 Today we will be having the 4th lecture of the series and let us see what we had done in the last class. We had in the last class briefly describe about 4 products electronic typical electronic products and we had seen almost all these products involved in terms of analog signal processing functions amplification which is multiplication by a constant filtering which is mathematically equivalent to solution of differential equation. It can be transient or steady state oscillation which is nothing but solution of a second order differential equation with the first order term being absent that generates time mixing modulation demodulation phase detection frequency multiplication we saw all involve the simple act of multiplication by 2 variables that digital to analog conversion also can be multiplication treated as multiplication one variable is digital another is analog pulse width modulation is another way of multiplying amplitude with time with A to D conversion is nothing but comparison of continuous time voltage or current with a reference so that output is high or low the digital today will be dealing with a topic which is normally considered as the prerequisite for analog signal processing that is in networks and systems so networks and systems course you had undergone and that is a prerequisite for analog signal processing. Let us consider the topics in networks and systems which become relevant in analog signal processing 1 port network and 2 port network passive we have to understand passive networks how they are distinctly different from active networks what is it that characterises passive networks from active networks so these are the topics will be touching upon briefly as a revision of the prerequisite course material. So 1 port network for signal processing is the first topic that will consider review properties and signal process functions of linear passive active 1 port and then we will discuss the same thing with 2 port network. Network elements the basic network elements okay are the ones which are going to be touched upon passive network elements are not capable of power amplification that is why they are called passive active network elements on the other hand are capable of power amplification. Let us see these aspects later in the characterising these 1 port network elements what are the basic 1 port network elements which you have been already introduced to resistors capacitors inductors and diodes these are the passive network elements that all of you have been introduced to earlier 1 port active network elements on the other hand that give power amplification that are capable of power amplification the passive networks are not capable of power amplification. So that is basically a negative resistance independent current and voltage sources are also characterised as 1 port networks. 2 port network elements are 2 port passive network elements are which are common okay these are transformers and guiders again we will try to characterise this in terms of 2 port network parameters later. 2 port active network elements these are the most important things in analog signal processing these are called controlled voltage source controlled current source they can be controlled by again voltage or current. So these form 4 typical amplifier structures that are possible in electronics these are called amplifiers so these control sources are mainly called by us amplifiers. So these are the important topics which we are going to cover common terminology. So comparators these are not linear because input is analog output is digital. Control switches again they are not linear right on off control okay these switches are also going to be discussed multipliers multiplication of 2 variables voltages or currents or current and voltage or they need not be analog both they can be one analog another digital or both digital so all these combinations can do multiplying operation right. So these are normally called modulators demodulators so phase detectors so these are the operations they can perform these network elements are the ones that are going to be covered by us. So these are obviously the basic definitions of what forms passive network what forms an active network passive networks obviously comprise of only RLC and diodes active networks on the other hand can have negative resistance additionally coming apart from the passive components that are already included in the passive networks. Two port passive networks similarly comprised of RLC transformers and diodes and guiraters two port active networks on the other hand will comprise of amplifiers that are control sources and multipliers guiraters and transformers and all the other passive elements as well it is now considered the linear one port network as two terminals okay one port has two terminals where we can have input voltage and input current flowing through it but one of this only can be the independent variable the other becomes automatically the dependent variable. So if current is the independent variable that means if you are pumping in input current by using a current source ideal current source then voltage gets developed by the one port network if you are applying a voltage by using an ideal voltage source then current gets developed by the one port network. So how to characterize the one port network okay in terms of volt ampere relationship is what we are going to cover. So it is characterized by what is called imitance imitance means either admittance or impedance now depending upon what is the independent variable what happens is that if for example I is the independent variable this I into the Z is the output if this is the input variable output is current into impedance. So this impedance character is the one port if on the other hand V is the input variable that into Y is the output current so Y characterizes the one port and that admittance Y is the admittance can be frequency dependent depending upon the network. So Y of J omega S equal to J omega when you put as some real part plus imaginary part this is the conductance this is the susceptance. So if the conductance is greater than 0 then this network that we are considering as one port is called a stable network if on the other hand this conductance can become less than 0 or equal to 0 then it is an unstable network. So for example if this is greater than 0 right it can normally be formulated using only passive network whereas if it has to become less than 0 it is possible that this can happen with an active network inside the port network one port network elements let us therefore try to understand the basic elements that can exist in one port the first of all the resistor what is it that the resistor does it converts a current into a voltage if you pump in a current through a current source then output voltage appears as R into I okay if on the other hand if the input is a voltage it is the independent variable then output is current and then you call this same resistor as conductance it is inverse of R in a linear network okay G is 1 over R G is the conductance in expressed in Siemens whereas R is the resistance in ohms let us see pictorially how we can depict this relationship is called volt ampere relationship of the element resistor. So this is depicted in terms of a line passing through the origin with a slope equal to R that is the resistance value. So this is a linear resistance as long as it is having the same slope all over okay then it is a linear element same