Mod-06 Lec-19 Concept of space vector

welcome back to this lecture series on pulsewidth modulation for power electronic converters so lets take a quick look at what the what are the various modules that we have covered so far in this course now so as you are aware initially we started with an overview of power electronic converters this is was probably one of the longest modules covered till now ah we looked at a various kinds of power electronic converters started from switches and then we went on to dc dc convertors buck boost etcetera and we looked at voltage source and current source converters multilevel converters etcetera then if we had a particular focus on three phase voltage source convertor and we looked at applications of voltage source converters and applications of voltage source converters would ah be like you know the most important application that we are concerned about here is motor drive that is an induction motor drive other than an induction motor drive you know there are other things that ah you are look look at ah for example we are looking at active frontend converters and you are looking at ah um power quality equipment such as you know static compensator which you can use for compensating ah supplying reactive power to the system and further you also use voltage source converters when you want ah to clean up the harmonics drawn by certain loads that is harmonic filtering active ah filters as they are called so these are the various applications we had a quick review on these various applications and you may remember that each one of these applications could actually be a course by itself and we had a very quick review on this now and once we had this you know we we the role of a voltage source converter we looked at pulsewidth modulation for a voltage source converter and we first look at the purpose of pulsewidth modulation and if you look at it the voltage source converter there are three things one is the dc side voltage then the ac side voltage then the modulation so the purpose of pulsewidth modulation is to make sure that on the ac side you get the desired fundamental voltage but in addition to the fundamental voltage you also get some harmonics so the trouble is you want to avoid those harmonics so the purpose of pulsewidth modulation as we saw was to control the fundamental voltage and to reduce the harmonics and their ill effects so these are the main purposes of a pulsewidth modulation and then with this we went into low switching frequency pulsewidth modulation where you barely have you know one or two switching angles very few switching angles in every quarter cycle i mean the inverter switching frequency is very low is just you know the you know a few times that of the fundamental modulation frequency like five times or seven times if you are taking a modulation frequency fifty hertz we could consider you know just a switching frequency of two fifty three fifty hertz etcetera so i talked about the practical relevance of this and why i mean why it is relevant from an academic point of view etcetera earlier and this is probably where the the last two this is where we came to you know like these are what you would call as now very important pwm methods and ah the triangle comparison method is what is ah probably most alone from a practical point of view if if somebody is familiar with many of these material then this is probably one where one would particularly focus on and in many of these practical converters we do have this triangle comparison based pwm ah the best known of this category is the sine triangle pwm where you sinusoidal modulation signals and compare them with the triangle or carrier and you produce pwm signals and to the sine you can add some third harmonic or you can add many other triple frequency components common mode components you know those this is what we discussed in the previous module now so today we are going to start up with a new module which is again about pwm generation so the earlier module was about generating pwm for a three phase voltage source converter thats following what is called as a triangle comparison approach now once again you are talking of generating pwm for the voltage source convertor but what happens is we are going to use a different approach and what is called space vector approach so we are going to deal with space vector base pwm in this module now so what is the space vector base pwm so these are the various things that we will be covering as part of this module so first we would look at the concept of space vector thats what we will primarily be doing today so once you know what is the space vector we know we understand the term space and we understand the term vector individually so what we mean by the space vector something we need to look at and then we would look at what is called a space vector transformation that is actually three phase quantities can be transformed into two phase ah vectors and thats what you call as space vector transformation you would look at this and from then on we would go into conventional space vector pwm method which is the most popular pwm method using this ah space vector approach ah so thats conventional space vector pwm i would say is as popular as sine triangle pwm i mean if not more popular and in fact its this benefit is you know compared to sine triangle pwm it can give a higher amount of ac side voltage for the same dc bus voltage i mean whatever advantage we got with third harmonic injection a similar advantage you can get with this conventional space vector pwm also so you will be discussing this conventional space vector pwm now so this is ah what i would call as the most important method using the ah space vector approach now and there are alternative space vector based approaches this bus clamping pwm you recollect that the discontinuous pwm or bus clamping pwm which we discussed in the earlier module triangle comparison methods we will now discuss in this module also how to produce this bus clamping pwm not by comparing triangular carriers and ah modulating signal but from the space vector approach this is what we would ah also be looking at from this now and then these two methods triangle comparison and space vector would look a little different from each other but actually they have a lots of similarities so we would look at the equivalence of triangle comparison and space vector approaches to pwm and most of the literature stops here there are very you know if you read through good part of the literature you would get a feeling that both these approaches are pretty equivalent but what we would like to stress in this course is space vector approach is a little more general than triangle comparison approach that is whatever can be done using triangle comparison approach whatever pwm waveforms can be generated using triangle comparison approach can also be generated using space vector approach on the other hand everything that is possible using space vector approach is not necessarily possible using triangle comparison approach i would call space vector approach as more general than triangle comparison approach and you know one category of space vector based pwm is this advanced bus clamping pwm these pwm signals can only be produced from the space vector approach and they cannot be produced from the triangle comparison method