>is the airdensity (Kg/m3), C is the dimensionless power coefficient and A thearea swept by the blades in m3.
Inequation 2.1 above, the power drawn from the wind is proportional to the cubeof the wind speed. This means that if the wind speed doubles, there is 8 timesas much power available from it [7].
Afurther important parameter is the tip speed ratio. The tip speed ratio isdefined as the ratio of the tip of the blade to that of the undisturbed windvelocity entering the blades [11]. The ratio is given by [7]:
/> (Eq2.2)
whereR is the radius of the blades, ωr is the rotor speed in rad/sand W the wind speed (m/s).
Multibladed rotors operate at low tip speed ratios of 1 or 2, where else, one, twoor three bladed rotors operate at higher tip speeds of 6 to 10. The powercoefficient in equation 2.1 depends on tip speed ratio as shown in figure 2.2. Fora particular wind rotor design there exists a tip speed ratio which willproduce the maximum value of power coefficient [11].
/>
Fig.2.2 Power coefficient Cp versus tip speed ratio [11] 2.2.2Permanent magnet generator
Usingpermanent-magnet generators for small wind turbines is very commonly used worldwide. Usually an AC generator with many poles operates between 10-100 Hz. Manyconfigurations use surface mounted three phase permanent magnet synchronousgenerators with a rectifier connected to the generator terminals. [16]
Asimple PM generator consists of the stator, magnet rotor disk and a shaft. Themagnet rotor disk is mounted on a bearing hub so that it can rotate on theshaft due to the rotating blades of the wind generator.
Thestator has coils of copper wire wound around them, which are accommodated inthe slots. Electricity is then generated when the magnets on the rotor disksrotate past the coils embedded in the stator. The magnetic field that iscreated induces a voltage in the coils [6].2.2.3 Rotor design
Thereare two types of rotor configurations commonly used world wide, these are thedisk and the cup as shown in figure 2.3 below [20].
/>
Fig.2.3 Disk and cup rotor designs
Theradius of the rotor primarily depends on the power expected from the turbineand the strength of the wind regime in which it operates [5]. 2.2.4 Tower
Themain function of the tower is to raise the blades and the generator to a heightwhere the wind is stronger and smoother than the ground level. The wind speedincreases with height because of the earth surface [9]. The towershould be high enough to avoid any obstacles such as trees, building, etc.Practical considerations such as expense, safety and maintenance limit thetower to between 10m to 20m [6] above ground level.2.3 Design of a wind turbine for Ga-Rampuru village
Inthis section a wind generator that is designed specifically for Ga-Rampuruvillage will be discussed. The generator will be designed using recyclablematerials such as car brake plates, cables and drums found in the village [Seeappendix A]; this will clearly ensure a cost effective design. The wind turbinewill be designed in such a way that the local people can easily assemble andmanufacture it themselves.
Allthe recyclable materials that will be used in this design will be discussedbelow and an artist impression of the wind generator will be sketched.2.3.1 The drum
Theoutput of the wind generator depend on the amount of wind swept by the blades,therefore the wind extracting materials in a wind generator are verysignificant. A plastic drum will be used in this design to extract the windsince it can be easily shaped and carefully balanced to run smoothly. Also, itis resistant to fatigue braking and has a very light weight.
Thedrum will be assembled as follows:
1. The top and the bottom part of the drum will be cut carefully byusing a knife or pair of scissors to make a cylinder with open ends.
2. The cylindrical drum is then cut length-wise into two equalhalves.
3. The two halves are then glued together similar to the drum shownin figure 2.4.
/>
Figure2.4 An S-shaped drum
Toprevent the over speeding of the drum, the permanent magnet generator shouldalways be connected to a battery or other electrical load. If this is not donethe wind turbine will become noisy and may vibrate so much that some parts comeloose and fall to the ground [6]. 2.3.2Magnet rotor disk
Aftera tour around the village neighbourhood dumpsites it was discovered that thereare many discarded loud-speakers that are no longer in use in the village.These loud-speakers have permanents mounted to their back. Since the PMgenerator requires magnets, these loud-speakers will be recycled and themagnets on them will be used in this design. Figure 2.5 shows one such magnet thatwas found in the village.
Thereare many factors such as heat, radiation and strong electrical currents thatcan affect the strength of a magnet [8], especially in suchdiscarded state. These factors will be discussed later to investigate exactlyhow much surface magnetic flux density these magnets loose in the dumpsites.
Andlater on in this thesis the performance of a PM wind generator designed usingstandard commercial magnets will be compared to a generator using the recycled loudspeakermagnets as substitutes.
Designinga generator using the speaker magnets will pose the following challenges due totheir shape and strength:
· How does one design a machine with these magnets?
· Do they have to be smashed and aligned to work?
· Or should they be used the way they are?
· How much flux density do these magnets have, in other word, canthey give out any power when used in the generator design?
· Can different magnet types be used on one machine? As this magnetsare picked randomly in the rural area.2.3.3 Rotor Disk
Acylindrically shaped rotor is preferred as it allows the proper distribution offlux over the armature surface as the field coils are spread over the peripheryof the cylindrical rotor. Hence, a brake plate from an old car like the one infigure 2.6 will be used as the rotor in this design to hold and house themagnets.2.3.4 Distribution cables
Allthe cabling that will be required in the construction of the wind generator wasfound in an old car in the village [See figure 2.7]. 2.3.5Artist impression of the wind turbine
Figure 2.8 below shows the artist impression of the windgenerator designed exclusively for Ga-Rampuru village.
