 Today we are starting with the dynamic behavior of the feedback systems that we have been covering so far static behavior has been covered already let us see what we have done in the last class that is the 12th lecture static characteristic of negative feedback system we have seen that in a negative feedback system output feedback always follows the input as long as the loop gain is very high now we also saw that if there is coder encoder form of loop inverse operation can be generated if one operation is already done by a block by putting that block in feedback path with loop gain be very high inverse operation can be generated these two are the points that have been covered earlier. Now distortion reduction is another aspect of feedback that has been covered distortion can be thought of as okay introducing something extra at the output apart from the linear output that has got assuming linearity so extra thing is got by non-linearity so that can be thought of as an output in I mean signal introduced at the output it causes extra signal so that signal effect is also getting reduced because of the loop gain being very high linearity comes about because of distortion reduction noise is again something that is introduced at the output let us see so similar to distortion introducing something extra at the output noise introduces something other than signal component at the output so that also gets reduced by the loop gain effect of saturation this is a drastic non-linearity that means the change in output is 0 for a change in input for the block that is defined as saturation that incremental change in output for an incremental change in input is 0 effect of that is the major thing that is causing limitation in the operation of the loop because the loop gain then becomes 0 and therefore the range over which loop gain is much greater than 1 is considered the range where all the effects of positive effects of negative feedback exist so the dynamic range of operation or the lock range of the system is in a range where the loop gain is much greater than 1 so dynamic characteristic of feedback loop is the topic that is going to be covered after we touch upon certain aspects of lock range already covered in the last class so let us now consider let us say MOS or bipolar transistor let us say the input let us say voltage which is the gate to source voltage and output current they are related by this relationship this we have already seen 3 by 2 Vi – VT whole square this is a N channel enhancement type of MOSFET we have taken with threshold voltage equal to VT okay then we have shown that this is the characteristic that is followed by the MOSFET above let us say this V naught okay for V naught okay greater than Vi – VT this relationship is valid greater than or equal to so that is the saturation current of the MOSFET now what is done is the full output current is fed back to the input so this is a feedback negative feedback so full output current okay is fed back to the input so what happens now this is a feedback system where the current gain is infinity and full output current is fed back to the input so output current change for an input current change okay this is the input current now okay so the voltage and current interchange their role the input becomes a current okay and output becomes a voltage earlier voltage was the input Vi and current was related to this okay using the square law relationship now V naught the voltage becomes the dependent variable current is the independent variable because full current is fed back I naught is equal to I I okay that change in output current for a change in input current is equal to 1 because of full feedback loop gain is infinity that current gain loop gain is infinity so the error current goes to 0 what happens now if you substitute I I equal to I naught okay this relationship gets valid so V naught becomes square root of 2 I I okay 2 I I by K plus ET so we had a square relationship as far as input as voltage and output as current is concerned we give full current feedback okay and therefore input becomes a current and output becomes voltage so that is inverse square root relationship V naught is related to I I in a square root fashion so this is what is if you use bipolar transistor instead of mass then voltage to current relationship is exponential okay and the current feedback makes current to voltage relationship logarithmic okay so we can write down that this is Vi and I naught is I E naught exponent Vi by VT so now when you give current full current feedback output current okay becomes same as input current okay and therefore this voltage is V naught it develops a V naught okay which is equal to VT log I I by I naught so inverse relationship that is exponential this is logarithmic okay this exponential is called data expansion logarithmic is called data compression same thing earlier it was doing data expansion square law and square root is data compression so is one of the most important signal processing encoding decoding technique now as far as dynamic range of operation is concerned here the current in this has to be unipolar that means it is always in this direction that means this current has to be always unipolar for this relationship to be satisfied and V naught has to be greater than this so here V naught is equal to okay Vi this is shorted so this is always the case so this relationship is automatically satisfied in this case when this is