 So we were looking other day about the maximum frequency at which an amplifier or an analog circuit can function and we gave a concept of what we called as the figure of very frequency which is essentially called FT okay and we did derive or at least showed you that typical FT is Gm by CGS related to Gm by CGS and if I put the values please so concept of a figure of married frequency which is called FT and we defined it by saying that it is equal to Gm by CGS the CGS term essentially coming from input capacitance if there is any additional capacitance it will be Gm by Cn actually okay but right now I am only using CGS. So for a device it is only that capacitance which I have written but for external circuit may have some more capacitances there and if I write this transistor FT which is 3 by 2 by mu by square BoV typically this may be much higher than gigahertz in terms of gigahertz typically it may be as high as 100 gigahertz and we always say that the concept of the maximum frequency at which a circuit can function relatively correctly or you can say correctly means our modeling which we did is based on the lump models of the device is valid at that point we discussed last time that typical frequency could be 10th of the FT. Now this is the other method I have suggested that okay you can take a geometric mean of the pole frequency and the WT and it become close to the value of 10th okay so it is not that the either one is wrong or right may be little more accurate than this but 10 is good enough approximations we also discussed that day apart this is what we did I am just trying to recapitulate in the FT valuations we always see that we use the model which is called large signal or larger channel model a long channel model for MOSFETs and there will say that larger the BoV as the function shows FT will be larger. However this was assumption that device is always in a reasonable inversion but if it is in very very strong inversion or a very weak inversion these assumptions are not valid and therefore FT relationship with BoV is suspect. In fact the problem starts now that whenever you actually go down to short channel devices okay the definition of short channel device also varies from people to people and device people are some other way of telling but for you can take from me a short channel device is called short I mean MOSFET device is in short channel if the channel length is of the order of the depletion layer widths okay if it is in that order then we say you have in a short channel devices or if you are a technology minded person typically similar scaling is done for all length and widths so one can say if the channel length is some way a similar numbers like the junction depth of the source and drain then you are also in short channel. So this is a definition which is arbitrary to some extent the major thing which we see in a short channel is okay and that actually changes the other parameters there are many short channel effects some other course they should learn about it there are n number of things which you can talk about from circuit point of view my worry is EY is now stronger and I kept on saying EX is too strong compared to EY which may not be now valid they may be comparable so both fields may cross there will be effect from both sides. The short channel width is only as I say the definition and take it in reality whatever the experimental results that is the short typically one can say 0.35 micron or even 0.5 micron device can be said to be in short channels so most of the analog circuits are now actually working on short channel devices but with the digital technology going to 90, 45, 65, 45, 22, 16 nanometers you are really going into short and short and shorter channel devices and the worries will become even much more than what we actually saw at 0.25 or 0.18 okay so this fact that we are looked into FT that in a short channel the relationship between FT as well as GM by IDS is a little suspect okay. So one should not really go too strongly saying that view is a good parameter of design which we have been talking very you know casually or that is the end of it once I know this I can do everything but in reality this is not the best parameter of the design okay so we will come back to that part again but here is some two figures which makes my point clear if you have the point which I am now making is interesting for a MOSFET amplifier therefore two parameters decide the performance one is GM by IDS the other is GM by CGS okay. Now if you that is the expression I wrote that is what point I am trying to prove again to you if I plot GM by IDS against VOV and also plot GM by CGS against VOV to you said these are two figure of merits okay so I plot as a function of VOV you can see the GM IDS is inversely proportional to VOV so it go it actually shows some more accurately does not follow linear it should look linear but it is not in that is what I am trying to say it goes partially like a square function or under functions and if you look at the GMCGS term yeah it does rise roughly linearly with VOV now if want to maximize GM by IDS gives what? GM by IDS gives you performance of the power okay that decide the power now if GM by IDS is one parameter of power as you are spec then there is an issue here if you want to have GM IDS can control and at the same time you want to higher bandwidth so you want larger GM by CGS now you have problem larger GMCGS will have lower GM IDS and larger GM IDS will have lower GMCGS that means there is some way optimization is going or optimal value is where they cross problem so this VOV dash is somewhere where both are reasonable now this is not too good a value in fact so can you push this car up okay if you can by some way then we say okay we have this VOV dash little better control right now if this is very small that control itself is very small however this has to be understood that