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Figure2.8 Ga-Rampuru wind generator
Thefollowing chapters describe the steps taken by the author to investigate theperformance of a synchronous permanent magnet machine constructed usingrecyclable loudspeaker magnets.
Chapter 3. Generator Design 3.1 A brief background
This chapterwill detail a simple procedure undertaken to design the wind generator fromrecyclable materials. Permanent magnet machines are preferred for thisapplication as they reduce the excitation losses significantly and hence asubstantial increase in the efficiency of the machine. In addition, permanentmagnet machines are simple to construct and maintain [10].
The mostcommon wind turbine systems are three blades rotating on a horizontal axiscoupled to an alternator to generate electricity, which could be used to forbattery charging. For a picture of a typical basic wind turbine system refer tofigure 2.1 in chapter 2.
Anormal two- pole synchronous permanent magnet generator will be designed andits performance will be analysed. Then recyclable loudspeaker magnets found inthe rural area of Ga-Rampuru village will be used to substitute the standardcommercial magnets in the generator. The performance of the new generator willbe analysed to understand the effect of the loudspeaker magnets on thegenerator performance.
Forthis investigation, matching the refrigerator load in chapter 1 will not be apriority.
Thischapter will start with outlining the desired generator specification and thenthe generator will be designed thereafter. To design the generator thepermanent magnet properties will be discussed to understand their effect on thegenerator performance and losses due to these magnetic materials will also beinvestigated. And then, all the variables that are necessary to construct anddesign a generator geometry will also be discussed.
Throughoutthis thesis the generator performance will be tested under no-load conditions.
3.2 Generator specifications
Inthis thesis, a generator with the following specifications will be designed andmodelled in FEMM, a finite element package:
· Output power = 36W @ 12V
· Number of phases = 3
· Number of poles = 2
Thechoice of the above dimensions of the generator was influenced by the followingconsideration:
·Induced output voltage, 12V is standard voltagethat is used in many applications. For example it is suitable to charge abattery. Batteries are suitable to power a wide range of rural appliances andinstruments especially in remote areas of South Africa [11].
·The generator must be easily assembled andmanufactured so that the rural artisans with little training can be able toassemble this generator.
Thefollowing design procedure will be followed:
1. A simple two-pole synchronous permanent magnet generator will bedesigned using available standard commercial magnets such as ceramics, alnicosand rare-earth magnets.
2. The effects of the above magnets on the performance of thegenerator will be investigated.
3. The magnets from a loudspeaker that was randomly picked in thevillage will then be used in the design and the performance will also beinvestigated.
Thedesigns above will be modelled using FEMM, a finite element package. The mainreason for using FEMM is to observe the output induced voltage of thegenerator. This will be the method of how the performance of the generator willbe monitored.
3.3 Generator basic principle
Themain function of a generator is to supply power to the load, in order to do so;voltage has to be generated at the terminals. The generator principle is basedon Faraday’s law of induction [10]:
/> (Eq.3.1)
wheree is the instantaneous voltage, />is the flux linkage and t is thetime.
Thelaw states that for voltage to be induced in a winding, the magnetic flux hasto change relative to the winding. This means that the flux linkage is changingand the conductor is fixed or stationary. The flux linkage is the total flux,/>, linking allconductors in a winding with N turns. Therefore the flux linkage is given by:
/> (Eq.3.2)
Togenerate voltage in practice, a mechanical motion and a source of magnetic fluxmust be present. The mechanical motion can be linear or rotational, in thisthesis the motion is rotational and provided by the wind turbine. The source offlux is permanent magnets. 3.4 Properties of permanentmagnets
Theuse of permanent magnets in the construction of electrical machines has lots ofbenefits. A PM can produce magnetic flux in the airgap with no exciting windingand no dissipation of electric power [14].
Permanentmagnets are known for their large hysteresis loop and B-H curves. These curvesare in the second quadrant of the loop called the demagnetization curve; thisis where the magnets operate. Demagnetization curves of the PM materials aregiven is Fig 3.1
Inall machines using permanent magnets to set up the required magnetic flux, itis desirable that the material used for permanent magnets have the followingcharacteristics [12]:
a) A large retentivity (residual flux density) so that the magnet is“strong” and provides the needed flux
b) A large coercivity so that it cannot be easily demagnetized byarmature reaction fields and temperature.
