 We are looking for fully differential opamps or fully differential systems. The typical differential opamp can be connected in this fashion. You can see from here, this is a, these are two inputs of a differential opamp and you are receiving V in 1 and V in 2 and there are two outputs V in 1 and V in 2 which are feedback in a negative fashion, negative feedback from both sides. And if you have only single ended, the output could have been in the centre okay which is what we have already done earlier. In this case, we have two outputs and we are feeding it back. Please remember both are negative feedbacks. Right now for the sake of this stability in everything and performance, I chose R1, R1, R2, R2 but they are different also can be solved a case in which there will some issues will come later. Now why differential? We already discussed it at last time. One of the interesting point we said there was anything noise sitting on these can be cancelled okay because if you take the difference of V in 1, V in 2, the noise which is super reading either inputs can also get cancelled and subtractions. This is what we said but there are few more interesting things which is of relevance and I think that should go first before we start noise. For single output op-amp, a typical amplifier shown here which is an inverting mode amplifier and the output looks something like sinusoid. Let us say if input is a sinusoid and you receive VO max, VO min. If there is no distortion then VO max will be equal to VO min and we define a output common value VO common or VO VCN as the common mode voltage at the output which is some of VO max plus VO min by 2 okay. The maximum you can reach is VDD, the lowest you can get to VSS and therefore some could be 0 or if VSS is 0 it could be anything else. So VCM is something which you can decide how much you want by deciding up to which these two will actually go. In case they are equal as the shown here then the VCM is a VOC is or may be VOC M I should say or VCM we shall say later is 0. Please look at it the system which I have shown you earlier just a minute before this is symmetric to this okay. This is symmetric structure is called balance structure. In case of balance fully differential system we can also figure out what will be VO1 and VO2 independently okay and if I subtract what can happen at the net output which I am going to get. So I repeat this is a single stage opamp or sort of single ended opamp now I am using it similar structure which I have shown earlier to have a differential mode opamp fully differential. So you may have for VO1 and VO2 because of opposite polarity they are they may look like this and if I subtract take a differential of them the peak values is V max minus min minus V max minus V o min and they are equal and opposite then twice of the earlier values you can attain. However the common mode which is still subtraction with 2 is still 0 because we are holding them equal this and this and this are same so the common mode is still average value which is still 0. So please remember that by making a differential I am still able to get VCM 0 which I wish if I can set any other value but we can get the output which is VO1 minus VO2 differential output is double that of the single ended ones. So one feature of the fully differential is you can get twice the value of outputs swings so what is this called the swings the swings in the case of differential opamps is larger compared to single ended system many applications require larger signal outputs but you do not want to reach saturation I mean this nonlinear zones in VOVI this may be one possibility in which you can get larger output swings is that point clear otherwise for larger output if input increases you may reach into a nonlinearity zone of VOVI characteristics and then you get distortions we will see that later. Now the problem which is why this kind of amplifiers are used one is of course swing the other is of course for most of the circuit which requires higher gains differential systems are very popular but as soon as I say higher gain we will see what kind of loads I should have compared to single ended opamp which we already discussed. So the issue let us say we need an amplifier which has a higher gain and we also request we also desire that such an amplifier has large signal to noise ratio for your sake of those who are not done earlier this a signal to noise ratio is a maximum signal output power divided by output noise power okay is the ratio of this is called signal to noise ratio SNR if V square signal is the peak value of this is the peak value of this square by 2 is the maximum signal and output noise power is gone VONs bar square or VONs square bar average value please remember if I use I solve this in the noise case these values but these are trivial one can solve that VONs square for a single ended case is 1 plus R2 by R1 square 4 KTR 1 into Fn and Fn is called the noise bandwidths this 4 KTR is something probably we will see soon in few minutes which is essentially the thermal noise of a resistor. So the output noise voltage square if it is always expressed in square terms the Y square anyone essentially represents power so V square by R if R is 1 we represent power by V squares where in the case of fully differential it happens to be twice that of this square. However you also see the swing in the case of single ended is just V max minus VOMIN where in the case of full differential it is twice that of you max minus VOMIN swing is also larger. So if I scale this I get 4 this is half so still SNR will be how much larger or smaller twice that of single ended of bands is that clear this will give you 4 this is half so 4 by 2 which means at least signal to noise ratio will double when I use fully differential systems. There is another term which we will see in the case of NLF designs which is called noise figure noise figures is explained SNR at the output divided by SNR sorry SNR at the input divided by SNR at the output some other time when we come to actual noise figure calculations. These