 Today we will have the 6th lecture of the series let us see what we had done in the last class imitance matrix this is an important property of the two port network this imitance matrix can represent Z, Y, H or G matrices which are the 4 possible matrices if you take one input term variable as independent and one at the output as independent and let us look at what we consider as self imitance at the input total self imitance at the input including that of the source imitance is Pi plus PS under certain condition like it may be short circuit at the output or open circuit at the output P naught plus PL is the total self imitance at the output again under short circuit at the input or open circuit at the input PR is the feedback so just look at it that we have introduced the most important concept in analog signal processing the feedback this is PR something from the output gets fed back to the input either voltage or current gets fed back to the input as voltage or current PR again either voltage or current at the input gets transmitted to the output through PR as voltage or current. So this in general therefore describes very nicely the property of a two port network and if PR is 0 it is the device is said to be unilateral that is transmission only in the forward direction and the determinant of the matrix this matrix which is this into this minus PF into PR so that is indicated here as PI plus PS into P naught plus PL into 1 minus loop gain this is another important concept let us now understand what loop gain is loop gain is something from the input gets transferred to the output through PF and that voltage or current okay forms either voltage or current by the total admittance or impedance at the output P naught plus PL so PF by P naught plus PL is the forward transfer gain or parameter so that is multiplied by the whatever is developed at the output gets transferred back to the input this is the feedback through PR and that develops the voltage or current at the input as PR divided by PI plus PS both with negative signs okay these are the forward transfer parameter into the reverse transfer parameter which is called the loop gain under the condition of load termination and source termination so that is called the loop gain which characterizes the determinant okay and 1 minus GL is an important factor okay in the performance factor of the imitance matrix that means input imitance by taking the source imitance at the input and load imitance at the output is PI plus PS okay into 1 minus GL right and P out is P naught plus PL total self-imitance at the output into 1 minus loop gain so it characterizes the entire two port so let us consider next what we discussed in the last class passive network we have demonstrated by suitable examples that all passive two port networks will have the forward transfer and reverse transfer the same in parameters Y and Z they are exactly same YR is equal to YF and ZR is equal to ZF in the case of G and EH they are different in sign GR is equal to minus GF HR is equal to minus HR that characterizes a passive network obviously active network is something that does not fulfill this condition so anything that does not fulfill this condition that YR not equal to YF ZR equal not equal to ZF forms an active network similarly GR not equal to minus GF and HR not equal to minus HR characterizes in active network amplifiers for example we will see ideal amplifiers are characterized by PI equal to 0 PR equal to 0 PI equal to 0 means either it is voltage control or current control PR equal to 0 means there is no reverse transmission and P naught equal to 0 means there is the either voltage source at the output or current source at the output and PF is the only factor which is in existence and we will discuss this in detail in today's class okay this is what characterizes amongst active devices ideal amplifier so this will be one of the topics of today's lecture. So linear signal processing functions of two port networks the two port networks can perform attenuation and amplification two port passive networks can perform mainly attenuation and active networks can perform amplification we will talk about ideal amplifier definitions later other linear functions include addition subtraction integration differentiation and filtering will be adequately covered later after discussing attenuation and amplification let us consider first attenuation we can have purely resistive attenuators or purely capacitive attenuators or purely inductive attenuators. Now a combination of resistive capacitive resistive capacitive inductive also can be had for ideal attenuators we will see how we can design these attenuators in the next few lectures now in the previous lecture we had considered several passive networks including these attenuators and these attenuators are going to be used in later lectures on feedback amplifier design so that is why these parameters are important so let us consider the resistive voltage attenuator first we have a resistive network that forms an attenuator a series resistance and a shunt resistance at the output series resistance at the input. So if you follow the definition of parameter to be chosen we want a transfer from input to output corresponding to voltage that means output voltage is V2 and input voltage is V1 so V2 by V1 is what is needed obviously the most important parameter of the attenuator is RA by RA plus RB which is the forward transmission factor GF open circuit okay. So RA by RA plus RB is the important factor consequently we select the G parameter see please look at the importance of parameters because the important parameter is transfer from input to output of voltage we have chosen G parameters where it is clearly defined rest of the parameters corresponding to this network or GI which is 1 by RA plus RB open circuit input admittance and G naught which is the short circuit output impedance RA parallel R and the