Foranalysis purpose, the magnet properties have to be known, the remanence fluxdensity Br and coercivity Hc. The magnets arecharacterised by a large B-H loop, high Br and Hc. Table3.1 summarizes the properties of some of the standard commercial magnets, thesewere estimated from figure 3.2 which indicate the demagnetization curves ofdifferent permanent magnet materials.Magnet Type
Br (T) Hc (kA/m) Rare-Earth NdFeb32 1.22 900 Alnico Alnico5 1.21 50 Ceramic Ceramic8 0.4 260
Table3.1 Magnets properties
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Figure3.1 Demagnetization curves for different PM materials
Theremanence magnetic flux density Br is the magnetic flux densitycorresponding to zero magnetic field intensity. High remanence means that themagnet can support higher magnetic flux density in the airgap of the magneticcircuit. While the coercivity Hc is the value of demagnetizing fieldintensity necessary to bring the magnetic flux density to zero in a materialthat is previously magnetized. High coercivity means that a thinner magnet canbe used to withstand the demagnetization field [10].3.4.1 Types of magnets
Thereare three main types of magnets that can be found, these are [10]:
1. ALNICO (Aluminium, nickel, cobalt, etc.)
Thesetype of magnets poses high magnetic remanent flux density and low temperaturecoefficients. The coercive force is very low and the demagnetization curve isextremely non-linear. Therefore, it is very easy to magnetize and demagnetizeALNICO magnets.
2. Ceramic or Ferrites (BaFe203 or SrFe203)
Aferrite has a higher coercive force than Alnico, but at the same time has alower remanent magnetic flux density. Their main advantage is their low costand very high electric resistance.
3. Rare – earth (SmCO, NdFeb-Neodynium Iron Boron)
Theseare one of the strongest types of magnets available. They poses high remanentflux density, high coercive force, high energy product, linear demagnetizationcurve and low temperature coefficients. The main disadvantage is the cost.
Highperformance rare-earth magnets have successfully replaced Alnico and Ferritesmagnets in all applications where the high power-to-weight ratio, improveddynamic performance or higher efficiency are of prime interest.
3.4.2 Factors affecting recycled magnets
Therecycled magnets that will be used in this thesis were randomly picked;therefore there is no indication on how long they have been in the dumpsites.The following are the factors that can affect the strength of magnets:
· Heat
· Radiation
· Other magnets in close proximity to the magnet
Ifa magnet is stored away from high temperatures, and from the factors mentionedabove, it will retain its magnetism essentially forever. Modern magnetmaterials lose a fraction of their magnetism over time if affected by the abovefactors [8]. 3.5 Generator losses
Thelosses in a synchronous generator consist of rotational loss (mechanical lossand magnetic loss) and copper loss in the armature winding. The rotational lossand the field winding loss are subtracted from the power to obtain the powerdeveloped by the armature. By subtracting the copper losses in the armaturefrom the developed power, we obtain the output power of a synchronousgenerator.
Inthis section, the core loss will be discussed since they are due to themagnetic flux variations. 3.5.1Eddy current loss
Thispower loss occurs in a magnetic core when the flux density changes rapidly inthe core. Because core material has resistance, a power loss i2Rwill be caused by the eddy current and will appear as heat in the core [13].
Theaverage eddy current loss is:
/> (Eq.3.3)
wherePe is the eddy current loss in watts (W), ke is theconstant that depends on the conductivity of the magnetic material, f is thefrequency in hertz (Hz), δ is the lamination thickness in meters, Bmis the maximum flux density in tesla (T) and V is the volume of the magneticmaterial in cubic meters (m3) [14].
Theeddy current losses can be reduced by [13]:
· Using a high-resistivity core material
· Using a laminated core, in transformers and electric machines theparts that are made of magnetic core and carry time-varying flux are normallylaminated. 3.5.2Hysteresis loss
Duringa cycle variation of current i, there is a net energy flowing from the sourceto the coil-core assembly. This energy loss goes to heat the core. The loss ofpower loss in the core owing to hysteresis effects is called hysterisis loss.
Bytesting various ferromagnetic materials, Charles Steinmetz proposed thathysteresis loss can be expressed as [14]:
/> (Eq.3.4)
wherePh is the hysteresis loss in watts, kh is a constant thatdepends upon the magnetic material and n is the Steinmetz exponent.3.5.3 Core loss
Thehysterisis loss and eddy current loss are lumped together as the core loss ofthe coil-core assembly, and given by:
/> (Eq.3.5)
3.6 Design Variables
Inthe following section, all the parameters that are necessary to design andconstruct a generator will be discussed and variables such as generatordiameter, length, etc. will also be calculated. 3.6.1Speed of the generator
Theannual mean wind speed at Ga-Rampuru is approximately 4m/s [11]. Therotor will rotate at the same speed as the wind turbine; therefore this meansthat the rotor will rotate at:
/> /> = 250rad/s = 2387.3 rpm
Therotor speed and the average frequency of the induced voltage are related by:
/> (Eq.3.7)
Sincea two-pole machine will be designed, the frequency is calculated using equation3.9 to be 39.79 Hz.3.6.2 Rotor and Stator Core
Acylindrically shaped rotor will be appropriate for this design as it allowsmaximum flux distribution over the armature surface as the field coils arespread over the periphery of the rotor. This type of design also accommodatesthe use of small cylindrical magnets [11].
Alow carbon steel core with low permeability will be used in this design as itwas found in the recyclable materials found in the village. This type of steelis cheap and mostly available. Compared with other better and expensive steel suchas silicon, cobalt, etc. this type of steel has a very high core loss. Thesteel saturation flux density Bsat is estimated from the BH curve tobe 1.5T. 3.6.3Rotor Diameter (D)
Therotor diameter must be greater than the rotor yoke height (Hry),shaft radius (Rshaft) and the radial magnet length (Lm) [10].