is what i am going to talk about towards the end of this module now so thats just to give you an overview of what you would expect in this module which we are hoping to cover in four lectures i am just getting started with todays lecture the beginning of this module thats the concept of space vector what do you mean by a space vector now lets start with an example this revolving mmf this is about an example of a space vector approach of of what is the space vector so this revolving mmf in three phase machines thats what i would like you to consider most of you should be familiar with three phase machines and even in this course we have been dealing with three phase machines on and off now you consider a three phase machine like a three phase induction motor or a three phase synchronous machine so what you have is you have a three phase winding sitting on the stator and what you do to the three phase winding you apply three phase sinusoidal voltages usually and when you apply a three phase sinusoidal voltages on to this three phase winding it produces a revolving mmf there is a magneto motive force this field this magnetic field produced and that magnetic field revolves in air gap this is what happens now and this revolving mmf is what where does it revolve it revolves in space so in in some part of the air gap at some angle you may find the peak the magnetic field the the the you know the flux density might be at its peak now it will gradually move around the space so you know this is a revolving mmf so thats its its actually a vector which really moves in space and so you can call that as an example of what is a space vector now so lets get a little more into this what we mean by a revolving magnetic field so a step towards understanding revolving magnetic field is a pulsating magnetic field a revolving magnetic field is produced by three phase winding a pulsating magnetic field is produced by a single phase winding so first lets take our time and slightly review this pulsating magnetic field ah produced by single phase winding and from then on build that understanding to a three phase winding this is many of you would have done a course on machines but it would be good certainly to review this here now so now you have this rotor its ah its clear so its marked rotor and this innermost circle stands for the rotor then this small ring that you have is the air gap and what you have here is the stator and its also written in a stator here and on the inner periphery of the stator you have a few conductors here and you have few conductors there they have a placed kind of diametrically opposite to one another which is the case in most machines ah you know sometimes the return path is not exactly one eighty degree away so what are called as ah you know short pitch coils and usually use full pitch coils so you know you can say that there are conductors going here and conductors going here for the conductor here the return path is here for the conductor here the return path is here for the conductor here the return path is here thats how we assume so its anyway those details are not very important for this purpose now let us say you have a coil what does this coil do it produces a magnetic field what is the direction of the magnetic field is an important question now well that depends on the direction of current flowing so lets consider some direction of current flowing ok what would be the direction lets say the current is a dot and the current here is cross i hope most of you would be familiar with this dot and cross dot basically means current is flowing out of the sheet its like an arrow coming out of this you just see a point so its like current coming out here and cross is like arrow penetrating into the sheet so arrow means current is flowing into this now so if you do this what is going to be the direction of this well you can hold on to your ah you know right hand and you can look at this and now for current flowing out through the top this is going to be the direction of magnetic field and this is what you would call as the axis of this magnet what is going to happen is this is going to produce magnetic field which really goes around the this things and this is what is really the axis of this magnet and when the current is constant this magnetic field is a constant magnetic field when this current varies with time certainly the magnetic field also varies with time but the axis is unchanged this is the axis of this magnetic field so if the current alternates then the magnetic field also alternates sometimes you know it is in this direction sometimes its in the other direction and you know the amplitude goes on changing its magnitude goes on changing so you might get a pulsating magnetic field so lets get a little clearer about the pulsating magnetic field lets do that mathematically so what are we going to do that to deal with it mathematically so lets say you have this mmf so what is this f f stands for mmf r stands for the winding its r phase winding we just considering a single phase and just lets call that simply r and one basically what our mmf is produced by a winding lets say like what you have here it would be an mm it would be some alternating waveform it need not be sinusoidal it wont be sinusoidal to make it closer to a sinusoid as why you do a few things you know you you design this you distribute the windings and all that but you know but still it is not sinusoidal but most of the machine analysis you would ignore the special harmonic components and you will deal with fundamental component there now so when i say this a one the subscript one refers to the fundamental part of ah the mmf so this spatial harmonics of this mmf are ignored and you know this what you have is fr one that is the current passing through this r phase winding produces some mmf so the fundamental part of that mmf is what is taken as fr one now so what is that that mmf should obviously be proportional to the current that is flowing through that right ir so it is ir there and k is some constant of proportionality and now this theta is the angle for example if you consider here let me say you consider like this this angle you can call as theta a e theta is because its an angle a to say that its air gap and e is the electrical angle i am talking of the electrical angle here i am just considering it as a two pole machine for simplicity so you know you just call it take it as theta ae this is what you have now so if the magnetic field is highest here as you go around that as you go around the the strength of the magnetic field will go on reducing and the magnetic field will be zero here and if you go here it will be negative peak here and then it will strength will reduce and will be zero and then it will be back to positive so you will get a distribution which is alternative and which is actually you can presume it to be sinusoidal distribution now and therefore you know what you will have is this is what you get here that is cos theta ae would be the mmf there at that angle theta ae now so i am just not getting into the details of k that would for one thing it would involve the winding distribution factors etcetera so i am not getting into that so there is a constant k lets simply leave it at that fr one is kir cos theta ae now what is that ir that is not going to be a constant current