VOV is if it varies we are really varying most of the performance of the circuit so this should not be treated as as good a design if you use even a third optimal third figure of merit which what I suggest sometimes if you multiply GM VGA GMCGS into GM IDS and call it both you want to optimal okay multiply it and call it third figure of merit okay and if you see now if I substitute those expressions of both GM by IDS and GM by CGS as to be a VOV and this it gives me interesting expression which says 3V by L square okay various technology okay various technology channel length is decided by technology nodes however that number you can change because I need not work on the shortest channel device or longest channel device length can be in my control now this essentially now trying to say that this figure of merit has nothing to do with the VOV okay so choice of VOV has now lost because you suddenly thought that if both are to be optimized then you need not worry too much about which is not true in fact how much current I will push will decide the gain okay so now the issue is that FM3 may essentially represent power and speed but it does not talk of gain okay so VOV as a parameter which we have been in so far talking in all my analysis so far VGS minus VT all that you may have to control is not the best stuff design parameters is that correct and because of that I always suggest that this is not a fair deal so let us see what we can do afterwards and this is an issue which I thought as a designer you must keep in mind that VOV is not the best design there is no great correct expression between VOV and GM ideas GMCJs into VOV same is there is no relation with strongly with any other parameters with VOV exact ones so this is like a first factor you are trying to fit somewhere and getting answers okay so what can be the other possible design spec or design parameter and we will declare it little later that which one should be actually use if possible okay okay so this fact which I said you that this is last time I should have showed you this is from the last slice I should have showed you but I just remember that this was not shown so I thought you should remember that my VOV is not really a great contribution so we will see more if I later that is that any other thing we can use as a parameter instead of VOV and that is why I am trying to show back this two slides which I is that okay everyone yeah but length I can always double it triple it I can do anything based on that okay I can have point let us say nanometer technology is 10 nanometer 65 nanometer I can pull 130 nanometers also links okay it is not that I cannot reduce then the technology note I cannot pull it down to say for 65 nanometer I cannot have 45 up to the 65 the channel length is not 65 some other name technology for say these numbers are nothing to the channel lengths okay 45 nanometer have actually 25 nanometers channel lengths 24 nanometers actually 12 nanometers has channel length so these numbers are also not very true but these are called roughly you can say okay these are channel lengths okay so you can always go up to the minimum channel length available for it but you can double it triple it that number is in your hand okay is that okay there are also issues in the short channel which I did not say so I thought that today I will finish that part in a short channel device as I just now said the lateral fields and velocity saturation effects both affect GM by idea stones okay it is known that approximate expression which we used earlier for long channel can be modified for a short channel analog circuit performances I write this current slightly modified by an expression which is 1 upon VOV by ECL where EC is actually called critical field and it is different for different technology node for example for 0.25 it is 600 to the power 6 volt per centimeter for 0.13 it is 1.5 to the 10 to the power 6 volt per centimeter so if you see this can you see from here ECL into VOV ECL into V what does that show which kind of expression it is showing if I substitute ECL here and I get this expression what is the showing parallel combination of two terms 1 upon ECL and 1 upon VOV so now there is an addition we are only looking for now there is an additional term in short channel has to be taken care which is EC by L EC into L is that correct now this additional term will modify and if I have done some calculation for a given value typically if ECL is 2 volt for a VOV of 200 millivolt the GM by ideas becomes 0.9 of the 2 by VOV term so in a long channel what you would have used 2 by VOV in a short channel you may have to multiply it by 0.9 this is for one specific case for a given technology for a given length this EC has to be actually obtained and then used to find how much this coefficient should be 0.85 0.8 0.9 and therefore you can see now the GM ideas available at long channel is not available to you in short channel is that clear so these are the design issues which as a designer we should keep in mind okay EC is a critical field where mobility starts actually strongly dependent on the fields okay strong dependence though we use always function of heat but a stronger dependence when we say is called critical fields okay yeah for technology nodes so for 130 nanometers this EC is 1.5 into that sort of a for a different technology load you will have to evaluate EC more accurately okay EC into L okay and that value you have to substitute there so you find this constant factor will change for different technology nodes okay is that no this is true this is to some extent till you go to very very strong inversion or very big inversion this expression is true okay but the at least it has taken care of increase electric fields laterally okay which earlier we are not looking so the additional field which we are talking about is now taken care through this term okay we are assuming only