 In the last class we were exposed to one ideal active device the ideal op amp we started with ideal amplifier and then went over to what an ideal op amp is. Then we covered the synthesis of ideal amplifiers using ideal op amp and we went over to another device which is a comparator. How the comparator is different from the op amp, ideal op amp has been made very clear to you. Today we will be covering the multiplier which is the darling of the communication signal processing okay and we will see a variety of communication applications with regard to this multiplier. Further we will be also covering the important active devices which we have been earlier sort of exposed to okay in the beginning itself. These things have taken a back ground role now and mainly used in ICs as basic building blocks in IC designs. These are the MOSFETs. We will also cover the BJT okay which is primarily used today in combination with MOSFET to improve the performance of integrity circuits. However most of the present day uses with reference to the MOSFET and also the power MOSFETs. So we will be covering these different topics in today's lecture. Let us start with multipliers. Multipliers provide multiplication of two input voltages or currents either two variables as voltages or currents can perform multiplication. And if you perform multiplication what are the signal processing activities you can do. One such activity is if one voltage is kept as control voltage the other voltage gets the input voltage then the multiplication makes it a voltage controlled amplifier. And we can use this for phase detection okay mixing modulating demodulating etc. Now the filtering and other activities are done by using normal RC components or LC components or the active filters. Signal generation in terms of facilitating frequency synthesis. If you have a signal that is generated how to obtain frequencies other than this generated frequency that is called frequency synthesis. So all these activities can be performed with the help of a multiplier. Let us see how this can be done. Or commercially available okay as voltage controlled amplifiers that is why multiplication need not really fight on us it is nothing but a voltage controlled amplifier which can be used as a multiplier. Current controlled amplifiers digitally controlled amplifiers even we will say let us see that DAC is acting as a multiplier where one variable is digital word the other is analog signal okay. So however the general analog multiplier can be given as V naught equal since there are two inputs that is now considered there are going to be two inputs VX and VY. So if output is a function of generally these two variables then whatever the device that does this function if it is non-linear then we can show that V naught is going to be a factor which is independent of VX and VY that is called offset if it is current output it is a current offset. So this is independent of the two inputs then one output due to only VX that is linearly related to only VX independent of VY. Another linearly related to VY independent of VX these are called feed through components okay. In a multiplier these do not have any place to appear the out wanted output is this primarily it is K naught into VX into VY that is important for us in multiplication of other terms in this which are for example functions of VX squared VY squared VX squared VY VY squared VX these are all called the non-linearities okay in a multiplier. The offset is the DC offset voltage that has to be made equal to zero and these feed through components also have to be made equal to zero after which K naught can be adjusted to a standard value in precision multipliers K naught is we will see is adjusted to something like one over 10 volts okay where VX and VY have limitation of plus minus 10 volts that is called a four quadrant multiplier okay. So this is the four quadrant multiplier I am talking about VX can take both polarities VY also can take both polarities output changes when anyone of these two okay chain polarities. So that is why it is covering the entire four quadrants of one of the variables VX where VY is the parameter that is varied okay. So let us say VY is kept constant then at 10 volts okay then output is equal to VX okay. So that is it gives unity gain okay slope okay. So that means by varying VY to different values you get amplifiers with different gains okay linear amplifiers with different gains you can even make it inverting type of amplifier by just changing the polarity of this DC voltage. So it becomes inverting or non-inverting amplifier and the slope can be varied by changing this voltage that is why it is voltage controlled amplifier basically okay. Now what are the methods in hardware you can actually fabricate this multiplier basically I would consider these two are the most important methods okay to obtain electronic multipliers V naught is going to be equal to mathematically we know that VX plus VY whole squared minus VX minus VY whole squared divided by VR that means you add the two inputs square it subtract the two inputs and square it and then subtract these two terms then you get this output as four VX VY that divided by VR because output is a voltage the dimensional of the constant K should