PMSM Dynamic and Steady State Models

okay so welcome to the the next the second session on MSM theory of brushless DC machine theory so we went through a couple of different concepts related to permanent magnet synchronous machines and I think we finally came up with oh stop at the rotor reference frame motor model we started with the rotor stator frame model which is sinusoidal or at least your parameters are varying with rotor position right that's the important part the motor parameters are wearing with rotor position depending on the type of the machine of course but most practical machines that is true with rotor position you will see the machine model change and then if you look at the variables such as voltages and currents or flux that goes into the machine or that is in the machine that is of that those are sinusoidal in nature and then the same theory that applies so that we discussed about the rotating MMF or the rotating flux that is still applicable to this permanent magnet synchronous machine because we have a round round or three phase bound state right the only difference is we have a permanent magnet or rotor in this case and then the voltages and currents are sinusoidal and we were able to show that using the reference frame transformation if you can transform into the rotor reference frame or the synchronous reference frame our model as well as the signals become much more easier to analyze the variables the currents voltages became DC light and then the motor model we will discuss it today in detail that becomes even more simple okay so when we look at the previous the state reference frame model Oh as you would look at it in you know if you have to look at it in the normal circuit diagram this is what you are going to see okay and then we look at what will happen when we do the transformation to those parameters right so that's going to that discussion here all right now last time let's we kind of showed what we Q as VDS and V 0 s you know the Q axis voltages D axis voltages and the zero sequence water just looked like and we had resistances currents flux linkages all in that one equation so let's write them out previously we had them in matrix form so let's expand them out to see what we get and we can start talking about more of that okay so we said vqs R and I'm going to use simple variables here because these are dynamic variables which can be varying you know with time you can see transients and all that RS IQs r plus lq v IQs are now we are just converting that variable P into a derivative term D the IQs are over DT plus omega r LD ids R plus Omega R under him okay so this is the Q axis dynamic voltage dynamic voltage okay qsl voltage after transformation and then if you look at the d axis voltage VDS are we are going to see the following RS s yes I did yes our lesson ideas our last LP the i TS t minus omega r l q fq s thank you okay so this is my this is my V DSM the voltage of D axis alright so we have Q axis and D axis what else we have zero sequence 0s we have here well you don't need our zero sequence RS I 0 s plus LS t 0 s so now when we look here we can clearly see the back EMF Kappa right prime X is due to the permanent magnet that is a long week you now let's go through the different elements ok when we walk through we see that we have our s IQs are plus lq the IQs are over DT ok so a resistant terms times the rate of change of current times inductance ok so we have seen this before we have seen this before we came into our AC machine discussion can you guys remember can you recall where we have seen this this is very familiar at least we should be very familiar to us right if we are to write this as s lq in laplace so this is a differential equation if you are to implement a motor model this is one equation these are the equations that you can use to implement your motor model in the rotor reference for you okay so this is s lq IQs are right and then this is RS IQ VESA in if you transform this equation to laplace domain so i'm not can you recall what this but this is similar to if you think of so if you think of DC machines right if you think of DC machines if you can recall that we had a resistance inductance and a back EMF by DC mode that's what we had so we had we said this is PI a common because all this VDC or VA this is RA this is LA and this is Eve right very simple terms right so now if you think of along the same lines you can see here see a resistance times the current inductance time the same current and then we have some different term here we can treat them as a voltage okay and that's why that's another reason we call them call this brushless DC because it looks like a DC motor in the transformation now the primary difference in the synchronous for the synchronous machine is instead of one equation so for a DC motor you only have one equation one differential equation here we have three equations okay and if you assume balanced condition you will only need two you don't even need three equations you'll have just the two equations okay and they have again they have the same nature or similar nature to the DC motor okay all right and then for this type of a machine if you don't look at the torque which is important right when you are building a motor model we need to be able to calculate