okay good afternoon um so last time we started talking about Power Electronics with three-phase elements we spent our time talking about just diode rectifiers and that's actually quite important because when you're building really high Power Systems very often that's the kind of thing you have at your front end between your AC to convert to DC um at the same time a lot of applications require high frequency switching power conversion right so the kind of things we were talking about single in Phase inverters the question is what do you do with a three-phase inverter and one possibility is just to in some cases is just to replicate a singlephase system right so you could imagine if I had say a three-phase machine with open-ended winding um maybe I would have you know something where I had you know a phase a and phase B and phase C and you know each of them have a pair of Terminals and I say okay I can go build a three-phase inverter basically by building three singlephase inverters right so maybe I come in from my DC bus BDC right and I could come in and say okay let me just build a singlephase inverter which would be say four power mosfets okay and I'll show the diodes on the power mosfets Just for clarity right and I could come out and here's one terminal and here's the other right and I I build this to Drive phase a and then I build another copy this is Phase a I build another copy Drive phase B and I bring it in from the same Terminals and then I build another copy and I drive phase C okay and I could again build in same DC bus and away I go okay so I have 12 devices I just have single phase inverters and I run it okay and in fact sometimes people do this okay and in fact actually at MIT right now we're building a megawatt system that basically has this structure it's sets of singlephase inverters that drive a three-phase system where the machine is a custom machine and has so-called open-ended windings like this okay so it works perfectly well however the vast majority of say three-phase Motors are not just separate windings connected like that right typically if for example I have a three-phase motor what I will have is either something connected in y right where this is one motor winding a second motor winding and a third motor winding right where I'll have a b and c okay I only have three terminals okay or maybe I have a d this is a y connection maybe I have a Delta connection where the machine is wound like this and again I have a b and c I just in this case I might have a neutral in this case I have no neutral okay so in this case I can't just arbitrarily apply voltages from three independent single phase inverters why because if I accidentally you know connected phase a to high with one Inver and phase a to low with another inverter I'd sort out my DC bus and everything would blow up right so if I'm going to drive drive something that's connected as a three-phase load in y or Delta I need a different plan okay so what do people do uh it's very much the structure that we saw with the three-phase bridge rectifier instead of having dodes however I will have active switches okay so maybe I would come up and build something like this um I'll have my DC bus come in so I'll have two terminals VDC right I will have a first Bridge leg like this and let me make sure and show the diodes again and this can generate VA a second one can generate VB and a third one can generate VC okay maybe what I'll do is I will consider the midpoint of this bus for reasons we'll talk about and I'm going to call this node V subr okay that's my reference voltage it's just the center point of the bus where I ground this thing you know we can discuss that but let me just pretend that this is exactly in the middle it's VDC over 2 from here okay we'll see where we're going to talk about that and that's my inverter and maybe I would connect it up to some three-phase load so maybe it would look like this I would have my three phases here's one here's two and here's three and maybe I would connect this point to what I call V subn or the neutral point right that's this neutral point if it was a y connected load or I could just have line to lines if there's no neutral okay so what's the advantage here well you'll notice that this structure only requires six cits instead of my 12 okay so it's simpler okay it also turns out that as compared to that solution let let's suppose I could connect my motor either way I wanted right I could connect it in y or Delta or I could connect it as three independent windings because this circuit like this phase leg whatever current goes out it returns through one of the other phase legs okay I get an advantage in terms of the utilization of the devices so I will either require less semiconductor area to get a certain conduction loss in this structure or I will have lower loss for the same semiconductor area lower conduction loss for the same semiconductor area as I will compared to the top solution okay so this solution in most three-phase applications is vastly preferred okay where is the exception to that it turns out that if I had you know I wanted to generate a certain Ripple in an inductive Lo load like a motor inductive uh back EMF kind of load it turns out that you end up