channel from the top now we say okay there is a additional source of charge can come okay so this fact could be taken care in our analysis please remember some of these expressions are more what should say analytical expression what does that mean that I have derived under certain assumptions every time okay so not all assumptions are true in all times okay so in reality what should we do we will see what can be done okay but these are reasonably way of doing it why I am saying reasonable because at the end of the day you will work on some cat tool which is like a spice for example so where to start at least you know correct guesses can be given to the input files okay but this at least gives you some idea of physics what has happened there is that clear some physics has been understood that as you base there is an additional term may appear which will reduce GM by ideas and this is very relevant in calculating why GM by ideas decide power for example so now you can see power may actually have gone down okay some or in some case game may go down whichever way we will look at we will see that part so essentially unknowingly we actually thought that long channel expressions are good enough but in 2012 I think that is not fair so I that is what I said few years ago I never showed this now I thought you should all know that in real life things are changing and you must know more physics to bring closer to what can the experimental results show however I will tell you about this when I finish my design here I will show you how exactly we should do in fact now I will actually give an example of a very simple amplifier which many times we are doing but some values may be now given since our course have no formal tutorial as such I thought at least some problems I will solve so that you will have idea of equivalent tutorial at least for 10% of my time let us say I have a simple MOS amplifier which is biased by a ideas current source and the data given to me is the new C ox value is 100 micro ampere these are all random numbers these are not specific in real life okay so these are some contrived numbers channel length is 0.8 microns so this is quite good long channel approximate 0.8 is sufficiently long in channels the only voltage is given to you as 50 volts the technology first parameter alpha is 1.25 the load capacitance is essentially 0.1 puff the bandwidth required for this amplifier is 2 megahertz and I am expecting gain to be higher than 200 trials but important so what are the specs I repeat the earlier it is 50 volt channel length is 0.8 new C ox is 100 microns per volt square alpha is 1.25 if alpha is not specified what value will choose 1 okay if alpha is specified use that value if not given just use 1 or just do not use alpha anywhere then essentially saying that okay CL is 0.1 puff bandwidth is 2 megahertz gain greater than equal to 200 okay this is a spec given to us for a given technology node and we want to find what can we find from there what is the thing which have not been given to us and which is what we are designing the size of the transistor okay that is the design length I know but if I know my width I get the design of transistor and therefore I get the all the specs to be available for actually performance okay. So that is fine so for this simple amplifier we can give it is equivalent circuit there is a CGS as the capacitance and CGD right now since I have not specified anything on the CGS my assumption therefore is 1 upon omega CGS of at 2 megahertz is large enough to be open circuited okay. So I did not worry too much about if I am given then use that okay if I give you what is the value of CGS you use that right now I want the simple solutions to show so I neglect because this frequency is higher so I am not looking for it is much more than 2 megahertz it will give bandwidth so I am not looking for that value okay this is a simple circuit perform this one can solve using Kirchhoff law as I say I like this neural equations because spies solve most cases the relationship by node equations but you can always use meshes okay if you feel comfortable using mesh you can do mesh analysis if you feel comfortable with node analysis you do node analysis this is why I do node analysis simply because spies does it so you can also do it I agree that I want to hold CGD CGS I removed but CGD I want to show that there is a feedback what is the issue on that typically CGS is higher compared to CGD okay so CGD is a very it is a series so anything in series troubles you most okay will come that that is not to be Miller theorem we will talk about that is anything in series feedback is an issue okay anything across many a times it can be neglected for the omega 1 around this may be large enough forget about it is that clear to you this is not true for actual case you may substitute and figure out keep everyone there is nothing that you have to remove or not to remove you solve full circuit terms will automatically get cancelled if they are smaller okay you cannot add one in 10 million so what do you add there so you that term you can keep there is nothing wrong to solve simple I slightly took some assumptions okay so if I solve this I start from SCGD into V in minus V o minus GNV in minus V 0 by R 0 these are all currents at this node I am talking I am talking here okay minus V by 0 minus SC 0 V 0 is 0 then I I collect the terms of V in and V 0 so when SCGD minus GM minus V 0 SCGD 1 upon R 0 plus SC 0 equal to 0 so you get an expression of gain which is V 0 by V in which is minus G 2 minus SC GD 1 upon R 0 plus SCGD plus C 0 is that okay simple second year nothing more and nothing less okay so I will be doing it we want to solve using there we know we gave a gain or a W values now I want to figure out if given this data what W I can get if I you can now see that for a final expression if I do this GM R 0 out okay what is GM R 0 minus GM R 0 is