be 1 over voltage. So we get here 4 VX by 4 VX VY by VR as the output so this is one way to obtain a multiplier another way is using squaring devices this is using squaring devices and adders this one is V naught is anti log of summation of log VX plus log VY minus log VZ this is log anti log multiplier it is called anti log of log of now VX into VY by VZ which is nothing but VX VY by VZ. Now this results in okay multiplier come divider so this is really very interesting here we have a multiplier come divider. So this is the log anti log device this uses log anti log device and adders to obtain multiplication or division. So these are the two principles which are even now used for obtaining multipliers MPY 634 okay is a commercial multiplier made by Texas instruments okay it is available as log anti log multiplier the same principle that we had earlier given so this is the chip that is used it is a differential input that is VX can be a differential input VY can be a differential input okay and then we have a VZ terminal along with an op amp okay in feedback it can give you what you consider as VX VY by VZ okay VZ is in this case V reference and that is equal to 10 volts okay so that is something that we can adjust in this. So it is characteristic bandwidth this is an important aspect of this multiplier so only about 6 megahertz however most of the communication activities take place at frequencies much beyond this and however we can understand the signal processing activity by using this chip very well okay slow rate of this is about 20 volts per microsecond output offset voltage is about plus minus 100 millivolts output short circuit current is about 30 millilitres these are the limitations of this multiplier. Let us see the application of this multiplier VX is VP1 sin omega 1T and VY is VP2 sin omega 32 we have here V naught okay equal to VX VY by VR VR is 10 volts in this case so we have VP1 VP2 by 10 into sin omega 1T sin omega 50 you know this trigonometric relationship can be therefore expanded as VP1 VP2 by 20 into cos of omega 1 minus omega 2 into T the difference component of frequency and minus cos of omega 1 plus omega 2 into T these are called as side bands okay actually therefore if you consider omega 1 as the carrier omega 2 as the modulating frequency around the so called carrier you generate omega C minus omega M omega C plus omega M that is two side bands. So this is what this is called as in communication double side band okay modulation is called balanced modulation that means this is a balanced modulator circuit under this role when omega 1 is the carrier and omega 2 is the modulating frequency. So this is one way of modulation which is adopted in communication let us see what happens when we simulate this with this is the this is called a mixer also it can be either called as a modulator balanced modulator on mixer okay. So F1 is 10 kilo hertz F2 is 1 kilo hertz omega 1 is okay 2 pi into 10 omega 2 is 2 pi into 1 radiance per second VP1 is 2 volts VP2 is 1 volt this taken as an example and this is the so called carrier frequency which we have chosen 10 kilo hertz and this is the modulating frequency 1 kilo hertz maybe audio frequency. So we have now the output which produces omega 1 minus omega 2 omega 1 plus omega 2 which is the double side band waveform you can see this waveform here. So the signal comes like this here and like that okay this is the signal modulating it. So this is the output of the mixer now I just multiply by using along with F2 I add a DC of 1 volt and then multiply F1 with DC plus F2 okay. So then what happens the carrier comes as a feed through component because when you multiply the VX with VY having an AC and a DC in added to it then we have the carrier coming through as feed through component that particular thing is called amplitude modulation okay. So this is what you get when have shifted the modulating frequency by 1 volt so you can see the DC of 1 volt added to it and then multiply it by the carrier this carrier comes as such and then it amplitude is modulated okay by the modulating frequency on either side you can see and this method of transmission is adopted in AM transmission in order to facilitate easy this detection here you can simply because you have transmitted the carrier also okay with large amount of power you can receive this and then detect the modulating signal by simply using a rectifier that is the easy detection method okay that is why this particular multiplier application is the one where you are having a feed through component at the output. So we have discussed amplitude modulation okay and double side band in communication the other one application is linear delay detection okay. Let us suppose that one input is a square wave another input is a square wave with a certain amount of delay let us say thaw okay that is applied to the other but both with same frequencies then when you multiply you can see this is plus 10 volts and this is minus 10 volts so plus 10 volts into minus 10 volts divided by 10 volts is minus 10 volts. So when both are plus it is again plus 10 volts again when it is changing this one is changing to minus 10 and okay this is at plus 10 it is minus right. So within one half period you have one full cycle