 Today we are going to continue from where we left off in the last class ahh we had seen this 2 port 1 port and 2 port devices we are supposed to understand out of which we have considered the 1 port devices in the last class resistor inductor capacitor and diode form the basic passive 1 port devices and the only ahh active device 1 port that we discussed is negative resistance. We had seen the characteristics of these ahh 1 port devices and also the applications in terms of how ahh the RLC network can be used for integration differentiation and also low pass filtering first order differential equation solution and second order differential equation solution and even designing oscillators using ideal L in shunt with ideal C the second order differential equation harmonic oscillator. In that discussion we had seen that the first order differential equation results in ahh if combined with negative resistance that is the active device can result in an exponentially increasing response okay with respect to time or decaying response with respect to time okay. These ahh decay and rise are governed by the RCE time constants of the network. So this kind of response if it has then it is ahh unstable right every system should have some kind of decaying response okay in order to ahh have stable response afterwards for the forcing function. Now this indicates a combination of RLC and ahh that gives a second order differential equation with response which is called ringing response right it is ringing. This is the initial ahh voltage for example of the capacitor 1 volt that ahh if it is shunted by an inductor and resistor is likely to decay the energy is going to decay in the tank circuit in the in exponential manner e to power minus t by 4 here I have given a typical decay waveform this envelope is e to power minus t by 4 and this cos 4 t corresponds to the frequency of ringing. Now the same second order equation can be therefore used as a band pass characteristic as a filter characteristic this also we had seen and then we went over to diode function generator which is useful in ahh simulating a nonlinearity in terms of piecewise linear approximations. Now we come to linear two port networks linear two port network is the topic of today's lecture and it is a very important topic in understanding analog signal processing particularly ahh the understanding of basic characteristics of amplifiers. Now before we go to amplifiers let us understand two port network parameters which are involved these are basically 4 YZ admittance impedance and the other two form what are called the hybrid parameters G and H. This metric parameters this can be all represented by just one ahh matrix parameter that is called immittance immaterial what you use Z, H, Y, Z or G. So characteristic of two port passive networks will understand in terms of what how passive networks are uniquely different from active networks ahh we had already defined ahh an active network is capable of power amplification and passive networks are incapable of power amplification. So how does it come into picture in terms of parameter ahh definitions for characterizing passive networks how it is distinctly different from that of active networks. Control sources are called active networks okay ideal control sources are classified as ideal amplifiers. So we will be having ahh these amplifiers characterized by different parameters in this course this particular figure depicts ahh two ports this is called the input port where the source that may be a non-ideal source which is represented by a seven inch equivalent as voltage source in series with the resistance or by its ahh northern equivalent with the current source shunted by the admittance load is represented as ZL or YL. So we have these two ports where input is fed to the input port and output is taken at the output port through the load. So input port can be actually having either the transducer itself forming the input and another possibility is that it is also the output of another two port network ahh input source can be either represented by as I told you this is nothing but thevenin equivalent okay and this is northern equivalent. Output can be connected to a load as represented in the earlier diagram or it can be output to the input of another signal processing network. Now characterization of a two port network we have in the case of one port we had either voltage or current as independent variable then correspondingly the current or voltage became the dependent variable whereas in the case of ahh two port we have input voltage input current output voltage output current four parameters out of which any two can be ahh independent variables and the other two automatically become dependent variables. Now we can therefore write down the equations governing the ahh input output relationship in terms of two equations okay because there are two independent variables and two dependent variables and that way we can formulate six combinations of these network parameters and these are respectively termed as Y which is called a short circuit admittance parameter Z which is called an open circuit impedance parameter and G and H are hybrid parameters and A, B, C, D and S parameters. If one is restricted to use one independent variable at the input and one at the output port since we are talking about characterizing amplifier it is imperative that one input at the input okay one variable at the