the DC gain please remember in analog circuit the DC gain is not really DC input gain is a frequency independent gain is that word clear it is called DC gain which means without frequency whatever is the value is called DC gain in fact there is nothing essentially I am not saying any input signal of AC I am getting a DC output it is not a statement that but it is called DC gain DC gain space at frequencies close to 0 whatever is the gain is called DC gain okay it does not say input is DC is that clear to you so please come and get to this so this is my DC gain into 1 minus SCGD upon this and I have this now you can see C 0 per C GD is the total load capacitance at the output which has been given to us as how much 0.1 puff so the C GD value could have been given C 0 value could have been given I just said that okay whatever the sum total this value is 0.1 puff I am trying to reduce my calculations in real life you will have to use the actual values given to them at that point is that okay this C 0 plus CL C GD I have used as a CL and I say this is the value for the solving purpose okay now assuming again to solve a very simple case I assume that the 0 SCGD by G1 is given me a 0 that 0 frequency is far off okay away from the pole so I just now even do not bother of it may be it is relevant in some values but now I am saying the C GD what he was worried about is such that the 0 is some 100 MHz or 50 MHz so 2 MHz is my bandwidth and my 0 is sitting at 100 MHz by then the gain has gone down to some 200 minus DB so I am damn caring about it okay so I am not worried about then when it start rising by the time it will come to 0 it will be million years okay so I am damn care if the 0 is far away I mean design and that is the game you will be in design C 2 that your 0 is far away from your any number of poles you are looking at what is the advantage of that that because it is going to give you some plus phases and you are trying to adjust phase margins in actually amplifier this should not be the bothering point for you okay and that is one spec I use that okay the 0 has to be at least 10 times this and I use this value itself to calculate okay so 0 is something which we do not want very much because 0 again is not for file so that fact should occur far away from us okay however 0 has an advantage what is the advantage if the same position 0 occurs at the pole then it will cancel so your game will become flat anyway so 0 is not that bad either okay but right now let us see what is going to happen so this is possible in most cases as I say 0 and this this is possible most cases as normally are 0s are much higher than GM's okay 1 upon GM's this value which I am saying this is a condition which is normally made so 0 may come normally far away but not necessarily because CG may decide where it will come okay as I said this is a solution for the sake of design and these are not any specific technology rules so now I have some expression from this I have neglected 0 so I have got a US is a V01 plus SR0 CL now that is the bandwidth from this expression 1 upon R0 CL omega minus 3 dB as it is called if you wish the definition in Bode's diagram okay then this is 1 upon R0 CL the DC game is minus GM R0 the game bandwidth product is GM by CL now these are the expressions which I can use in using my design for evaluation of what the width all that is width my interest is in width is that expression you know these are the expression I use the bandwidth which is 1 upon R0 CL the DC gain is minus GM R0 and the bandwidth is again bandwidth product is GM by CL now using these expressions we can now solve our problem what data I have given I could have directly started on this but I thought I will give you expression first and then show you where do I substitute okay is that 3 expressions clear only 3 things I am talking about game bandwidth and gain that is what it is okay so if I substitute now it is a constant current source as bias which expression I should use I am fixing IDS to beta IDS a under root of that is that see that chart in which I showed which one I am fixing Keras which is a center one I am using so I am now using GM is that clear that they are gave you table for free fixing in this case this is the minimum which I use in which IDS is getting fixed okay so GM is 2 upon 2 beta by alpha if alpha is 1 you may not use it but here I have given you so 2 beta by alpha IDS and we also know R0 is early voltage divided by IDS okay so these are additional expressions 3 first I said and these two more important expressions I have to use in my design since I said the bandwidth is 1 upon R0 CL what is the bandwidth 1 upon R0 is the frequency which is 2 pi f is the angular frequency so actual frequency is 2 megahertz given to you so you say from that expression R0 is 2 pi the bandwidth into CL I know bandwidth given to me is 2 megahertz and I am given the load capacitance of point one puff okay so I substitute it everything here so I get a value of R0 so what is the value of R0 I got 0.8 mega ohms is that okay this is 2 pi bandwidth into load okay that expression we just wrote we are just substituting values now so the first thing I figure it out that the output load for my case is 0.8 megahertz and they mega ohm coming from which parameter the bandwidth since we are given bandwidth to me I figure it out that it should have this much as R0 now I want to evaluate IDS okay because IDS is going to decide GM so I must know my IDS but I know IDS this IDS is nothing but early voltage by R0 okay which is 50 by from where this expression is coming the slope into this is the voltage so this is the from the slope characteristics so I evaluate IDS equal to 62.5 microhertz as I said these numbers are not sangro sign these are very arbitrarily chosen values please take it the real numbers may be very different it may be actually mini amps or mic 10s 100s 800 micro amps kind of