reverse transfer in all these networks will be if voltage is the forward transfer please note that short circuit current is the reverse transfer which remains the same as the voltage transfer but opposite in sign minus RA by RA plus RB this is the macro model of these 2 equations okay. Now the same network if you interchange the input and output becomes the current attenuator and then the parameter is the dual of G parameter which is H parameter where the forward transfer is the short circuit current which is the same as the earlier voltage transfer but opposite in sign because the current when it enters here it leaves here so the sign change occurs right. So again the H parameters of this network are given here indicating clearly the forward transfer is same as reverse transfer but opposite in sign and this is the network used in later we will see current amplifiers the voltage amplifiers and current amplifiers use respectively current attenuator and voltage attenuator. Instead of resistive attenuator we can also go for capacitive attenuator where the ah capacitance CB and CA form an attenuation of CB divided by CB plus CA as the attenuation factor. So it is particularly suitable at very high frequencies to use these as attenuators because low valued capacitors are naturally there and then you can exploit these capacitors for attenuation at high frequencies. So again the parameters are naturally given in the same form as the resistive attenuator except for the fact that capacitive impedance replaces the resistance. Now we can form for example inductive attenuators ah current attenuators you can have voltage attenuators also but for this example we have taken the current attenuators where H parameter determines ah these performance factors LA by LA plus LB is the reverse transmission of voltage and minus LA by LA plus LB is the forward transmission of current that is the macro mode. Now this is an important thing we can combine a resistive attenuator along with a capacitive attenuator ah giving the same attenuation factor at both these points nodes V02 and V01. If the attenuation is the same for all frequencies at all times at every instant of time then we can as well short circuit this and the resultant circuit is RB in parallel with CB and RA in parallel to CA and the attenuation factors being equal we will have a condition that gets satisfied okay for these capacitive resistive attenuators RA CA equal to RB CB this time constant RA CA should be the same as RB CB which means if this is Z and this is N times Z then whatever be the nature of Z okay the attenuation factor is independent of ah the time and frequency etc. So it remains the same okay and that concept is used in these combination attenuators same thing can be done with the inductor and resistor or inductor resistor and capacitor just given an example of this this is the example of a practical probe the oscilloscope probe which is commercially available this is the impedance at the input of typical impedance at the input of an oscilloscope 1 mega ohm centered by some few picofarads then picofarad then we have a provision normally for an attenuation factor by if by 10 then obviously RB has to be 9 mega ohms okay in order to provide RA by RA plus RB equal to 1 over 10 so RB becomes 9 mega ohm and since CA RA should be same as CB RB we have CB becoming equal to 1.11 picofarad this attenuator is what is called as compensated compensated attenuator. So there is a provision actually wherein you can vary this capacitor in the probe so that this condition is exactly satisfied RA CA is made equal to RB CB exactly what is done is there is a square wave that can be applied available at the output of the oscilloscope itself and you try to see a square wave exactly attenuated by 1 over 10 at the output. So you can adjust the capacitor for this kind of compensation then the condition satisfied is this CA RA becomes equal to CB RB. Let us consider another passive network which is a two port network which is commonly used in electrical engineering whether it is low frequency or high frequency this transform is pretty common. So this transformer N1 turns here in the primary and N2 in the secondary gives a stepping up factor of N2 by N1 for the voltage right we know that N1 I1 the ampere turns is equal to N2 I0 N1 I1 or Ni I is equal to N2 I0 and input power is same as output power an ideal transformer has a power gain of 1. So we can get V0 as N2 by N1 into VI from these two relationships power gain of an ideal transformer is unity used primarily for impedance matching for maximum power transfer to the load from the source this is represent this can be represented only by G matrix because either voltage gets transferred or current gets transferred that is how you can view the transformer as. So G is equal to 0 self-imitance at the input 0 self-imitance at the output and forward transfer is voltage N2 by N1 and reverse transfer is current minus N2 by N1. So forward transfer is same as reverse transfer but for the opposite side here the same thing can be represented also by H parameter which is the dual of G parameter and that is nothing but minus N1 by N2 the current transfer and N1 by N2 the reverse voltage transfer okay. So this is the impedance matching for maximum power transfer from the source to the load that happens when the source impedance RS is equal to H in equal to N2 by N1 square into RL. Let us now consider a ladder network as a voltage attenuator this ladder network finds itself very commonly in application like digital to analog converters. So this is an important application of attenuators so this is called R2R ladder network R2R R2R and 2R termination. So if you do that this 2R comes in parallel with this 2R to form effective resistance R so the voltage here gets attenuated by half from here to here transfer. Then this R combines with this R to form 2R so this 2R in parallel with the net 2R results