input is converted to a dependent variable at the output. So in such combination if you select one as independent at the input and one at the output then only Y, Z, G and H are of interest are possible the other two parameters are primarily used for studying transmission line characteristics and microwave networks. So we will be concentrating on Y, Z, G and H in the forth for going discussion. G and H are hybrid parameters involving admittance, impedance, voltage ratio and current ratio Y on the other hand has all admittance parameters and Z similarly has all impedance parameters matrix representation of the two port network. So we can see here II and V naught are the dependent variables VI and I naught correspondingly are the independent variables and then this matrix is called the G matrix and you can note that GF is defined as the forward transfer parameter GR is defined as the reverse transfer parameter GI is defined as the input self admittance G naught is defined as the output self impedance. So similarly this H parameter is the dual of G parameter where whatever is there on the right side goes to the left side as dependent variable and these are the independent variable II and V naught are the independent variable and you will note that the forward transfer parameter is nothing but output current related to the input current I naught is equal to HF into II so and the reverse transfer parameter is a voltage ratio whereas in the case of G parameter forward is the voltage gain reverse is the current gain okay. HI is the input short circuit impedance H naught is the admittance open circuit output admittance. Now coming to Y parameters input current and output current are the dependent variables independent variables are voltages VI and V naught so all parameters are Y parameters YF is the forward parameter YR is the reverse parameter admittance YI is the admittance at the input self admittance Y naught is the self admittance at the output these are all short circuit parameters on the other hand Z is the dual of Y VI and V naught are the dependent variables II and I naught are the independent variables and ZF is the forward transfer impedance ZR is the reverse transfer impedance ZI is the open circuit self admittance at the input Z naught is the open circuit self admittance at the input and at the output. Immittance parameter that means all these 4 parameters can be generally represented by P immittance immaterial whether it is admittance impedance ratio voltage ratio current it does not matter what it is it can be generally represented by P parameters input current relating to input voltage is input self immittance input current or voltage relating to output input current relating to input voltage is input self immittance PI that is either admittance or impedance output current relating to output voltage is known as output self immittance P naught again it can be impedance or impedance input related to output is called okay forward transfer parameter this is the most important thing and output related to input is called reverse transfer parameter that is something at the output resulting in something at the input okay so that is called reverse transfer parameter input related output is called forward transfer parameter okay characterizing a 2 port passive network the 2 port passive parameter network has forward parameter okay same as reverse parameter this is an important statement forward parameter is the same as reverse parameter it is exactly same as in the case of Z and Y these 2 parameters are exactly same both in magnitude and sign whereas in the case of H and G only in magnitude they are the same but the sign is reversed so this is what is to be remembered as a characteristic feature of any passive network 2 port now we come to the ideal control sources ideal control sources are known as amplifiers there is an important statement ideal amplifiers have zero input power and they deliver finite output power to the load therefore the power gain of ideal amplifiers is infinity this is an important point to be noted power gain of all ideal amplifiers is infinity for the input power to be zero it should have either zero input current to the amplifier that means zero input admittance open circuit or zero input voltage to the amplifier which means zero input impedance short circuit so this simply means that this is zero input admittance open circuit this is zero input voltage to the amplifier or zero input impedance that is short circuit so this is the characteristic of ideal amplifier input volt is voltage controlled or current controlled output is either a voltage source or a current source voltage source means zero output impedance for it current source means zero output admittance so we can have these as voltage source or current source so it can be voltage controlled or current control this way we have four combinations of these possible voltage controlled voltage source current controlled current source voltage controlled current source current controlled voltage source four types are possible theoretically no more no less so resulting in four types of amplifiers voltage control voltage source current control current source or current amplifier this is called a voltage amplifier this is a current amplifier voltage control current source which is called a trans conductance amplifier current control voltage source which is called a trans resistance amplifier. So these are the 