you are going to be injecting an alternating current im cos omega et is there omega et is the fundamental angular frequency if you are talking of a fundamental frequency twenty or fifty hertz this is two pi into fifty a hundred pi that is so radians per second that is omega e so you ah and im is the amplitude of that co sinusoidal component so you have ir is equal to im cos omega et therefore you put one into the other you have fr one is equal to kim cos omega et cos theta ae therefore you find that this mmf special mm its a its a function of special angle theta ae and it is also a function of time t so its a function of both space and time now its a product of two cosines you can always write it as sum of two cosines like this so it is f max by two this kim let me give a term called f max for that let me just call that f max for convenience and this f max by two i can say thats it is cos theta ae minus omega et which is basically the difference between these two arguments plus cos of theta ae plus omega et which is the sum of these two thats what you have now so i can write this as this fr one as two different components theta ae minus omega et theta ae plus omega et so now what really happens is this itself is one component which rotates this is a another component which rotates now one rotates in the positive direction or the a anti clockwise direction the other one rotates in the negative direction or the clockwise direction so you can write this fr one as some fr plus and fr minus the clockwise component you can write it as fr plus and fr minus thus you can resolve a pulsating magnetic field into two i mean revolving magnetic fields as many of you might be aware in fact this is what is actually used in an index single phase induction motor also you produce two magnetic fields and you know this wont be able to start what you will do is you will do some trick you will slightly we can one magnetic field and strengthen the other one so strengthen one relative to the other and you will get the um machine running so thats what you do in a single phase motor as many might be aware so you most of you should be aware of the fact that the pulsating magnetic field can be split into two revolving magnetic fields one rotating clockwise and there other rotating anticlockwise so and this is the math this behind what you can i mean behind this statement now right so now instead of one winding what you have is you have three windings so you can say that this is the old r phase winding we are talking about these are the corresponding written conductors now then about the y so for y what i have do i have i have used green color here so this is the y phase belt and the corresponding belt is y here then the b phase i have used blue color here and this is one that is one now and if you look at the various axis what you can find is the r phase axis is like this so you have these windings and you have these windings the r phase axis is like this as we saw earlier this is the r phase axis how about y phase axis you have these windings in these phase belt here and y phase is it goes like that this would be your y phase axis in a machine and what it is the axis of the b phase winding you have here and here this would be the b phase axis these are the axis of the three windings of this machine now so lets look at what happens when they are excited these three phases now i am just the same three phase windings i am putting them in a different form i am just indicating the three axis here calling them ryb as i said before and we pass certain current through this r phase winding what is that current ir is equal to im cos omega et so sinusoidal current i am taking that the value of sine they mean that is equal to zero at time t equal to zero so i am writing this as a cosine function here ir is equal to im cos omega et if that is the current then the y phase current is im cos omega et minus one twenty and the current through the b phases im cos omega et plus one twenty degree right so this is what we have and from this we move on what we are doing is three phase windings we are you know we are passing three phase sinusoidal currents through that through three windings now so this produces a revolving magnetic field how first we had this earlier case fr one is kir cos theta ae as we had said before now if you had considered the y phase axis y phase mmf would also be the mmf produced by the y phase winding will also be you know some proportional to the y phase current so it is kiy and cosine of now we are looking at the mmf at angle theta ae but the y phase itself is at an angle hundred and twenty degrees from the r phase and you know this is theta ae measured from the r phase axis so this is theta ae minus one twenty cosine of that is what would be the component along y phase axis so fy one will be proportional to you know cos of theta ae minus one twenty then you have the mmf at that angle produced by the b phase winding so the current flowing through the b phase winding is ib so this is fb one is equal to kib cos of theta ae minus two forty thats because the b phase winding is two hundred and forty degrees away from the r phase winding and hence you have this now and what are our ir iy and ib im cos omega et im cos omega et minus one twenty im cos omega et plus one twenty or this is also minus two forty so these are the three phase currents that you have now so plug in ir iy ib into the previous thing you get a similar expression as we found earlier each of them produce a pulsating torque i mean pulsating magnetic field ah as ah we discussed before fr one is f max cos theta ae cos omega et so you know f max is now this k into im thats what is taken in that way similarly fy one would be f max cos of theta ae minus one twenty and iy is substituted by you know for i mean ah iy ah is proportional to cos omega et minus one twenty so you get this cos omega et minus one twenty and if you take fb one that is f max times cos of theta ae minus two forty multiplied by cos of omega et minus two forty you see the difference between theta ae in omega et and you again see the difference between this argument and this argument the difference between theta ae minus one twenty and omega et minus one twenty is the same theta ae minus omega et as before similarly theta ae minus two forty omega et minus two forty is also the difference is still the same so lets write down the equations further what you can say is fr one fy one fb one they can also be written as f max by two multiplied by something sum of two cosine terms the first term is cos of theta ae minus omega et that is that is the case for all the three phases the second term would be cos of theta ae plus omega et thats basically the sum of these two is basically the sum of these two terms and the sum of these two theta ae plus omega et in the next cases it is theta ae plus omega et minus two forty which is plus one twenty and in the third case the sum of the two is theta ae plus omega et ah you know plus one twenty so this would be plus two forty here so this is what is here now so the r phase y phase b phase all the three of them produce pulsating magnetic fields and each pulsating magnetic field can be split into one component revolving in the anticlockwise direction and another component revolving in the clockwise direction now so the ones revolving in