 Today we are going to have the last lecture of this series it is the concluding lecture summing up analog system design. Let us see I would consider that every designer should know particularly analog designers are recommended that they should understand macro modeling very well. It refers to modeling a given system to the level of detail of concern to the application on head. So illustrating this for example let us consider an active device it did not be an op amp at all times but it could be transistor also. So in the first understanding of the op amp or transistor for example one has to understand about the most important thing in op amp is the feedback. So the op amp gain is of concern this gain has to be made okay very high in order that op amp can be usefully used. So input VI becomes equal to the output V naught because in this feedback system the loop gain is A okay. So V naught by VI is 1 by 1 plus 1 over loop gain so that 1 over loop gain says that it should be 1 over A. So this basic principle of voltage follower is fundamental to the aspect of most of our feedback systems. So the output is fully fed back to the input then output follows the input. So output follows the input is the basic principle output follows the input. So what is it that we have done we are now able to isolate the input structure it may be some source or transducer and this may be connected to the load. So any power can be delivered okay to the load using source which is not capable of driving that much power into the load. So this is a power amplifier. So the model used here is a simple voltage controlled voltage source okay or current controlled current source where it is a current follower with a certain gain that is just DC gain. So this makes us understand all about follower systems which are the characteristic systems that are used throughout the analog system design. It could be if it is current II and current II and output current okay is fed back and this is the current amplifier okay this is let us say current amplifier the output current. So again if this is A times I in output current by input current becomes equal to 1 independent of A. So the desensitization aspect of negative feedback okay linearization aspect of negative feedback all these things can be explained. The actually it is also next step is to improve the model so that you understand other concepts what is it even if this relationship between the input voltage VI-V0 and the output voltage in this active device is not linear that means A is not really a linear relationship between output and input voltage of this voltage control voltage source even then this follower action is produced that means we can next bring about a non-linearity okay in the device that is used as voltage control voltage source and show that in spite of that non-linearity this relationship is valid for incremental changes around any operating point and prove that over the entire range where the loop gain is very high this relationship of output following the input is valid. So that is my macro models are important okay and there is no point in including in the first instance itself the exact model for the device here and causing confusion to exist in terms of analysis etc. So the simplest model is the one that has to be used first in the system to good enough to make the basic principle get understood what is the basic principle? The basic principle is as long as the loop gain is high okay output is very nearly equal to the input. So this is making a system become linear desensitize it with most of the characteristics of the active device used inside okay. So the macro model therefore can be linear or non-linear it could be frequency dependent 0th order, first order, second order frequency dependence. So higher order frequency dependence can be progressively introduced to see the effect of higher order models on the performance of the system. In most op-amp applications which means in negative feedback applications of the active device is sufficient to use a model of VCVS with this kind of voltage gain. Now another thing is for example if an op-amp uses let us say 10 to 15 transistors and some resistors etc. Each device if it is modeled by its most complicated model including non-linearity frequency dependence etc. Then the number of nodes in the whole system keep increasing enormously particularly in VLSI systems this is going to be the case that micro model level of modeling is to be just used only at maybe it may not be necessary at all but if it is to be used it is necessary to be used only at the end after you understand okay each of the parameter of the active device influencing the system by bringing it one after the other. So for example for understanding the basic principles of negative feedback initially it is enough if you use a linear model okay and then understand and then see the limitations of the linear model and then improve the model by including the effect of let us say gain banded product for example instead of bringing in the effect of frequency in terms of large number of poles associated with the complex device like op-amp one could just bring in a first order okay frequency dependence and see the effect of the gain banded product on the performance of the feedback stage. So what happens in that case again in the case of op-amp one sees that the op-amp is approximated as an integrator with a gain of GB bias instead of A equal to A naught DC gain and then one sees the effect of this on the op-amp with feedback. So in the