of this output okay and then again this whole thing gets repeated and that way we have produced once again okay the double the frequency because it is omega here and omega here double the frequency with a DC what is the DC average you just take this area DC is corresponding to the area okay total area over this time period. So this is minus 10 into thaw okay minus 10 into thaw this is that area plus 10 into T minus thaw divided by the half the time period T by 2. So this is the average voltage of this output okay. So you can therefore see that this becomes DC which is proportional to the average which is proportional to thaw by T. So that is how it can detect the delay okay in terms of a DC. Now I am also going to show you this simulated you can see that average is what we have just now computed is 10 into that is the DC voltage 1 minus 4 tau by T. So for example if the phase shift is or the delay is by T by 4 okay this becomes 0. So that is what I have done I have applied VY here and this is VX which is delayed by T by 4 okay and then the output here is just this that we had already shown in the earlier diagram and if you take the average of this that is 0 okay because positive area is same as negative area and it has produced twice the frequency. So this is this delay detector is the one that is used in phase lock loops okay and in any place where you want a phase detection or delay detection. Now that we are going to repeat it for sign waves you can see I am applying two sign waves instead of two square waves of the same frequency. The frequency is 10 kilo hertz and the phase shift is 90 degrees. So one is a sign wave the other is a cosine wave okay. So VP1 is equal to VP2 is equal to 10 volts. So then we get 10 into 10 divided by 20 that is 5 volts okay and double the frequency with the cos 5 because now the omega 1 minus omega 2 because the two frequencies are the same with the phase shift okay the first term is cos 5 the next term is 2 omega t cos 2 omega t plus 5. So this is the cos 2 omega t plus 5 okay 5 being pi by 2 it again becomes a sign wave okay and the average is 0. We will do this for a phase shift this phase shift of 90 degree is called quadrature okay. So if you have a sign and a cosine applied to the multiplier you get double the frequency as the output with 0 DC. So now it is the same frequency the phase shift is adjusted to be 60 degrees okay so cos 60 is half. So we have VP1 equal to VP2 equal to 10 volts. So it will be 10 into 10 divided by 20 cos 5 which is half okay which is therefore going to result in voltage of 2.5 volts as the DC average okay. So this is the twice the frequency component emerging with a DC shift okay of 2.5 volts. So these are the applications of this particularly here I can show that this is also used in electrical engineering as an analog wattmeter for example if VX is the proportional to the line voltage using the voltage transformer and VY is proportional to the line current using the current transformer you can actually get the output average as nothing but proportional to the real power okay which is going to be the voltages multiplied and cos 5 okay. So I can also measure the power factor okay and that is why it is almost equivalent to a wattmeter okay. Now the fourth example that we are taking for the application of the multiplier is nothing but sine wave generation from a triangular wave. So the triangular wave okay can be generated in terms of for example sine wave sine X is nothing but a polynomial in X okay Taylor series expansion of H is X minus X cubed by factorial 3 plus X to power 5 by factorial 5 so on and so forth. That means any periodic wave form can be generated from the what is that triangular wave form okay using this polynomial series for expansion. So X is nothing but the original triangular input then you have to cube this and divided by factorial 3 that is approximated to this sine wave. If X is a triangular wave form it is possible to create a sine wave by using X minus X cubed by 6. So this is what is done you can see that this is the input triangle this is cube by using first squaring by feeding the same input to the of the strangle to one multiplier you get squaring after squaring the squared output is again multiplied with the original triangle and you get the cube and the cube is then subtracted okay from the original triangle as X which is omega t so we just have this it is omega t minus omega t cubed divided by 6. So that is what is done okay we will therefore see that okay this is what is done omega t minus omega t cubed by 6. If you want to get more accurate this thing you can actually remove the higher order harmonics by using a low pass filter. Now we come to this important point that is the transistors. Let us get introduced to this in a very continuous manner from what we discussed earlier input output relationship of an ideal device amplifier okay active device is output matrix related to the input matrix by means of this in a two port okay that we had already seen imitance matrix PF is the important parameter from input to output the transfer parameter forward transfer parameter. So but most of these devices have limited dynamic range that means they get saturated in the value that means this value okay really is