having to switch this bridge at a higher frequency than you do the single set three sets of single phase so in a system where it's a really high frequency output and you're switching losses are heavily dominated the top solution actually ends up starting to be advantageous so in very specialized high-speed machines for example you might go for the top solution of independent phase drive if you're allowed but the vast majority of three-phase inverters or three-phase DC to AC converters power could flow either direction where you have active control is with this three-phase bridge structure any questions about that so far okay so one thing I wanted to notice about note about this is that I want to pay attention that that these devices are especially if the load's going to be inductive so I don't actually control instantaneously say what the direction of that load is I usually have devices with reverse diodes on them or I use mosfets which have it built in because that way when I turn one device off I can always guarantee there's a path for the current so these are single directional blocking bir directional carrying switches so this is a kind of voltage source inverter right there are also three-phase current source inverters just as there are with single phase but again this is vastly the most common just to um illustrate this I'll show you this guy here uh this is actually an inverter out of an older model Prius okay you'll see the two in here that's the DC bus coming in that's where the capacitors and everything else would hook up then go back to the battery and then there's phase a phase B phase C output okay each of these what you'll see is a pile of devices in parallel in order to provide the required power and when you look closely at this you're going to see that there are essentially three terminal devices that go back to gate drivers those are the igbts in this older model today they use silicon carbide fets and in anti- parallel with them because the igbt can't carry reverse current they have actual diodes right so in in their design the actual diode is separate from the actual switch okay okay um and so I'll pass this R you can look at it and by the way if your Prius happens to be missing an inverter I swear I had nothing to do with it so what can I do with this inverter okay keep in mind oh yeah go ahead Jack I a question about the inverter why would they choose to do igbts with di instead of mos Ah that's a very good question because if you think about a mosfet a mosfet's on state looks like a resistance whereas an igb's onate looks like a constant drop and it turns out that a time at the time that was built for a very high voltage device you could get lower forward drop for a given device area with an igbt than you could with a fed and so you know maybe until five years ago or maybe 10 uh it was most common to use igbts especially at high voltage more recently uh with wideband gap devices people are starting to move over to uh sometimes igbts but but mainly kind of fet kind of structures because you can get away with that and it turns out to be lower loss so whichever switch gives you less loss is ultimately what you want to choose to use so let's think about what we can do with this and keep in mind we said one of the big advantages of um having three phase is that we get natural cancellation of all the triple n harmonics right we're going to see that with this inverter structure but essentially if I think about I have three Bridge legs okay and um each one of them either the top switch around can be on or the bottom switch can be on right so it's sort of like there's 2 to the3 or eight possible States for this inverter these inverter switches to be on assuming one switch is always on in each Bridge leg okay so eight possible States um if I come and show you what those States look like they look like this right so if I think about the three inputs I can have you know bottom bottom bottom top bottom bottom Etc there's eight possible States okay and these are what you get for a br and CR normalized to the DC bus voltage and then this is what you get for a V vbc and VCA the Line to Line voltages and this is what you get for the line to neut uh line to neutral voltages on this load okay and I want to emphasize that in this structure I'm not proposing that we connect this reference voltage to the neutral this neutral if it's a y-connected system we're typically leaving this neutral floating so it's not connected to anything or an a Delta connected system if if the you know I I drew this as if it were a y connected system if it was a Delta connected system there wouldn't even be a neutral okay so the neutral is not connected to this reference voltage all right so I can how do I figure out what the voltage say between VA and the neutral is so this would be I'd call this V a n well if I imagined that my load which might be a motor acts like three identical impedances then it's just a voltage divider right so if I have one bottom switch on and two top switches on VN goes to sort of 2/3 of the way up if two bottom switches are on one top switch is on it's 2/3 of the way to the bottom if all the top switches are on VN goes to the top right so the neutral just sort of goes up