D= 2 Hry + 2 Rshaft + 2Lm (Eq.3.8)
Inthis design, D is restricted by the magnet arc radius of 25mm. Therefore D willbe 50mm. 3.6.4Rotor and Stator Yoke heights
Theminimum rotor yoke height Hry is the same as the stator yoke heightHsy. The height should be large enough to avoid saturation. Thisalso has advantages of reducing core loss and reluctance.
Theminimum yoke heights are given by [10]:
/> (Eq.3.9) 3.6.5Airgap Length
Theairgap length has a minimum value limited by the manufacturing tolerances; thisvalue is typically in the range of 0.3mm to 1mm. In this design 0.5mm is chosento be the airgap length. 3.6.6Generator Length
Thegenerator length is estimated to be 95mm; this is approximated from fluxrequired to give the output voltage of the generator.
3.6.7 Airgap Flux Per Pole
Ina radial machine, the flux per pole is given by:
/> (Eq.3.10)
whereB is the average airgap flux density, D is the rotor inner diameter, L is thegenerator length, Kst is the lamination stacking factor and p is thepole pairs.
Forthis design the average flux density per pole Bgav is equal to thepeak flux density Bg since the magnet arc is close to 180 degrees. Thereforethe peak airgap flux is estimated to be 0.5T at the airgap and Kstis assumed to be 0.97.
Theairgap flux and the lamination stacking factors were estimated from thefollowing dimensions of the loudspeaker magnet:
· Magnet arc = 180 mechanical degrees
· Inner radius = 8mm
· Arc radius = 25mm
· Magnet radial length = 4mm
· Area of one pole = 706.8 μm2
Fromthe above magnet dimensions, the flux per pole in the machine is then estimatedto be 1.16 mWb this value is calculated from equation 3.10. 3.6.8Windings
Thestators of most synchronous generators are wound with three distinct andindependent windings to generate three-phase power [14]. A simplelayer winding was used in this design. Slot per pole per phase was chosen to be1 and the winding pitch was full pitch.
A. Types of winding
Thepreferred type of winding is distributed winding as it reduces harmonics andmakes better use of the stator or rotor structure. The mmf induced in theairgap is not sinusoidal, to get a pure sinusoidal mmf the number of slots haveto be infinity. This means that the distributed winding is expected to havesome harmonics.
Inducedvoltage for the distributed windings is:
/> (Eq.3.11)
Kwis the winding factor and depends on the winding arrangements and has a valueless than unity. Distribution factor Kd and a short pitch factor Kpreduces the winding voltage magnitudes but also reduces certain harmonics inEMF and MMF waveforms.
/> (Eq.3.12)
Distributedwinding configuration has one slot per pole per phase and its winding factor isequal to 1.
B. Winding arrangement
Singlelayer winding, where each slot contains one coil side, will be used in thisdesign as it is economical to manufacture and has simpler end connection. Emfand mmf can be modified to reduce harmonics. In a three phase system evenharmonics do not appear due to the winding symmetry, the only visible harmonicsare the belt harmonics.
C. Winding Pitch
Shortpitch is the most commonly used type of winding pitch. It reduces thedistorting harmonics and produces a truer sinusoidal wave. The length of theend connection is also reduced thereby saving copper and reducing copper lossin the coil.
Thedrawback of short pitch winding is that the induced emf in it is smaller thanin a full-pitch coil. The reason is that the total flux linking the short-pitchcoil is smaller than that of the full-pitch coil.
3.6.9 Number of turns
Thenumber of turns per pole is estimated to be 60 turns from equation 3.11.
Thedesign parameters discussed will be modelled in FEMM in the next chapter toinduce the output voltage and flux of the generator.
Chapter 4. Modelling the design in FEMM4.1 Introduction
Theinvestigation that will follow focuses on the effect of substituting standardcommercial magnets with recyclable speaker magnets that were collected from adumpsite in the village, to compare the performance of the generator in eithercase.
Inthis chapter, the two pole generator geometry discussed in chapter 3 will bemodelled in FEMM to analyse the output induced voltage and the flux of the generator.The lua-script in FEMM is run and the rotor is rotated 360 electrical degrees,for the lua-script refer to appendix C1.
Initially,a choice was made of three typical commercial magnet grades.Neodymium-iron-boron NdFeB was chosen from the rare-earth magnet group. Alnico6was chosen from the Alnicos and the last type was barium ferrite from theferrite or ceramic group. Then the machine will be modelled using differenttypes of commercial magnets to investigate the performance of the generator.
Theauthor then proceeded to investigate the magnetic properties of the loudspeakermagnet. This was done so that the parameters can be modelled in the finiteelement package.
Finallya design using the loudspeaker magnets was modelled to explore the recycledgenerator output. 4.2 Two pole geometry
Table4.1 below summarizes the generator specifications that were discussed inchapter 3. These parameters will be modelled in FEMM to view the output inducedrms voltage and the flux.
Quantity Value Frequency 39.79Hz Poles 2 Connection Y Diameter of Rotor 50mm Machine Depth 15mm Air gap length 0.5mm Turns per phase 80 Stator slots 6 Steel Core 1020 steel
Table4.1 Data of designed PM machine
Thedesign is modelled in FEMM and is illustrated in figure 4.1 below.