are the specification for a given amplifier they will specify you typically what is the SNR needed okay so the kind of thing which I am I was trying to talk to you that for higher gain 1 and this second is higher SNR. So if you are fully differential one can say is twice that of single ended amplifiers and larger the SNR greater is the amplifier is that clear signal to noise ratio larger means lower noise higher signal in a ratio okay. So it is better to use differential amplifiers fully differential amplifiers instead of single ended opens if you are looking for higher SNRs where single ended the problem is slightly different in this case single ended since the one end of the circuit we are only putting diode connected load. So the load at the one end only is RO3 parallel RO4 in a single ended opens whereas in this case now I will show you I can have larger ROs on the both sides and that by increasing the gain as well as increasing the SNRs okay this the equivalent circuit of a differential amplifier both at the input and output I will post it on the website you can check it it takes a little time one can express inputs for the differential stage one I will show I will please read it in case you do not I will come sometime then I can get some relationship between V I D I I C V I Cs by similar argument I can put an equivalent T network for the output of a different fully differential system and again I can get VO1 VO2 VOD and VOC accordingly this is what has been done and using this we will be able to calculate ACM and ADM for a fully differential amplifiers okay please remember I am I am sorry I am not able to spend time on this because there are not many things to be done but I am going to post it so you can look all of it for example fire before we go ahead the VOD is VO1 minus VO2 which is called ADM times VID and for common mode VOC is VO1 plus VO2 which is ACM times VIC so we can evaluate both VODs and VOCs and what is important for us in this particular this common mode voltage to be fixed and let us see why I am so much worried about fixing the common mode voltage the theory about this please read it on my website in case there is an issue there may be some other day I will again explain to you just figures then write right now because it will be available to you we keep saying to maintain a good common mode voltage we have said that normally fully differential systems are used for higher gains okay now if you see if you are this is your standard double ended okay one at VO1 and other is VO2 and to make it common mode or rather to make it simpler to first understand I have connected output of this to the input of this in normal case what will be here the resistance output to input R but right now I chose R to be even feedback can be near a feedback differential feedback so this is a differential feedback which I kept this is your bias circuit that these devices receives a potential VB which decides the ROs of M3 and M4 is that point clear M3 and M4 receives a potential VB which decides the RO3 and RO4 of this is not diode connected load this fact has to be appreciated this is through P channel and the lower current ISS is governed by this N channel mirror shown here now the problem which is very me right now is this if everything goes well what is the output resistance of this arm RO3 parallel RO1 what is the output resistance of this arm RO4 parallel RO2 so GM times that is the voltage which I can get then I can get individually my problem is now something more my assumption is this current is ISS is governed by this device and this potential is governed by this P channel device which is also a current source let us say for reasons variety of reasons sizes abilities for a reasons these two currents are not equal in a case of love one forces the current should be equal but let us say it starts with the two equivalent saying I may say okay to put it non-equivalent say I put a small resistor in between which probably can then control the current and then drops here are not drops here now this issue which looks to be very odd is very interesting for to understand if this current which is essentially flowing in this current in ideas 3 or ideas 4 is not same as ISS by 2 and ISS by 2 we want them to happen but let us say for variation purpose it does not occur okay now what can happen there are two possibilities can happen this current is larger than ISS by 2 or this current either way this or this this current is less than ISS by 2 because of the P source current is not same as N source current is that clear to me now that happens let us say what are the problems which we can get I repeat this is essentially please just see the figure why am I am saying equivalently saying this is given in the various book this is I 7 P maybe right now I can say this is I 6 N and across them there is output resistance ROP and RO for this sources R1 if these are not equal in this arm there will be IP minus IN which is whichever is higher minus plus sign will appear and since the at this node these are parallel trans resistors so there is this at this node I have IP minus IN times ROP parallel normally what should have been there 0 drop across there are two current sources no drops there should have their balanced it out but it may happen that P channel current may not be exactly same as and channel current and because of that this node voltage is not at 0 is that clear now this is a very which is real life occurs okay it is not something which I am thinking about as a new thing like that happens I have said the other things I will explain normal open business so there is nothing more to write what is related feedbacks purpose to stabilize a system I am stabilizing so I believe that with this any change should have actually been managed but to my surprise I find this negative feedback which is actually a differential feedback does not solve the problem which I am going to face normally what is a negative feedback purpose is to stabilize the system should pull down or pull up to get to a normal mode here I am now