the output tow now for a DC motor it was K T times the current the armature current right now for a synchronous machine this becomes a little bit more complicated I mean it's in the rotor reference frame it's fairly simple but that becomes the following so this is the number of poles poles by 2 okay we have to have that factor times lambda M prime R okay lambda M prime R and I'll show you how to find this variable this is associated with the back EMF the permanent magnet flux linkage okay lambda M prime R I q sr q axis current right Q axis current + LD - lq IQs are five ESR so the product of the two currents times the inductance now what is this LD and L Q let's go back to our equations here okay now when we look at these two equations this is the Q axis voltage equation this is the D axis voltage equation right and if you look at the inductances the D axis inductance has a different name and the Q axis inductance has a different name lq and LP right lq and LD and depending on the design of the machine l q + LD can be the same and lq + LD can be different okay and we talked about this when we talked about salience e right when we talked about salience e let me take up that very straight for example we had a stator like this and then we said we had we have a rotor and we have magnets like if we let's think of an interior permanent magnet right so this is the rotor and we have magnets now when you consider along this axis you see different material therefore your inductance is affected right so you will see a different inductance along this axis versus this axis right so that same principle applies and if you have a surface mount magnet right surface mounted permanent magnets which is like this right so you're gonna see a north and a south like that so that means regardless of which axis you consider you are still going to see the magnet materials for that surface mount right there for your lq + LD will be almost equal okay for surface mount p MSM so LD is almost equal to calc you for IPM interior permanent-magnet l LD or L Q is not equal to LD j we just talked about that so that's that's very important piece of information to keep in mind now when we go and look at our coke generation right in the torque equation we can see that foreign for a pump as SMP MSM right surface mount magnet surface mount pit permanent magnet synchronous machine our electromagnetic torque equation will simplify because your LD minus L Q becomes negligible this will go to zero therefore this equation becomes 3 over 2 PI over 2 and M prime R IQ vessel okay and if you look at IPM in theory a permanent magnet this will be valid and now what you can see here is you can even use your D axis current to generate talk okay now remember the axis is the direct axis that is what that's we said that is for a certain reason right we said we have a quadrature axis and a direct axis direct axis is going to be along aligned with the flux the the rotor flux right so if you think of a think of a synchronous machine permanent magnet synchronous machine I'll use the blue for the stator back there so for the stator vector with the stator windings you can have two exits right two axis for the flux I'll call this Q and D we are not aligning we are just labeling the axes okay if we think of the rotor you think of the rotor I'll use red for the rotor it's going to have its own Q and D axis right it's going to have its own Q and D axis depending on the orientation of the rotor Q and D axis right which is determined by the magnets right so if you direct D axis is the direct axis along the flux so our note on the rotor is going to be here South is there so you have your direct axis along the flux now when we generate flux using the state it is based on this reference frame right so we want to orient that reference frame to be ninety degrees out of phase such that now we apply the state of flux along Q axis quadrature axis quadrature means it's 90 degrees away from the direct axis of the rotor flux okay so that's how you will generate flux and this is going to rotate however when you generate in a IPM machine you will you can even apply direct axis current which will help you generate talked because of the difference in the inductance okay so if that's a little bit too much to handle for now forget about the IPM the toke generation think of the SMP MSM very simple you have an equation similar to KT times ie right KT times I like a DC motor very simple very easy to understand this can be you know it may be once you have a good understanding of this you can think about the use of using the IPM machine and this ID current now this comes down to the optimal torque generation or maximum tow per ampere in P P a so this is a research topic that has been studied extensively maximum tow per ampere because that's what you care about right when you generate talk in electromechanical systems you care about maximum torque per unit current now in the surface mount magnet case the only talk generation current is IQ so you can basically find that relationship directly but when you have IQ ID and such a nonlinear relationship this becomes a little bit more complicated than the surface mount magnet case okay so but this is the electromagnetic torque equation alright good any questions so let's let's