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Figure4.1 The generator modelled in FEMM 4.3 Commercial magnets
Toinvestigate the performance of the generator, the author began by modelling thegenerator with standard commercial magnets with the properties given in table3.1. The output rms emf and flux of the generator is tabulated in table 4.2with different magnets that were used in the design.
Referto appendix B for the graphs of the outputs. Matlab soft ware was used to draw theoutput rms emf and the flux, matlab code included in appendix C2.
Table4.2 Generator output with commercial magnetsMagnet Type Flux (Rms) EMF (Rms) Rare-Earth NdFeb32 0.0459 9.4262 Alnico Alnoco6 0.0186 5.1619 Ceramic Ceramic8 0.0175 3.6075 4.4 Recyclable magnet found in the rural area
Themagnet that was used in this section was from a loudspeaker that was foundlying in one of the dumps at Ga-Rampuru village. To start with the magnet shapewas not of concern. The author aimed to investigate the performance of themagnet on the speaker if used as it was found. The properties of this magnetwere investigated and a design was modelled using these magnets. The magnet isshown below in figure 4.2.4.4.1 Background on the characteristics of loudspeaker magnets
For speaker applications, the amount ofpermanent magnet required is directly proportional to the rated output power ofthe speaker. In other words high power speakers are often made using thehigh-grade magnetic types like the rare-earth. But since the speakers found inthe dumpsite were from low power appliances their typical magnets are normallyfrom the ceramic group type. In addition unlike Alnico magnets, ferrite orceramic magnets are not easily demagnetised magnetized and hence find wideapplication in such appliances.
4.4.2 Properties of the loudspeakermagnet
According to its nameplate the speakerthat used the magnet in figure 4.3 had a 0.5W rms and an impedance of 8 ohm.The magnet type on the loudspeaker is a ferrite [Refer to appendix D1]. Themanufacturer of the magnet on the speaker is traced in order to find the B-Hproperties of the magnet on the speaker.
Appendix D2 indicates TDK datasheet forferrite magnets FB series. These notes were used to find the magnetic, physicaland mechanical characteristics of the magnet. The properties of the loudspeakerare summarized in table 4.3. Magnet Type
Br (T) Hc (kA/m) Ferrite FB5N 0.43 214.9
Table4.3 Summarized properties of the magnet speaker4.4.3 Output EMF and flux of the recyclable generator
Theproperties were modelled in FEMM, and the generator outputs are tabulated intable 4.4. Refer to appendix B2 for the graphs of the outputs.
Loudspeaker Magnet Flux (Rms) EMF (Rms) Ferrite 0.0171 3.4987 Table 4.4 Generator outputwith the loudspeaker magnet4.5 The estimated outputpower of the generators
Theoutput electrical power of a generator is given by:
/> (Eq.4.1)
whereV is the terminal voltage of the machine. The power factor is assumed to beunity for these calculations since all the simulations and investigations aredone at no-load.
Fromthe rated power of the generator which is 36W. If the rated voltage is assumedto be 12 V then the rated current of the generator can be calculated fromequation 4.1 to be 1A.
Table4.2 and 4.3 above gives the results of the simulated induced voltages and fluxobtained from the generator with commercial and recycled magnets. Using the 1Aabove as the rated current, the output power of the generator using commercialmagnets and recycled loudspeaker magnets is summarized in table 4.5 below. Theoutput power in all the cases is calculated from equation 4.1.Magnet Type Output Power Rare-Earth NdFeb32 28.3W Alnico Alnoco5 15.5W Ceramic Ceramic8 10.8W Ceramic Speaker magnet 10.5W
Table4.5 The output power of the generator
Chapter 5. Analysis of the generator outputs
Inthis chapter the author first began by analysing the output power of the generatordesigned with commercial magnets and the one with recycled loudspeaker magnets.The author then explored the factors that may have affected the outputs fromthe recycled generator.
Theterminal voltage induced from the recycled generator is also explored to viewif it can be used in any applications in the rural village. This is done sothat the voltage can be evaluated if it is useful or not
Lastlythe loudspeaker magnets are investigated to view how they can be used in therecycled generator design; whether they should be smashed and aligned to bere-used in the generator design or if they should be used the way they arewithout being smashed. 5.1 The estimated output power of the generators
Theoutput power of the generators is estimated from the output induced voltages ofthe generators. Consequently, this means that the higher the terminal voltageof the generator the larger the output power.
Fromthe theory of magnets it is clear that the induced voltage is directlyproportional to the remanent magnetic flux density Br of a magnet. In otherwords it is expected that rare-earth magnets which posses higher Br will alwaysinduce high voltage when used in generators. Therefore it can be said that thetype of magnet used in a generator is very important as it determines theoutput power of the generator.
Ascan be seen from the results, the induced voltage of the generator with NdFeBmagnets from the rare-earth magnet family is higher than that with the AlNiCoand ferrite magnets. This was expected because of the different B-H propertiesof these magnets.