trying to show you it does not even if there is a differential negative feedback this does not return to equivalence okay so here is the issue which you should look into you can write down if you wish the last line is what is bothering me that the current in M3 or M4 may be larger than ISS by 2 or less than ISS by 2 if P channel currents are not same as N channel currents now if this happens what can happen to us please remember ISS is controlled by whom 5 M5 ideas 3 and ideas 4 are current controlled by M3 and M4 is that which are governed from 6 and 7 mirrors okay M6 M7 mirrors you first write down then we will take the two cases and we will happen what is going to a problem if this happens this and we say Raj what could have happened you say it is larger ideas 3 is larger than ISS by 2 what could happen you cannot hold this situation so if you want to make it equal then what should happen to something some transistor must change the status otherwise two currents cannot be equal is that correct that change the status is worrying me the most because change the status means the resistance of that device will change and if resistance changes my gain changes okay so they my common mode I mean differential mode feedback was there but I still be changing outputs but let us see you finish this then I will discuss this the two issues of interest is let us say the current dream current of N3 or M4 is larger than ISS by 2 or less than ISS by 2 these are not issues which are trivial this is in real design these issues keep coming at least in fully different states this is what we are elevating and that is why that they become very popular open systems for many of the applications fully differential but what is the problem cost extra kuch karo pay for that okay when I already shown you fully differential can always be used as single ended look at the signal end only and do not worry about the other one okay so there is no issue if we really use on this other single-ended amplifier ideas 3 4 is the current drain current of M3 and M4 is that correct ISS by 2 is the current in M1 and M2 half of okay as I said this is been given from reserve is booked though my normal pleasures are slightly different from them and the way I have written they might not have written so that is the only difference but the content is same as reserve is booked this is under the section in open call common mode feedback CM FB is it okay so case take the first case ideas 3 4 is larger than ISS by 2 please remember I ideas ISS is controlled by 5 for 5 transistor current which is coming from N channel mirror whereas ideas 3 4 is coming from and 7 mirrored P channel now these are greater we like to restore ideas 3 4 equal to ISS by 2 that is the natural in a case of law in a one arm positive to negative end one current only can flow but in reality right now it was instant and really did not which means we are 2 or we want to be same in normal cases however current in M3 M4 then must reduced I must say what is there let us say this is my current in M1 M2 ISS by 2 this is coming from ideas 3 4 which is following because of the current in the drain current of MC M3 or M4 if this current has to reach here what should I do it should traverse back to reach this point and what is this point then will make 7 3 and 4 enter into because that Vx value if I have to fix a V1 V2 I have to same then M3 M4 must come out of saturation and try to enter linear mode is that clear you said that current is larger but the current which N channel M1 M2 has received by 2 this was larger so it has to reduce itself so that the currents are same so that is valid and if that happens M3 M4 comes out of saturation what is the advantage of the system we were discussing high gains did we now have the high gains because M3 M4 would be in a linear zone which is a smaller around therefore gains will actually fall is that correct gains will actually fall now this is an issue if ideas 3 4 is greater than ISS by the possibility of this internal current may be larger than P channel so opposite can also occur okay so to say ideas 3 4 is less than ISS by 2 ISS by 2 is coming from which device M1 and M2 or M5 is that correct M1 M2 receiving current from the tail current M5 which is ISS which is coming from N channel M6 transistor is that okay so let us as everyone thought about it so let us go to the next case if ideas 3 4 is less than ISS 4 ISS which is supplied by M5 now we say ideas 3 4 is here and M1 M2 which is receiving half the ISS current from I5 is following this IV law okay now what will happen if this is to be made equal then M5 will now come out of saturation what will happen to M5 if it comes out of saturation or these RF5 becomes very small do you recollect what will happen to other some other parameter of open the common mode gain is proportional to one upon RE if that becomes common mode gain will increase so what will decrease CMRR so by if it does happen that the other current is ideas 3 smaller than this the amplifier may remain same V1 V2 but device will now show you much lower CMRR is that clear so case which look very trivial otherwise has actually then we have a differential feedback we are connecting gate to the output so we have a feedback and in spite of that such a situation is all it is not returning back it is actually reaching one end okay this is something is not correct so if I can make something common mode constant then the my CMRR will not be dependent on this situation so let us see further is that okay 5 transistor current will reduce so that it becomes equal to ideas 3 4 ideas whatever it is which means M5 will come out of saturation and therefore by doing this it will actually reduce the increase the common mode gain or reduce the common mode rejection ratio okay now these issues looks to be you know many people believe this never happens but it is not so it does happen and you need to put a circuit which is called common mode feedback which retains the common mode if something increases it will reduce if something goes down output it will pull it