in again an impedance seen here which is R. So again the attenuation at every node okay respectively occurs by a factor of 2 okay. So we will see that this attenuation from here to here is by a factor of 2. Again from here to here it is by a factor of 2. So at all these nodes the reference generates VR by 2 VR by 4 VR by 8 the binary weighted voltages and you can have the binary weighted currents okay coming through these. So this is an important network generating the binary weighted voltages or currents in the network which is used in the conversion of digital to analog information. The same network if it is changing its input and output you can give current as the input IR and get at various points currents corresponding to the binary weighted currents IR by 8 IR by 4 IR by 2 and IR right. So IR IR by 2 okay IR by 4 and IR by 8 will be the currents that get attenuated. This illustrates the principle of forward and reverse transfers being the same for voltage and current in the case of this kind of attenuation network. Let us consider another example here of attenuation. Here I am applying a voltage here to attenuate the voltage by some factor in this case we want it to be one fifth the input voltage. What is the resistance R that is to be connected here in order to have one fifth the input voltage of this appearing here. The second question is if I have a current of different frequencies okay pumped in here what is the short circuit current of this coming to this point through this voltage the current it will be the same because forward voltage transfer is same as the reverse current transfer the short circuit current due to this current is the is by a same factor as the voltage getting attenuated from here to here that is demonstrated by this problem. What is the current through 10 kilo ohm resistor due to the current source here okay. First part is V naught by VI is to be made one by five for the voltage here. Now what is the attenuation here this we will illustrate by taking this network consider this network this is a method of evaluating attenuation factor in any generalized network like this okay how to write down the attenuation factor straight away. I am trying to tell you the method by which this problem is solved easily okay. So let us consider this network comprising of three sub sections like this attenuating the voltage. So if this is V naught this is VI I can straight away say that V naught by VI okay is going to be equal to if all these are open circuits okay that is if YA is equal to zero YB equal to zero YC equal to zero the output current is output voltage is same as input voltage okay. So these are going to the existence of these will cause the output voltage to become definitely less than the input voltage. So that means this is the factor which is responsible because of the existence of these to reduce the output voltage to less than input voltage. So it should be one by one plus this voltage is converted to current if it is only this network that is present and these two are open circuits then the attenuation will be you can say ZA by ZA plus Z1 it is just the same thing that way which can be written as one by one plus Z1 which is the series resistance into YA this is a very simple thing. So Z1 impedance in series into YA admittance in shunt causes the attenuation to be less than one okay by this factor. If you have now this open and this alone existing that is going to be causing a factor of Z1 plus Z2 coming in series into YB as the attenuation factor. If these two are open and this alone exist then it will be Z1 plus Z2 plus Z3 into YC. So this is the effect of individual admittances on the attenuation factor. Now if you combine two at a time the same thing can be continued further plus let us consider these two sections being present two at a time. One at a time we have finished all the effects two at a time if you consider this into Z1 Z2 YA YB these two are attenuators successively coming okay. So Z1 into YA is one factor Z2 into YB is another factor plus let us consider this one and this one Z1 and this being open okay Z2 plus Z3 into YA YC plus now these two to be consider this being open Z1 plus Z2 is the series thing into YB Z3 into YC so that they has taken care of two network combinations. Finally all the three attenuators coming into picture Z1 YA Z2 YB Z3 this is the total attenuation factor we have straight away written just by looking at the network the transfer function from output input. So this is an easy method and this actually takes effect of all the admittances and impedances into a card and we have applied this to the problem there you can see let us work it out it is V0 by VI is required to be 1 by 5 V0 by VI 1 by 1 plus this impedance 10 K into conductance 1 over 5 K 10 by 5 you can see. So that is first order effect with this open with this open it is 10 plus 5 15 see so simple into 1 by R 15 by R now both attenuators acting 10 by 5 okay it is going to be 10 by 5 into 5 by R okay that is the second order effect. So we have 1 plus 10 by 5 okay plus 15 by R plus the second I mean double network causing attenuation 10 by 5 into 5 by R so that is how we got this 5 5 gets cancelled we have this so R is equal to 12.5 this technique can be extended to RC networks or RLC network anything and quickly one can write down the transfer function without going through the Kirchhoff voltage and current loop equation solutions. So now we come to the most important application of this section amplification attenuation is pretty simple passive networks carried out very efficiently so even to this day we use for attenuation only these passive networks amplification on the other hand is done by active devices later on see what are these active devices however these active devices are of no consequence in terms of technology for this discussion what according to us is the basic definition of amplifier ideal amplifier in terms of