not constant at all operating points okay it keeps on decreasing on either side of the maximum. So generally all these systems will be having PF tending towards lower and lower values on either side of the maximum value okay that is the non-linearity. So ultimately it may go to 0 in what is called the saturation region okay it is really a small variation of the output caused by a small variation in the input it is not perfectly linear. So this non-linearity we will consider only the first order non-linearity okay itself can be such that PF is a function of one of the two variables input or output. So we are having only two variables here one is the dependent variable another is the independent variable. So PF is nothing but the incremental change in output caused by an incremental change in the input. So this itself is not constant when it is non-linear so this can be a function of the input variable or linear function of the input variable or a linear function of the output variable that is the kind of non-linearity that we are first considering if that is so what happens to this nature of the non-linearity. This is a very interesting thing let us therefore consider such example wherein in electrical engineering we have only these variables as current and voltages we are therefore considering ideal trans conductance amplifier that we have already discussed the voltage control current source. So the output variables are the currents okay and the input variables are the voltages and this is the trans conductance GM we will call it as it is normally called input to output the transfer of input voltage to output current is occurring by the trans conductance here it is normally termed as GM okay this GM is a relationship between output change in current for an input change in voltage it can be therefore a function of VI and I naught these are the only two variables of significance here. So this is equal to either K times VI when it is proportional directly to the input variable VI or the output variable as linear K times I naught interestingly the first relationship gives us this is delta I naught equal to K VI into delta VI we integrate this we get I naught as K into VI minus some constant VT whole square. So this VT according to us you can easily recognize this as square law relationship of the famous field effect transistor. So VT is known as the threshold voltage so we have not bothered too much about any particular device we have started with what is possible as a first approximation to the non-linearity either it is directly proportional to the input voltage or directly proportional to the output current and we have got a device which should have this non-linearity and that is there with what we call as field effect transistor. Next let us see interestingly the other relationship leads us to nothing but the other device that is important we will see delta I naught in this case divided by I naught because it is proportional K times I naught GM okay so that I naught is brought to this side this is equal to K into delta VI. So we get here log of I naught is equal to K into VI after integrating and I naught is equal to IS into exponent K VI minus 1. So if you include the constant also this ISFO is known as reverse saturation current in the case of a bipolar junction device. So this results in the bipolar junction transistor relationship so you can see how beautifully these 2 relationships where the trans conductance is proportional to the input voltage or trans conductance is proportional to output current result in respectively field effect transistor with square law and bipolar transistor with exponential law. So these are the and even in field effect transistor this if you are operating in what is called above threshold region it is going to follow the square law and below threshold region it jumps into the other device characteristic that is exponential that is the beauty of these bipolar and MOS devices they are forming a complete set by themselves most probably if you therefore are looking for completion of a set you would have naturally got these relationships first and actually invented the devices later okay. So semiconductor devices that exhibit these relationships are respectively FETs and BJTs FETs exhibit a square law relationship in the region above threshold voltage and exponential relationship between input and output in this up threshold region BJT exhibits exponential relationship okay. Now something of history see basically the field effect device was predicted okay much earlier to the bipolar junction device okay that is important only thing is it was very difficult to fabricate this field effect device okay and it took lot of time because it is a surface phenomena okay and it took lot of time for people to have technology that facilitated fabrication of MOS FET okay JFET however was got the moment one could really fabricate a BJT right. So one could fabricate a junction okay and you see that this is history it was shockly with the help of Badin and Retain who really got us the first theory and then actually fabricated the device when they were really trying to look for the device action in terms of field effect they stumbled upon this okay effect first and that is why BJT denominated