 So, today the 15th lecture we will be having design of feedback amplifiers in a systematic manner. In the last class we had discussed negative feedback systems wherein output variation follows the input variation only if the loop gain is very large in negative feedback therefore right output follows the input at the point of feedback voltage follower and current follower were discussed and their application as buffers and regulators were pointed out. Phase follower and frequency follower also on similar lines were discussed and the inversion property of the feedback in terms of FM detection and FSK application followed thereafter. Then we had seen that the lock range of feedback systems the range over which loop gain is kept greater than much greater than 1 and output follows the input at the feedback point and that range is the static characteristic of the feedback system how to evaluate that range in the case of amplifiers in the case of trans conductors and trans resistor in the loop in the case of frequency lock loop as lock range were discussed. Dynamic operation of the feedback system was then considered in terms of discussing the first order system and what is called as gain bandwidth product of the feedback system as a quality of measure of the feedback system dynamic response was also discussed. 3 equal to 1 in a second order system for high speed operation of the feedback system was the ideal value of design of all such systems whether it is FLL or an amplifier with the unity gain or gain of K that Q equal to 1 is the criteria for making it have minimum time taken for coming to steady state Q equal to 1 over root 2 is considered as the best value for steady state response to be maximally flat. The earlier lecture lecture 8 on nullator nullator concept okay in synthesis of ideal amplifiers facilitated the synthesis of topologies for voltage amplifiers current amplifiers trans conductance amplifiers and trans resistance amplifiers these were actually indirectly okay feedback amplifier because output is finite and that is how nullator nullator functions as a pair and input to the later goes to 0 exactly similar to the control system that is feedback system design wherein the input to the system goes to 0 the error goes to 0 and output follows the input at the feedback point okay so this is the same concept that is followed okay in synthesis of all these ideal amplifiers then the feedback as also being considered okay in network 2 port network theory as a parameter that comes okay of due to interaction of output variable with input. Now that is what is going to be considered here that feedback system in circuits will be discussed in terms of two port parameters feedback in systems are presented earlier used ideal amplifiers in a lateral amplifiers and even the feedback was in a lateral here the feedback network is a bilateral network in circuits that we are likely to use and that is what is going to be considered in today's lecture. So let us get on to what are ideal amplifiers it is just a revision of what has already been discussed earlier in two port theory so signals sources cannot deliver directly power to the load because that power that is generated by the signal source is normally very very low so a power amplification is resorted to by amplifiers amplification is required to enhance the signal power amplification is one of the major signal processing activity in analog and there are different types of amplifiers dealing with different power levels so the ones which are dealing directly with sensor input okay are the called pre amplifiers these are the pre amplifiers which deal with low power levels they can be tuned that means they can have a selective circuit at the input so as to reject the noise outside the band and select only the signal based on the signal bandwidth so they are called tuned amplifiers so the filtering is a major operation of improving the signal to noise ratio right at the input okay and power amplifiers deal with again they may be tuned okay so or wide band amplifiers and they are the efficiency of delivery of power matters so based on different criteria we have to make these designs however the basic amplifiers are four types based on input and output variables voltage control voltage source voltage control current source current control voltage source current control current source but commonly these are called voltage amplifiers trans conductance amplifiers trans resistance amplifiers and current amplifiers now what is the criteria for division of these two these four we have discussed this in the earlier lecture however let us repeat that okay this we have already discussed pre amplifiers deal with low power enhancing signal to noise ratio is the primary objective of this power amplifiers deal with signal levels which are pretty high okay and efficiency is the primary consideration ideal amplifiers are zero input power and deliver finite output power that can be done two ways one is VI can be zero which is primarily called VI equal to zero means this is primarily having current as the input variable so current control and II equal to zero means this is voltage control voltage is the input variable at the output it can be current source or voltage source so we have the four types of ideal amplifiers ideal current source or ideal voltage source so ideal current source means output impedance is infinity ideal voltage source means output impedance is zero now the types of amplifiers in terms of commercial availability for active devices op amps MOSFETs JFETs BJTs we have understood the basic functioning of all these devices these were introduced earlier and these are devices which mainly use MOSFETs and BJTs and JFETs okay so these are basically circuits which have already been designed and biased automatically when you use dual supply or single supply whereas these require special care to bias these devices and therefore these discrete circuits made out of these are now becoming very rare in major applications of analog circuits and ICs have dominated the scene types of amplifiers operational amplifiers itself are available as operational voltage amplifier, operational