shorted to this so V naught is always greater than Vi – VT is satisfied so it is always in current saturation region in this mode so I I okay and V naught are related by square root relationship now what is the loop gain it is nothing but GM the small variation of current resulting due to small variation in voltage that is defined as trans conductance divided by GDS that is the gain of the stage loop gain GM by GDS so when it is a loop you can either call it voltage gain from here to here right our current gain so it does not make any difference it is the same factor GM by GDS this around the operating point at all operating points if it is much greater than 1 then this is satisfied at all operating points that is why even for last signal this is satisfied as long as this is satisfied for every operating point okay in the range from for I I the range is from it should be greater than 0 and it can be going up to its maximum value whatever is the case with the MOSFET what is the maximum current it can tolerate depending on the power maximum okay so that is the lock range the lock range for this therefore is from 0 to IMAX in order to fully utilize the lock range it is necessary to offset the incoming current along with an offset current constant value which is equal to IMAX divided by 2 so we can have a DC current of IMAX by 2 superimposed over a sign current okay which can swing as much as IMAX by 2 on either side coming all the way to 0 on one side and going all the way up to IMAX on the other side that is what is called biasing that will elaborate further later when we discuss the the individual transistor circuits okay here as a system we consider that means if you have this restriction of unipolar current flowing in okay then we must bias the device at IMAX by 2 in order to take advantage of the full swing that is possible for the signal on either side of this DC current so consequently it is going to operate at V naught Q this called the cohesion current so this comes about in almost all feedback systems that is the lock range coming about because of the thing going to its limit of operation the device going to its limit of operation and that can come about due to saturation or cut off of the device or devices used in the block now let us consider the automatic gain control that we had all earlier consider if VP sin omega t is the input the output is VP O sin omega t and therefore the gain is VC by 10 this VC can at most go up to 10 volts because of the limitations of the IC that is multiplying so most of these ICs are designed for certain specific maximum input voltages so this is a multiplier with voltage inputs where the voltages here can go up to plus minus 10 volts maximum and output also can be plus minus 10 volts because of these limitations of saturation limits for these inputs and outputs of a device we have limitation coming about for the entire lock range within which the loop gain can be made very high compared to one so outside the lock range it is obviously having loop gain which is coming to 0 so it cannot get locked the control system won't work or the loop is not going to work as expected by making the error going to 0 outside the lock range so what is the lock range of this that is the range over which okay the input of this is limited to 10 volts in magnitude and output of this is limited to 10 volts in magnitude and another thing is that since this is a square that is used in feedback loop this output is always positive so that means actually this reference has to be negative so it therefore cannot work with an inverting amplifier configuration that means this voltage control voltage should not go negative at any time of the operation of this that means it should remain always positive that means limit for VC is going to be 0 to 10 volts so because of this limitation due to VC one can see that for a given V reference this has to create a V reference if it has to work satisfactorily so that output follows the input in magnitude but opposite in sign because we are summing at the error point okay so this V reference has to be developed and this V reference is nothing but VTO square by 10 to sign square to omega which is 1 minus cos 2 omega t by 2 so we get a DC of VPO square by 20 equaling to V reference at which point this keeps working so the VPO is going to be square root of 20 for a V reference chosen here of let us say 0.8 volts this is going to be 4 volts at which it is going to lock itself as far as output amplitude because this is constant this has to be constant because this is constant this has to be constant and because this is constant okay the gain has to adjust itself so as to make this a constant output of 4 volts that means as far as the gain of this stage is concerned it is limited to a maximum of 1 that means if you are expecting an output peak of sort of 4 volts then input of this should be such that it is never expected to go above 10 volts that means gain should not be greater than 1 volt which means VP less than 4 volts this system won't work because then the gain has to be greater than 1 in order to maintain the constant value that is not possible so for VP less than 4 volts the system is not going to work as AGC so the range over which this can work is 4 volts to 10 volts because of the limitations of the multiplier outside the long range it cannot work so this is the phase lock loop within the lock range we can see that this works now this dissimulation