Therecycled generator in this thesis was designed using loudspeaker magnet that isfrom the ferrite family. These types of magnets are cheap and readilyavailable, but their disadvantage is that they posses low surface flux density.The induced voltage was therefore expected to be much lower than the voltageinduced in a generator with NdFeD magnets. 5.2 The rms output flux ofthe generator
Themagnetic flux density in the gap of PM generators is limited by the remanentmagnetic flux density of PMs and saturation magnetic flux density offerromagnetic core. Hence, the simulated value of output flux is directlyproportional to the remanent magnetic flux. In addition, permanent magnetmachine cannot normally produce the high flux density of a wound pole rotor.5.3 Factors that may have affected the recycled generatoroutputs
Thereare many factors that should be taken in consideration with regards to theinduced voltage from the recycled generator. Some magnetic deterioration mayhave occurred after the magnets were thrown into the dump. But, due to themagnet’s magnetic permanent properties, these magnets are expected to stillhave some surface flux density when found in the dumpsites.
Thisis evidence that any permanent magnet that is found in the dumpsites can bereused in a generator design to induce some voltage, of course depending ontheir B-H properties.
Theestimated properties of the speaker magnets that were used in this thesis werefound from the loudspeaker manufacture, clearly these properties will not bethe same as the properties of recycled magnets that were found in the ruralarea of Ga-Rampuru. These recycled magnets have been affected by differentconditions such as temperatures, climates, etc.
Theexact properties of the recycled magnets can only be found by testing thesemagnets in the laboratory. For this thesis the author was unable to take theloudspeaker magnets found in the rural area of Ga-Rampuru to the laboratory. 5.4 Applications of voltage from the wind turbine
Theinduced voltage of the generator will vary with the wind speeds experienced inthis village. The generator can be connected to a battery to store the powerwhich can be utilized when there is little or no wind.
Ifmore power is required, the voltage can be boosted by using any economicalbooster that can convert the output of the recycled generator to at least aminimum of 12V. The voltage from the booster can then be put through a cheapelectronic regulator that will only charge the battery if the boosted voltagefrom the wind generator is sufficient to produce at least 12V direct current.
Topower the refrigerator in chapter 1, the store owner in the village will haveto purchase an inverter that will convert the DC voltage to AC voltage. Theinverter will convert the low-voltage from the battery (12V) into mains-type230V alternating current.5.5 Design using speaker magnets
Finally,the author investigated how speaker magnets can be used in the generatordesign, if they have to be smashed or used as they are.
Asalready investigated, loudspeaker magnets are commonly from the ferrite magnetsfamily. Ferrite and rare-earth magnets are by nature very hard and brittle.Although they can be cut, drilled and machined this should only be done by individualswho are experienced with ceramics. If the magnets get over about 300 deg F,they will lose their magnetism permanently [17].
Therefore,it will be very difficult for rural artisans to cut these magnets and use them.Due to limited time the author could not investigate if these magnets can beused as they are in the machine.
Inthe next chapter the author attempts to assemble the wind generator in thelaboratory.
Chapter 6. Practical comparison of thegenerators 6.1 Introduction
Thefollowing chapter outlines the steps that were taken in order to assemble thepermanent magnet generator discussed in the previous chapters. This is done inorder to compare the practical outputs of the generator with the simulatedones. The other reason is to investigate the performance of recyclable magnetswith irregular shapes.
Thisinvestigation will only concentrate on assembling the generator part of thewind turbine system.
Forthe construction of the PM generator in this thesis two options wereconsidered, the first was to collect readily available off-shelf materials toassemble a small generator. And the second was to convert an AC induction motorto a PM generator. Both options are discussed in this chapter. 6.2 Materials required toassemble a PM generator
Themain idea is to build a portable generator that is easily assembled andconstructed in the laboratory. The author first begins with highlighting allthe materials that are needed in the construction of this generator. Figure 6.1gives the schematic of how the generator will look like.
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Figure6.1 Basic wind generator design
Fromthe generator illustrated above it is clear that the following materials willbe required in the construction of the generator:
· Magnets
· Stator and rotor
· Rotor mounted on a rotating structure
· Structure mound
Inthe following sections the author will outline steps taken and the challengesfaced in collecting these materials. 6.2.1Magnets used in the generator
Inthe investigation of the performance of the generator, the author was to beginby designing the generator using standard commercial magnets, which were to be latersubstituted with recyclable magnets. The recyclable magnets are picked randomlyin the dumpsites of Ga-Rampuru village.
Findingcommercial magnets for this investigation was a major challenge since for thistwo-pole generator the author needed to purchase two NdFeb32 magnets, twoAlnico5 magnets and two ceramic8 magnets. 6.2.2Stator and rotor
Therotor rotates with the structure mount while the stator is fixed and mounted toa support structure. Since all these investigations were to be carried outunder controlled laboratory conditions a drive and a frequency inverter whichare readily available in the laboratory will be used to rotate the rotor at thedesired speeds.
Thedrive will rotate the rotor and the induced voltage from the coils on thestator will be monitored by a voltmeter in the laboratory. Figure 6.2 illustratedthis type of drive.