full feedback case it becomes now 1 by 1 plus 1 over loop gain which is going to give the transfer function between V naught and VI as 1 by 1 plus S by GB. Now that is a measure of the bandwidth up to which this system gets used. So the loop gain has direct influence on the bandwidth of the feedback structure that is what we are concluding from that. So it becomes a measure of the range over which the device is useful frequency range over which the device is useful. Now apart from this the saturation states where the op-amp no longer can be considered as an active device because the gain is 0 that is for a change in input voltage output change is 0 that is what the saturation state is. So that model has to be brought in to understand the effect of saturation states step by step that will limit the region over which this V naught following VI is valid or it will give the dynamic range over which this is functioning okay so called lock range right. So this is the case with most of the feedback systems that we are going to consider right it is going to work okay with all these equations getting valid only if within the lock range okay within the range of the dynamics possible for the circuit. Now apart from that non-linearity we have another functional limitation of the active device so that is nothing but the maximum rate at which output can change because these are using capacitors etc may be at the output or in feedback okay the rate at which the voltage can rise across the capacitor is limited by the maximum bias current okay that can go through the capacitor so that is limited. Therefore the op-amp can only work up to the maximum rate which is delta V naught by delta T max this low rate the consequence of this is that an output signal which is let us say should be sinusoidal okay for a sinusoidal input okay the output let us say is VPO sin omega T okay then the delta V naught by delta T expected is going to be omega VP naught cos omega T. So maximum rate at which output has to rise in order to be free from the distortion is omega into VP naught. So omega into VP naught in all these circuits must be less than or at most equal to the slew rate that means omega up to which this system can be used okay with the peak sin of VP naught is slew rate divided by VP naught or F max is always equal to this has to be verified before proceeding with any experiment okay. So this is the precaution that one should be that this if this is not incorporated in the model okay particularly at high frequencies maybe much less than the gain by the product of the op-amp itself this limitation will come into picture and people may just not notice this distortion happening and measurements will go become inaccurate one such a classic example is one if one is making a measurement of the gain bandwidth product itself. So the signal output should not swing okay at any frequency okay to an amplitude greater than what is limited by this. So that means at any frequency you are likely to calculate this VPO should be less than or equal to slew rate divided by 2 pi F okay corresponding to that. So this is an important limitation which normally people tend to forget it is enough even if you are dealing with higher order systems is most of the time enough okay will come to the restriction to second order in overall system design. All analog signal processing systems can be designed using active second order analog slash digital subsystems as building blocks okay. Now digital subsystems may or may not come into picture okay we will discuss digital by that we mean there may be certain amount of switching etc taking place okay in an analog system itself because like switched capacitor filters okay pulse width modulators etc. So second order analog subsystem is realized using op-amps as active devices and resistors and two capacitors as independent capacitors that means most of these systems whether they are analog systems or complex mixed mode systems or even electromechanical systems all systems today can be modeled as a sort of cascading of second order analog subsystems which use may be op-amps and as active devices and resistors and capacitors okay along with the op-amps. That means basically this analog computer modeling which was prevalent before the digital computer committee existence particularly in simulation of anonautics okay and mechanical systems okay as made a comeback in the form of its availability in terms of op-amps being made available okay and the facility of state space filter in simulating a simple second order system okay. So the analog computer simulation which has been used for long for even non-linear differential equation solution etc has made a comeback in the form of the availability of op-amps and systems like state space filters being made available. A digital subsystem is made up of ADC DSP and DAC then that means one can incorporate such a digital system okay in order to control the analog front and back end like filters okay and analog multiplexers can be controlled by DSP okay and the backend power amplifiers pulse width modulators switch mode power supplies all these things can be controlled okay by using DSP and such subsystems of course take a lot of time for the control so such digital control of analog system is what is prevalent today okay because of the ready availability of DSPs at low cost which are conveniently usable for controlling these analog subsystems. And the other advantage is the controllers integral controllers etc that request large valued capacitors