4 basic amplifier topologies that are possible and this can be represented only by one of the 4 parameters that we have already defined. So voltage control voltage source is represented only by g parameters ideal voltage control voltage source or voltage amplifier. It has 0 g i 0 g r 0 g naught okay and g f finite. This is the only forward transfer parameter that is important as far as the voltage amplifier is concerned. The it is an open circuit at the input and output impedance is 0 because it is an ideal voltage source and reverse transfer parameter which is a current transfer is 0 that is no feedback. So ideal amplifiers have no feedback current control current source is to be represented only by h parameter ideal current amplifier it has h i 0 h naught 0 h r 0 no reverse transfer voltage possible only current transfer h f is defined. Voltage control current source as y parameters so it converts voltage at the input to current at the output by this factor g f and all the other parameters by parameters are 0. CCVS has only RF as the transfer parameter and all the other z parameters are 0. So before we go ahead trying to do design we should know how to characterize ideal amplifiers. So voltage amplifier is characterized by g parameters current amplifiers by h parameter and trans conductance amplifier by y parameters and trans resistance amplifier by z parameters. So let us generalize this ideal amplifiers are always represented by an imitance matrix now imitance means it can be z h g or y in all this p i the self imitance at the input is 0 p r is 0 p i 0 means either it is voltage control or current control p naught 0 means either it is voltage source or current source p r 0 means it is unilateral that means there is no feedback transmission occurs only from input to output that is an important unilateral device. Ideal amplifier is unilateral there is no feedback they all have 0 reverse transmission parameters p r. So this is the characteristic of ideal amplifier. So let us now consider 2 port network parameters in terms of y parameters this is what is called as the macro model this is the macro model of any 2 port active or passive. So how do we represent this macro model v i at the input is transferred to the output as a current source y f into v i this is the trans admittance parameter and this is the self admittance when I short this effect vanishes so it is just nothing but the admittance at the input and why not is the admittance at the output port shunting this current source which is the transferred voltage appearing as a current at the output. Similarly whatever happens at the output the voltage at the output gets transferred to the input as a feedback as y r into v naught. So the resulting equation is the total current at the input is summation of the current in this this current is dependent upon v i v i into y i plus y r into v naught that is an equation governing the input port i naught is equal to v i into y f this current plus the current through the admittance at the output v naught into y naught this is the macro model just take an example of a passive network I have put an admittance y f between input port and output port this is linking the input port to the output port this is the common terminal. So the only way this can participate in transferring something from input to the output is in terms of voltage getting converted at the input okay to current at the output if I short this now this participates and the short circuit current is v i into y f since current is going to flow outwards it is minus whereas currents flowing into the port is considered as positive. So minus y f into v i is the transfer so this is capable of transfer from input to the output as current similarly if something happens at the output if I have a voltage at the output it gets transferred to the input as a short circuit current by the same admittance here. So the forward transfer admittance is same as reverse transfer admittance that is what you will net is minus y f here minus y f here the in short circuit admittance also is y f here and output also it is y f so all the parameters are determined by y f okay y f here minus y f minus y f this is the y parameter of this simple network and we have illustrated that forward transfer admittance is same as reverse transfer admittance for this network. Let us take an asymmetric network here we see that R1 is the resistance coming in series at the input and R2 is the resistance coming in series at the output and this is a common capacitor between the input port and the output port. So this is responsible for feeding something from input to the output and output to the input so this is a common network. So what is the result of this if I evaluate the y parameter of this okay the short circuit admittance here I short this and you get the admittance as this 1 plus SCR2 by R1 plus R2 plus SCR1 R2 that is this y i of this network. When I short this at this input port and look at it as an admittance here then it will be 1 plus SCR1 divided by R1 plus R2 plus SCR1 R2. If I apply a voltage here and get the start circuit current here which is the forward transfer that is nothing but minus 1 by R1 plus R2 plus SCR1 R2. If I on the other hand apply a voltage here and get the short circuit current here then it is the same it is minus 1 by R1 plus R2 plus SCR1 R2. You will see that forward and reverse transfer parameters are the same even for the asymmetric