the anticlockwise directions are adding up now the ones revolving in the anticlockwise direction you add these three they will sum up to zero so here you can call this as a fr plus fy plus and fb plus they add up they add up to what three into f max by two cos of theta ae minus omega et on the other hand these three you know this is some cos of x this is x plus then there is a another one twenty degree added here there is another two forty degree added so these three cosines add up to zero fr minus plus fy minus plus fb minus would add up to zero means it essentially becomes equal to ah you know it it should have been three times fr plus so you know just you make a change into that so this would be simply be three times ah fr plus is what would be your ah some of that and that is nothing but ah f f ag one is three by f ah f max by two cos of theta ae minus omega et and now this you can see is a a revolving magnetic field so thus you get a revolving magnetic field produced by three phase windings excited with three phase sinusoidal currents so this is a revolving magnetic field and now if you want more on this revolving magnetic field you can actually take any book on electric machines so one of the references i would suggest is ah ae fitzgerald kingsley and ah umans electric machinery by tata mcgraw hill two thousand three well there are also different additions of this book available i mean there are also older additions this and there are several books in the subject of electric machinery so in you can actually take any textbook on electric machines and that would be able to give you a very good picture on ah the revolving mmf ah to greater extremes and particularly the design of windings and many things if you want you can actually look at those books there now our main interest in saying this is you know revolving mmf is one example of a space vector so we just first trying to understand that thats what we are trying to do now the next thing is from this we have to go into gradually understanding a space vector in a more general ah sense and ah coming up with what really would be space vector transformation now let just say one thing now you have three phase windings and what does the three phase winding do it essentially produces some mmf which rotates like this this is what it does this is done by three phase winding now the question is can it not be done by two phase windings as shown here that is i can also have a two phase winding you know that is when i can call as alpha the other one i can call as beta and this alpha and beta are ninety degrees away from one another they are separated in space by ninety degrees and i can also excite them with currents which are shifted ninety degrees in time and by doing this i might be able to produce the same kind of revolving mmf using this two phase winding now so for three phase winding you could consider an equivalent two phase winding which really can do this job of producing a revolving mmf now lets go little further now i have shown the three phase axis ryb simply as a lines here and this is the revolving mmf that i i i had pointed out earlier so this is the revolving mmf so corresponding revolving mmf can be produced here lets write down the equations if you really want to do that what should you be doing the two mmfs are supposed to be equal what is what are the two mmfs one mmf is produced by this current i alpha excuse me so there is alpha winding and through this there is some current i alpha flowing and through the beta winding through which some i beta is flowing so this n times i alpha should produce a component along the alpha direction and this i beta that n times i beta is a component along the beta direction now so n times i alpha should be equal to what now the r phase is the one that produces the mmf entirely along alpha axis so it is n times ir and a component of the mmf produced by y phase is along the alpha axis that is niy cos one twenty again another component of the mmf produced by the b phase winding is along the r r phase axis or the alpha axis so this is nib times cos two forty so you need to satisfy one condition n i alpha equals nir plus niy ah cos one twenty plus nib cos two forty this is one of the conditions that you really have now right this is one condition and how about the current ni beta so ni beta is the mmf along the beta direction now r phase winding does not produce any mmf along the beta axis it produces ah its a its aligned along the f alpha axis therefore it doesnt produce along that so what is the mmf produced by the this three phase winding along the beta axis it is niy cos of you know there is something you know this this should actually be niy times you have cos of thirty degrees let me just change it here so if you look at it it is going to be cos of thirty degrees some sorry there is a mistake here so it is cos of thirty degrees now and what is this again this is nib and that is going to be this b beta axis is one fifty degrees away from here right so this is thirty degrees and you have another hundred and twenty degrees so this is going to be one fifty degrees so this is going to be cos of one fifty degrees now cos thirty degree is root three by two cos one fifty degree is minus root three by two and this is actually its a plus sign here you can say plus nib cos one fifty so this is what you really have so you can produce whatever is the revolving mmf produced by a three phase winding ryb can also be produced by a two phase winding alpha and beta whose axes are separated by ninety degrees provided your i alpha and i beta satisfy the two equations given here so thats what i am writing down in the next slide i alpha should be equal to ir times cos one twenty plus ib times cos ah two forty and this should be ir minus iy by two minus ib by two and further your iy plus ib is equal to minus ir because your ir plus iy plus ib is zero therefore iy plus ib is minus ir therefore you get this as three by two times ir so this is what you have you have three by two times ir so ir plus iy plus ib then how about your i beta its a same mistake that has carried over here also so this is going to be cos thirty degree and you can say plus and this you can call this as cos one fifty degree and this will be a root three by two times iy minus ib so this you can call as so you you can you know the mmfs are equal that is the mmf produced by three phase winding excited by currents ir iy ib is equal to the mmf produced by two phase winding alpha and beta carrying currents i alpha and i beta if i alpha is equal to three by two times ir and i beta is equal to root three by two times iy minus ib and this is what is space vector transformation of three phase currents thats what is reproduced here i alpha is equal to three by two times ir and i beta is equal to root three by two times iy minus ib so we can just simply write this in a matrix form you can write i alpha i beta and this is three by two zero zero zero and the first zero and zero and first axis is zero root three root three by two and minus root three by two you multiply by this by another matrix ir iy ib so i alpha is three by two times ir i beta is equal to root three by two iy minus root three by two ib so this is space vector transformation of three phase currents i have just given this in a matrix form here just for convenience