and down instantaneously based on voltage division from these individual values okay now that I'm not saying you could not tie the neutral back to the reference the center of the bridge in fact that would be this just this circuit with half Bridges instead of full Bridges it's just not common to do because you lose the three-phase uh hormon the three-phase triple end cancellation if you do that you lose a lot of the benefits if you if you don't let the neutral float okay um so this just shows sort of the voltages I can synthesize two of the eight states basically short out the load right because if all the top switches are on basically I'm tying all these three nodes together so they basically just short the three terminals together if all the bottom switches are on I do the same thing and any other switch configuration um I get some other set of States on my inverter output okay any questions about that so where's the voltage div between again so for example um suppose I took let me draw it this way here's VDC right suppose I turn the top switch on here so that means I tie this guy to the Top If the top switches on here I tie this guy to the top and if the last guy has the bottom switch on I tie him to the bottom right so if I have two to the top and one to the bottom this node voltage if if these green boxes are all equal impedances will be 2/3 of the Way to the Top by voltage division does that make sense okay so that's the way you tend to think about it so one way I could run this is very much like the inverse of the diode rectifier I talked about last time right if you remember last time uh as the AC waveform happened and drove the diode Bridge the individual diodes turned on if you remember it was like 1 2 2 3 3 4 4 five 5 6 61 right so it just went around in this six-step pattern okay well I could actively control the the these devices to follow through those six patterns ignoring 0000 and 111 okay so those are the two I wouldn't use and this is what it would look like if you did that okay you would see that each say each half bridge in this inverter would be on half the time so a might be on exactly like you know a square in Phase with a sine wave B would be on but shifted by 2 pi over3 and C would be on with high shifted by another 2 pi over3 okay so I'd have three square waves VA VB and VC with respect to the reference okay what would happen at the load well um if I looked at VAB VB vbc and VCA that's these next the next three here and I apologize it's not the prettiest diagram are the Line to Line voltages okay so all I'm doing is instead of looking at you know what this is with respect to the reference I'm just looking at the difference between VA and VB okay and what you can see is that 3 three suddenly take on this six-step waveform which just looked just like what the diode was rectifier was doing okay it's just that now I'm imposing a DC voltage with that six-step pattern to say the machine or whatever it is I'm driving any questions about that now if I do that um because the Line to Line voltages you know these square waves which is the line to reference voltages have third harmonic right but because I'm taking the difference between two of them to get the line to line voltage shifted by a third of a cycle all the triple n components disappear so the Line to Line voltages have no triple n components right so whatever my motor sees it's Line to Line If I'm driving say a Delta connected machine it will see no triple nend drive and that's just free because I have this three-phase inverter structure if everything's balanced it can't generate three-phase Line to Line any questions about that okay um what would it do as regards the line to neutral voltage well if I did that we said this thing's basically a voltage divider right but I could think about you know it basically being the neutral voltage is being sort of the the average of va VB and VC right and what essentially happens is that each of va VB VC has a fundamental and then it has for example a third harmonic right because it's a square wave the fundamentals are 120° out of phase but that means that the third harmonics are exactly in Phase so the third harmonics of each of these square waves add at the neutral and you get a neutral voltage that Bops up and down at the third harmonic okay so this has the the neutral contains the third harmonic and actually all the triple triple n harmonics all right and so as a result just as the Line to Line didn't have any trip n harmonics neither do any of the line to neutral voltages so if it's y connected machine the line to neutral voltages which is really what's driving current in this thing no triple n harmonics okay does that make sense to everybody so this kind of operation is what's known as six-step operation okay I just each AC cycle I I'm sort of switching my devices as minimum as I can and I just go through and you know step sequentially around in a cycle and in fact the uh the Tesla egg demo that we we brought that's basically what it was doing it had six switches and it was just boom boom boom six stepping around to generate a rotating magnetic field which spun up the egg okay that's a fine way to operate and in fact it generates the largest fundamental component