up okay so that always came on our remains constant so is the gain the output resistance remain as high as they can be okay so now we say average of open output is defined as common mode output Cm CBCM which is V0 plus plus V0 minus by 2 let us say the full swing we get V0 plus go to VDD V0 minus goes to VSS and let us say VDD is 2.5 VSS is minus 2.5 then what is VCM 0 okay VCM is 0 but let us take a similar supply of amps in which VDD is 5 volt and VSS is 0 in which case VCM is 2.5 I can actually decide to have any of the VCM values between VDD and VSS of my choice okay which I can fix so it is necessary for a fully differential amplifier to be stable just differential feedback does not help you and one needs what we call as common mode feedback or Cm FB as the word goes and this allows you to fix a value of VCM because we are coming out of the current so if you can fix them then the outputs will be always VCM will be always adjusted to V0 max plus VOMIN please remember it need not be VDD it need not be it can be any value in between 0 to VDD okay minus VSS to VDD you can fix anywhere choose any value average of best those two will be the VCM value which is DC value which you want to decide on okay. Now how do I decide normally I will prefer if it is double ended I should keep VCM 0 if it is single supply rail then I should have to fix VCM value to a fixed choice of my choice how do I then give a feedback here is something as everyone noted down these are something not necessarily useful directly in the real spice also fails many times in not having if you do not have Cm FB circuit your amplifier may not function properly it starts actually ringing too much for a long time output never settles okay these are one some issues which you will see in real life so you must put Cm FB circuit to stabilize that out okay a typical Cm FB circuit is shown here this is your defam and this ideas 3 ideas 4 are coming from some BB and mirror as we did I actually do one thing I figure out what is VO1 and VO2 so I must first sense them okay so there is some circuit which is called common mode sense circuit it figures out what is the VO1 and VO2 values okay then from those two values it generates what we call as old Cm FB voltage VCM then we compare it with the desired value which is chosen VCM which is called reference voltage in a comparator if it is larger or smaller correspondingly the control voltage call it V control if you wish will be larger or smaller minus or plus or whichever values you fix this value can control the voltages to either these two or these two I have shown you both ways some people you actually control these currents some people control this current anyway we have to bring it down to equal from either on the top or from the bottom so please remember we sense VO1 and VO2 outputs sense them generate this compare with reference create a control voltage which controls the currents in either ideas 34 or ideas 34 ideas 3 or ideas 4 or ISS one of them can be controlled so we either we can change the tail current or we can change the load currents why when the circuits are shown this is the principle behind the Cm FB circuit which we are going to show and in that case the way circuit which I have I am controlling the load currents okay but same can be used for IC I5 current also ICS current if I use Cm FB circuit which is also to some extent an amplifier is a kind of system what can it create it can create stability issue you are increasing another RC time constants with it is that clear to you it may have a system may become unstable so in a normal case of open how do you stabilize that by actually putting a miller capacitance in series to a resistance pole splitting or directly miller and poles okay. So that the phase margin is relatively 50 degree around so you have to do full analysis with 200 completely supported by RC network and figure it out whether even with this it stabilizes okay so there is a it is not so trivial but typically if the loads are larger it happens to be stabilized so you do not have to do it but I do not want to say a priori unless you see what let us say I put 10 pop yes it will stabilize this is natural phenomena but some other time okay is that okay sedan so here is a circuit which is very simple and popular please remember I am only generating VCM VCM FD this is only generating the common mode feedback voltage how to generate it which includes the comparator so it is essentially control itself I am generating I have two transistors M1 and M2 which is my sense which is receiving signals V1 and V2 or VO plus VO minus is that okay Santosh M1 and M2 are my sense devices each receives VO plus VO minus or V1 V2 whichever names we have been following then there are another two transistor M5 M6 which are so connected that the source of this are our enchants okay the source of this is connected to M1 and source of this is connected to source of M2 is that okay M1 source is connected to M5 and M6 source is connected to M2 M5 and M6 receives the value of PCM which you want to set for that is the value set there then this M7 M8 M9 are essentially our current sources because they are given from a fixed supply VB1 VB2 may be banned from the band gap references okay so that these are constants so please remember these are current sources you can also ask me I could have done one single current source think of it why I do it in series okay think of yourself if not maybe I will ask later I could have done with one series trans one bigger transistor of the L length is larger which I have tried what is larger length helps the lambdas are higher and therefore you get alumna are smaller therefore get larger are so this is some kind of pseudo cascades okay it is not really cascades but it is called pseudo cascades okay these are current sources and same currents are flowing here now let us say Vo plus and Vo minus are such that the average value of them is larger than the common mode voltage this is the first possibility these are larger that that average value of them is larger than the VCM