okay for causing dominant poles to exist in the loop okay can be simulated using digital processors and delays cost in the processing okay can be cumulatively added to form large delays and lags. So these are all parts of modeling that we can take into account in understanding system design any analog system is functionally a filter. So particularly these linear systems can always be modeled as a filter that means an amplifier a wide band amplifier by itself is going to do filtering because of the fundamental limitation of what is a gain variation with respect to frequency right gain always falls at higher frequencies. So a second order system can be represented by a transfer function with a second order polynomial in S as denominator and the numerator that can be constant S term S square term and or the combination of all these terms in the numerator. So basically therefore we have most often this can complete all systems are basically looking like low pass filters. Some systems can be high pass filters of second order right and some will be okay band pass filters of second order right. So this refers to let us wide band amplifiers okay and also narrow band amplifiers okay. So and all these wide band amplifiers and narrow band amplifiers or draw band selective amplifiers all these things can be divide designed by cascading okay successive staggered peaks okay if it is low pass for example that we had already seen a maximally flat characteristics for this low pass can be got by cascading a first order low pass with a second order or higher order okay. So this is the way you can build higher order low pass systems. Again a band pass systems can also be built in a similar fashion you can build them using higher order and make them maximally flat okay by cascading together these staggered tuned amplifiers. So so on and so forth. So that is basic understanding in all these cases is that right it should be having a basic building block which is clearly a second order with a first order maybe. So these are other things that same thing is applicable for a band stop right you might have okay a thing like this notch or if you want broader notch right you may be having it something like this. So so on and so forth right higher order. Now it has also been seen in the present day technology that the analog parameters like resistance value or capacitance value okay. These are fundamentally sort of components whose value varies considerably as poor tolerance depends upon temperature etc. Right only ratios can be perhaps used fairly accurately okay to an accuracy which is better than that of the absolute values. So one must try to design systems which depend only on ratios of components rather than absolute values. And that is the attempt in present day design as far as analog systems are concerned otherwise further accuracy can only be obtained by digital signal processing okay and the analog system today is primarily used for high speed signal processing activities like in communication okay. If one has to use precision at that point of time precision is mostly locking and lacking in these particularly at high frequencies because parasitics become dominant at that frequency and maybe one has to use parasitic components themselves as components for signal processing application in which case one might have to tune these systems to the accuracy that we need by proper sort of feedback tuning control systems which are primarily DSP based control systems. So digital tuning of analog subsystems okay become absolutely necessary. So the initial need design need not be very accurate but when that should be provisioned to vary the various parameters involved by a wide range in order to tune precisely to the accuracy that we need. That is why tuning procedures for for example filters was thoroughly discussed and self tuned filters were also understood in the context of precision that is required. Higher order analog systems passive components in high frequency analog systems can be in the range of parasitics that is what I mentioned. So cleaning becomes part of day to day activity in signal processing. As the parameters drift with temperature there might be requirement that this tuning circuits themselves might be part of the system which is integrated into it okay. Now negative feedback is one important concept that is conveyed in analog signal processing very okay. Negative feedbacks makes the performance of the system less sensitive to variation in values of parameters associated with the active device okay. In which case actually the full feedback okay the full output being fed back to the input okay and output following the input is the procedure that we resort to in most of the feedback systems wherein we are not able to trust the accuracy of components used within the loop okay. It may be having two blocks okay or three blocks it does not matter. As long as the output is fed back to the input output follows the input if the sensitivity of the blocks within product of the sensitivities of the blocks within is what is called as loop gain is much greater than one that is all that is required. They need not be linear they need not be independent of temperature they can be frequency dependent. But output follows the input as long as the product of the sensitivity factor remains in all these regions of operation okay much greater than one. So voltage followers current followers phase followers frequency followers okay digital data followers okay you can have a D2A converter and A2D converter. So