Thesize of the rotor in this thesis was constrained by the diameter of therecyclable speaker magnets. Therefore steel with this shape had to be found orcut to this shape. After finding the relevant steel, the cylindrical steel hasto be drilled at the center. 6.3 Converting an inductionmotor into a PM generator
Dueto the challenges faced in gathering the materials needed to assemble thegenerator the author then decided to find an alternative method to investigatethe performance of the generator using recyclable magnets. A company calledMagnetag that supplies motors and generators was approached and after some negotiationsthe company was willing to donate an AC induction motor to the author.
The idea was to convert this AC induction motor into a permanentmagnet generator. This was going to be done by stripping the motor down andreplacing the wound rotor with recyclable magnets. This looked like anattractive option since recyclable magnets with any shape can be used in thegenerator to explore its performance
Theauthor was unable to complete investigating this option in detail. This isstrongly recommended for further work most probably at MSc level.6.4 Challenges faced during the construction of the PM generator
Themain challenge in the construction of this wind generator was cost. For thelaboratory investigation of the PM generator, a lot of materials, like themagnets and coils on the stator were found to be very expensive. This inadvertentlygives more support for the use of recycled materials.
Therewas a lot of machining needed for this project, the rotor and the stator neededto be cut and shaped to the desired diameter and drilled in the centre to fiton the mount structure. Time was the major constrained since a lot of thingswere required to be done in the limited time given for this thesis.
Howeverthe framework of how to proceed in constructing and assembling the windgenerator is already well laid out in this thesis.
Chapter 7. Conclusions
Basedon the findings of the report, the following analyses and conclusions weredrawn: 7.1 There is an urgent needof electricity
Dueto the number of people living without electricity in rural South Africa it isclear that alternative means of supplying these areas is essential. Accordingto ESKOM, all house holds will eventually be electrified, but the problem is,what is happening in the meantime? Where are children’s medicines being stored?Therefore this makes the electrification process extremely urgent.7.2 Resource assessment 7.2.1Recyclablematerials in the village
Anextensive assessment on the rural village of Ga-Rampuru was conducted. Thereare plenty of recyclable materials including old milling machines that are notin use. These materials can be recycled to clean Ga-Rampuru village. 7.2.2 Rural artisans who can assemble the wind turbine
Sincethere are many local artisans who fix cars, electrical appliances and do somemechanical work in this village, manpower should not a problem. An engineerfrom the government or Non-governmental organization could educate these localartisans on assembling the wind generator. This will have a positive impact onGa-Rampuru village as it will encourage people to work and be creative. There are many old wind mills used for pumping water inGa-Rampuru village, most of these wind mills are working perfectly wellsupplying sufficient water. This is a clear indication that there is a reliablesupply of wind in the village.
7.3 Simulation results
Ithas been shown that a reasonable amount of power can be realised from agenerator using recycled magnets from the dumpsites 7.4 Cost involved in thedesign
Theoverall cost of assembling this wind generator system will be very costeffective since all the materials are recycled from the village, and the entiresystem will be assembled by local artisans. 7.5 Validity of this thesis
Smallpower that can turn on small lamps will really be appreciated in this villageas children will be able to study after sunset. This will clearly have a widerange of positive developmental benefits on this community.
Chapter 8. Recommendations
Basedon the above conclusions, the following recommendations were drawn:
1. For a more accurate recyclable wind turbine design, all itscomponents such as the drum, the tower, rotor disk and cables must be exploredin depth. The following must be considered:
· Investigate how to extract maximum power from the wind using thedrum, and how to prevent the drum from over spinning.
· How to use other irregular recyclable magnets in the village inthe generator design.
2. Investigate how a permanent magnet generator topology can bechanged or re-designed to accommodate the design of a generator with the shapeof the loudspeaker magnets.
3. Look in to how the magnets can be removed from the speakers, sincevery strong clue is used to mount them, how this can be done in a costeffective way.
4. The axial flux permanent magnet topology should also be lookedinto to compare it to the radial flux type.
5. The exact costs of assembling and maintaining the recycled windturbine should also be incorporated in the design procedure.
6. With the little output power generated in this thesis, thisproject must definitely be taken further to alleviate the electricity problemsin South Africa.