sometimes you might find that convenient now so what is applicable for three phase currents now we are looking at three phase voltages so you can also extend this idea to three phase voltage where also you have the same condition vrn plus vyn plus vbn is equal to zero if you are considering a balance star connected load that is i am connect considering a star connected load so n is the load point neutral ok so now vrn plus vyn plus vbn is equal to zero into this condition what you can say is you go by your same thing as v alpha is equal to vrn plus vyn cos one twenty plus vbn cos two forty and that goes to add up to root three by two vn the same mistake is perpetuated here also so you can probably call this as cos thirty this is plus you can write and you can write this as cos of hundred and fifty degrees and you can see that this is going to be a root three by two times vyn minus vbn so it is the same transformation that you are getting here also so you know that so this is the transformation for three phase voltages it its very similar to that v alpha is three by two vrn and v beta is equal to root three by two times vyn minus vbn so this is the three phase transformation of ah i mean a transform a space vector transformation of three phase voltages so three phase voltage gives converted into an equivalent two phase voltage so what you see here is you first have a situation lets say ir plus iy plus ib is equal to zero this is from a kcl so this is a three wire load and the sum of the three phase currents is zero similarly you can consider in the load neutral is not connected its a three phase ah three its a three wire load and you have balance indeed you you have vrn plus vyn plus vbn is equal to zero this is the kind of three phase voltage that you have now so both these equations if you see they are of the form x plus y plus z is equal to zero in the cartesian coordinate system so what does x plus y plus z is equal to zero represent in the cartesian coordinate system its the equation of a plane so its its a three phase quantity but you see that the three phase quantities are they are not three independent quantities because they are equated you know they are they they have a relationship so a similarly here vrn plus vyn plus vbn that this three phase quantity doesnt mean the three phase the they are three independent quantities they are actually they represent a plane and how many dimensions does a plain have only two dimensions so a two dimensional space can be spanned by simply two orthogonal vectors so a a three phase quantity like this can be represented by just a two phase quantity i alpha and i beta similarly a three phase quantity such as this can be represented by two phase quantities namely v alpha and v beta thats the moral of this the three phase quantities sum up to zero and they can be represented only i mean by only two independent quantities now so for further reference you can look at this book by holmes and lipo on pulsewidth modulation for power converters principles and practices you can particularly look at chapter one of this book where there is a section on the concept of space vector where this is discussed further right so this is our idea of a space vector so now we have come up with whats a space vector and you know a hard space vector transformation and we have got a transformations for three phase voltages and three phase current stuff so now what we are going to look at is we are going to look at the specific case of a voltage source inverter so you have voltage source inverter it has a dc voltage vdc it produces some ac side voltage so ryb this ac side terminals these voltages can be measured with respect to some point or the other the preferred point is o this is what we said in the previous lectures also so you can have vro vyo and vbo that would be the output what is the value of vro vyo vbo that would depend on the inverter state for example if the r phase top device is on then vro is plus vdc by two if r phase bottom device is r then vro is minus vdc by two the same thing about y phase and b phase and you totally have eight different combinations because this top or bottom can be on y top or bottom can be on b top or bottom can be on so two n multiplied by two multiplied by two so there are eight possibilities which are called eight different states of the inverter and ah for this eight different states of the inverter there are you know basically the inverter produces eight sets of output voltages how many sets of output voltages eight sets of output voltages it produces now so for these eight sets of output voltages we have to find out what are the corresponding voltage vector so these are three phase voltages right so these three phase voltages we should be able to transform into ah you know um ah two phase equivalent two phase quantities lets come to that a little gradually lets first look at the load what is the nature of the load lets say the load is a star connected load but it is a three wire load this r is connected to the midpoint of r phase this y is connected to the midpoint of y phase leg b is connected to the midpoint of b phase leg so ry and b are connected to the midpoints of the three inverter legs and this load neutral n is not connected anywhere particularly it is only ryb are connected this is a three phase three wire kind of a load thats what we are assuming is what is true in case of most induction motors so you go on with this now the load neutral n and the dc bus midpoint o are different thats what we need to bear in mind so your vro is the pole voltage what we mean by the pole my pole we mean the midpoint of a leg why because every leg is like a single pole double throw switch and its midpoint is the pole and on the two ends of the legs are ah the throws so we saw that its a single pole double throw switch the midpoint is a pole so this is the midpoint potential of a leg measured with respect to the midpoint of the dc bus so vro can be plus or minus point five vdc depends on whether the top device is on or bottom device is on in the r phase like this is what we just discussed a while back similarly vyo can be plus or minus point five vdc depending on whether y phase top devices on or y phase bottom device is on similarly b phase also if the b phase top device is on it is plus vdc point five vdc bottom device is on it is minus point five vdc so these are the three possible values of that and you have eight sets of a three phase outputs possible now now what is your space vector transformation v alpha is equal to three by two times vrn this r n this is the r phase voltage at the midpoint of the leg which is the load terminal but measured with respect to n the load neutral and this is not the dc midpoint this n is different from o ok so you have vrn plus vyn plus vbn is zero but your vro plus vyo plus vbo is not equal to zero thats an important point you have to bear in mind you take any of these values here say point five vdc point five vdc point five vdc why how do they sum up they sum up to one point five vdc that is not equal to zero similarly you take any arbitrary values here here here they will never almost never sum up to zero whereas vrn plus vyn plus vbn always sums up to zero so now the space vector transformation is v alpha is equal to three by two times vrn what is this vrn you can express this in