the other case could be they are smaller so that their value average value is smaller than VCM so two possibilities if they are larger what will happen to currents of these if Vo plus and Vo minus are larger M1 and M2 will draw larger currents VGS will increase with them but they are actually getting currents from the current sources these are of course diode connected loads okay these are I am pushing fixed current please look at it I am pushing fixed currents is that okay but these two start drawing larger current so which will draw smaller current M5 at M6 will draw smaller currents because we are that is what we are added actually okay that together gives me these two currents so if Vo plus and Vo minus or Vo1 Vo2 average value is larger than VCM these will draw larger currents but for that these two should draw smaller currents now if these two draw smaller currents then the voltage I have dropped across this will be smaller and VM voltage will increase if I increase the voltage I can correspondingly change the load resistance currents or I5 currents is that correct now take the next case if these are smaller what will happen they will draw lower currents compared to this they will start increasing the current so VCM that we will actually go below so it will increase or decrease as per requirements of increase or decrease of Vo plus and Vo minus and average value will be then retained to VCM Vo plus Vo minus either way it will bring it back to its VCM value once the common mode feedback is fixed I mean settles that means your common mode rejection ratio is permanently fixed by you this is something which CMFB allows few more things I do not want to say more these two currents are actually fed to the defense stage what we have seen with other cost code stage down at the single stage cost codes so it also divides the current in defend as well as the single gain stage I am not showing the circuit these currents can be chosen such that they feed both to difference as well as the cost code stage or gain stage change in those currents will decides if the defense stage is going down it will actually boost the cost code stage if the cost code stage goes down the defense stage game will increase okay so that the output gain remains constant with the CMFB which I have not shown here but that is why it retains higher gains and constant common mode value which fixes higher CMR SNR higher gains and of course for stability I repeat you may have to do a RC network for your both ended up ends is that clear that you cannot avoid that this additional this network is going to create additional poles and zeros so the system has to be stabilized so you must actually do more RC network analysis to get whether your system is still stable but remember if the double ended outputs have larger load to drive which is the one of the major purpose of this then compare to all others that they will be the dominant poles and they will actually govern all the bandwidths and no others instability issue will come okay but this is something guesswork which I am telling unless you do one doesn't know so I do get the point why I did common mode feedback because this is one of the major circuit which you will use in all actual designs whenever there is a issue of stability or ringing starts look at it somewhere here gains I mean the currents are not matching this matching problems can easily be handled by on common mode circuit prices additional power lower bandwidths and extra area so of course all this is at the cost of getting perfectly good highest highly stable and highest in our amplifiers is that clear this fact has to be understood in actual designs because these are not normally used by everyone we can do a million times your circuit may function without that but if it does not the solution please remember every circuit I show does not mean you have to put it if you design and it works God saved you okay but if you got is not ready to save you can save yourself okay that is the trick behind all of it this finishes everything related to opens of course there are few more things I am ending here okay few more things you must remember whenever I want gain stage to hire must cascode this is something which we learned day one cascode it if I want to improve the bandwidth situate that the first dominant pole is at least far away so that the other pole shift further so the shifting of poles is one of the best is null if you know it your dominant pole is in your hand okay so try to use null techniques but nulls are not very easy to attain but as much as you can okay these are tricks which are as much as possible if you know higher SNR higher gains you always double ended fully different system the biggest advantage they gives the noise is eliminated even what is the else is eliminated the offset are also eliminated what is an offset voltage the output is 0 but input differential is still there so which means you have to upset input to get output 0 this value also automatically gets cancelled because differential this output also equally equal so both cancels in different differentials okay so these are very ideal amplifiers okay okay so this finishes essentially the open base systems please read is all the boys do whatever books you have so we start now another area of interest which is actually not necessary to analog but in analog it becomes very worrisome problem though the word which all of you are so familiar noise why I brought it before other few things left because I thought that this is an issue which is also needs to be taken care during the design itself okay just now I talked about SNR so this that preceded this but actually it should follow after this so we will look into noise with following points first what is the noise and what types of noise and then the mass model for the noise because we are interested not on all other devices we are more interested in mass circuits so we will look more mass models and finally for given mass circuits how do I evaluate the noise okay the word which we use in the case of amplifier is