the scene thereafter okay until MOS FET became a reality once IC fabrication set its foot okay. So now what are the situations regarding present status of these devices it is seen that bipolar junction device was very popular okay before MOS VLSI tried to replace most of the front end components with bipolar devices and VLSI circuits. So the bipolar has a disadvantage primarily it is a minority carrier device and the leakage current is important as we saw in the expression itself it figures in predominantly reverse saturation current which doubles for every 10 degree rising temperature and therefore it is highly sensitive to temperature that means actually power device particularly as it gets heated the device current keeps on increasing drastically and then it might go into what is called thermal runaway. So it might destroy itself unless we take care with heat sinks and other things whereas in the MOS FET is a very interesting things if you put a BJT in a curve tracer and put a soldering iron on the top of it right you will see the characteristic simply rises up okay on the other hand you replace it with MOS FET and put the same soldering iron to it the entire characteristic comes down that as temperature increases the currents decrease therefore the same conditions okay that means actually there is a safety mechanism in MOS FET and it is the MOS FET devices which have also become powerful units in power electronics today. So BJTs have almost vanished okay in as even discrete devices altogether okay however people are trying out because of the great advantage of large value of GM for the same geometry an order of magnitude higher GM is possible with BJTs compared with MOS FETs it is nearer to the ideal than the corresponding MOS FET and therefore people are trying out okay BJT in combination with CMOS it is called by CMOS technology for some of the analog functions. Now this is the some and substance of electronics today and therefore particularly today the devices using discrete small signal low power bipolar devices have vanished altogether in industrial usage. So because of the fact that MOS as a device has this weakness that the gate oxide which is coming right at the input between the gate and the source is very sensitive to the voltage right. So because of this that very high input impedance possible with MOS it cannot be directly used for that high input impedance because one might destroy the gate oxide by just touching it. So even though inside integrated circuit is well protected as a device by itself it is not a rugged device unless you put protection circuit between the gate and the source which again will bring down the high impedance it has okay. So these are for IC applications right for them to be rugged the input device it better be a bipolar device okay at the output you can use MOS devices and therefore that is why by CMOS technology sometimes is adopted for small scale integrated circuits. Now let us consider the field effect transistor which is a 4 terminal device source drain gate and substrate it has. So the gate substrate voltage controls the current between the source and the drain the source and the drain are separated by the channel this channel can connect source and drain if there is a channel existing earlier or it can be created by applying suitable voltage to the gate. This kind of thing where the channel already exist is called depletion type of MOSFET where the channel is created by applying a voltage it is called enhancement type of MOSFET. Then again we can have N channel and P channel okay type of MOSFETs so we have variety of MOSFETs available to us apart from that MOSFET we have the JFET okay wherein the gate and the channel are separated by a reverse bias junction okay and that is also can be N type or P type. The MOSFET there is an insulating oxide layer and therefore the impedance at the input can be pre high. So now we come to the preferred devices in VLSI enhancement mode of FETs are predominantly used why because source and drain are not connected initially they get connected only if a suitable voltage above threshold voltage is applied to the gate with respect to the substrate and therefore this is normally off device that if you do not apply any voltage it is off right if you apply 0 voltage it is off right. So this normally off devices are suitable for digital design and therefore it is going to be made on only if an applied voltage is crossing the threshold voltage. So because in today's world 99% of signal processing is carried out digitally and the digital technology is what governs what you use so enhancement type of MOSFETs have been universally used in VLSI. On the other hand if there is a choice for analog processing depletion mode of FET is suitable mainly because with 0 voltage as bias voltage you can directly apply a signal and it is working in the active region that means gate voltage can be made to go positive or negative it still works as an active device. Because the enhancement mode of technology is used mainly by digital even analog circuitry has to go along with that okay and therefore in today's activity of ICs it is primarily enhancement type of MOSFET which is CMOS that is primarily used. So let us now consider the FET characteristic