terms of vry and vbr vrn can be taken as one third of vry minus vbr when you talk of balanced loads so you ah your three by two vrn becomes half of vry minus vbr so once you have this vry is vro minus vyo similarly vbr is vbo minus vro so if you do that if you plug in those you get it as half of two vro minus vyo minus vbo this is your v alpha so this is now v alpha the original space vector transformation is in terms of vrn now we are trying to express that in terms of vro vyo vbo now you see that your v alpha does not depend on vro alone it depends on vyo and vbo also then how about v beta v beta is root three by two times vyn minus vbn and vyn minus vbn is same as vyo minus vbo its basically the difference between the potential at y the midpoint of y phase leg and the midpoint of b phase leg you know whether the potential is measured with respect to n or o the difference is the same so it is vyo minus vbo so here you have v beta this first part of the space vector transformation here the you know the voltages which are the load ah phase to neutral voltages have been converted into the pole voltages measured with respect to the dc midpoints so now you have v alpha and v beta available in terms of vro vyo vbo so thats what we have now so for the eight inverter states you know the values of vro vyo vbo you can plug them in to get your v alpha again you know the values of vyo and vbo you can plug them in to get your v beta values thus for all the eight ah inverter states you can get the corresponding voltage vectors thats an exercise which we would do you can actually plug in the numbers and go and play income things related now so what i would like to do is i dont want to plug in the values here and show what is what i can very easily tell you that first consider like something like plus minus minus so this is vro that what i mean by plus minus minus this is an inverter state which represents to a situation where the r phase top leg is on and y phase bottom leg is on and the b phase bottom leg is on so there are three signs given they correspond to ry and b respectively in that order ok and r is positive meaning the top device is on y is negative meaning the y bottom device is on b is negative meaning b bottom device is on so vro is plus vdc by two and vyo and vbo are equal to minus vdc by two since vyo and vbo are equal to minus vdc by two you can see from the previous slide that v beta is zero so it is only v v alpha and you have vro is equal to vdc by two therefore this becomes vdc here it is all minus vdc by two and minus ah vdc by two so it adds to two vdc and half of two vdc simply equal to vdc so it becomes a vector like this and this vector is as got a magnitude vdc i have not indicated this here but i have indicated this in a later slide so this inverter state plus minus minus leads to a voltage vector of a magnitude vdc at an angle zero that is v alpha is equal to vdc and v beta is equal to zero is what it leads to similarly you can do for all the other states one after the other now lets understand this a little better and then now thereby we can see things ourselves so what do we see here we see that when the r phase alone is connected to the positive bus and y phase and b phase are connected to the negative bus you produce a voltage vector of magnitude vdc along the r phase axis now everything is symmetric the machine is symmetric you know the inverter is all symmetric now so instead of r phase alone connected to positive you have y phase alone connected to positive on the other two are negative see that is the case here your y phase alone is positive and the other two are connected to negative so what you should say by symmetry you should be able to say now that you know the the resultant voltage vector will be along the y phase axis just as it was along the r phase axis it should be along y phase axis and the magnitude of the vector should be same like vdc for this transformation that we are considering here so you will get this now similarly you can look at a situation where b phase alone is connected to the positive bus and y r and y are connected to the negative bus this will produce a vector of magnitude vdc aligned along the b phase axis so that is how you can say that these are easy you can plug in the values you know we had the expressions of v alpha beta in terms of vro vyo vbo you can plug them in and come up to this now i am just trying to give a feel for that excuse me now if this is your vro vyo vbo then you invert you say plus minus minus you go to minus plus plus excuse me yeah so instead of plus minus minus you have minus plus plus where all the three are inverted so vro vyo and vbo are inverted that naturally means v alpha and v beta should be inverted in sign it is going to produce a vector aligned the negative r phase axis here the r phase terminal is alone connected to the negative dc bus and y and b are connected to the positive dc bus and the resultant vector is along the negative r phase axis the same arguments are valid for y phase and b phase so you can consider there are six possibilities for each of this possibility you will have a vector aligned along the positive r phase axis or negative r phase axis or positive y phase axis and negative y phase axis and so on one or the other axises you get all the vectors are of magnitude vdc now this is accounted for six there are two other states which are indicated here minus minus minus and plus plus plus minus minus minus means all the bottom devices are on that means the load is shorted here plus plus plus means all the bottom devices on the load is shorted we have discussed this earlier we know that this is a redundancy that is there in a voltage source inverter because of both of this what happens the resultant voltage vector is equal to zero you can say that here all vro vyo and vbo are equal and you know the line to line voltage is vry vyb vbr are all zero therefore the line to neutral voltage vrn vyn vbn are also zero v alpha and v beta are zero these two are called zero states because there is no transfer of power during these zero states between the dc bus and the ac side so these are called zero states the other six states are called active states because during this process there is transfer of power between the dc side and the ac side whether the power flows in this direction at that direction depends on the operating condition we will look at that at a later stage but there is a power flow whenever active states are applied whenever the zero states are applied the load is shorted its simply freewheeling you know the dc bus is opened out and the load circuit is opened out so there is no power really transferring between the two now so thats what is also there are eight different inverter states there are eight sets of output voltages six sets of output voltages produce six voltage vectors of magnitude vdc we call them as active vectors and here we can call them as a zero vector other two states produces zero vector now this is what we haven and go here now you go to the next level so what do you have here one two three four five six i am just reproduce the same thing now so i have indicate with this magnitude vdc here just for clarity so this is our for this space vector transformation that we have defined you come to this now sometimes you should remember that there are