we can see dissimulation here the input input has been first adjusted to be having a magnitude which is greater than 4 volts it is within the lock range so for VP equal to 8 volts for example we can see the simulation analysis transient you can see this is the control voltage so that control voltage is going to be so it is getting adjusted such that the output is you can see automatically adjusted to be 4 volts right you can see the pointer so this is very nearly 4 volts and the peak value of the input is nearly at 8 volts the control voltage has got adjusted to meet this output come to 4 volts so it automatically gets adjusted now it is going to be tried for another situation okay where input is going to be adjusted to let us say less than 4 volts let us make it 2 volts so this value can be adjusted to 2 volts and at that point of time you can see the sort of waveform of simulation okay we will connect this so this is the phase lock loop before that again outside the lock range what happens is that this particular thing fails to work within the lock range right outside the lock range this sort of scheme does not work so frequency lock loop that we have all so considered earlier let us now consider the lock range of this frequency lock loop it has been designed VCO has been designed to have this kind of linear characteristics 10,000 this is called the free running frequency plus K times VC when VC is equal to 0 it runs at 10,000 hertz so first we consider within the lock range that means omega I Fi is going to be 10,000 at that point of time both are of the same frequency okay so the phase shift should be automatically adjusting itself to cos Fi okay that means cos VP1 VP2 divided by 20 cos Fi is the average because of the low pass filter so one gets VP1 VP2 has been made 10 volts that divided by 20 okay into cos Fi is the average so for VP1 VP2 equal to 10 volts let us say this is 10 sin omega t this is 10 sin omega t plus Fi so we get it as 100 divided by 20 cos Fi as the output average which is Fi cos Fi so the maximum voltage deviation that can happen here is 5 volts another side of 90 degrees so when this frequency is same as this frequency free running frequency 10,000 the cos Fi should adjust itself to be such that it is 5 into 0 so it is 0 that means the average should go to 0 because nothing has happened here this output should continue to be at 10,000 okay that is what happens okay and you can see it this is the average which is 0 and it is having the double the frequency here so when incoming frequency same as free running frequency the phase shift adjust itself to be this is sin Fi sin omega t the other one is cos omega t 90 exactly equal to 90. Now as within the lock range we go to the upper limit so let us do the transient you can see the control voltage getting adjusted and we have gone to the upper limit where the phase has from 90 degree change to 0 degree both output and input okay they are of the same frequency but the phase is exactly equal to 0 output has merged with the input okay and the DC voltage is automatically going to the limit of about 5 volts there see therefore from 0 to plus 5 volts it has gone that is the limit of lock range on one side. Next we can consider the other limit you go to the lower limit of the lock range the phase only can adjust okay frequency is all the time same okay so I have adjusted the incoming frequency now to be about 5100 on the other side of 10,000 so automatically you can see the thing has gone from 0 to negative value okay and let us see the output so the phase shift has changed okay from 0 on the other one extreme to 180 degree on the other very nearly 180 degree it has the added component of double the frequency which is predominant not filtered much okay causing the waveform to be distorted this way. So what is the DC voltage it has gone to very nearly 5 volts okay on the other side okay negative voltage that is where it has got locked these are the two extreme limits now let us consider the range going over to the lock range beyond the lock range. So beyond the lock range what happens is it simply goes outside the lock range you can see you will run this no there is no control voltage okay control voltage has come back to 0 it cannot control okay it has gone beyond the lock range the loop gain has gone to 0 and you can see output does not follow the input output is at one frequency and input it is at all together different frequency output is at is free running frequency and input is at the incoming frequency which is 5000 okay. So you can see in this case it is I think the other end 15,000 right so you had has gone out of lock range and this is no longer a DC so this is a bit frequency which is possible because of the two different frequencies no longer a DC as before okay. So this is the complete description of the dynamic range of different systems AGC we have covered and phase lock or frequency lock loop is also covered this is when it is out of lock that is the beat frequency which is the lower frequency noise distortion and offset reduction in feedback okay these things can be treated as some extraneous component getting added at the output. So for this kind of thing we have already shown that the noise component okay getting added results in an output with the original noise component divided by 1 plus loop gain here current is getting converted to voltage voltage is converted to current and this is a trans resistor okay