IDS that is drain to source voltage okay the drain to source current IDS is equal to K by 2 we use K earlier but that will put as K by 2 into VGS – VT whole square this is the saturation current in the what is called active region that is called the current saturation region as long as VDS is greater than VGS – VT this is valid for the N channel enhancement device. For VDS less than let me just see there is a mistake there VGS – VT right it is IDS this is for IDS is equal to K into VGS – VT into VDS – VDS square by 2. So VDS less than VGS – VT IDS is equal to okay this is the LF IDS that is called the triode region that is before it reaches current saturation this is the region where it is normally used as a resistor voltage control resistor this is the region where it is used as an active device as an amplifier and this is the physical appearance of a integrated circuit MOSFET where we have N as source N plus source and drain and by applying voltage here okay which is positive you can make this surface go into inverted mode compared to the substrate which is P. So that links the source with drain with a channel of these minority carriers electrons come towards this and make this young. So it can be made young plus by applying higher voltage. So that is why it is called enhancement mode of MOSFET where this conductance here is enhanced by applying a voltage here. So this is the micro model of MOSFET for large signals you have a VGS applied okay and the current IDS is equal to K by 2 VGS – VT square that is the last signal model that is all it is a transconducting amplifier voltage is converted to current in a non-linear fashion. The micro model of higher level this is the micro model for small signal see where delta VGS is the change in input voltage okay and it is called GM into delta VGS what is this value of GM you can simply consider it as a change in output current for a change in input voltage delta IDS by delta VGS that is equal to if you differentiate that relationship square law you get it as it was earlier K by 2 VGS – VT whole square is equal to IDS so delta IDS by delta VGS is K because this by 2 gets cancelled with 2 of this coming here into VGS – VT that means directly proportional to the input voltage VGS this is what we started with right. So the GM is directly proportional to the gate to source voltage through this relationship or it is also equal to root of you can replace this VGS – VT by so we have IDS we have therefore IDS IDS equal to K by 2 VGS – VT whole square so you differentiate this you get this 2 here it becomes K into VGS – VT so VGS – VT itself is root of 2 IDS by K if you substitute it here okay you get it as square root of 2 IDS into K in the region where VDS is much greater than VGS – VT this we had already mentioned earlier okay now higher level model for this it is just that the current is not constant at the saturation value throughout it is IDS into 1 plus lambda times VDS again this lambda is known as channel length modulation it increases slightly that it does not remain constant at saturation value with VDS variation it is dependent upon VDS the second order error okay it keeps on increasing instead of remaining constant at IDS it keeps on increasing so this is an important fact that you should remember instead of remaining constant at with respect to VDS it increases that is defect of channel length modulation. So GDS which is the variation in IDS with respect to VDS okay IDS variation with respect to VDS at the output it is called output conductance of the device okay it is not 0 it is delta IDS by delta VDS which is lambda times directly proportional to IDS is roughly okay so this is the second order effect that there is a finite not non infinite output resistance the high frequency equivalent circuit we have these between every electrode okay two electrodes you have a capacitor so between gate and source between these are all overlapping capacitors they are called gate and drain and then drain and substrate okay so we have these capacitors in addition to the earlier circuit okay bipolar junction transistor is a three terminal device emitter base and collector now you see the number of process steps increase here that is why it is not a preferred device okay in integrated VLSI schemes mainly because the more the number of steps less will be the yield in any technology right. So the MOS device request the least number of steps compared to the bipolar device that is why bipolar device has been totally rejected okay as a device for VLSI structures. So BJT is a three terminal device emitter base and collector okay this is the collector base and emitter this should be normally N plus region so made very rich okay in electrons here so that the total current is IE is made up of only these electrons in N plus region so that is the transistor action here simply means by suitably forward biasing emitter base junction okay and reverse biasing collector base junction the entire current IE is made to appear as collector current that means IC is very nearly equal to IE okay and that is a factor called alpha alpha is very nearly equal to 1 in a good bipolar junction transistor. So then since it is a forward biased junction between emitter and base IE is equal to IE