References
1. Socio-economic rights project, “The rightto affordable electricity” copyright @ community law centre 2002
2. IDASA, www.idasa.org.za
3. Department of Minerals and Energy, WhitePaper on the Renewable Energy Policy of the Republic of South Africa. August2002
4. Department of Minerals and Energy, WhitePaper on the Renewable Energy Policy of the Republic of South Africa. November2003
5. Sathyajith Mathew, “WindEnergy-Fundamentals, Resource Analysis and Economics ” © Springer- VerlagBerlin Heidelberg 2006
6. Smail Khennas, Simon Dunnett and HughPiggott, “Small wind systems for rural energy services”. ITDG Publishing 2003
7. Kevin Reeves, “The design andImplementation of a 6kW wind turbine simulator” University Of Cape Town, SouthAfrica, Oct 2004
8. FrequentlyAskedQuestions www.magnetsales.com/Design/FAQs_frames/FAQs_2.htm
© 2000 Magnet Sales & Manufacturing Company, Inc
9. R.C. Bansal, T.S. Bhatti, D.P. Kothari,“On some of the design aspect of wind energy conversion systems” BirlaInstitute of technology and science, Pilani, September 2002
10. Jacek F. Gieras, Mitchell Wing,“Permanent magnet motor technology-Design and Applications” 1stedition. Marcei Dekker, Inc. 1997
11. Prof E. J. Odendal, “Design,construction and testing of a small wind generator with electronic controllerfor domestic use”. University of Natal, Durban
12. Jacek F. Gieras, Mitchell Wing, “Permanentmagnet motor technology-Design and Applications” 2nd edition. MarceiDekker, Inc. 1997
13. P.C. Sen, “Principles of electricmachines and power electronics” 2nd edition. John Wiley & Sons
14. Bhag S. Guru, Huseyin R. Hiziroglu,“Electric Machinery and Transformers” 3rd edition. Oxford UniversityPress, Inc. 2001
15. Dr. James Livingston, “MagneticMaterials Overview”
16. E. Muljadi, C.P. Butterfield, Yih-HueiWan, “Axial flux, Modulator, Permanent-Magnet with a Toroidal winding for windturbine applications”. Cole Boulevard, Nov 1998
17. Magfag, 2003 by Force Field
18. M.A. Khan, P. Pillay, “Design of a PMwind generator, optimised for energy capture over a wide operating range”
19. Joe Naylor, “Speakers with Alnicomagnets vs. speakers with ceramic magnets”
20. Hybrid (Wind/Solar/LP Gas) Systems forRural Community Development, “Electrifying South Africa for prosperity anddevelopment”. Upper Maphaphethe by Mike Wintherden
21. Danish Wind Industry Association,‘Guided Tour’ online htt://windpower.org/en/tour/wres/betz.htm
22. Lysen, E.H., ‘Introduction to WindEnergy’ CWD,2nd edition, p.p 51-73
23. Ripinga Nonkululeko, “Comparison of gridand off-grid rural electrification, based on the actual installation in LimpopoProvince”. University of Cape Town, Oct. 2005
24. Alfred Still & Charles S. Siskind,“Elements of electrical machine design”. 3rd edition. McGraw-HillBook company,inc. 1954
AppendixA
Graphsof output rms induced voltage and flux of the generator
1.Commercial Standard Magnets
a) Ceramic FLux_RMS = 0.0175
EMF_RMS= 3.6075
/>/>
b) Alnico FLux_RMS =0.0168
EMF_RMS= 5.1619
/>/>
c) NdFeB FLux_RMS = 0.0459
EMF_RMS= 9.4262
/>/>
2. Loud Speaker Magnet
FLux_RMS = 0.0171
EMF_RMS = 3.4987
/>/>
AppendixB
Matlabcode for sketching the output emf and flux of the generators
%EMF calculation from FEMM
%ByMaribini Manyage
clc
clearall; close all;
P= 2;
w= 1912; %mechanical speed in rpm
freq= (w*pi/30)*P/(4*pi); %frequency
XA= load('flux_link_A.txt');
XB= load('flux_link_B.txt');
XC= load('flux_link_C.txt');
beta= XA(:,1); % angle between Is_r and d-axis [elec degrees]
alpha= beta — beta(1,1); % Rotor position in [elec degrees] from Zero
time= alpha*(pi/180)/(2*pi*freq);%*1000; %time
flux_link_A= 2*XA(:,2);
flux_link_B= 2*XB(:,2);
flux_link_C= 2*XC(:,2);
%Perform spline in order to differentiate flux linkage vs time
pp_flux_A= spline(time,flux_link_A);
pp_flux_B= spline(time,flux_link_B);
pp_flux_C= spline(time,flux_link_C);
%extracting piecewise polynomial coefficients and derivation
[hgt,wdth]= size(pp_flux_A.coefs);
clearAA;
fork = 1:hgt
AA(k,:)= polyder(pp_flux_A.coefs(k,:));
end
dpp_flux_A= MKPP(time,AA)
[hgt,wdth]= size(pp_flux_B.coefs);
clearAA;
fork = 1:hgt
AA(k,:)= polyder(pp_flux_B.coefs(k,:));
end
dpp_flux_B= MKPP(time,AA);
[hgt,wdth]= size(pp_flux_C.coefs);
clearAA;
fork = 1:hgt
AA(k,:)= polyder(pp_flux_C.coefs(k,:));
end
dpp_flux_C= MKPP(time,AA);
%backemf
emf_A= ppval(time,dpp_flux_A);
emf_B= ppval(time,dpp_flux_B);
emf_C= ppval(time,dpp_flux_C);
figure(1);
plot(time*1000,flux_link_A,'r-');
holdon;
plot(time*1000,flux_link_B,'b-');
plot(time*1000,flux_link_C,'g-');
title('Fluxlinkage — under noload');
xlabel('Time[ms]'),ylabel('Flux linkage [WbT]')
grid;
figure(2);
plot(time*1000,emf_A,'r-');
holdon;
plot(time*1000,emf_B,'b-');
plot(time*1000,emf_C,'g-');
title('BackEmf — under noload');
xlabel('Time[ms]'),ylabel('Back EMF [V]')
grid;
x= length(flux_link_A);
FLux_RMS= norm(flux_link_A)/sqrt(x)
y= length(emf_A);
EMF_RMS= norm(emf_A)/sqrt(y)