you can generate out of this inverter why because each halfbridge is basically generating a square wave okay the largest fundamental component you can generate when you're not tying the neutral back to the reference point I should say okay that's not the only way you could drive this you could actually start thinking about doing things like um harmonic elimination waveforms right instead of generating square waves in you know between the top and bottom switching you could start thinking about putting notches in those waveforms okay now if you're going to put notches in the waveforms you wouldn't you wouldn't try to eliminate the third harmonic cu the third harmonic is going to go away you might try to eliminate the fifth and possibly the seventh harmonic right so the same kind of games we talked about playing with a singlephase inverter you can play with three phase inverters it's just you're doing each phase and then you're shifting that by a third of a cycle to get to the next phase okay any questions about that so those are the kind of games you would play if you're at sort of very high outputs where you don't want to switch your devices very much but in a large part of the operating range for Modern Electric machines um you don't have to do that you can do a much nicer job of synthesizing a waveform that would look very close uh you know that would have a fundamental content that follows very closely for example a sine wave if you were driving a machine that wanted sinewave voltages or that would generate um three-phase waveforms if you wanted to make a three-phase uninterruptible power supply that would synthesize sort of a three-phase output that would look very much like sinewave outputs right how would you do that okay let's think about that what would I do well what I could do is I could make each of the phase outputs thinking about it independently generate something that looked like a sign with respect to the reference on average right well how would I do that the same way we did things with DC to DC converters or DC to AC converters uh and single phase DC AC converters maybe I would take and say okay suppose I wanted V to follow some sine wave I could create some triangle wave that's a carrier wave okay that does something like this and I'm going to exaggerate the ratios between the switching frequency and the line frequency just for visualization okay but here I have a triangle wave okay maybe I'll call this minus VT Max and VT Max right we might think of those as one volt plus or minus one volt right and this is zero okay and this we might think of as t- switch T switch is equal to one over the switching frequency that I'm going to switch at okay and then what I could do is I could create a signal let me call this um I'll call this V ref this is the reference voltage to which I want within a switching cycle VA to follow with respect to the reference okay so what would I do i' then put this into a comparator right if I called this sort of a v triangle right so I could then just create a comparator and say okay into this terminal I'm going to put V ref into this terminal I'm going to put V triangle and then out of this comes q1 of T which is equal to which is equal to 1 minus Q4 of T and this is q1 so q1 of T goes here and Q4 goes here right so whenever whenever this switch is high this whenever this switch is on this switch is off and vice versa okay okay so what what would I why would I do that if you think about it if VRE is all the way at the top that means the switch the top switch is always on if VA a is at near the bottom that means the bottom switch is already always on if the reference is right here in the middle this triangle wave goes from you know it's plus and minus I would have it switching with 50% duty cycle between top and bottom and on average with respect to VR this would be zero so I can kind of linearly change the average voltage synthesized by VA with respect to VR over one switching period based on this triangle intercept pwm does that make sense to everybody now I should say I'm showing this with balanced triangles you could do this with any other kind of waveform like a Sawtooth waveform or something like that where at least there's a proportionality in time between between sort of the height and how long the comparator trips for but balanced triangles tend to give you better switching harmonic content than non-balanced triangle waves does that make sense everybody so suppose I had a machine and I wanted to uh create some waveform that would look like something like V on average so I'm going to draw this bar meaning average over one switching cycle I would like this to be v subm s of Omega line T for example where I'm assuming that Omega line is much much less than 2 pi F switch that is this is a very slowly varying sine wave right so you know what I'm what I'm saying is you know I have you know some triangle wave like this maybe I'll draw it bigger this is my pwm wave form and I'm trying to generate a sine wave that maybe there's 100 100 switching Cycles or something like that per line cycle right so this is you know uh 2 pi over Omega switch and this is 2 pi over Omega line versus time does that make sense to everybody and as long as and I