encoder with trans conductor in a loop with current feedback this we had already discussed the voltage limit translates itself as current limit at the input. So what happens now with this as far as the lock range is concern okay now that we have already seen the current range comes as the lock range for this loop because of the voltage limitations. Now in simulation what we are considering is I am giving both a voltage input here and a current input so this can act as a voltage follower from here to the output and a current follower from here to this. So it is a voltage follower for some signal and a current follower for some other signal so that is demonstrated clearly by selecting voltage as for example signal and then current input here at this point is the noise or vice versa you can current signal as the mounted signal and voltage signal as the noise. So what happens we can see that the output if you take okay at this point as voltage and at this point as current okay you can only see the corresponding current or voltage signals we will see this in simulation so the same circuit is simulated here so this is the current signal which is the input is going up to 1 milli ampere peak and output is of negative sign inverted and of the same amplitude and nothing of voltage signal is seen here whereas voltage of 10 volts has been fed there at the input and voltage of same 10 volts magnitude is coming at the output and nothing of current input is seen here okay. So this is the beauty of this circuit it is illustrating the distortion reduction noise reduction and operation of the same loop both as a voltage follower and a current follower. So this is a powerful technique of signal processing which can be exploited for linearity noise reduction distortion reduction etc. The last topic about the dynamic range distortion caused by non-linearity now when non-linearity is not so much then we can define distortion as if you apply an input which is a sign wave to the system okay output should be if in a linear system should be purely a sign wave it may be some change in magnitude and may be phase. However there is no place for the harmonics whereas the distorted output has presence of this so called harmonics. So distortion is then defined as square root of square of all this peak magnitudes of harmonics summation of the squares divided by the fundamental peak so into 100 is the percentage distortion. So how to estimate this roughly so this has been considered for G10 in a feedback loop of 100 G2 equal to 1 and the amplitude of input A1 as 1 volt. So then X naught is equal to strictly speaking XI minus 1 by 6 XI cube that is the saturation non-linearity which has been sort of approximated as hyperbolic non-linearity. So this is the tan hyperbolic XI approximation okay around XI equal to 0 this is the approximation right. So substituting that and getting the writing down the earlier error expression we can see that X naught now becomes with feedback equal to XI minus 1 by 6 XI cube by 100 cube so that means it drastically reduce the third harmonic distortion caused by this. So this is the evaluation of the third harmonic distortion that means the whole thing becomes perfectly linear. So dynamic behavior of feedback system is what is discussed and negative feedback system reproduces the input signal at the output in the presence of noise parameter variations and non-linearity. Dynamic behavior is characterized by frequency dependent behavior time dependent behavior and dynamics of the loop gain that is GL equal to G1 into G2 significantly influences the performance of the feedback system that is what we are going to the next discuss right. Please understand that all these things can be taken as variation in the loop gain caused by different parameters like temperature variation like signal magnitude variation that causes distortion okay and the gain variation with signal level okay gain variation with frequency is what is discussed in the for going later chapter and let us therefore consider linear feedback systems get modeled as differential equations and Laplace transforms okay frequency dependent behavior is explored true Laplace transforms feedback system is treated as first let us consider as a first order or second order or at most third order system beyond that it is not worthwhile discussing the feedback system at all negative feedback system because we will later see there will be problems regarding keeping it stable. So practical systems have to be discussed okay which are either first order or second order at most their third order they have brought they have to be brought down to second order the best system we will see later from dynamics of the working of the system is the second order feedback system and therefore one has to be very familiar with a second order feedback system or one has to always make a system become second order so that it is optimally working. So let us consider the first order system which is having this block G1 is now frequency dependent and normally most of the systems have this transfer parameter or sensitivity decreasing with frequency most of the systems are like that output always tries to lag the input so first order lag system this is so G1 can be therefore written as G10 independent of frequency divided by 1 plus S by omega 1 so it has one pole this is called the pole G2 is a constant let us say independent of frequency so that the slope gain now is a first order okay