thing so right now do not go by this oh you said 60 60 some arbitrary number I also know my AV0 which is GM R0 as the magnitude I just calculate GM which is game divided by R0 you have been told that okay instead of 200 I use higher you said so I this 240 only to divide by it nothing great you can use any other value so 240 by 0.8 10 for 6 this gives me GM of 300 micro Siemens okay is that okay please remember you said greater than 200 so 240 is greater than 200 it is not 300 so it is not too high also of course you can use 200 value here and get whatever but that is a minimum point you are higher I used it so if GM I already done the written expression what is GM expression to see ox by alpha into W by L into IDS is that correct under root of course so from here I can figure it out what is the value of W by L so a W by L is GM square alpha upon 2 mu C ox IDS is that okay just manipulate substitute all values GM square 1.2 as alpha 2 into mu C ox is 110 to power minus 6 and IDS is 62 microns okay so what is the final result coming out of this all calculations W by L is 9 roughly accurately you may find but this is roughly I am very sorry okay so since the length is 0.8 check it whether that is 0.9 also okay I mean I just roughly did this and maybe wrong also but just so essentially W is now figured out for what are the values we met the bandwidth of 2 megahertz gain greater than 200 driving the load capacitance of 0.1 puff with a given technology of channel length of 0.8 with mu C ox beta value given to us as this numbers so I have been given a specification and I have designed the amplifier value okay this is how you design so it is the inverse process as I said start looking what is given start looking expressions substitute properly and get the what is not known to you okay yeah biased strong fixed current source okay I can put something to bias by VGS also I put two resistors across the gate and I can give a gate bias as such okay I can do meaning I can also push from the lower side the source current sin current which is through mirror okay so there are number of ways in which I can play this game but as of now I say okay this is the okay in this calculation you must have seen in real life if this resistance of the current source is not infinite where it will appear in the circuit this if that has some RO base another RO problem will come from the load side okay this has to be understood in this case constant current source with ideal it is used as I said the idea around 0.35 down you should use short channel effects okay but people have used long channel devices models even at 0.25 also but why we are not worried too much about because when you go on a spice for a given technology the models are available for that technology in the spice itself so I am not really looking into what model I but analytically up to where I should use expressions so I say okay in most cases up to 0.25 you can use these analytical expressions at the end you are finally going to simulate on spice and that for a given technology node will be able to take care of the actual models of that technology okay so we are not too much worried about models but since I do I want to know where do I start I must know reasonably good initial models so that I can give you a good initial guess is that clear to you so this issue has nothing to do with actual spice simulations okay because spice will take care of different models at different nodes okay and then it will use different analog models it will use different RF models it will use different digital models so it has all kinds of programs which it will take care okay so my opinion for this design the GM by IDS is 4.8 per volt now let us say someone says that I want a low power design for the same amplifier okay so what does that mean I want to reduce IDS is that clear if someone says I want to design an amplifier for a low power comparatively right now what is the power this IDS into VDD which I am not specified but that is the power dissipation so I say okay I reduce power means I reduce the IDS value itself now what will happen if you see an expression now the width is proportional to 1 upon V O V I have given those expressions please I have given enough expressions for W's and everything and W is directly proportional to GM by IDS okay now this if I use more IDS which is what low power people are asking me I need from this expression width will be larger is that clear is the design issue clear if someone say reduce IDS essentially I am saying increase widths okay is that okay so what is the penalty I am paying area but that person may say no it should have a smaller area then I have an issue because you say I want a design which has low power and low area is that point issue clear I reduce IDS W may increase I may meet the spec but now he is saying you know I cannot give area so what do I lose in case I do not want you area either okay if I reduce IDS GM I W is proportional to GM by IDS and fixed GM okay the GM has to be fixed the reason why I want to GM has something to do with that so I do not want to hit GM otherwise IDS is going to hit GM as well okay you can see IDS is under a beta into IDS so if I lose IDS this will GM will go down but I will adjust W by L to get the GM so I am fixing GM and they say okay I reduce IDS so that the power is only related to IDS so I say okay pull down that as soon as I pull down that I will have to increase W because I am to beta IDS if IDS reduce beta must improve that means width must increase is that okay since width increases with area increase but specs as well reduce area you are missing the design part what I am going why I am showing you this because of your designer these are the issues which will come before you okay this can be actually done by an optimization you can keep changing to values till you get reasonable area and reasonable power okay how are the models