current amplifier, operational trans conductance amplifier, operational trans resistance amplifier these are all designed for a variety of applications and may develop these are the major manufacturers of op amps and most of the analog ICs Texas instruments natural semiconductor analog devices linear technologies maximum inter-cell etc non idealities of op amps the primary non idealities are finite DC gain finite bandwidth finite gain bandwidth product this is most important of all right. So in the dynamic operation of most of these op amps what matters is primarily the gain bandwidth product of the device that decides the quality of the op amp offset voltages and currents offset drifts more than offset voltages and currents the drift of these voltages and currents with temperature is causing a signal at very very low frequency and therefore when one is dealing with very low frequency signal these matter a lot becomes difficult to isolate this noise okay matters okay a lot in terms of interfering with signal finite input impedance and output impedance okay. So these are the secondary non idealities slew rate this is one of the most important non idealities in power sort of output of the integrated circuit okay that means what is the power bandwidth of a system that is decided primarily by slew rate that is the maximum rate at which output is capable of rising current and voltage limitations obviously voltage limitation comes about due to power supply used and current limitation comes about due to the device having a maximum power handling ability finite common node rejection ratio is an important aspect of design of a front end of most of these stages okay where differential input signal is facilitating rejection of common node noise and acceptance of differential mode signal so that is one way of improving to improving the signal to noise ratio straight away parameter dependence on temperature and supply voltage are the main causes of what is the variations interfering with design and therefore any design should therefore have the design independent of the active parameters that vary with temperature and supply voltage these are typically some of the popular op amps 741 where everybody knows about this op amp it is having a gain bandwidth product of about mega megahertz input impedance of about 1 mega ohm output impedance of few hundreds of ohms and slew rate of 1 volt per microsecond and TL 747 is a dual op amp TL 081 is unlike this 741 which is bipolar input this is a fat input stage and therefore it has a higher source flow rate it is about 13 volts per microsecond and the bandwidth of gain bandwidth product of 3 megahertz this is a single dual and power now like that there are a variety of op amp may available and most important I would like to mention that these are compensated to operate from unity gain to higher gains whereas this is uncompensated so when one is deciding to make it operate even for unity gain what happens is it has been made to have a queue of 1 at that point of time in its dynamic operation whereas for higher gain it is luggage whereas depending upon the gain for which you are designing the op amp you can use 748 which is exactly similar to 741 but for the fact that compensating capacitor is not put internally it can be compensated externally that means the design can work as Q equal to 1 for any higher gain that you are using it for so this is a better proposition than these compensated op amps in actual design so parameters of TL 081 you have whatever I have mentioned here you can note that the offset voltage is 20 millivolts this is a typically the kind of offset voltage with that input stages whereas bipolar input has typically 2 to 3 millivolts as input offset laser timber matching of resistors can improve the order of offset by one order of magnitude further but then they become costly processes. Feedback in 2 port active networks now consider this that a general 2 port network in Y parameters this Y I is the self admittance Y R is the component from coming from output voltage back to the input as current this is the feedback so feedback has already been touched in networks course for you in terms of 2 port parameters and this is the feed forward VI gets transferred to the output as VI into YF and this is the self admittance at the output and you are going to connect a load at this point okay there may be a source shunt admittance here. So effectively you can have these parameters getting modified by source and load admittance that you can have YS of the source and YL of the load modifying these parameters as YS plus YL and this then becomes IS where IS is the source current so that is in general how you represent the Y parameters and you can look at this that input VI gets transferred to output as current and appears as YF into VI and that flows through this load of Y naught plus YL so develops a voltage V naught which is minus YF VI by Y naught plus YL so I can see the forward voltage gain therefore becomes minus YF by Y naught plus YL then this output voltage comes back to the input as YR okay and develops an output voltage of YR minus therefore it becomes plus into YI plus YS so this particular thing can be considered in general as loop gain equal to GL term this always as loop gain this is got by taking the determinant of the matrix what is the determinant this into this minus this into this so YIA plus YS into YOA plus YL minus YFA YRA so this is something that you can take out then rest of it YIA plus YS into YOA plus YL into 1 minus GL that means determinant of this matrix is product of these two into 1 minus GL so let us remember that that the determinant of the matrix is product of this self total self admittances at the input and output and into 1 minus GL so let us now consider feedback in two port active network if the two port network is an active device which is assumed to be unilateral the reverse transfer parameter YRA has to go to 0 so we are using an amplifier with YRA equal to 0 then rest of the parameters YIA, YOA are finite and small compared to YFA YFA is the large factor okay transferring something from the input to the output input voltage is converted to