should say what I'm trying to do is I I need my if this is V subm sin of Omega l T right I need V subm uh scaled such that the at least the I'm sorry this is VA this is the reference voltage is less than VDC over two so uh let me let me clarify how we're going to do this and I apologize that's not probably not very clear what I'm going to do is I'm going to make V ref equal to V subm / VDC over 2 times V sub triangle and then I'll make that s of Omega LT okay so this thing this thing is the thing this is VA ref that I'm going to compare to VT Max so this is I'm sorry this is VT Max right so if VT Max was one volt just for Simplicity I'm going to make V subm over VDC over2 the height of my reference waveform okay so I can create if I do this then what I'm going to get is uh a v as a result of running this thing into this comparator Network right because this is my Triangle Away From My Controller the peak of my voltage is switching between plus VDC over two and minus VDC over two right so what I'm going to get is uh V is going to be equal to uh uh VM over VDC uh it's going to be equal to uh VM s of Omega LT okay where as long as V sub is less than VDC over two right so that me can I is that clear or should I try to explain that better I can modulate this thing up and down such that what would be the biggest sine wave I could create without Distortion with this scheme or at least with with what I've told you so far would be if V subm was actually equal to VDC over 2 which would mean uh the top of this waveform would exactly touch the top of the triangle wave so I wouldn't have any period in which I'm exceeding Dy ratio of one right the pro the problem here is right if my if my reference waveform exceeds uh the height of the triangle wave then I'm going to get some time period where I stop switching and I just sort of have uh the top switch on constantly or the bottom switch on constantly and then I get some Distortion okay by keeping my reference always below the height of the triangle wave I kind of keep the um no Distortion relative at least over a switching cycle averag over a switching cycle any questions about that so when we do this okay and I say okay I'm going to do this for phase a I do the same thing for phase B except by shifted by a third of a line cycle and phase C is shifted by a third of a line cycle then at least as far as the local averag is I'm creating a voltage a with respect to reference that follows a sine wave a voltage B that follows sine wave voltage C that follows the sine wave and yes each of these nodes is Ping up and down but if this you know load is inductive it will filter that and then the currents will look very sinusoidal does that make sense everybody so I can really generate something that looks like a sinusoidal at least on average output just the way I would do it with a singlephase inverter so this is the same trick of triangle intercept pwm that we talked about before any questions okay so that's nice right th this technique is very good when we are trying to synthesize a waveform with a lot of Purity right the six-step waveform I showed you before had a ton of harmonic content a ton of low frequency harmonic content in it this pwm waveform really only has content as long as you stay where where your reference never exceeds the height of the triangle um really only has sort of harmonic content near the switching frequency and it's harmonics right there's some some indifference with the line frequency and the carrier frequency stuff but it's all sort of pushed up to the switching frequency so if I have a high switching frequency I get a very nice Purity waveform at least averaged over a switching cycle okay but I said said what's the largest waveform I can synthesize that way at least with what I've told you so far is I can make this voltage V subm up to VDC over2 so this inverter okay can switch and it can synthesize I could think of I could think of this is synthesizing v r vbr and VCR which is going to also give me the same thing for Van an vbn and vcn because there's no triple n content here of being you know V subm sin o Omega line T so long as VM is less than or equal to VDC over2 okay now if I'm going to synthesize a sine wave like that we often Define something called M or the modulation index sometimes this is called the depth of modulation also which is essentially um V subm / VDC over 2 right this is how big a sine wave am I generating you know how how big is this sine wave relative to VDC over2 or relative to the maximum I can do without distorting okay if V if if at least with what I've told you so far I make uh M greater than one or V subm bigger than VDC over two then what will happen is I'll stop switching for power to the waveform and I'll start distorting okay any questions about that well uh that's fine and through this whole period at least averaged over a switching cycle if I'm doing this sinusoidal or Tri s triangle p M this neutral point over average over switching cycle always has zero average voltage in other words equal to the reference voltage okay halfway between the bus voltage so I'm not getting any third harmonic swinging around on the neutral okay why do I think about this because if you think about an electric