available to your only long channel models and we say these are not the ideal models okay so how do I optimize that is the actual design is that point issue but CL has nothing to do with W by L but that CGD is very small compared to the output load the other load is driving something which I do not know I just put it point one value see the load is not the part of the circuit load is from externally you are putting it can be any value any driving you have to do open may drive any load for example the speed rate may change but that is the way it is I may design for worst driving situation but load can be anything it is not it is plus CGD I do with you but CGD is a function of W that I appreciate but this other values of external load see why I am going to put can be any value okay and to save myself I am saying okay even if that varies the CGD is stronger than the CGD increase because the next W where I do not know the next stage which I am driving I do not know what is it what is it you say A to D inverter input or a space switch capacitor filter input I do not know what W by is they are asking for their input side so I am only saying that whatever it is equivalent capacitors I have to drive it so much is that clear I am not trying to say that this is the final answer as I am just trying to give a method of looking things is that as a designer how do I look into it okay okay this can be actually done by optimization however long channel models may not then suffice that is our issue is that okay so just as I said if you have a design for a smaller area and smaller power we are at a lot of situation available for lower power we reduce ideas as we say hence for a fixed team that means gain constant GM by ideas increases because if ideas reduces GM by ideas increases since we have is one upon be is proposed to GM by DS we will occur reduction in view we will occur at lower ideas but W is inversely proposed to view V so W increases so increase of ideas whatever you look you always see the way is increasing now how to get read of such situation because this models are so I worry is this linearly going or is it some other light is following okay if that is so then it will not proportionately increase is that clear so I want to see is that model which I use for V of V is an ideal model though I may not have better model immediately but at least can I do some shift to actually get read of this situation is that my W is what I will do W is increasing okay so this all that I said earlier is the statement written by here nothing more than that okay whatever I said earlier is the written part an actual language part I wrote here so that at the end what I said is available for you to read so what is the actual what is the accuracy you should build to doing this is what my next is in design so what do we do now in real life do not use VAV as your design parameter use GM by ideas as your designs parameter this is slightly different from most books which they talk I am not saying all books most books they always go by VAV terms or they are called VAXS some call VGS VT VGT books name different but this is slightly different but please remember they are connected it is not that I am going physics out so I say okay I from the expressions I have for a real device for a given technology nodes 0.25 0.35 90 whatever it is I actually do spy simulations but different GM ideas I evaluate different ideas by W okay and plot it is that okay this it can be done even experimentally by technologies and by simulation by designers sometimes this technology people will give you this graph what is that I am plotting ideas by W as a function of my design parameter knowledge GM by ideas is that clear to you so I some random numbers are not really I know this is typical values for these technologies so 0 to 20 feelings and I plot micron by micron micron on this side so I say we made smaller the GM by ideas ideas by W with smaller and larger this I get W is that clear to you so this figure I actually can either using spies are using technology data if provided to you evaluate this this is a major start work before once you use this also use it for different channel lengths you create this graph for you is that clear which is your base design this you do not do any times for a given this you do once both for different channel length if you are going to use multiple channel length devices for each of them you create these graphs okay instead of plotting this graph in that fashion if I plot logarithmic 0, 10, 100, 10 to power 1, 10, 100 on a law of scale against GM ideas it may look more like linear graph you must know how logarithmic term bears not exactly linear but close to linear now for that other problem which I said I have solved that GM by this was 4.8 per volt okay so I go on this graph for this GM by ideas I actually figure out what is ideas by W is that clear so for example in this some number I chose which may not be accurate again let us say for 4.8 this values 40 micron per micron this I do not say exactly some number it can be 45 it can be 38 it can be 30 but some number so for GM ideas of this I have figured out ideas by W is this number but ideas we are having I will give us 62.5 micro amp from that design so I got now it is only 1.56 microns so if you realize that the decision of getting actual W is if you have only used VOV expression you have over designed the chip W is not a lot of this may be not one point it actually then may be 4 microns 3 microns is that clear to you this number which I am only putting from Fungra but this may certainly always be lesser than that earlier we have a model so the first time I told you that do not use VOV model use GM ideas model okay and if you use this your bits will be much more controllable or much more mean to you which may be having layer areas then what you thought is that correct so if up go first time in this class we are not