transfer function where S the complex frequency and omega 1 is the corner frequency in radians per second omega 1 is known as the bandwidth of G1 in this case upper 3 dB frequency of G1 okay so in such situation what happens so we can therefore see that again X naught by XI as before is nothing but G1 divided by 1 plus G1 G2 this is what we have seen or it can be written as 1 by G1 divided by 1 plus okay 1 over G1 G2 that is called the loop gain okay. So here we have G1 to be substituted as G10 divided by 1 plus S by omega 1 therefore this becomes equal to 1 by G1 which is independent of frequency okay divided by 1 plus 1 over G10 G2 into 1 plus S by omega 1 after substituting this there so G10 G2 is assumed to be much greater than 1 so this is approximately equal to 1 by G1 which you can term as the gain of the system okay without with feedback okay without frequency dependence 1 plus S by the loop gain G10 G2 omega 1 so one can see that that is what the transfer function is is what is written okay so you can see that it is 1 over G2 divided by 1 plus S by omega 1 G10 G2 omega 1 G10 G2 is nothing but the DC loop gain G10 G2 is what is called the loop gain without frequency dependence into the bandwidth of the system. So bandwidth of G1 so is the bandwidth of the feedback system now so the original system had under open loop a bandwidth of omega 1 and now with feedback the bandwidth got extended by nothing but the loop gain into the bandwidth of the gain G1 block. So what happens to that so this can be translated as for a step input the output is going to be 1 over G2 times the step that has been given at the input into 1 minus e to power minus t G10 G2 omega 1. So that is the modification factor of the delay of the system okay delay gets reduced okay or bandwidth gets improved by loop gain. So delay gets reduced originally the system had this X naught okay now therefore depends upon okay the loop gain of the system. So if G2 is equal to 1 over 3 okay this is going to be 3 times 1 minus e to power minus t by 3 and if G2 is 1 okay this is going to be X naught equal to 1 minus e to power minus t ultimately it will reach the same as input in this case it will reach thrice the input okay because G2 is 1 over 3. So however the rate of rise you can note is the same this is purely dependent upon okay the loop gain into the DC loop gain into bandwidth which is G2 okay G1 into omega 1 okay. So that means the gain bandwidth product which is nothing but the gain of the system 1 over G2 into the bandwidth G10 G2 is practically independent of G2 it is nothing but G10 into omega 1. So that is the gain of the forward block into the bandwidth is bandwidth so it is independent of G2 same thing since the gain bandwidth product is independent of G2 we can see the rate of rise is also dependent only on this gain bandwidth product and it is independent of G2 magnitude of the transfer function you can say is 1 over G2 divided square root of this 1 plus omega square by this and therefore this is for frequency is much less than the gain bandwidth product this factor becomes very small so it is nearly equal to 1 over G2 whereas in the frequency range within the gain bandwidth product okay there is a certain amount of phase error okay magnitude error is almost negligible that is demonstrated here so the phase of GF is tan inverse omega divided by this and that for small angles is going to be roughly equal to omega by G10 G2 omega. So that error is one that has to be considered within the band gain bandwidth product and you can ignore the phase error so the gain varies okay in this manner from 10 it falls earlier okay and gain equal to 1 it falls later that means this is the gain bandwidth product okay and gain into bandwidth of this always is equal to gain bandwidth product. So if the gain is increased 10 times bandwidth decreases by 10 times in a first order system this always true if the gain is increased by 10 times the bandwidth decreases by 10 times so that the gain into bandwidth remains constant this is the phase characteristic. Now the next topic of interest is the second order system so in a second order system we have two poles as against one pole in the first order system causing as we say at high frequencies instead of the original 20 decibels per decade decrease in gain now it is going to be decreasing at 40 decibels per decade that is the asymptotic fall at high frequencies for a second order system so 40 decibels per decade fall off of gain. So transfer function of the system with feedback now what happens so the same expression is written now as you can see G1 by G1 G2 is going to be written as 1 by G2 divided by 1 plus 1 over G1 G2 as before now G1 G2 you substitute here G1 as G1 0 divided by 1 plus S by omega 1 into 1 plus S by omega 2. So what happens is 1 by G2 divided by this again this 1 by G1 0 G2 can be neglected compared to 1 so we have S coefficient which is 1 by omega 1 plus 1 by omega 2 again decreased by a factor of the DC loop gain. S square also gets decreased by the same loop gain okay so that into omega 1 omega 2 is the coefficient of this. So what happens almost all S coefficients in any system get reduced by loop gain. So what is the consequence of this in a second order system is what we are considering a second order system is generally written as normalized 1 plus 2 zeta into S by omega n plus S square by omega n square where omega n is called the natural frequency of the system that