then digital data at the input is followed at the output. A classic instance of this is where digital is converted to analog analog is converted back to digital okay. So that A2D converter is nothing but one bit A2D converter is nothing but a comparator okay and this can be an integrator right. So this is the summing point at the integrator so that is the classic topology of this ignore delta modulator. So higher order feedback systems have always this issue of instability coming into picture and therefore if one has to have feedback to desensitize the performance factors with respect to the parameters associated within the loop then it better be having only an order which is second order or if it is higher order the attempt is made to tune the feedback loop such that it becomes second order that is called frequency compensation. So all higher order the second order systems when they are coming to be there may be control systems the like voltage regulators okay current regulators or the speed regulators or what is that AGC, AVC or amplitude stabilization or phase lock loop any of these okay the queue of the system if they are thought of as second order systems the queue of the system is kept equal to one if it is transient response that is of interest. So that there is just one peak okay and it quickly comes back to steady state if it is a step input for example. So this is followed in voltage followers current followers or phase followers or phase frequency lock loops okay. So that is the case with these control systems if it is steady state response frequency response that we are concerned then we should not let it peak if it is a second order and therefore it is constrained to be maximally flat that this queue is equal to 1 by root 2. So if it is not the magnitude response which has to be flat but it is the delay that has to be flat queue for a second order system is 1 by root 3 right. So these are the variants that are possible okay when actually we are designing filters also. So whether it is filter design okay or amplifier design or control system design the same rule of thumb is what is followed okay. All higher order analog systems are designed through cascading second order system that we have already explained. Now particularly very high frequency high speed systems like RF and microwave systems always require total avoidance of feedback intentional or an intentional right. So that is for them to work satisfactorily okay otherwise there is instability. So please remember that you design high speed systems by avoiding feedback we come to this point of low pass filters for example. So a low pass filter is basically first order low pass filter is in R and C. So it is transfer function is 1 by 1 plus SCR this we topic we have already discussed in filters but most of the systems to a first order frequency dependence can be treated as a low pass filter that is even an amplifier is a first order low pass filter first without feedback okay. So with this feedback this remains the same as first order but bandwidth of usage increases. A second order filter they had seen so the parasitic resistance which is normally associated with the transmission line okay between one node to the other node you can consider it as a resistance centered by a capacitance at the output okay. So this is the cumulative effect of a transmission line from one node to the other. So this is the interconnect the approximation to the interconnect is an R in series and a C in short like a low pass filter. So probe of course we have a resistance attenuator here and it will have a capacitance there. So we have lot of practical design applications of these filters okay in the actual design of electronic systems. Then we have also the attenuator which is nothing but a R to R ladder scheme which is a passive component okay and the parasitics there will cause delay. Now this important approximation in the modeling of such interconnect pattern okay it can be modeled as a higher level higher order system with R and wire inductance being taken into account and this hand capacitor. So it is an RLC low pass filter of second order and second order transfer function is already known to us. That means it might for a step input like a digital word coming here might cause that digital word okay to ring like this. And the ringing frequency and Q these are determined by this same omega naught and Q that we have earlier used in second order low pass filters. So the amplitude of this ringing may be such that this is going to trigger our digital circuit many times intentionally right. So this is the problem with this kind of interconnect response. So what has been therefore taken in the modeling is primarily the delay involved from transmission of the information from one node to the other. That is normally thermal as modeled as almost delay. So this is the thing interconnect is any connection between the two points in an electronic system. So that is modeled for example A to B RLC B to C another value of RLC depending upon the length and okay R3 L3 C3 like that it can go on. So how to model this? The ideal delay is modeled always as E to power minus S thaw where thaw is the delay. And this is important for the digital design particularly in high speed digital design this becomes very important criteria in designing such digital systems. So the analog man might like to approximate it better but the digital man wants to primarily know what is the delay caused by this complex network. This E to power minus S thaw is approximated