naught reverse saturation current exponent BB in divided by VT this we had already seen for a diode okay the emitter current is very nearly transported to the collector current if it is suitably forward biased between base and emitter and reverse biased between collector and base the emitter current is very nearly transported to the collector current this results in IC is equal to alpha by 1 minus alpha times IB this is called beta factor and beta is going to be very high typically of the order of 200 or so in the case of bipolar junction transistors times base current if you express this in terms of base current as the input then VB is equal to VT log IE by IE naught this is the input relationship from which you can find out delta VBE by delta IE and that will be the input resistance okay common emitter configuration of the transistor IC is equal to alpha IE naught exponent VBE by VT GM which is delta IC by delta VBE therefore is equal to alpha IE naught exponent VBE by VT okay divided by VT if you differentiate this you get this so that is equal to alpha IE okay divided by VT so this is the GM VT is typically KT over Q okay about 26 millivolts T equal to 300 degree room temperature is what you should always remember and this is the equivalent circuit of the BJT in common emitter you call as the impedance 1 over GM appears as beta plus 1 okay that is the difference between the emitter current and the base current okay beta plus 1 into 1 over GM and this is what is called RC this is the second order effect okay this is coming as an impedance across the current source so again it is a transconducting device so this is what I have already mentioned to you right and relationship between alpha and beta is this typically for bipolar junction transistor working at 1 milli ampere current okay this will come out as GM will come out as please remember this 40 milli Siemens at 1 milli ampere at room temperature. Now we come to model of 3 terminal transconducting devices if GM tends to infinity what happens it again becomes an operational amplifier type of device therefore you can just see that this is a 3 terminal device now where the transfer parameter goes to infinity if originally all the other parameters are 0 and only GM exists if GM is made to go to infinity for finite output again the other parameter that you have assumed to be finite also goes to 0 it becomes a nullator at the input port and no rate at the output port so we are now developing the nullator nullator model for the bipolar transistor as well as MOS both MOS and bipolar transistor have the same equivalent as nullator norator in the following fashion so when GM goes to infinity between the base and the emitter that is the input voltage okay that goes to 0 and base current goes to 0 okay anyway gate current was originally itself 0 so we have this becoming a nullator both for MOS as well as bipolar base and emitter region is replaced by nullator gate and source is replaced by nullator and since this source current now cannot flow here it will flow through the drain or emitter current will flow through the collector so no rate comes in series at the output nullator comes in series at the input that is the model if emitter is common between input and output or source is common between input and output so both these transistor lead to the same model nullator norator model in three terminal okay now we have synthesized several such structures using nullator norator voltage amplifier with gain 1 this is the earlier synthesized model and that can be replaced by a transistor like this the junction being emitter or source this being collector or drain end of norator this being the end of nullator base or gate if you replace this you get a common collect this particular structure okay there is a mistake again this is emitter this is the collector so same thing this is source follower this is emitter follower okay so next one is current gain with gain equal to minus 1 so nullator comes in shunt norator comes in series here so input current is same as output current but flowing in the opposite direction to I0 okay so gain is minus 1 so that is again replaced by bipolar transistor here and the MOS transistor there resulting in this is nothing but common base or common gate which is known to have gain equal to minus 1 then this is the trans conductor amplifier that we have synthesized again replacing the transistor with either MOSFET or bipolar result in this kind of trans conductor amplifier design and you can see these realizations okay or nothing but this is called emitter degeneration topology this is this is the source degeneration topology this is the emitter degeneration topology this is the last one trans resistance amplifier here so we have this RF coming in shunt here nullator in shunt norator in shunt resulting in a current getting transferred to the output as voltage II into RF with a negative sign this is the topology in summary today we have discussed the multiplier application is primarily a communication application and then introduced to you two devices which complement one another in terms of non-linear relationship okay square law and exponential law devices how they can be used for designing near ideal amplifiers also has been indicated.