output current. So all the parameters of active device are sensitive to temperature time bias voltage supply voltage and have poor manufacturing tolerances using a suitable passive network two port network with the active device it is possible to make the resultant system as an ideal amplifier let us see how we can mathematically achieve this right and then realize it physically. So we have the ideal amplifier okay shunted at the input and shunted at the output by a passive network because these are all short circuit parameters the moment I short this or this the feed passive network gets separated from the active network okay so the total system now has Y admittance which is summation of the Y admittance of the amplifier with the Y admittance of the feedback net that is how we can modify a feedback amplifier system okay the ideal amplifier coupled with the passive feedback network so YIP YRP YRP YOP are the feedback Y parameters of the passive network YFA is much greater than YRP and YRA is 0 these are the things which are known to us so now composite Y parameter of this network is going to have I the currents adding at the input port currents adding at the output port forming the resultant current II at the input and I not at the arc the voltage is common both at the input and the output okay for both the networks so Y parameters of amplifier network add with Y parameter of the feedback network this week now this feedback is now totally determined by this what is that passive network and the feed forward is totally determined by the active network this is close to very small value compared to this or that is how you have to select the feedback network so that this dominates then let us see what have admittances increase in the input port as well as the output port admittances increase means impedances decrease that means at the input it becomes current control by the modification and at the output it becomes a voltage source and that is what we want to have current control voltage source as the design that means it is possible to have this going towards 0 this going towards 0 this going towards 0 in the inverted Y matrix of the composite network and this is the one Z and Z has to become independent of the parameters of the active device let us see whether it happens when we invert the Y matrix of this composite network so this is the inverted matrix Y matrix here represented in this manner total admittance at the input is YIA YIP plus YS total admittance at the output is YOA plus YOP plus YL and this is dominated by YRP alone and this is dominated by YFA now inverted it results in this divided by determinant of the matrix minus of this divided by determinant so everything gets reduced by the factor decided by the determinant of the matrix so that means a modification of the element values of the matrix take place here now that determinant is equal to this into this minus this into this so if now as pointed out earlier this into this divided by this into this we have defined as loop gain please remember this this has been defined as loop gain GL so if GL is negative it is negative feedback if GL is positive it is positive feedback so this is the best way of identifying whether it is negative feedback that you are adopting or not so if GL is negative that is what is considered now it is negative feedback and magnitude of GL must be much greater than 1 for us to have effective feedback so under that situation the determinant of the matrix can be approximated as this alone because this by this is much greater than 1 so this is the resultant determinant and that modifies all these element values so let us write down the element values modified element value everywhere we will see that this is what happens that YFA gets cancelled that means the forward transfer parameter of the composite network is now totally decided by the passive network 1 by YRP it is 1 by G2 what we discussed in system design 1 by YRP inverse of the feedback factor of the passive network and all other 3 elements are reduced by YFA which is huge and therefore they go towards 0 and that is what we are proving that this now becomes a current control voltage source with a transfer parameter totally independent of the active device and the load okay and the source im- imitances and this is the network now okay making use of an op-amp this is RIA input impedance this is ROA output impedance of the op-amp this is the voltage gain typically ADC gain is 10 to power 5 to 10 power 6 RIA is 1 mega ohm ROA is 100 ohm let us consider a problem with RS as 10K RL as 1K load RF as 1K because let us say we pose a problem saying that I want to design a current control voltage source or trans impedance amplifier with the trans impedance matrix as 0 0 0 and – 1K so that means actually I have to use feedback resistance the passive network between input and output which is 1K okay so in each case let us see what happens 1 over 1K or 1 milli C means okay is the Y parameter of this I will indicate it here in terms of composite Z matrix of this current control voltage source 1 over RIA for the amplifier 1 over RF due to the feedback resistance 1 over RS due to the source all the self admittances appearing at the input when the output is shorted and this is 1 over ROA 1 over RF and 1 over RL at the output when the input is shorted and the amplifier network composite network feeds from VI to the output A times VI a current which is A times VI divided by ROA current coming into the port and therefore it is positive so that is the Y parameter of the amplifier that is used and due to the feedback network okay there is a feed forward of – 1 by RF when VI is applied VI by RF flows out so similarly when V naught is applied and input is shorted VI by RF flows again out so there is a negative this thing so effectively inversion of this results in the modified matrix Z parameter which is – RF as desired it is totally independent of A ROA RI and we have the input impedance which is RF by A going towards 0 into small factor here and again output impedance RF by A going towards 0 again multiplied by a small factor and reverse transmission is reduced considerably 1 over A ROA so