networks irrespective of the technology we may keep on shifting from one device to another for actual implementation of this amplifier. So this is what everybody should know about amplifier ideal amplifiers have zero input power and finite output power okay therefore the power gain is finite output power divided by input power which is infinity now how do you make input power zero either you should have input current to the amplifier zero which means it is an open circuit or input voltage to the amplifier zero that means it is a short circuit at the input so either it is voltage control which means open circuit at the input or current control which is short circuit at the input that means one of the parameters is dependent already zero only the other parameter can be the independent variable so voltage control means voltage is the independent variable at the input current control means current is the independent variable at the input other parameter is already dependent and zero it should be an ideal amplifier at the output which is ideal voltage source or ideal current source this is to facilitate any wanted power at the output for any load. So for this to be possible we have to have ideal voltage source or ideal current source at the output in which case we can actually have four types of amplifiers in practice this sets are well defined in networks and there cannot be any deviation from these for ideal amplifiers and we should know what ideal amplifier characteristic is before going ahead with designing these amplifiers types of ideal amplifiers as two port networks first one I would consider is voltage controlled current source which is called as trans conductance amplifier trans conductance the parameter of importance is conductance trans conductance CCVS current controlled voltage source which is the dual of VCCS this is known as popularly trans resistance amplifier voltage controlled voltage source which is commonly called as voltage amplifier current control current source which is commonly called as current amplifier. So these are the only four possible amplifier which are ideal and what are the characteristic of these ideal amplifiers the parameters associated with the ideal amplifiers will be trans conductance GF for voltage controlled current source trans resistance RF for current controlled voltage source voltage gain GF for VCVS and current gain HF for CCS and the corresponding parameters in the case of current control current source it is uniquely defined only in H parameter voltage gain VCVS is uniquely defined only in G parameter trans resistance amplifier is uniquely defined in R that is Z parameter trans conductance is uniquely defined in Y parameter. Now there are certain things that we should understand out of these four types of amplifiers which are the ones which are more basic that means it is enough if you try to make all the attempts to realize only two of these not necessarily all the four then they can be the basic building block for any of the electronic functions that you use amplifiers for. So therefore voltage controlled current source and current controlled voltage source happen to be the most basic fundamental amplifiers and the other two which have been really made popular in the earlier decades or so that is voltage amplifier and current amplifiers are not enough to sort of function as amplifiers okay if you do not have the trans conductance and trans resistance. Therefore on the other hand we can also show that current control current source and voltage control voltage source ideal amplifiers can be realized by cascading the current control voltage source with voltage control current source and vice versa that means if you put voltage control current source followed by current control voltage source then you can generate ideal current amplifier or voltage amplifier with current gain and voltage gain both equal to GF into ZF right that means the conductance into the impedance will form the voltage gain or current gain in the respective ideal amplifiers that you are realizing. So it is enough if you put in lot of effort to realize trans conductance amplifiers and trans resistance amplifiers not necessarily have effort put in for voltage and current amplifiers. So let us look at the macro model of these ideal amplifiers. So we have a voltage control current source this is open at the input VI is the input voltage and current is YF which is nothing but GF into VI and please remember that you have to keep it shorted in order to dissipate no power otherwise it will be dissipating if you have a load it will be dissipating power at the output. If you do not want any power to be wasted you can short this okay. Current control voltage source is the ideal of voltage control current source is called trans resistance amplifiers only now with IC design etc these amplifiers are becoming available okay and people have realized the significance of this and high frequency utility and it is a short circuit at the input short circuit is always useful because you know the capacity effect is got rid of a short circuit is not going to have any voltage rise at this point. So it is very fast here okay in responding so and it is a voltage source at the output which is ZF into II ZF is equal to RF this is the unfortunately popular amplifier today voltage control voltage source but this is not enough if you have only this kind of amplifier even if you have the dual companion current control current source it is not a complete set it is voltage control current source and current control voltage source which can form a complete set okay of active devices this is the current amplifier where HF is the parameter of importance transferring input current to output again this has to be kept shorted please remember if you open the current source or short the voltage source ideal then you are causing instability it does not know whether to sustain zero voltage or