slightly different transformations are used basically they use a scale factor you can use some scale factor some you know root of two by three certain other factors can be used for some reasons and this may not be vdc it may be some k times vdc so that depends on the scale factor used but for the transformation that we defined a while back these are all going to be vdc now so now what is going to happen if this is only to show that there are eight of them like this and the next thing is about the reference vector we will come to this anyway little later what we do here in space vector based pwm is we use a revolving voltage vector as the reference what we do in sine triangle pwm we use three phase sinusoidal signals if you are transform three phase sinusoidal signals into the space vector domain it is basically a revolving voltage vector and therefore you have a revolving voltage vector here which is shown like this during some time it will be here later on it will cross here and go into this it will go here and it will circulate around all this now is what we will do ok so now let us say you consider a particular instant or you know in interval a small interval of time ts during which the vector is here now can we produce this vector is the question now what we did average pole voltage is recall the concept of average pole voltage your pole voltage can be as high as plus vdc by two or minus vdc by two and therefore you can produce a an average pole voltage anywhere between plus vdc by minus vdc by two the same way here is the concept of average voltage vector over a sub cycle ts now what you try doing here is now this is what i have to produce a vref vector so if i apply this active vector one for example for certain interval of time t one of the you know sub cycle time ts so it is applied for a fraction t one upon ts so then i am applying this vector which i can call as v two vector for some fraction of time t two upon ts now and for the remaining time i am applying the null vector so what is going to happen is i am going to get a vector like this another component this is parallel to v one vector and it is t one by ts times v one vector and this component is parallel to v two vector it is equal to t two by ts times v two vector these two are added up to get your v reference vector this is the average voltage vector now thus you can produce an average voltage vector whose tip can lie anywhere within this triangle so you consider that these are two sides of triangle you join these two tips by another straight line so that will give you a triangle so you can produce any v reference any average voltage vector whose tip is anywhere within that that is what is possible by averaging so earlier you are doing an averaging of the pole voltages two instantaneous values plus vdc by two and minus vdc by two here now we are not looking at three individual phases we are looking in them as collectively as three all the three together as inverter states or the voltage vectors now we use these two voltage vectors on the null vector to produce any average voltage vector within this triangle thats what is written here now if you want a vector v ref ts that v ref into ts v ref vector into ts will be equal to v one vector into t one plus v two vector into t two plus vz vector times tz so this is what is called as volt second balance now this is the reference volt seconds this is the applied volt seconds over a sub cycle reference volt second should equal the applied volt seconds and this is given in an alternate form this vz vector is a null vector so this term is a null vector term so effectively you have v reflector is equal to v one times t one plus ts plus v two vector times t two plus ts thats what is illustrated in the figure now and what is v one vector v one vector is this vector which is vdc angle zero and what is v two vector is the same magnitude vdc but the angle is sixty degrees now so this is what you have as an average voltage vector in a sub cycle that is by applying this vector for t one this vector for t two and this null vector for tz you can produce some reference vector like this thats what we are trying to say right so this is what the ah volt second balance equation is given here so for a particular value of v reference vector which is basically some v ref an angle alpha which basically is a magnitude v ref and angle alpha what should be the values of t one t two tz you can just do this you can decompose them along the alpha axis and beta axis and you can derive these expressions what is what we can call as dwell time t one is the time for which vector v one should be applied t two is the time for which vector v two it should be applied tz is the time for which the null vector should be applied now so this t one t two and tz t one and t two can be calculated like this they depend upon the magnitude v ref and angle alpha of the reference vector so from there you can just derive this that is from the previous page you know you can for example these are is a vector equation you can decompose it along the alpha axis and beta axis for example and you can come up with these values for t one and t two you will get that t one is v ref sin sixty minus alpha times ts divided by vdc sin sixty similarly you will get t two as v ref times sin alpha multiplied by ts divided by vdc time vdc into sixty degrees and tz is the remaining part that is ts minus t one minus t two for the remaining portion of the time ah i you apply the null vector tz so this is how you calculate the dwell times we will be doing more on this in the next few lectures this is just to show that an average voltage vector can be synthesized using ah these ah two active vectors and one null vector now so what we do in conventional space vector pwm is we try to apply these vectors now for example this is a null vector and this is a active vector one this is a active vector two and again this is null vector how exactly we apply here we go to the inverter states minus minus minus and plus minus minus and so on and and do this here now if you start here you can apply this minus minus minus to produce null vector and from here you can switch the r phase and go here and that would be plus minus minus and then from there you can switch y phase and go here to go to plus plus minus and from there you can switch b phase and go to plus plus plus so that is what is shown here you you you apply the different vectors and here you are applying the same vectors in the reverse direction plus plus plus all all the legs are positive then here b phase is switched then y phase is switched and finally r phase is switched you can see that you know r phase switches like this y phase switches like this and b phase switches like this now so what we have is you have two active vectors applied in the middle and the two zero states are applied at the end so the two zero states in conventional space vector pwm what is done is this is two zero states what is called as t is zero and seven they are applied for a duration is t zero and t seven t zero equals t seven is equal to point five tz they have to be totally applied for a duration equal to tz so you are dividing that equally between the two and that is what is done in what is called as conventional space vector pwm which i mentioned ah while earlier that it is the most popular method we will