this if A is made large all these 3 factors go towards 0 and the loop gain is GL equal to – A divided by 1 plus some small factor and a small factor here okay and therefore this has to be in magnitude much greater than 1 okay and it has a negative sign indicating it is negative feedback and the determinant of this matrix is A by ROA RF okay so mathematically it is understood very neatly as this a problem has been solved here so design a trans resistance amplifier with a Z matrix which is 0 0 0 and – 1 key this is the design problem for a source resistance of 10 K RS is 10 K and low resistance of 1 kilo ohm consider an op amp with input impedance of 1 mega ohm voltage gain of 10 to power 6 and output impedance of 100 ohms typically similar to 741 the op amp is represented by this Y matrix 1 over 1 mega ohm 1 microsecond this is 1 over RIA this is 1 over ROA which is 10 millisiemens 1 over 100 ohms okay 0 here no unilateral no feedback and this is nothing but A by ROA positive A is 10 power 6 ROA 1 over ROA that is 1 by 100 so it is 10 power 4 Siemens added to the feedback passive network which is 1 millisiemens 1 millisiemens – 1 millisiemens – 1 millisiemens that is the network that we have considered here RF between input and output port so one has if you put 1 K here you get 1 millisiemens 1 millisiemens – 1 millisiemens – 1 millisiemens as the Y parameters of this feedback network that is added to this along with the source and the load and then inverted the Z matrix now becomes 1.2 milli ohms 0.11 milli ohms 0.1 milli ohms see how close to 0 it is compared to the RF which is 1 K which is the forward transmission that has occurred so current control voltage source with input impedance of 1.2 milli ohms output impedance of 0.1 milli ohms and that voltage source converts the current by this factor 1 K so this is the transfer impedance so this is the complete design so this is the mathematical part of it you can see that the loop gain is very high compared to 1 actually tends to be indicated that this is minus okay this is getting multiplied by this divided by this okay so this is the final network RF is equal to 1 key RL is equal to 1 key this is 10 ohms 10 kilo ohms and IS IS into RF is the forward output voltage with a negative sign because this IS now flows through this totally this is a nullator remember and therefore the current flows through this this voltage at the output is simply IS into RF so using this concept of nullator we have already synthesized this structure and now the mechanism of how this gets altered because of ARI and RU has been demonstrated by taking the two port network and by using the appropriate parameters for this so how do we apply this technique in general to any of these four that means we have to take request to immittance matrix the ideal amplifier that is to be realized that should have the transfer parameter alone becoming independent of the active device okay that is PF here and these three parameters becoming zero PF is finite and is the chosen design parameter should be independent of parameters of the active device used so let us see how it can be in general dealt with so I am considering an amplifier appropriate amplifier with PIA finite P O A finite zero as the reverse transmission and PFA very large PFA is very large all the three elements of the immittance matrix have poor manufacturing tolerances can vary widely with temperature time and by supply voltage then I couple it with a passive network which is having its own parameter getting added PIP, PRP, PFP, POP the distinction being that PFP is small compared to PFA and these two are same in magnitude and may be opposite in sign depending in H and D and same in sign in Z and Y. Now it gets modified by the determinant of this matrix in its inverse so the input and output variables interchange okay because of inversion okay and this is the inverted matrix and we have here the inverted matrix written in terms of delta delta is the determinant of the matrix which is this my into this minus this into this. So it can be written as this into this into 1 minus GL that is what has been done in the next step so once you do that you get this inverted matrix as composite inverted matrix a 1 by PIA plus PIP into 1 minus GL 1 by P O A plus POP into 1 minus GL and minus PRP divided by this into this into 1 minus GL. So all these terms are affected by 1 minus GL and since loop gain is very much greater it is negative and much greater than 1 in magnitude what happens is that it is primarily much greater than 1 by PFA and therefore this PFA is a large quantity okay is going to be affected only by GL here okay and you therefore have only the reverse transmission of the passive network coming as the forward transmission of the feedback network and here all these things are going towards 0 so 1 minus GL is the factor by which all the things get modified suitably to reach the idealization corresponding idealization. So for available active device okay choose a suitable passive device and combine this together suitably to form the ideal amplifier design that you desire after inverting you get this modification. So in conclusion design of feedback amplifier involves the following activities right. So the voltage amplifier the topology of voltage amplifier let us consider what should happen actually you have an active device block how will you connect it you will connect so that it becomes voltage control that means actually it should be connected in series because impedances should increase right and it becomes a voltage source at the output this is a voltage control voltage source so it should be having output impedance decreasing that is possible only by shunting. So this is the topological arrangement for a voltage control voltage source that means this is easily going to be a modified element of the matrix by shorting the thing here okay and opening the thing here the parameters add they are coming in series here okay and instant at the output. So that is easily done if you use H parameter that is okay things in series add at the because impedance HI is impedance so HI adds okay it is a short circuit parameter and HO adds okay when it is