as 1 by 1 plus S thaw plus S thaw square by 2 factor and so on. That means this is approximated okay as 1 by 1 plus S thaw. The coefficient of S whatever be the complexity of the circuit involved has been assumed as almost delay. So for example when the network is from note to note approximated as this kind of RLC network then the almost delay thaw the coefficient of S can be quickly calculated. This C1 into R1 okay as far as SC1 is concerned as far as C2 is concerned the C2 into R1 plus R2 the resistance connecting input to the node C total resistance R1 plus R2 then C3 into R1 plus R2 plus R3 so on okay. So you can quickly conclude that the almost delay thaw when such cascaded networks N such cascaded networks are there simply C1 R1 plus C2 to R1 plus R2 plus C3 into R1 plus R2 plus R3 plus CN into R1 plus R2 plus R3 plus Rn. All higher order systems can therefore be simply approximated as equivalent to giving a delay to the digital word by an amount thaw equal to this. So this kind of approximation helps a lot in understanding a complex system design and this is what has to be learnt from analog or this is nothing but the networks okay knowledge but it has to be emphasized in the beginning itself to the student and these are the topics of relevance today in electronics that must be emphasized in the beginning itself. So band pass filter so basically an interconnect or a connection from one node to the other as a resistive component and an inductor okay and then there is a shunt component coming there and therefore all these things can act as a resonator block and resonate and may therefore cause okay resonance at omega naught equal to 1 over root LC that is really dangerous because something that looks like a short circuit particularly let us say electrolytic high valued polar capacitors which may act as basically inductors at very high frequencies with some resistance in association with it can act as a resonant circuit and they may not act as a short circuit at all they will act at that frequency as open circuit in contrast to our assumption that this is the case with our power supplies which require huge bypass capacitors or holding capacitors okay which may turn out to be inductive okay at high frequency that is why most of these capacitors are further bypassed electrolytic capacitor has to be always bypassed by non-electrolet good quality capacitor which truly acts as a short circuit at high frequencies. So these are the things that we learn from learning to understand second order systems okay. Now amplifiers with negative feedback we have seen can be considered as basically second order system instead of first order systems and we can also decide about the second pole okay properly so that the queue of the system becomes equal to 1 this is called frequency compensation of the feedback amplifier and the compensated amplifier for delay compensation if an amplifier is going to be used in feedback the most important thing is not the flatness of magnitude or anything okay or may not making the queue equal to 1 or anything it may be that the delay component can be compensated for by having the same delay in the numerator or 0 in the numerator so that the order of the system remains the same but effective delay is minimized for this plot. So this is what is done when an amplifier is put in a negative feedback loop as part of the component it is delay compensating the amplifier which itself is used in negative feedback. Now filter yeah in filter design again the emphasis is on understanding the second order very well and the second order building blocks like state space filters are the most common building blocks of complex filters in analog today. Oscillator is nothing but a second order system simply with queue going to infinity and it oscillates at omega not equal to 1 over RC. So again stabilization of amplitude is another feedback system which can itself be first order or second order and can be optimized for its own control loop performance. So one of the complex systems that we have designed today is after all these classes we have designed quadrature oscillator okay and made it voltage control oscillator and then cost stabilization of amplitude of that oscillator. So we have involved ourselves with a very complex feedback system why for quadrature oscillator we needed integrators two integrators. Integrator itself is a negative feedback system that means basically this can be a second order system. So this second order system cascaded with further such systems was used in negative feedback to generate an oscillation that means this is a second order system this is second order system this is a third second order system. This has become a second order because there is a first order effect due to capacitor and there is the second order effect due to the GB. So this also is a second order this also is a second order. However when it is used in this loop we have understood how this has to be approximated as a first order with an error delay. This also has to be approximated as a first order effect with delay. This also is having a delay and cumulative delay is what causes Q enhancement of the system okay. So that is what is thought of in this system and we had made it voltage controllable by putting multipliers voltage control and then we had used squarers to add these outputs okay and get a DC dependent upon the amplitude of oscillation and then control the feedback factor here okay. So this is where we had introduced a multiplier and control the feedback and