now that which level we connected input reflect noise is what we are interested and not at the output noise not at the output but because of the output what input noise you will receive is of relevance and we will see why I look into that little later what is the noise noise is a anything which is unwanted or undesired signal which couples with desired signal of this is termed noise so noise is not something very odd or bad or something you are none of you are from anyone from communication background here all micros okay in communication we will learn later the noise is not all that bad there are algorithms which use noise to improve the signal to noise ratio you actually AGM noise you add and you actually get a better figures better images at least in image processing so do not go by my statement that noise is bad okay noise is generated due to a random process okay and limits the minimum signal level that a circuit can process with reasonable but acceptable quality noise is generated due to random process and limits the minimum signal level that a circuit can process with reasonable but acceptable quality the problem why I am interested before we do some VCOs or others is that in a normal analog design noise has something directly related to power dissipation it is related to bandwidths and it is also related to linearity okay and then they also are connected to each other as well okay they are also connected to you just we have seen hexagons so the problem is noise cannot be eliminated in design because they may affect any of the basic parameter design which you want to do so a prior itself you should have an idea how much noise I have and how much I can tolerate okay that is the game in this so please remember noise is something unwanted and therefore and also random so what is essentially random some processes are called stochastic statistical barriers so we do not know noise is how it behaves let us take a xt function which is periodic and then we can deterministically say that if this is your function at any given time t1 t2 t3 I can find the value of x which is xt1 or xt2 or xt3 if this is periodic well-behaved function however if the signal is random in nature as shown here I cannot predictively say because I do not know the nature of the function that at any instant of time what is the value of xt I have okay so that is my major worry that it is indeterministic there are also other issues in noise and this is why I put it called correlated noise many of you probably are aware I do not I do I forgot the name but in a stadium when people come they are in 40,000 to 100,000 people but that noise in the background hardly it is heard by you okay but what is that way that they call it when all of them stand a maximum if that is so happens it makes a huge noise so if the noise is correlated it can have a bombastic effects okay uncorrelated noise normally distributes out and an average value is not those that strong okay so please remember that noise have many features to understand but we are only concentrating more from analog design side the issues of other noise are also equally important okay. Of course basic idea of noise will remain same irrespective whether I do an analog digital signal processing or whatever area noise is a noise okay so do not get too much worry is that okay Siddha it is even though the noise is random most noise performance if you see they at least show some average value it is not 0 some average value okay but of course that decides if you do it averaging for a long period of time very short period may become 0 or it may be very large or very small so if you actually average it for a very large time then the noise will have an average power so for example I have a resistor and I connect a VT source then the power delivered is V square T by RL so if I use the average power for any random source VT then it is 1 upon T tends to infinity large infinity does not mean really infinity we use infinity because many of the integrals can be easily solved if I put infinity okay so minus T by T to plus T by 2 which is a period in which I am going to monitor V square by T by RL DT is the average power delivered for a resistor by noise of VT please remember this T has to be large to get reasonable averaging so for example if this is your function and you take a square of that this will be something like this and you can see there is some average value I can get that correct as soon as I square it I get some value which is reasonable enough and then I can get some average value which is V square averaging okay in normal all analysis RL is chosen 1 ohm okay and so VT square is defined as watts looks to be but we say V square where it should have been in vast power is in vast but since R is always chosen one one on per say then the power average of a noise is essentially expressed in V square that is voltage square okay that is the method of expressing if there is additional load you may multiply it to get the actual noise value is that correct this is a RMS value if you take under root of base then this called voltage per load if you wish or just voltage but in normal case this is time domain analysis if you look at that the if you take the frequency spectra for this we find out that at different frequencies the power is not same okay this is called private distribution with it or also called PST power spectral density PST okay so if we actually if you look at the average power we should look for spectral density for noise because at different frequencies noise have different values okay so that is the definition of noise spectrum if noise average power defined in frequency domain we get what we call as noise spectrum which is called power spectral density or PST okay and it is defined as SXF F is for frequency average power is carried by XT in a let us say I have I have a noise which I pass through a band pass filter if this bandwidth is 1 hertz okay band pass filter of 1 hertz bandwidth if I pass through and square it okay I get XF1T square and then what is the range of frequency I use