infinite current right. So actually it breaks down at that point practically it is impossible to have infinite current when you short a voltage source or infinite voltage when you open a current source breakdown occurs. Now let us go through some of the examples to understand this very well it is a very simple concept and therefore you should become very familiar with utility of these parameters what is the voltage gain of a network whose Y parameter is this. Now Y parameter is given to you this parameter corresponds to YF that means voltage at the input gets transferred to current at the output. Now you can see VI into 10 millisiemens okay corresponds to I naught output current which is 10 times VI milliampere this current will flow through now there is no the output conductance zero output conductance only this must be the load 0.1 millisiemens is the load which corresponds to 10 kilo ohm load 1 by 0.1 okay. So 10 kilo ohm load this I naught current 10 into VI milliampere flows through 10 kilo ohm load to give you 100 times VI. Since the current is flowing inwards output voltage is going to be let us therefore look at this the model can be given as VI we have 10 times VI milliampere flowing through this this is 10 key so this is the macro model of this okay and this current is flowing in this direction this is plus and this minus so V naught is negative and minus 100 times VI that is what gets developed here. So gain is minus 100 so this is all this way let us take the second example on active devices design a voltage amplifier with the voltage gain of 1000 we want a voltage amplifier with gain of 1000 only using trans conductance and trans resistance amplifiers. So voltage amplifier with gain of 1000 can be got by cascading voltage control voltage source having this Y parameter let us say 10 milli siemens and a CCVS which is having 100 kilo ohm as the transfer resistance. So transfer resistance of 100 kilo ohm into 10 milli siemens results in a voltage gain of 10 into 100 which is 1000. So again we can put down this macro models this way you have VI and we have 10 milli siemens this current will flow through this short circuit which is the output of this and we have here a voltage source which is 100 K times this current this current is flowing like this okay and this voltage will be let us say minus here plus here okay. So we have a voltage gain of 1000 here the current will flow like that okay in this. So okay the third example this example here illustrates knowledge about all the things that we have learnt today in this lecture what is a passive network we had learnt in the previous lecture and what are the active amplifiers which are sort of possible that is also included in this transformer another passive to port is also included in this. So this let us take Y parameter of this is given here you will notice that this is minus 1 and this Y parameter is same as this this is as YF and that is YR. So YF is equal to YR that means automatically irrespective of these okay these can be anything right as long as they are positive okay and magnitude wise greater than this right these are okay right these are this should be in magnitude okay greater than 1 milli Siemens that is because otherwise there will be a negative ahh real part that will come into picture right. So this we had seen that the impedance seen at any port should not have a negative real part okay if it is to be passive. So this are passive networks because these are positive real parts at the input okay. So these two are the same therefore this corresponds to passive network so that we have identified the second one G parameter is given as 0 0 10 – 10 okay. So this is same as this and opposite in sign that means obviously this corresponds to a transformer this is the voltage step up voltage okay and this is the ahh current which is transferred from output to input okay. So ideal transformer is the next one Y parameter is given this way 0 0 0 1 milli Siemens that we have readily recognized as a trans conductor amplifier voltage control current source finally we have 0 0 0 100 as HF H parameter okay ideal current amplifier. So that is how we are able to understand what we have learnt today in terms of matching it properly. So concluding today we have learnt an important part of analog signal processing attenuation okay activity of a two port network using either resistive network or a capacitive network or an inductive network or a combination of these and how to design these attenuators ahh which independent frequency and time okay giving the same attenuation at all instances of time okay ahh what is the basic principle of such a thing called compensated attenuator okay all these things including that of the DAC okay. So we have learnt about ahh probe these are the and designing a probe and also designing a DAC D2E converter. Then we have understood the most important property that a passive network should have forward transmission that is GF equal to minus GR or HF is equal to minus HR or ZF is equal to ZR YF equal to YR anything that violates this active okay. Amongst active network we have learnt about ideal amplifiers and the most basic ones are trans conductor amplifiers and trans resistance amplifiers and these are not so basic but form part of the whole list of ideal amplifiers voltage and current amplifiers. And all of these have ahh immittance matrix okay which is given by 0000 and PF they are amplifiers which have ideally power gain equal to infinity and can deliver any given power to any load. So these with no feedback and also an important part that look gain of the system so we have already been introduced to feedback and feed forward which are equal okay in a passive network in the network goes itself so feedback is not ahh something unfamiliar to you right. So we will see you will exploit this okay nature of feedback network in amplifier design feedback amplifier design later on.