 This is what we did last time we were looking for a common source amplifier and we say if it is driven by a fixed current source biasing then for each transistor there is a equivalent noise current source ion square and since this is ion square the impedance seen at this node is the RO of this parallel of RO of this but since I is the good current source with RO infinite so this only RO of this will appear so we say it is ion square time RO 1 square is the V out noise square term will appear and we know this term we already figured it out is 4 kT 2 third gm into RO 1 square this is what we calculated last time please remember it has a unit of volt square per hertz or if you take under root of that as a noise voltage then it is root volt per root hertz and many a time this number may come in minus 9 or minus kind of thing so it is normally expressed in nano volt per root hertz it is not necessary really values may be differing and so do not say oh it came in picot or it came in milly it may come in even ohms I mean in the volts so please do not take it but generally noises of the order of nano volt per root hertz so this is what we did last time so the noise here we assumed everywhere was essentially because of the randomness of the nature and we say it is thermal noise okay so we say it is but as we discussed in the few first few slides there are other noise factors this was done last time this is just to recapitulate where we were the another noise of interest which you have to worry about is called one upon f noise popularly known as flicker noise most believe that this noise is due to random carrier trapping at interface states this most word is very important because some others do not okay so obviously I have to say most many of them feel that this is good enough explanation to get whatever value one monitors so this is may not a very absurd thinking but there are many other devices where there are no interface states so then how do you say that if there are no interface states so why there is one upon f noise so obviously it is not the only reason okay but in most industries probably interface is so strong between silicon and silicon dioxide or any insulator and substrate and the interface states are large numbers in fact those who know something about devices now at least most device course or most technology course the typical order may be order of 4 or 5 10 to power 10 per centimeter square this is the lowest one gets 2 at best normally it may even go to 11 10 to power 11 per centimeter so it's a very large numbers are seen actually and they may actually carrier trapping can occur because carriers are moving at the surface channel is at the interface in the silicon so trapping is very strong now this noise because of randomness in trapping once say there is a noise associated and this noise was said to be called I has showed a nature that it decreases with frequency and linearly goes down so we say this one upon f noise now if you if you are noted down maybe I'll move this as usual we can model this as either voltage source or a current source whichever way you feel like you know we are so whichever way you wish you can always this or V into G is I either way is that okay move in general as I say we can noise voltage or noise currents it can be monitored as one upon f noise voltage is k upon c of w by l into 1 by f and if you look at currents then k upon c of w into gm square by into 1 upon f now k is some kind of a technology constant related to a most technology you work with and therefore different and different technology nodes okay c ox is the oxide capacitance per unit area w is the width of the channel l is the length of the channel and one upon f is the f is the frequency and since for a given current by saying everything rest remains constant you can see the current noise source is proportional to one upon f however if you are looking for spectral density you must integrate it because that is always used per something so now integrate on f1 to f2 df by f and since this gives you some df by so here is another way of expressing this which is I intentionally brought you can also represent it in a different form that was a voltage gm form this can be written in ids form you know 2 beta ids is equal to gm square is that clear so adjust the terms okay then this k will not be same take some term in the k and then new constant can be given because that w will then cancel and l square will appear so this is how new expression can be derived why I am deriving it will soon will be obvious to you so now for a given ids biasing one then can immediately evaluate instead of finding gm one knows this is the current so you know what is the noise associated from this this is the method one uses often in actual designs okay so so I just want to tell you that some books or some papers if you see they may be using this expression rather than these expressions they are identical okay remember this case are not same they are some constants have been picked up there in k plus some into something a new name has been given now if I integrate this term which is df by f term is only frequency term there so it is log f and therefore I can say it can be k1 by l square ids by c ox ln f2 by f1 if you look at take some more constant out of that what is the constant I am taking 2.303 also if I took out then k2 by l square ids by c ox log f2 by f1 so what is it trying to tell why why this expression has been shown if you see the spectral density for this it shows essentially telling that the larger the bandwidth okay larger is the noise okay is that clear so that is something you have to understand that there is some relationship we are looking at noise with the bandwidth so when I am defining bandwidth for my amplifiers or my circuit please remember it also influences your noise voltages or at the end we say signal to noise ratio may deteriorate okay this is something which obviously not known earlier but now you can see that there is a dependency coming from here we also know there is a thermal noise so if you really want to say that 2 noises are simultaneously occurring you can see this is constant there is no frequency term here okay so if you see this noise volt currents somewhere this value will become equal to the noise voltage due to current noise due to 1 upon f noise in that point clear 1 upon f noise is decreasing and let us say thermal noise is much it may be I show you figure first and then come back the 1 upon f noise starts decreasing as frequency increases thermal noise is you can always tell that this is essentially not here only it is like this but it is constant so above this value where the 1 upon f noise has same value of thermal noise the 1 upon f noise will continue to decrease and therefore thermal noise will start dominating beyond this called corner frequency of noise which is fc at that frequency onward thermal noise will dominate below that 1 upon f noise may dominate is that clear to you so this frequency is very relevant for us at what frequency we are operating will decide whether to use one of them somewhere here both are equivalent terms and therefore normally you may put two terms all the time if one is smaller the it will take care the second term will mask the other anyway in numerical numbers so for us we need not worry too much