machine and this is a typical application for this kind of thing right the backing F tends to be proportional to speed and how much current I want to drive in the motor you know if I want more current I try to drive More Voltage which will drive more current into the windings right so very often the amplitude of the voltage if I thought of about some you know some machine which is just sitting here connected like this right and maybe I'm just trying to generate I'm trying to use my inverter to generate essentially you know three voltages which are then going to drive current into these phases right um at low speeds the voltage the the back EMF voltage of the motor is low and I want low or I want low current I want these voltage waveforms these are now at the line frequency these are at you know V subm sin of Omega line T right I want these voltages to be small but as speed goes up in the back end F internal to the machine goes up I want a bigger V subm for example right so very often I can basically by changing the V subm by changing the amplitude of my reference I change these voltages amplitudes and then I change how much current I'm driving into my load so I often want to modulate that and typically as I'm going up in speed or I'm going up in power or both I'm trying to make vabm bigger that make sense everybody now the challenge is eventually I run out of Headroom suppose the internal back EMF of these motors is getting really big so I need a pretty big voltage V subm to drive current into my load okay once vmm equals VDC over two suddenly I'm going to start distorting right I'm not going to be generating just sine waves I'm going to have low frequency harmonic content in my output so the question is is there a way you can extend the range over which you can generate sort of if you will pure sine waves um at the output of your motor and generate like no low frequency like line frequency harmonic content and the answer is there's a trick for doing this it can be done in a variety of different ways but I just thought I'd mention it because uh in the motors world this is a very common trick and the idea is this uh suppose you know if I thought if I thought um if I thought of what the duty ratio if I thought of this switch q1 if I thought of this is q1 the average value of q1 over a cyclist being the duty ratio of this switch right so modulation index of one the duty ratio of the top switch goes to one right with what I've told you so far so I could talk about the duty ratio of this half Bridge just like it was a buck converter okay where I'm talking about the on state fraction of the top switch um the way this would look if I did my S triangle pwm would be D1 of T would be equal to 12 plus m / 2 s of Omega line T right right so if I did my sign triangle pwm if M was one duty ratio goes to one when the sign wave as this peak and it goes to zero when sine wav is at minimum does that make sense everybody and then if I was looking at D3 it would be the same three actually it's not D3 I forget what the middle switch top switch it is D3 D3 would be would be equal to 12+ m / 2 sin Omega L tus 2i over 3 for example does that make sense everybody and D5 which would be the third Bridge leg would have 12 + m/ 2 sin Omega l t + 2i over 3 okay so this is what would result with this triangle intercept pwm done on the three phase okay and what we said was you know basically the average voltage is proportional the the the average voltage at that node with respect to ground is proportional to the duty ratio with respect to the midpoint there's this one half of the bus voltage in there okay the question would be um I have a trouble that I'm limited to making M less than one cuz when M exactly equals one the peak of this drives the duty ratio of one just touches the top if I tried to make M 1.5 what would happen is for some part of the cycle Duty ratio basically saturates at one the top switches on and I get a bunch of distortion is that clear to everybody so here's the trick people use it's kind of a cool trick there's different ways to implement this trick um so this isn't the way you have to do it but suppose I did this I could take uh and add M2 s of 3 omega line T to each of these waveforms why would I do that well because when I think about sign right if I looked at this Duty ratio right it would look like this if I just took you know here I would have a half and I'm doing something like this right um what I can do is I can add in a third harmonic component what does a third harmonic component do it does something like this so the sum of sort of this piece which is this and this piece which is this has some distorted shape which does something like this and what that means is I can turn up M just bigger than one and it turns out to be about 1.15 before basically the peak will hit VDC over two or before my duty ratio will saturate at one okay and so that gives me an extra 15% of voltage I can synthesize and the way I'm synthesizing it is I'm adding third harmonic Distortion in okay so I'm adding some third harmonic to V to vbr and VCR okay but why doesn't that matter because the third harmonic that third harmonic component that I drew in yellow just appears at the neutral and the Line to Line