key GM ideas you have a actual design parameter Hona chai it is not VOV all these years I have been teaching of course it is a VOV VOV so this year either I will change my track and I say okay do not use VOV use GM ideas but then I figure out I do not have ideas but actually plot okay and from that plot you can get your correct bits is that okay so this is a better way of looking at design than the VOV models now in this case I am not very much worried because spice will have models okay so these graphs can be more accurate to a given mode then what the actual VOV models I have derived okay so in that sense I am more correctly doing things which probably will come into real life okay so as a designer what is the best way of designing copy someone's but that is plagiarism don't do that what else you can do do this is that okay so this is I keep telling you that once I said you that every year I change my track this year I am changing my track to saying it is ideas GM ideas as a parameter of design they please remember 2 by 2 ideas by VOV is GM so I am not really going out of physics relationships are similar but not the same as I just showed you is that clear because it now is taking care of the real life situations in the device is that clear that that then you don't have to worry too much because this will take care of all such things which could probably occur is that clear okay the second amplifier of my interest the first amplifier I said the game is GM times R0 and game bandwidth is GM by C is that correct so if I want to increase the what is GBW game bandwidth product which is GM by C so if I now take a system in which I want a larger game one method is I can design a single amplifier with larger game that is also possible but the problem there is it will have two huge values of W bias okay it is called the aspect ratio will be very bad for you in actual designs so what is the most common practice of being designs is to put cascade of 2 amplifiers 100 make 2 10 gain amplifiers and cascade them cascade means output of first is given to the input of the next okay let us say each has a gain of a 1 a 2 which is GM 1 R1 GM 2 R2 so the game is GM 1 GM 2 R1 R2 and if they are equal GM square R2 is the game however the bandwidth is game bandwidth divided by a 0 so what does that mean if I increase the game by bandwidth is going to be lower now okay so cascade amplifier has yeah I do increase my game but I have a problem of losing the bandwidth across however that is what we do not need in most designs what will require or most system what will require I do not want to lose my game bandwidth okay I want to keep my bandwidth same okay but I want to lose the game this game bandwidth product is called figure of merit why it is called for a given this GM by C for a given ideas given capacitors that this is fixed it is like a FT value equivalent of that okay so that is a figure of merit now this figure of merit is that means Sankro Sank technology KLA power supply KLA KLA fixed but designers cannot go by saying that I cannot increase gain without losing bandwidth if they say so then there is no design left okay so the alternative to this cascading the word we use is amplifier we use is called cost code amplifier so is not to reduce bandwidth but to improve gain cascade doesn't do we figure out the another way of looking the same thing and we call call that as a cascode amplifier which allows us to retain bandwidth but push the games what does that mean you have broken the technology figure of merit gain bandwidth product is constant you said now I am breaking that you said okay now I am trying to see I will boost the gain but I will not lose my bandwidth is that clear a typical cost code amplifier is shown here just forget about M2 first so it will be a normal amplifier M1 is the driver and ideas is the bias current source so if I have M1 this and I interpose a transistor M2 in series though cascode amplifiers have its their own problems which we shall see later but there are car folded cost code opams available OTA is available so this process is very nice what is this what is source is grounded so which amplification it is doing common source to this M2 I am given a fixed DC bias what they may have fixed DC bias means for AC what is that terminal ground fixed DC means ground so what is that amplifier is what kind common gate amplifier so I have a common source amplifier getting in next stage is common gate in cascade I have both common source the first time you change from common source to next stage is common gate is that okay what is the difference between the two both stages in cascade was common source here common source is followed by common gate and that helps to improve gain but not lose that is exactly what cascode is all about if this is what your circuit is equivalently saying okay assuming right now sees are internal sees are not used equivalently saying I want a cascode amplifier which has a current source as GM effective VN shunted by our effective and of course the load capacitance is that okay this is what I am looking so that every 0 will be GM effective into our effective now if you change is what will change the bandwidth will change again I will worry about very worry on that where if GM does not change but only our own friends then I have boosted the gain without getting out of bandwidth issue I am getting out of the issue is that clear so I may like to do a design for this amplifier in which GM may remain almost constant almost a little bit this but all this as much as I want that is what the theory of cascode amplifiers is the one as a designer which will solve this problem soon is something like this in analog circuit what is the guarantee of each transistor region all theory you are derived on what basis the transfer is always in saturation is that clear to you so there will be a VD sat here or VD drop VDS drop here there