try to look at our voltage equations that are rotor reference frame voltage equations but from the point of a circuit so that way it will give us a little bit more clarity on what's going on [Music] yes yes Alec so the Katie value the Katie value if you look at here this is only depending on the machine design machine design it is not something a control engineer or an algorithm can control for this class of machines there are you know there are new machines coming out where you can modify your lambda based on certain currents certain mechanisms that you can use but in general if you are looking at an SMP MSM your your your KT P is going to be fixed in general okay now if you are if you have a method that we'll talk about later that you can modify lambda M they are essentially modifying KT right and we had a name for this when we talked about DC machines if you are not sure if you can remember that you guys remember we talked about a certain method to modify KT which will affect eventually influence ke right and we said with that we can operate at higher speeds we can improve how do yes we that field weakening control right field we can in control so the same concept will apply if we can control lambda M will we get into that discussion later I don't want to jump you know jump ahead of the topic here but this is the basic idea yes the KT is in general machine dependent and then you know the Machine the properties the magnets the winding structure the materials used all that gets accounted for in KT right of course over time if the magnets become weaker the winding resistance change you know you don't generate as much as current so then certain things get influenced but they're not going to go into those those subtleties and at least for now okay so let's try to take a look at what these voltages look and the circuit would look like if we are to draw a circuit for the the voltage equations okay the representation we'll call that the representation of PM in a sense this is basically the dynamic model okay because we have V M dynamic model we are modeling the differential equation okay differential equation you can talk about the steady state model - in a little bit okay so first of all so we have we talked about three differential equations and guess what that means you can have three three circuits right three voltage equation differential equations however you want to call them but you're going to have three circuits so first of all we have one for VQ we have a resistance you have an inductance then you have two voltage sources right and we'll name them shortly so we'll call we call this V a new simple letters you have EQs our we have RS we have lq we have omega r LD ids are so this is due to the d axis current this is called cross coupling cross coupling because it's coupling between the axes okay plus minus Omega R lambda M prime R okay and then we have a current IQs are okay so this is the Q axis circuit and then with let's look at the D axis circuit gonna have a similar circuit inductance then you have just the cross coupling term no back EMF so VDS r or s lv because this is d axis right look at the polarity make r lq ids are and then we have a current IDs are now actually this is a good point where we can think about a little bit about the field weakening now see here IDs are is the current going through this circuit right the VD voltage equation now by controlling the VDS voltage we can essentially control the IB current right you can control the ID current now if you look at this equation here we can by changing the current here we can influence our vq circuit voltage which is technically responsible for the talk generation IQ right so which is telling us that you know hey if you need to go to high speed or some kind of field weakening that you want to achieve you can do that by controlling the VD forty so this cross coupling is helpful okay and there are control algorithms that control algorithms that get rid of this cross coupling to okay and forget that I'm big I think it's called cross axis decoupling control so they kind of mitigate the coupling through control strategies if we have time we can talk about that too all right okay but do you see that relationship so let's say we have a negative ID here like if you have a negative ID current that means this is essentially going to flip in polarity and now you have more voltage on this circuit right so you have V Q voltage plus a little bit more coming from the v q VD axes that can help the the top generation oh if you want to go to a higher speed okay you can go to a higher speed because now you can control that voltage alright and then let's look at the zero sequence here we have just a resistor and we have the leakage down here so we had a VZ Rho s and RS 0 s and then this is our leakage term so we were able to break down that three-phase winding representation into this DC motor like circuit diagrams which will eat oh you know which will simplify our analysis and the voltages v QV d and IQ ID those are DC like voltages in steady state and then no longer sinusoidal all because of the rotor reference frame transformation okay let's see what else we can talk now we know that we have Omega R right we have our motor speed Omega R Omega R is the rotor speed and then this is