network. Now let us consider the effect of finite source resistance and load resistance when we put sort of passive resistance here between input and output. So this is equivalent to having a source with shunt resistance of RS and a load of 1 over RF. This has y parameter corresponding to RF here 1 over RF here 1 over RF here minus 1 over RF here minus 1 over RF here. This is similar to the example that was taken earlier. So if you now evaluate II is equal to VIYI plus V0 YR. I0 is equal to VIYF plus V0 Y0 and V0 is also defined again in terms of this RL as V0 by RL is nothing but minus I0. V0 is equal to minus I0 by YL. So I0 is equal to minus V0 into YL so we can therefore replace this I0 by this V0 into minus V0 into YL and we get V0 by VI which is called the voltage gain of this network which is minus YF divided by Y0 plus YL in terms of 5 parameters. This is what is called the voltage gain output voltage by input voltage which is called the voltage gain of this network. This is one of the characteristic properties of the two port networks. So we substitute that value of that V0 into YR V0 getting replaced by VI using this expression. So you will get this equation getting modified as this. YIVI minus YF by Y0 plus YL into VI into YR. So what do you get? We can actually now get what is called II by VI or IS by VI as 1 over RS plus 1 over RF plus RL as the total admittance seen at the input port. So using this expression we can say that II by VI is equal to YI minus YF YR by Y0 plus YL. Please remember this okay it is easy to remember. YI is the input self admittance minus okay whatever comes from output corresponds to YF by Y0 plus YL times VI okay. So that into YR is the effect at the input. So this is the effect due to the feedback YR coming from the output back to the input. So the input admittance gets modified as YI minus YF YR by Y0 plus YL for any load. If you do this for the output similarly you will get I0 by V0 at output admittance as Y0 minus you see I is replaced by O minus YF by YI plus YS that is the transfer occurring from input to output YF by YI plus YS and that into YR is the total input output admittance at the output. So if you just change I to O, L to S you get the output admittance from the input admittance. So these are the transfers you can clearly see that voltage gain of the two port network as discussed earlier is nothing but RL by RL plus RF that can be got from the expression that we have derived. Current gain is I0 by II which is nothing but the voltage gain minus A voltage gain into YL by YN okay. So that again happens to be equal to minus 1. In a port like this you can readily see the input voltage and output voltage both are the sort of same if I mean the input current and output current are the same. So current gain is 1 and output voltage is RL by RL plus RF that is readily seen here but using these parameters you can evaluate it okay in a general manner okay. Going to Z parameter, Z parameter is nothing but open circuit parameter where VI and V0 are the dependent variables, II and I0 are the independent variables. So VI at the input is equal to the drop across this self impedance II into Zi plus whatever comes from output current okay as a voltage source at the input okay this is the feedback coming from output to input output current to input voltage it gets transformed as I0 into ZR. Similarly the input volt current II gets transferred to the output as ZF into II this is the forward transfer plus that due to the output impedance the drop I0 into ZR. So this is the macro model of a two port network in Z parameters example is just an impedance in common to the input and the output. So effective Z parameter will be just Z Z Z Z again you see the feed forward is same as feedback the open circuit impedances these at the input and at the output. If you make it an asymmetric network using a Pi network like this then you will see that open circuit impedance at the input will be R1 parallel R2 plus R3 and at the output it is R3 parallel R1 plus R2 and if you apply a voltage at the input and try to find out the open circuit voltage at the output the forward transfer is R1 R3 by R1 plus R2 plus R3. Similarly if you apply a voltage at the output and find out the feedback voltage at the input that will be R1 R3 by R1 plus R2 plus R3. So these two parameters are again the same. Similarly as we had derived for the Y parameter the input impedance of the two port network for any load impedance can be derived as Zi minus ZR ZF by Z naught plus ZL. Zout can be derived as Z naught minus ZR ZF by Zi plus ZS. So this is including the self open circuit input impedance then ZF by Z naught plus ZL is what is transferred from input to output okay then it comes back to the output in terms of ZR. So this is the feedback effect and similarly Z naught is the output impedance open circuit again ZR divided by Zi plus ZS is what is fed back and that is appearing at the output in terms of ZF the feed forward effect. So current gain is I naught by I I it is minus ZF by ZL plus Z naught and voltage gain is AI into ZL by ZN okay. These are the four important parameters like we found out Y in Y out AI AV for the Y parameters. Now consider the effect of source and load with this acting as the two port network. So instead of Z we are replacing it by R then we can evaluate all the characteristics of the two port network using the Z parameter that we had already evaluated the