therefore that was a control system which itself can be a second order. How to understand this total system therefore taking care of voltage control frequency and stabilization of amplitude etc is only by making the models of all these things simple when we are considering the whole system. Then understanding first maybe stabilization okay then going into VCO operation then the oscillation of the system with Q equal to infinity. So these are the things to be done step by step bringing in complexity only step by step. The PLL has been modeled as again a frequency follower with transfer function equal to 1 by 1 plus S by omega naught Q plus S square by omega naught square. This again can be compensated using the same technique that we have used that means the low pass filter if it is just used okay the Q may be too high okay then one can introduce a 0 okay apart from this pole a 0 can be introduced here okay and the 0 is going to be just 1 plus S by omega Z divided by 1 plus S by omega P omega P is now equal to C into okay 1 by R Z plus R P omega Z equal to 1 by C into R Z. So one can therefore locate the 0 in the loop gain and make the Q equal to 1 if the original Q is very high that is what is called frequency compensating the phase lock load okay noise will therefore come into picture that means the double the frequency component for example will increase in its amplitude the moment is 0 comes into picture. So these are the basic design techniques of most of the systems that we have discussed so far finally okay what we can summarize is starting from the passive filters going to the amplifiers active filters oscillators all other negative feedback systems like DC regulators AGC ABC amplitude stabilizer oscillators PLLs and FLLs are designed as second order systems. So it is not something that has not been sort of said earlier but one has to realize the fact that these are all basic things which can be taught at fairly early state or assimilated early by our undergraduate students. So next comparators are used okay this is another category. So far okay whatever we have discussed we have discussed only basically analog systems comparator the moment comparator comes into picture comparators are used as interface components between analog and digital systems. Positive regenerative feedback is used in comparators to bring about hysteresis. So what is this hysteresis the hysteresis make sure that the active device is always used as a switch. Now this whether is a BJT or a MOSFET that active device is used as a switch that means particularly in power circuits we would like to use the active device okay for efficiency sake as a switch. Then it is necessary to drive such a circuit using comparators with hysteresis. So that is regenerative positive feedback speed trigger. A comparator is a 1 bit 8D converter an integrator with comparator can form a function generator loop that is the other mix this is the most important mixed mode circuit that we have discussed as a function generator. So it is used in almost all test oscillator circuits right and also in most of the VCOs okay currently used this is the topology that means current charging a capacitor okay and that is an input to a speed trigger with regenerative positive feedback. So this in fact if we have an analog input here right the output is going to be pulse width modulated here because of the analog input here okay. So this is similar to the basic concept of sigma delta modulator which is used particularly in data converters. So this kind of modeling okay this has to be modeled most of the time in time domain because of this non-linear element which is a regenerative comparator input is analog output is digital high or low. So comparators are also used in pulse width modulators duty cycle generators DC to DC converters and class D power amplifiers and switch remote power supplies where the active device is mostly used as a switch okay. So this aspect of understanding both comparators and op amp and multipliers. So these are the 3 basic building blocks along with as may be power devices the mass power device okay. So these are the ones that have to be given as basic components in laboratories to understand all these concepts very well. Where do we go from this course it is necessary that the basics in analog and mixed mode have to be thoroughly understood taking the topics and the components that we have listed earlier and analog IC design which is going to become increasingly specialized job okay and designing op amp designing comparators designing multipliers and designing the various ICs that are commonly used okay that has to be restricted as an elective or a higher level higher semester course okay and a follow up activity okay is particularly in power electronics for a switch mode power regulator design for large power and good efficiency and reliable operation is an important aspect of power supply management. So that has to be understood very well so then specialization for communication most of the analog signal processing is going to be restricted to communications. So one has to understand the RF electronics very well in understanding communication circuits and also microwave circuits again this is part of communication area. So these are higher level courses which may be electives I hope that this definitely must be made compulsory these three things can be kept as electives that is a recommendation right. Thank you very much for giving me an opportunity to say something about what must be done in analog today.