only 1 hertz XF can be expressed in terms of old square okay per hertz old square per hertz and voltage is expressed in under root of average so we say it is bold per root hertz so noise is expressed as bold per root hertz okay at different frequency I actually calculate like I calculated F1 please remember this one heart which I sorry I should have I made a mistake the central frequency is F1 around which the bandwidth is 1 hertz in a band pass you have to say where okay so I calculate similar thing at different frequency F2 F3 Fn okay and then plot this PSD versus frequency at different frequency each will have a different value of this and then I plot essentially that is called the spectra is that clear please note down this then I will show the figure so what is my next stage at different frequencies I evaluate XF2 square XF3 square every time I have band pass of 1 hertz around the frequency of my choice okay so if I do this SXF at different frequencies okay average value of that so I get a spectra this is called noise spectrum or power spectral density SF XXF have with F is something of of course this is random this is not that for everyone same curve will come this is some random figure which I draw this is the kind of spectrum we get for noise actually and since SFXF has a unit of X square under root of that as a unit of bold square by under root is volt per hertz per root hertz so noise is essentially expressed as volt per root hertz assumption is RLSR unit loads but normally it is not used there is a very famous noise what is it called white noise so let us see what is the spectral density for that or spectrum for that this is taken from the rise with any other figure is good enough okay so all that we did is find XF3456n with 1 hertz bandwidth for every point and then plot okay is that okay a white noise has a spectral density which says that minus infinite triplet and it is constant that is why it is called white noise okay of course in a normal spectrum that minus infinite and plus infinity will have some larger bounds so F1 to F2 is constant okay then definition it is minus infinity to plus infinite frequencies the noise is constant spectral value is constant now there is interesting part that I can modify the content of white noise if I pass through a transfer function HS okay which is HF is HS2 pi JF so if I avoid noise I pass through a transfer function of this nature which is HF square so if I multiply this essentially I mean I am going to get a band limited SYF that means this noise will get limited in the range of transfer function so choice of transfer function allows me to get spectral density of my choice you use this function as you want and white noise will convert into a pattern of your choice is that clear that is the fun part in designing these circuits okay also normally this noise spectra is minus frequency to minus to plus it is always shown spectra is somewhere here because transfer function may be of square Anna it can take minus value and plus value square with the same Mojica so it will be a method like if between these two bands of this if it is SNX okay you want to see the last one please have it okay one sided set to two sided say one side jatama cab old day integral of minus infinite two points minute twice zero to infinity same techniques is used so just double the amplitude okay that is how moist spectra are shown one sided after we are doing averaging averaging means integral okay so so two sided spectrum can be converted to single sided with double the amplitude so this is the statistical behavior of noise this was the first concept we wanted to give that what is essentially noise is all about and how it is you can actually get the pattern of your choice which device or which system which is called a mixed signal circuit which is one of the most popular I do not know it should be called analog or digital which actually requires noise shaping what is it called ad adc d ac a to d converters and d to a converters requires noise shaping to the maximum okay you may have one loop two loop just to shape the noise so that signal to noise ratio is very large there are other term which is called signal to noise distortion ratio or sndr some mixed signal course some other time so noise in analog is very very important parameter because if please remember what will happen if noise is larger you will require larger power to actually dissipate okay so your first hit will be power dissipation okay okay is it okay figure trivial okay so now let us see the three terms many of us use very unknowingly that they are as if they are same there are three famous words which we use in the case of systems one is called the device or circuit related noise the second term is distortion and the third is interference okay all are bad but they are different okay so let us look the three differently because we are more interested in the first part but in systems you will require even the other two as well if the output waveform is which is a function of input time some transfer function and if that value which you now see in reality is not same as if it should have come then you say output is distorted distorted from the ideal value you thought you should have okay like you have your vain characteristics from these two inputs dv0 by dvn is constant okay so to say any value here or here I can predict the output by just multiply is equal to mx c0 here so so I know the slope that is the gain in this case so I know dv0 by dvn however if I exceeds v in one or v in one either side then the view vi characteristics does not seem to remain linear this essentially means you are entering a zone which is called non-linearity zone okay and as soon as you enter non-linearity zone you can see dv0 by dvn is a function of Vn so it is a non-linear term so if you expand it by Taylor series you will get x cube xb x square c okay higher order terms okay which means some power in the output is now deleted to one frequency second frequency third fourth and highest power delivered generally is the third harmonic