voltages don't have it okay so that's a technique that's sometimes called third harmonic injection pwm you can do it by adding third harmonic onto your um you know if instead of doing this you add some third harmonic onto that thing and then you get an ability to expand your duty ratio range or expand the fundamental voltage you can synthesize without Distortion any questions about that yeah Jack can you explain again how it's like VDC is fixed so how is this increasing like overcoming that enough just right well let's assume the back EMF is the fundamental right so what I have to worry about is how much fundamental can I create here right the back Aus at the fundamental how much fundamental can can I create here and the problem with not doing this third harmonic Distortion trick is that if I try to turn up the duty ratio more I get a lot of low frequency harmonics I get third fifth other content in here which is then going to drive low frequency content in the machine okay this trick What it lets me do is I basically am going to create the fundamental plus I'm going to create some third harmonic here third harmonic here and third harmonic here but the third harmonic elements all cancel they just show up with a neutral and the neutral goes up and down by the third harmonic so I can sort of keep this component pure because I'm not saturating my comparison anymore I'm not making this signal go out of the range just for that little bit for that extra roughly 15% of modulation index over one so I can go up to M of 1.15 does that make sense now that gives me a little bit more headro to synthesize a little bit more vol voltage and we often we're often constrained by how much voltage we can synthesize with our inverter what do I do if I'm not happy about that well I could just keep turning up M Beyond one what happens if I turn up M Beyond one well if I turn up M Beyond one right and my reference waveform was going to kind of swing up here like that what I would do is I would get a bunch of pwm switching until the middle of the cycle or something where I just flat top and so my pwm waveform for each phase might look like this you know it's p pwm and then it'll just sit at the top and it'll start pwm again right and if I keep pushing it up and I keep making my reference waveform bigger and bigger and bigger the limit of that as M goes to Infinity is just that I'm just going to generate a square wave he's going to be on for half the cycle he's going to be on for half the cycle and you're back to the six-step waveform that I talked about before right and in that case then I really do generate the maximum fundamental right the maximum fundamental is basically V is a square Wave It's VDC over 2 * 4 over Pi uh is the fundamental of a square wave of amplitude plus or minus VDC over 2 okay that's the biggest fund I can synthesize and that sort of sets an upper bound of you know how the fundamental voltage I can drive here the only problem I get with doing that is that waveform has third harmonic which will cancel but it also has Fifth and seventh and 11th and 13th and I will get harmonic currents flowing in these phase windings when those harmonic currents probably aren't helping me drive the machine okay so I can synthesize something that gives me nice effectively at least on a local average basis average over switching cycle sine waves up until the amplitude equals VDC over 2 right if I use my third harmonic injection trick I can get it up to VDC over2 * 1.15 okay and then if I don't care about Distortion I'm going to let a bunch of Harmon low frequency harmonics exist I can just drive up the modulation index even further and get up to 4 over Pi VDC over two okay and people do that in fact in the Prius the the Toyota guys eventually published a paper saying yeah we ran out of Headroom and we wanted to run our motor faster for higher speed so we just let it saturate and and generate the harmonics and the motor still worked and we were okay but you pay you pay a price for that okay so so people do do this but if you want your waveforms to be pure and not have low frequency harmonics and generate those that noise in your load whatever it is that low frequency content in your load you got to keep your modulation index down any questions yeah is it possible to step up DC with a BL stage before absolutely and in fact that's what people do right so why why is this a concern because if you're in a Prius right you got this bus voltage and that bus voltage is your battery voltage right and that might vary over some range and then you're trying to deal with this motor whose back EMF varies over some range you need to drive it if you don't like that then you put another stage in between your battery and your converter and you can control the DC bus voltage to go up higher and then you're only limited by you know how high voltage these switches are right so you're you're always limited by that but yes people do actually you know control this bus voltage from the battery voltage for example in an electric vehicle to be something higher and they might control it dynamic Ally if they had to depending upon the operating condition they want all right well have a great day um we will pick up a new topic next class