we will consider why it is called natural frequency of the system and zeta is called the damping factor. So these are the 2 things that are important damping factor that is zeta and this one is omega n the natural frequency of the system. So let us now consider okay this in terms of these basic definitions zeta is therefore by comparison comparing the coefficients you get this as 1 by 2 square root of the loop gain so everywhere you see the loop gain enters into our discussion this is the DC loop gain. So square root of DC loop gain into square root of omega 2 by omega 1 omega 1 by omega 2 omega 2 is higher than omega 1 that is omega 1 is still first corner frequency omega 2 is the second corner frequency higher 1. So now we have 1 over 2 zeta which is defined in communication terminology as Q quality factor of a second order system Q okay that is equal to the DC loop gain GL divided by square root of omega 2 by omega 1 plus omega 1 by omega 2 omega 1 is always less than omega 2 omega 2 is greater than omega 1 that means if the distance between the second pole and the first pole that is considerable then this dominates the scene and this becomes negligible. So ultimately Q becomes the loop gain divided by the distance between omega 2 and omega 1 that is omega 2 by omega 1 square root of that. So let us understand with this portion of it has already been taught to you in networks a second order system the time behaviour and the frequency behaviour it is going to be a revision for you. So in a system like this X naught is equal to 1 by G2 1 by omega n square D square XI by DT square plus 1 over Q omega n DXI by DT plus XI the previous equation has been translated to time domain in terms of a differential equation from the after transfer okay when XI is equal to a step function this is the solution okay for we see Q this is the solution for Q greater than half a simple so or Zeta less than 1 Zeta less than 1 this situation is called under damped system Q greater than half or Zeta less than 1 is called under damped system okay. So this is what happens in the case of under damped system we are simulating this okay using a model this is a two time constant model okay or two pole model where omega 1 is decided by 16 K into 1 micro farad omega 2 is decided by 160 K into 10 micro farad hopefully this is not loading this much okay so we can individually consider this so omega 2 is almost 10 times or that is 100 times omega 1 okay in this case omega 2 is 100 times omega 1 that is how we have chosen it so the Q of the system has been adjusted to be 1 okay so Q we have seen is nothing but the DC loop gain which is 100 divided by square root of omega 2 by omega 1 square root of omega 2 by omega 1 okay has been adjusted to be 10 okay square root of loop gain is also 10 so Q is equal to 1 so for such a system you can see there is just one peak Q can be estimated as the number of visible peaks in the step response of the system what do you mean by visible peaks if you consider first peak as 1 volt okay you go up to 110 that value 0.1 volt and count the number of peaks which are greater than 0.1 volt. So this is a easy way of measuring Q of the system and that has been also simulated for Q of 10 you can see 1 2 3 4 5 6 7 8 9 10 beyond this it is less than 1 10 this so Q is equal to 10 it is called ringing okay the system rings and comes to steady state much later right that is this ringing itself is unwarranted in high speed systems. So what we want is a high rate of rise okay that means our high rate of rise Q has to be high right however if the Q is too high the ringing persists and prevents the system from coming to steady state early. So the best value of Q of a second of the system is typically 1 where the rate of rise is high okay compare this to Q less than 1 the rate of rise is going to be slow it is going to be exponential there will be no ringing okay. So this is the time response of second order system okay in particular we can note that this has been discussed for Q less than half okay and Q equal to half you can see it is the fastest coming to steady state and then Q less than half Q equal to half is the fastest without any peaking Q greater than half there will be a peak okay but it comes to steady state also early. So it has rate of rise which is faster than Q equal to half that is why Q equal to 1 is what is normally chosen in practice. So these are the conclusions that we can draw about a second order system Q less than 1 the rate of rise is lower Q greater than 1 the rate of rise is higher but there will be no peaks and result in higher settling time larger settling time for example for a typical Q of 10 there will be 10 visible peaks. So this can be generalized for Q equal to N also there will be N visible peaks the most desirable step response of a feedback system is corresponding to Q equal to 1 the response is characterized as good rate of rise with one small peak in this case right. So that is what we have concluded will be considering the frequency response characteristic of a second order system in the next class that is also the dynamics that is the earlier one that we have discussed the time response is considered as the transient response right the frequency response is considered as what is called the steady state response that is what is needed okay the transient response should be taking the least amount of time coming to steady state response as quickly as possible in any system.