and therefore it is called third harmonic distortions okay second harmonic in many circuits we tune it that is we reflect it back but third cannot be therefore the distortions are normally strongly related to third harmonics okay we also feel as if it is due to only active device no non-linearity come because of variety of other reasons one of course is active device non-linearity it is not linear okay so this non-linear distortion is essentially because of the non-linearity in the transfer characteristics okay so we say the distortion occurs due to non-linearity then transfer function of active devices but there is a possibility which is not just just there it is always there even in passive components there is a non-linearity for example cables you know cable it is like a transmission line okay it is an LLC circuit okay which also have different responses at different distances okay because the Z0 change there or Z reflected will be different so even in a cable you may get a distortion okay so you always say do not increase length so much put repeater somewhere even in fibers it starts distortions okay there is also in emergencies in the path the cable LLC may be varying all through which may not be a constant or in a fiber the multimode fiber becomes somewhere only few not multi but smaller numbers so much of the area is lost and therefore much is radiates okay so any inhomogeneity in the path also may lead to distortions so but we are more interested in devices because that is what we use often but in systems anywhere noise can I mean distortion can occur the next very for me is the interference see if you have one spectra that is one signal spectrum and you have another signal spectrum which is closed by then they may overlap okay this essentially is called inter modulation simplest inter modulation thing can be understood without even a receiver you have a two transmission lines or two signal lines if they are closed by due to just mutual inductance coupling signal in one way connect to the signal in the other this is called crosstalk okay this is called crosstalk when is crosstalk highest two lines are moving when the signals are in opposite direction the crosstalk is peak of both like differential so highest crosstalks okay so but in RF receivers because of the antenna input you receive there is an image which actually interferes and creates higher harmonics okay and that needs to be further filtered all filters leak DCs so there is major value there some other time okay so the interference is a very common in most RF circuits or RF systems okay like an antenna you have two two lines on a any dielectric has you been closer the radiation pattern of one will interfere with radiation pattern of the other so inter modulation will start and the pattern which will see at then at the antenna may not be single lobes may be a multiple lobe with diffractive diffractivity almost lost okay it may go scattered out okay so everywhere the inter modulation is very common thing we are not so much right now worried about these two the distortions we will probably but we will always you will remain in linear so that is the reason why we kill analog circuit as linear circuit because they want to remain in linear modes okay otherwise it could have been nonlinear circuits okay okay so here is the third and the most important among them is the noise the noise noise okay electronic components produce combination of some noise with the spectrum shown here if it is constant then we say it is a quiet noise and there are some noise which are inversely proportional to frequencies they are called one upon F noise and there are certain noise which are inversely proportional to one upon F square this rarely you see in the books so I added this for you it is called popcorn noise what popcorn noise is of course it is another name if you are in a communication area it is called burst noise okay since the burst occur when the data rate is much higher than with the line can hold then the noise starts picking up so it is called burst okay and the burst is essentially like popcorns you know they pop up so it called burst okay so it is called burst noise or popcorn noise yes larger the frequency smaller is the noise no but some other noise must be taking over there so this is one kind of noise which are the frequency dependent noise okay no it is limited by layer that is what one upon F square is telling however they are another class of noise which are generally generally thermal noise as the word goes however other than thermal noise also there are other kinds of noises available and these are named as short noise okay short I put my brother Mr. Johnson is some nice only when I but the most commonly known thermal noise is essentially named Johnson noise the first research was done by Johnson on every noise which I talk is essentially Johnson's papers okay and these are almost 100 year old papers still stands they do research 100 years which we think are there but never bother maybe you are still time to do that it is more money dependent you need large equipments the third type of noise which is popular is called GR noise and the last and the foremost which is very very troublesome in actual analog design is called KT by C noise okay this is very very tough to handle okay so you can see from here KT by I will come back to it KT by C noise is generated from the resistor but there is no R here so that is the fun that I have a resistor which is creating noise but my noise is essentially independent of that resistor okay so there is some issue which will see in KT by C noise which is very very troublesome in analog design you can see from here before I quit if I increase C KT by C will go down assuming T is constant so that is the worry capacity is capacitor so see now that is the problem we will see this C C is in an output capacity okay now anytime you see you see you have a problem okay and we will see now so this evening we will start with the definitions of each of them and go up to device and to a circuit how to calculate noises