input impedance output impedance current gain and voltage gain. It is a trivial example but it illustrates how nicely the two port network parameters are applicable to this. Now let us come to the two port network with G parameters. So what is important in all these things is that the G parameter the input voltage VI gets converted to output voltage through the parameter which is called the forward transfer parameter GF as GF into VI another voltage source. So this is the forward transfer this is what is called the voltage gain GF is the voltage gain and as for the feedback is concerned it is the current feedback current at the output gets transferred to the input as a current source which is GR into I naught. So these equations at the input port II the dependent variable is equal to GI into VI the independent variable and GF into I naught. So I think there is a mistake here it is GR into I naught which is what is shown in the figure right. V naught on the other hand voltage at the output is GF into VI in series is the drop here G naught into I naught. So these are the G parameters of a two port network. Now let us see you want to transfer using G matrix a voltage at the input to a voltage at the output. So the transfer parameter of importance is R1 by R1 plus R2 okay. In such an example therefore we had seen that this is a voltage gain or attenuation in this case it being a passive network. So the reverse transfer parameter is a current gain it is a short circuit current gain you short this then the current transferred from input output to the input is R1 by R1 plus R2 with negative sign because current going in results in current going out. So that is illustrated by the negative sign but these are okay magnitude wise remain the same. Then this is nothing but the open circuit input impedance which is admittance which is 1 by R1 plus R2 and short circuit output okay admittance which is 1 by R1 plus 1 by R2. So again we notice that forward transfer is same as reverse transfer but opposite in sign. This is characteristic of an ideal voltage control voltage source or amplifier where this is 0 this is 0 and this is 0 and this is finite. So that is how an ideal active device like voltage amplifier is characterized. We can just go back a little bit as far as this is concerned this is characterizing parameter for an ideal trans conductance amplifier where this is 0 this is 0 this is 0 this parameter alone exists okay. Then similar is the characteristics of an ideal trans resistance amplifier with this is 0 this is 0 this is 0 this is Rf okay. This is that of the voltage amplifier. Now we come to the last parameter H parameter. H parameter is characterized by input voltage the dependent variable II and V0 are the independent variable HI into II this is the drop here HR into V0 is the voltage source which is the feedback parameter okay coming from output voltage as voltage source okay. So feedback is a voltage ratio HR into V0. I0 at the output is summation of two currents one is HF into II transferred from input current so it is a current amplifier okay and then H0 into V0 is the additional current at the output. So the H parameter characterization is the dual of G parameter. So the approach here now is that it is the dual that means I have G parameter relating II and V0 to VI and I0 okay is this G matrix. So when we take the dual this goes to this side this goes to this side and this box here will contain the inverse of the matrix as far as G is concerned. So that is how we are going to treat it as so inverse of the matrix. Now how is it going to look like we had GI, G0 here GF, GR here. So inverse of that will be G0 by delta G delta G is the determinant of the matrix which is nothing but GI G0 – GRGF. So this is a matrix inversion is known to everybody and this parameter is GI by delta G and this is – GR by delta G and this is – GF by delta G. So this is how you can inverse the matrix. It is important the determinant can be rewritten as GI G0 into 1 – GRGF by GI into G0 okay. Now please note that – GF by G0 is the forward transfer admittance parameter and – GR by GI is the reverse transfer impedance parameter. So this product is called the loop gain here. This is the loop gain forward transfer parameter into reverse transfer parameter okay. Let us consider the G parameter of this network. In a G parameter okay the transfer forward transfer that is of importance is the current at the input is getting converted to current at the output by the network that is formulated here. So II at the input we have R1 and if you short this you have R1 by R1 plus R2 with the negative sign indicating outward flowing of current. So – R1 by R1 plus R2 is the forward transfer. The reverse on the other hand is nothing but the open circuit voltage gain. So if you feed a voltage here the voltage appearing here is R1 by R1 plus R2. This is a short circuit input impedance HI and this is open circuit okay output admittance 1 by R1 plus R2. If you open circuit the resistance seen is R1 plus R2 inverse of that is the admittance and HF is equal to – HR. An ideal active device which can be represented by H parameter is a current control current source okay or current amplifier it is called. It is having zero input power okay that is because VI is zero and HI is zero. The current at the input II appears as a current at the output HF into II. So we can really summarise this in this contest in the following manner. In