will be a VDS drop here and if this current source is applied by similar to P channel transistors there will be 2 VDS drop upwards also okay so from the power supplied to the ground how much voltage now allowed to drop for VDS if I want it in more I may put another series in that I may require 6 VDS the minimum VDS we shall see later is close to VT to make transfer in saturation which essentially means that now you are closing VDD either you improve your VDD further so that all remain to this or some transistor will come out of saturation so cascode started fantastical in theory but in real life when we started as chain increase in series okay this is like a NAND gate situation if you are larger fan in kind of NAND gates the issues is too much capacitance comes and too much current drops have to be given because that each will have to be then turn on okay speed any way goes away okay so it reduces all your other hardware okay so the NAND gate function is similar to what here we are now saying that too many in series I will please take it every time I push it by gain I put another one I will put another gain on that okay but if I put too many my worries are the transistor will not remain in saturation for a given VDD okay so there will be some limitation of maximum GMR 0 which I can attain even in cascode for a given technology nodes okay so it is one that you can increase this to any number okay you are limited by so 4 or some modification cascode okay so we tricks okay but it is not small okay as much you put additional power supply voltage either is required or you can no more than this chain is possible okay so there is a limited gain increase but the gain advantages your bandwidth gain bandwidth product is retained okay that is exactly what we are trying to do in cascode for an AC case this is my V in this is grounded this is V01 this is V02 this is equivalent of AC now what we say let us say the current flowing through this small AC signal for a V in is smaller okay so we say for this transistor M1 and this transistor M2 so let us say for M1 now I is equal to how what I should write this GM1 V in okay plus P01 by RO1 equivalent circuit 2 current sources 1 GM other V by R so 2 current sources in parallel that the net current in the drain side is that now what I do is I actually want to do something in which I would say if I say GM and I say I upon V in at V1 is 0 for the first case but if I do for the other let us say this is I1 I2 same so GM1 okay similarly I can write I2 what is I2 GM2 into how much no no just think of it how much is VGS there minus V01 0 my please remember the VGS for this is this so 0 minus V01 is the VGS for that so 0 minus V01 is that okay what is the current in this R2 plus GO2 into V02 is that clear is it okay so now I figure out oh sorry sorry you are right you are perfectly right VDS I am very sorry okay I perfectly right RO is GO2 okay 1 upon GO so now let us say if I had these expressions and I can now say for any amplifier GM is defined as if I ground VO2 okay if I ground VO2 then what will happen ground means for AC so the current is only proportional to V in which means GM is I by V in please remember current in a circuit is still same is that clear current in a circuit cannot be 2 so we say I is constant which is flowing through both M1 and M2 so now GM is only gain back for VO2 if I say GM when I say I should not say GM effective GM effective is I by VM when V0 is shorted by similar argument what is GO effective will be yes I by VO2 when VM is grounded when input is shorted input and see that the output node is the output impedance is that correct so all that now I have to figure out a relationship between I1 and I2 from this such that I can derive GM effective and GO effective and once I get what is the thing I am getting GM R0 is the net cost code gain is that clear to you and that I will GM by C if GM effective comes to be same as GM1 let us say then my gain bandwidth is not changing because GM by C is that clear but the same time gain is boosted because gain will become okay so I will tell you RO2 or RO effective will be GM1 RO1 times RO2 roughly what is this what is GM RO1 the gain of the M1 multiplied by the output resistance of the second will be the I will derive this is I am just giving you hints on that because this time is running out so I just did not want to derive it so the gain in cost code is that I somehow want to push RO higher the retail GM as what it was okay so I will beat that technology constraint and I will say okay I have got what you are really looking for in theory what we are doing is the following normally what we will do is if this is your amplifier okay this is your gain bandwidth product if I if I increase the gain really I will get this curve if I boost the gain A0 to A0 dash my this point G gain 1 will move to gain bandwidth 2 which is smaller this is what will happen in cascades okay so what should I watch what do I am now really looking for I may do this okay I am trying to do this I still reach the same point I improve the to some extent the bandwidth and I also improve my game this is what cascode does okay this is what cascode is trying to do is that clear that is why cascode amplifiers are ideally suited for analog applications but what is the catch in cascode as I said we will discuss in next more detail that the VDS adjustments becomes very very crucial okay typically VT is not now equal to 5 times VDS is not 5 times VT earlier it used to have 1 volt VT 5 volt VDD now 0.4 volt VT and 1 volt VDD VDD so only 2 and half times so if I want 4 VT drops it is 1.6 volt supply is required where I only 1 volt or 1.2 volt supply so now I am constrained that I cannot use cascode at random because my VDS situation may be not very conducive to me is that clear to you otherwise theory wise it seems to me I should just forget cascading and use everywhere see you then next time