but if you know jolly well you are working at these frequencies I mean one need not worry too much about 1 upon f noises but then a priori you should know for given technology node what is the fc for you to know fcs you need some constants to be known that is case so it has been found for for example okay this is essentially for 0.25 micron technology whereas the k value which I got from the Radhavi's book is essentially for 90 nanometer devices so obviously one can see why this is different in different cases why I said because as technology scales interface properties are getting difficult or in some cases even better either way there are different technologies like if I put high key I will be worse in that silicon dioxide I'll be much better now actually but as we scale down we may have to shift to half nem oxide half nem oxynitride or zirconium oxides or as we are working at lanthanum oxides or european oxides very large numbers so there as you scale down the case becoming you can see larger and larger what does that mean that means noise will become higher and higher as you scale down is that point clear so we very started now as we went down below 0.13 or 0.18 micron technologies we figured out the first hit was our noise itself because it was just increasing and increasing as the frequency node start going down okay you can also see the term l square in the denominator there is that clear so smaller the channel length you use larger is the noise you create is that clear to you so some issue has to be understood that why now I am talking so much about it because you have already gone to 22 nanometer process may be soon 16 will come and 11 may come and 0 will come but the point is that the noise will start then such a dominance that signal is not there only noise is there may happen so let us see what is the situation where this may occur or may not up so elevation is one way of trying to reduce as long as we get signal to noise ratio higher it is fair enough only noise increase then you have a worry is that correct or somehow you must try some filtering has to be done such that some way this number can be controlled okay that is something we will do later so is this constant values are varying you understood why they are different because they are different nodes the interface states are not same and therefore and the channel lengths are different so obviously the constants are and the total noise also is different and different nodes so as we now move towards this fc calculations we equate these two currents or noise voltages as per whatever it is this is noise voltage I am now taking 4 kT2 3rd GM is equal to okay this is square square currents only and then we get a term which gives me K GM fc put f is equal to fc and solve this equation what is it trying to say that again you can see it is function of w and l or l square as the if you take ideas terms k so larger the k larger is fc is that clear so the 1 upon f noise will start dominating if your k is larger smaller the channel length again the fcs are higher and higher so now 1 upon f noise is now seen even in rf circuits is that fine clear everyone ask us oh we are talking of gigahertz so what's the problem of 1 upon f you can see numbers is now going towards as many as gigahertz in some cases normally it will be order of 100 kilohertz to few megahertz fcs typically at the order of few kilohertz also it will be different for p channel and n channel device p channel have lower fc and channel will have higher fcs okay this also an issue which you must appreciate okay the GM square term is coming because of the resistance which is coming resistance of a transistor is seen from the lower side is only 1 upon gm so resistance the current is something by 1 upon gm goes above 4 kT by r is the current source r is 1 upon gm and r square if you do it is gm square is that clear to you a little same method okay so in nutshell about this 1 upon f or whatever it is I said already I repeat k1 values are larger for n compared to bmoss fc for the order is larger than that for bmoss fc is a proportional to 1 upon l square fc increases with technology scaling finally fc is also proportional to 1 upon gm by ideas which gives some understanding that if you have a larger gm your fc goes down okay larger gm but larger gm has other problem what is larger gm problem let us say I hold gm larger then I will proportionately will increase ideas because I will maintain gm by ideas larger ideas means what power dissipation so the first hit I got is I saw that oh I reduce power but I hit on my power dissipation which was decided only by earlier by slew rate the net current available to you at the ISS for defams and it is also decided to some extent by the heat sinking possibility in the device okay so the power if somewhere was not connected to noise and now we realize that yeah power has some relationship with noise lower noise can be obtained if you have a larger power dissipation is that clear to you so this is another feature which has now being added in analog design that the current you cannot just do something you like if you do something better for x you can you can see you want to maintain bandwidth let us say 1 upon rc if you increase c to keep this you will have to increase a decrease r by same amount or in our case is 1 upon gm so in some way you always find that you will hurt to the power immediately as soon as you go to the real device is that point clear to you this is an issue which normally many books are not telling earlier because this was in 5 micron to 1 micron or 0.8 or even 0.18 people thought this everything is okay manageable now this manageable is now becoming unmanageable so please remember the issues in 2010 or 12 are different from what 2000 we had so in my earlier first course in 98 I would not have told you anything this because I myself might not be aware of this but as I started teaching ahead I realized things are happening now much more stronger worries are started than what was happening in many years so please you have a more problems than what we had when I started my master thesis was on TTL TTL was a that time very interesting logic and it was very nice to work on TTLs if I now say TTL some some may ask what is TTL okay so I hopeful it may not but so things have changed over the years coming back to quickly on this noise in amplifier I will do quickly few cases let us say if I have a resistive RL as the load biasing is otherwise done then there will be two thermal noise sources two sources across the two components one is due to the RL and the other is due to the transistor itself okay so to calculate the output voltage that is the noise output voltage what is the method we apply use superposition take one noise source find output take second noise source find output okay or to say sum them out and then actually figure it out so for this we all know that this transistor has 4 kT 2 3rd GM this is what is this this is 1 upon f noise term and this term 4 kT by RL is RL term and multiplied by what is the load here RL RL parallel RO is RL so RL assuming RL is much smaller than RO so RL square this is I I square please remember this is I square by RL square so you can say I1 square R1 this I2 square RL square is the net output voltage at the noise you will get okay substitute this now