shunted it is a bit else okay. So you have feedback okay where H parameters add so I call it H feedback so this feedback tells us that H parameters add and the inverted H is the ideal amplifier parameter that is G feed parameter. So this is called H feedback and the matrix that is aimed for is the inversion of H and that is the G matrix that is what becomes 0 0 0 and then the GF and modifying factor is due to the loop gain okay. So the loop gain is to be made much greater than 1 is the criteria for it should be negative. The current control current source is the current amplifier wherein we start with a topology where in order to make it current control it should be the dual of this that means it should be shunt at the input this is for voltage control voltage source current control current source it should be shunt at the input and series at the output. So amplifier and feedback so actually G parameters add here and current control means the impedance should decrease here and current source means impedance should increase here so it should be series here and shunt here. So you have G parameters adding so we call it G feedback to achieve current control current source ideal and its parameters are represented by HF ideal H parameter okay similarly if I want a trans conductance amplifier trans conductance means voltage control current source I start with voltage control means actually I should have things in series so that it becomes voltage control impedance is increased and current source it should be in series at the output. So series at the input and series at the output that is Z feedback I call it Z parameters add because it is coming in series at the input and series right it is open circuit parameter. So series at the input and series at the output and it is converted to Y parameter and ideal Y parameter will become this okay. So this is amplifier this is the passing network. So this is for voltage control current source the deal of that is current control current source where both at the input and output it should be instant so that the input impedance decreases output impedance decreases because it is a voltage source at the output because it is current control here right so we have amplifier feedback network so this current control voltage source. So these are the topologies which naturally emerge and this is the most appropriate name for this okay cause the parameters add it is easy to modify the parameters okay and add here G parameter add Z parameter adds Y parameter adds and you convert this Y to Z and you get 00 and Z as the ideal parameter okay. In the next lecture we shall be discussing the design of the we have already discussed the design of current controlled let us say I think this is current control voltage source so this has been already discussed as starting from Y parameters we have obtained the desired Z parameters so we have this one okay and this one and this one in that order voltage control voltage source that is voltage amplifier has to be discussed. So we will start with the edge parameters adding and inert it and get the G parameter composite G parameter and when you get the composite G parameter you will simultaneously get to know the output okay the input conductance output impedance okay as also the transfer parameter exactly how close it is to the ideal value that is decided by how high loop gain is okay compared to 1 so the error is 1 over loop gain that is the error in the transfer function that you are designing that means higher the loop gain better is the approximation to the idealized value. So this is the effect of negative feedback now the same thing can be discussed also in terms of device op amp being replaced by transistors for example if instead of the op amp that is being used as the active device transistor is used just take the MOSFET. So this is nothing but let us say consider you can treat it as a voltage controlled already because it is open circuit here let us say this is VI and you get GM times VI as the small signal change in current at the output here. So we have that flowing through its own output impedance which is let us say 1 over GDS okay is going to cause the gain to be minus GM by GDS that is the value of the gain that means minus A equals to this minus GM by GDS. So it is equivalent to an op amp okay that is having a gain of GM by GDS right. So you can incorporate in the feedback network appropriately so as to get a current amplifier or a voltage amplifier or anything of that sort. So one way to do it is just to sort of have let us say full current fed back to the input we have already discussed this or synthesize the structure in the MOSFET okay when the full output current is fed back to the input the current gain is 1 that is the diode connected MOSFET. Now the loop gain in this case is since full current is fed back to the input okay you can treat it as VI as input GM by GDS as the output voltage fed back in the direction so that it is minus A okay which is fed back to the input so the loop gain is A right in negative feedback so the actual gain of this current gain or voltage gain has to be okay 1 by 1 plus 1 over A. So that is the gain and A being equal to GM by GDS. So this is a very simple single MOSFET current amplifier with gain equal to 1 which has a loop gain of GM by GDS just like we do it in the case of op amps it can be done in the case of FETs and bipolar transistors okay. However this can be postponed to IC design okay at system level it is better to use near ideal devices as active devices and op amps are the best ones okay which will have least amount of parasitics coming into picture okay in causing deviation not only that in an op amp you can create a structure with high loop gain okay whereas in the case of a transistor it is limited to GM by GDS which is not going to be very high. So worst of the understanding of feedback and its effect can be well illustrated using op amps instead of the bipolar transistor or field effect transistor and the single transistor devices are very rarely used in system design okay and ready made ICs are made available to you okay which can work very well okay up to very high frequencies today.