be discussing this in much greater detail in the ah following lecture so what you are doing is both the zero states are you applied for equal durations of time thats kind of conventional wisdom you can apply a null vector i using either this option or that option by keeping all the bottom devices are on or keeping all the top device on what do i do do it both equally you know apply this for half the time apply that for half the time is the is a simple logic and you just do this and thats conventional space vector pwm as i mentioned we will discuss more of this later now so what you can do is you can apply an average voltage vector like this for some time and next you can go out to the next value if you have applied this over some ts now and over the next interval t is excuse me you you can apply another voltage vector like this then over the next sub interval a sub cycle you can apply this now here the angles i have shown is fifteen degrees you know between these two i am taking my sub cycle time to be equivalent to fifteen degrees or omega ts where omega is the fundamental angular frequency multiplied by ts i am taking it as fifteen degrees here which is actually big in in many practical inverters of of you know the power level of kilowatts or so ah this sub cycle duration could be much ah lower than that it can be just a few degrees now ah its ok this is taken for an illustrative purpose now so what i am doing is i am lets say i am applying this average vector for fifteen degrees and and after that fifteen the next fifteen degree interval i am applying this average voltage vector how i am doing by applying vector v one vector v two a null vector for appropriate durations of time t one t two and tz the next time i will go here so what do i do here i will apply the same v one v two and null vector for different durations of times such that i produce this average the next sub cycle i will apply t one t two tz for another you know appropriate set of appropriate values that i produce this average vector thus in every sub cycle i produce an average vector like shown here like shown here around that now so what i am doing is i am going close to a rotating voltage vector what am i supposed to produce i am supposed to produce three phase sinusoidal voltages and in space vector domain three phase sinusoidal voltage is nothing but a revolving voltage vector i am supposed to produce a revolving voltage vector so thats what i am doing i am producing a vector like this which stays there and after short interval it just moves by a small step it stays there after a short interval it moves by that so that the jump you mean the step by which it moves is equal to the you know the is equivalent to the interval of time for which it stays there so actually what is happening is circular motion is approximated by you know sampled circular motion just as you know sign wave is approximated by a sampled sign wave and a sampled sign wave is not very different sampled and held sign wave you know thats not very different from an actual sign wave provided the sampling intervals are close enough is the similarly these you provided these you know the sub cycle durations are substantially smaller than the fundamental cycle you know this is a very valid approximation of a revolving voltage vector that is you can thus you can produce a revolving voltage vector using a three phase inverter and this can be applied to a three phase induction machine so you you know you when you are able to produce a three phase voltage vector it simply means that you are able to produce three phase sinusoidal voltages so this is about space vector pwm so what are we doing what are the essential differences now we are using this idea of space vector and what is the space vector three phase quantities can be transformed into two phase quantities you can so now you can see this is only a two phase quantity now and there is a three phase sign and a three phase sign becomes a revolving voltage vector of constant magnitude and a uniform angular velocity so if its uniform angular velocity you sample it at equal intervals of time you are going to get samples at equal intervals as shown in this particular figure and so this is what you go by ah you know doing this so now what you can do is now this is how can you produce this so the circular motion is being approximated by like this a vector which is stationary here for certain amount of in for an interval of time omega ts after that it jumps by omega ts and it stays there for that and it goes on doing like this now and how can you produce this vector for example this vector can be produced by time averaging of this v one vector v two vector and null vector and if you are talking about this vector for example this can be produced by excuse me time averaging of this vector this vector and the null vector if you are talking about this vector it can be produced by time averaging of vector v four vector v five and null vector thus you know wherever you are you can call all these as six sectors and in each sector you use two of the neighboring active vectors on the null vector to produce the average voltage vector i am going to deal with more of space vector base pwm particularly conventional space vector pwm in the next class this is only to give an idea of what you could call as you know the concept of space vector and space vector transformation and some introduction into space vector based pwm and a more detailed discussion on that would is reserved for next class now here i have a few references for you please have a look at this the first two references one is holtz you know thats and on the other one by you know van der broecks skudelny and stanke these two i would call as ah you know ah probably the seminal work in the area of a space vector based pwm so there are other there were a few works in the middle ah mid eighties these two i would say are probably in the second half of a eighties so these two papers ah reported a space vector base pwm ah um you know they are early papers in this area of what you would call as space vector pwm so for further you know your knowledge and understanding you can probably look at these things and ah another one is you know this book this dg homes and ta lipo is a book it covers many things about pulsewidth modulation and all that so it also takes about talks about space vector concept transformation and a space vector base pwm so thats a book which can also be used as a reference now then the last one indicated here is varma and narayanan that is space vector pwm has a modified form of sine triangle pwm for simple analog or digital implementation so this is published in iete institution of electronic ah um and telecommunication engineers journal of research this is a tutorial paper this is particularly written from the point of view of ah understanding them easily so these are some references i would like to give and i would give you more and more references as i would go on relevant references and ah you know i hope that you found this lecture interesting and we would continue this in the next lecture and we will discuss conventional space vector based pwm in our next lecture thank you very much

Channel: nptelhrd

Views: 49,102

Keywords: Concept of space vector

Id: xcxfDlz1bEs

Length: 57min 8sec (3428 seconds)

Published: Tue Jan 21 2014