the Y parameter we have seen taking the source admittance and load admittance into a current the following equations can be written II is equal to this, I0 is equal to this and I0 is also equal to – YL into V0 okay. So we can represent it this way this equation therefore eliminates okay V0 from this okay V0 can be now obtained in terms of VI which gives you voltage gain as okay – YF by Y0 plus YL V0 by VI okay and because of this Y in is equal to II by VI is YI minus the voltage gain actually so voltage gain is this into YR because it is YR into V0. So V0 is replaced in terms of VI this way and then you get II by VI as this similarly why now out can be got this way. So this is applicable to any parameter that is what we want to generalise here. So the input immittance and output immittance take the same nature in all parameters that we are illustrating here okay. So current gain on the other hand is this I0 by II nothing but the voltage gain into YL by VI same thing is illustrated in using H parameters okay. So if you write down those equations and I again I0 by HL is equal to – V0 so eliminate this V0 by using the expression V0 by II in this case which is going to be – HF by H0 plus HL okay. So this is the forward trans impittance input impittance takes the same nature look at this HI – HFHR by H0 plus HL. So in all parameters it is of similar nature H out is H0 – HR HF by H0 plus HL. So these two expressions remain exactly identical in terms of parameters. So forward trans impittance I0 by VI is HFHL by HIN into H0 plus HL. So they are similar in nature okay and therefore we can generalise saying that input impittance in any parameter is always P in equal to PI it could be YI, ZI, HI, GI it does not matter minus PRPF it could be YR, YF, ZR, ZF, HR, HF, GR, GR divided by P0 plus PL it will be Z0 plus ZL, Y0 plus YL okay, G0 plus GL, H0 plus HR. So total input impittance therefore which includes the source impittance also is PS plus PI minus PRPF by P0 plus PI or PS plus PI into 1 minus another way of writing PRPF divided by P0 plus PL plus PS plus PI which is what is called as the loop gain. So because of the feedback the self total impittance at the input both PS plus PI gets modified as PS plus PI into 1 minus the loop gain will illustrate this in several examples later on also but this is a general trend due to feedback. Output impittance PR is just change I to O P0 and L to S rest of the things I to O has been changed L to S has been changed PR into PF okay R to F you just change it does not make any difference it remains the same. Total output impittance therefore is PL plus PR which is PL plus P0 minus PRPF by PI plus please just replace I to O and S to L okay you get the output impittance again it is PL plus P0 which is the total impittance self impittance at the output port into 1 minus GL same thing total at the input port into 1 minus GL. So you see the effect of feedback in all these things this is the sum and substance of evaluation of all important parameters associated with any one of the 4 parameters put in a common form using impittance matrix parameters. So summarizing we can say that forward transfer impittance is always minus PF by P0 plus PL forward transfer impittance I mean one can be voltage gain the other will be current gain one can be impittance and the can be admittance okay. So these are the various characteristics in various parameters so it is PF PL by P in into P0 plus PL forward transfer impittance minus PF by P0 plus PL reverse transfer impittance minus PR by PI plus PL the loop gain is a product of these 2 PRPF by PI plus PL P0 plus PL negative if this product is negative it is called negative feedback if this product is positive it is called positive feedback again in terms of system level discussion will do this negative feedback and positive feedback in the next few lectures the composite impittance matrix therefore which is involving source impittance and load impittance there is a correction here this should be load impittance is this and therefore the determinant of the matrix that we had just shown is PI plus PS into P0 plus PL minus PRPL and what is the determinant it is the product of the self-imittance at the input self-imprint set the output into 1 minus the loop gain 1 minus loop gain characteristic polynomial of the network okay input imittance is PI plus PS into 1 minus GL output imittance is P0 plus PL into 1 minus GL it is easy to remember in this form right you do not have to keep on rederiving it so the conclusion today regarding the two port let us summarize we discussed two port network general two port network which can encompass active passive and the combination of these is characterized by okay H G Z and Y matrices parameters all of which can be represented by imittance matrix P and we had seen that passive network has the reverse transmission equal to the forward transmission only in the case of H and G sign is negative okay one is negative of the other okay and in the case of Z and Y the same okay this is important as for active devices are concerned ideal amplifiers are characterized by 0000 and PF to say that 0 input power can be got by either the input of the amplifier being voltage controlled or current controlled that means either II is 0 or VI is 0 and output is an ideal voltage source or a current source and it is unilateral that means transmission occurs only in one direction that is characterized by no feedback.