basically the rate of change of theta R the rotor okay and then we need to apply voltage to drive this three-phase motor right and we can measure current later but we have to apply voltage to drive it so let's say we apply a three-phase set of voltage ABC okay so if I applied voltages sinusoidal sinusoidal then you know we a s gonna be I'm gonna say V s now we as cosine II V okay Tita Evie is is a co today if it can be written in the following way V has cost time Omega E the key plus let's say V Prime okay now the importance here is we have a frequency for our voltage so this is essentially this is going to be our stink Rinna's frequency right because this is a synchronous machine and we can now introduce a phased into our voltage which we can control when we apply our voltages we can decide to apply it the in phase with our back EMF or out of phase with our back EMF and when we say out of phase it can be any phase angle or at any phase shift there is an optimal angle that we'll talk about later but just we need to have the concept that we can control how we apply our phase voltages to the machine okay so this is a little this is a generic representation now if we applying V is like that to a plot to be balanced our VBS vs is got VBS will have the same amplitude right then cosine theta V minus 2 5 over 3 and then VCS same amplitude cosine theta V minus 4 5 over 3 PI over 3 okay so we have that now what will happen if we transform these two rotor reference frame you see you know like a position measurement right yeah we are accurately measuring the the position of this of the theta e V is being accurately measured right if you can do that what we can do is we can we can determine what these voltages would transform into when we look at them from the rotor reference frame okay so any thoughts what will they look like fronts we transform them so if we do if you perform a rotor reference frame transformations wrote a reference frame transformation we will see that we have the Q SR which is equal to vs cosine if you do LS realign with it you're going to have what we call this field V okay and then we're going to have VD as are we yes let me see I think was negative right sine and then what is fee V right this is the angle V V is basically the angle between theta E V and theta R so if there is a difference between the rotor angle or rotor position and the applied voltage angle that will introduce the phase shift now once you transform it becomes a magnetic right so it will either it will be a factor that will apply to V Q and V D so basically if you have to think of let's say V s is 10 right let's say yes is 10 volts so for sinusoid right 10 volts and then depending on how you choose the phase shift phase shift to be for example if it was 45 right if if this value 3v was 45 degrees then what is cosine T V but cosine 45 cosine 44 pipe cosine fortified what's cosine 45 10:45 one over two but if this was 60 where is this part of least 60 course across 960 half-right cosine 60 so this is V Q is going to be 10 times 1/2 and then V DSR this is going to be negative 10 times R sine 60 root 3 over 2 right so now you can see that how the the voltage will split between Q and D axis depending on how much offset you introduce to that voltage this is called that voltage angle okay this is going to come up again when we talk about field oriented control which is very useful okay so let's just keep that in mind and then Omega E is our electrical frequency and we can say that this is Theta evey over DT yes so this is our electrical frequency all right now if you introduce such an offset of course you can apply voltage to Q axis as well as the axis but if you properly align you can apply all the voltage to Q and no voltage to D so you have the ability to control how you apply the voltage to the Machine okay so previously we talked about the dynamic model right we looked at the dynamic model of a simpie M permanent magnet synchronous machine which had differential terms okay but often times you don't need the full dynamic model to analyze machines you can basically use a steady state model steady state okay and we are assuming a balanced machine balance machine yeah if there are imbalances you know the model is going to get a little bit more complicated and and it that depends on the imbalance - sometimes it gets a little bit not a little bit but a lot complicated now I'm going to use capital letters if V Q SR is equal to RS IQs are your lq indicus disappears we are considering steady state steady said plus omega r lb ids are plus Omega lambda am applying for and then VDS r becomes RS IDs are minus omega r lq thank you sorry okay so very nice very straightforward equation and the generic torque equation it doesn't change whether you are looking at the dynamic model or the steady state model 302 - where i'll tried it a little bit differently here let me take IQs are both sides so then you're gonna get lambda M prime R plus the minus Q I be as thank you okay so this is your steady state motor model now this is much more easier to implement then if you are to implement that three-phase model okay which is still possible you know if I have time I might record how to develop state a frame model in MATLAB and Simulink maybe later but you know you can develop either one and they're going to perform all the same at least for the balance keys in balance key is you have to be a little bit more careful okay good any questions okay look so we have the models now right now when you get a machine we need to be able to you know characterize as a machine that's the other important topic that you have to think about characterization of a permanent magnet synchronous machines okay have you had have you had to characterize a machine you know for a project thank you big body no so do you recall how we characterized DC motors DC motors how to characterize a DC mode although we characterize a DC motor [Music] but do we characterize the DC motor not necessarily characterization characterization where there are no poles at least we don't consider that for DC mode that's right characterization of the synchronous machine how what are the parameters of interest for a DC motor you guys recall if you can remember for a DC motor we had the in de resistance right the winding resistance that we looked at right and similarly here we are going to have our phase the phase resistance and then we're going to have inductance right in this case we're going to have lq and lb2 inductance values and then we are going to have our back EMF constant right lambda M prime are like back EMF constant and then one more thing we shouldn't forget number of poles but not a lot of people talk about this but number of poles are also an important especially when you try to align with you know your precision sensor you need to know how many poles you have for your calculation to divide your mechanical position okay so how do we find the number of the poles how do you find number of poles so papi this is false right number of poles [Music] it's straight forward I mean relatively straightforward and there are a couple of different approaches you can follow if you have a position sensor attached that you can take a measurement from that will be ideal but it's not very practical in most cases you might not have the position sensor attached omma you might need to power the position sensor to take measurements so so that way it might not be that practical but typically I'm trying to think what I would do is try to rotate the motor right so if you think of the motor poles take a very simple Porter here okay and we have let's say we have four or to do pole pairs of four paws right so you have a north-south north-south right then okay so this is the rotor rotor only right and what you can do is you can spin this motor shaft with with a with some device a DC motor maybe or you can use a drill a power tool I had to you know take good care when you use a power tool like that you know but spin it relatively slow but because what we are trying to do is we are trying to look at [Music] [Music] [Music] Oh get it get it right there yeah in here yeah I can hear but did you have the same issue did it drop out yeah that's good at what point did you did I lose you you were talking about the voltages okay so how to measure the back EMF yeah how do okay Alex can you hear me yeah okay good where did I lose you you were saying something about that the these two lines that you're pointing at right here I did be open for okay so these are the phases right ABC phases and you have to have them open because we are going to look at the face to face back EMF okay so if you want to visualize this based on our little circuit diagram right so you're gonna have duck turns resistance source inductance resistance so basically we are looking at so if this is a and B and C we are looking at B and C this voltage while it's not connected to any device and you are basically measuring this back EMF back EMF is going to have the influence of poles because your magnets are going to cut through the windings and the number of back EMF cycles per revolution will give you a way to find how many poles you have okay so that's why we that's one way we can find the number of poles so if you spin the motor this way and then if you record for one revolution if you don't have a sense that you have to come up with a method to find that one revolution you see that for one revolution for this motor you're going to see two cycles alright you're going to see two cycles because you're going to see a not a South a not a South peak a Valley a peak a valley right so depending so first of all record that count the number of cycles per one mechanical revolution one mechanical revolution right and that's the number of pole pairs you have if you want to know the poles it's basically you know how many peaks and valleys so we have one two three four peaks and valleys that means this is a four pole mode for for water for two pole pairs to forecast okay so this is how you will find the number of poles in a permanent magnet synchronous machine okay good alright so we have to talk about three things next we have to talk about flux linkage and the inductance okay I want to hold off because I don't want to stop in the middle we have about ten minutes and what we can do is we can talk about the rest of the characterization process next time okay good so do you guys

Channel: Sandun Kuruppu

Views: 1,962

Keywords: motor control, AC machines

Id: 8Mc-nsQpXHI

Length: 49min 37sec (2977 seconds)

Published: Fri Jun 05 2020