thing is depicted in terms of a conductance here okay where the input variable is a voltage and I is equal to G times V again the slope of this is indicated as G it only changes one variable into the other form current is converted to voltage or voltage is converted to current so it is just transforming one variable to the other so it is not significantly doing any great signal processing activity it is used primarily for attenuating voltage or current and also in later we will see data conversion operations in converting reference voltage into set of different voltages okay which are binary weighted capacitor this is one of the most important elements in analog signal processing we will see why this is the most important element it is doing an important operation if I is the independent variable then integral IDT is the charge collected by the capacitor that charge divided by C okay gives you the voltage across the capacitor. So it is doing what is called integration operation and integration operation is the fundamental operation involved in analog signal processing on the other hand the same capacitor if B is the input voltage to the capacitor then the current through the capacitor which is the dependent variable is CDB by DT it can be used as an integrated differentiator inverse operation of integration. So depending upon the input variable it can change its activity into integration or differentiation so this is the capacitor it can store energy one take resistor is only capable of dissipating energy okay in the form of heat whereas capacitor is an element that can store energy in the form of half CV square where V is the voltage across the capacitor. So this is a storage element inductor which is almost looking in terms of signal processing activity exactly like that of the capacitor only difference is V and voltage and current interchange and their roles if I is the independent variable then LDI by DT is the voltage across the inductor on the other hand if V is the independent variable I is equal to 1 over L integral V DT again it can be therefore used as differentiator or integrator based on what is independent variable and what is dependent variable. LI is the flux linkage in the inductor and half LI square it for stores energy in the form of electromagnetic nature and half LI square is the energy stored normally in present day electronics inductors are not preferred because for base band signal processing you will see that inductor size is enormous compared to the active devices which are may be transistors. So inductors are presently out of use in base band signal processing totally inductors exist only as sort of signal processing units at very very high frequencies of the order of diga hertz diode is a control switch will understand the diode characteristic as a switch presently current I is the independent variable in the forward direction when the diode is forward bias current is forced into it this way then the voltage across the diode which is the dependent variable is invariably 0 right. So the voltage is 0 in the case of a diode I greater than 0 V is automatically 0 okay V less than 0 in the reverse direction this is in the forward direction forward bias this is called in the reverse direction reverse bias direction V is less than 0 I is always equal to 0 this is what characterizes an ideal diode that is pictorially depicted here this corresponds to I greater than 0 and this corresponds to V less than 0 this is the characteristic of an ideal diode. So how is it a switch if you want the switch to be closed then we must pass through this diode f current in this direction if you want the switch to be open then a voltage has to be applied in the reverse direction so applying a voltage in the reverse direction opens the switch and applying a current in the forward direction closes the switch so that is why it is called a control switch two terminal element now we come to the important component which is active the previous four components are passive this component is the only component which is active in terms of one port what is it it is a negative resistance what is a negative resistance in the earlier positive resistance we had seen that the slope of the resistance was this way in the first and the third quadrant okay it is going whereas the negative resistance falls in the ideal negative resistance falls in the second and the fourth quadrant the slope is minus R so that is defined as an active device same thing can be considered if it is the voltage which is the input variable then minus G times V is the output current so we have this as negative resistance except that in practice if this negative resistance is embedded in a network comprising of positive resistances okay we will see that it can become positive the overall resistance can become positive this way that is it can become positive this way or it can become positive this way this is nothing but n type of negative resistance this is nothing but s type of negative this kind of thing we will see when we go to practical devices but this is the only way in a network when it is embedded with a negative resistance effective resistance can become positive this way or this way this we will remember one is called s type this looks like a letter s and this looks like a letter k so this is in summary what the negative resistance is capable of it is also negative conductance in c means now let us look at some of the practical utility of these basic elements forming a network one port network signal processing functions of one port networks now there are certain things which we are not supposed to do an ideal voltage source should not be shorted obviously if you try to have V as the independent variable you can create a current by having an impedance Z and Z is 0 okay means current goes to infinity so we have in this particular case V as the voltage source independent variable okay Vi that divided by Z when Z goes to 0 current goes to infinity so this is not allowed short circuiting a voltage source causes instability the current source I likewise if it is having an admittance as one port okay and the admittance goes to 0 then the voltage goes to infinity so the voltage going to infinity or current going to infinity okay is an unstable situation so you are not supposed to do this so obviously your tenance theorem is not applicable okay in terms of I mean converting source practical source from voltage to current or current to voltage okay the Nordern equivalent of a voltage source okay practical voltage source is not existing in the case of ideal voltage source okay and ideal current source as only Nordern equivalent no tenance equivalent so resistors convert voltage to current or current to voltage resistor or conductor they can be used in practice for building attenuators now practical source has a voltage in series with its internal impedance which is called the source resistance RS so we are connecting this resistance R as one port to it so then voltage at the terminal V naught becomes R by R plus RS and this N ratio attenuation is always less than 1. And if it is a current source shunted by an internal resistance RS then when R is connected to it RS by R plus RS is the attenuation of current and that is always less than 1. So it is useful as attenuator now the capacitors we have seen can be used for either integration or this is the integration okay as you pump current into the capacitor the output voltage is integral of current into DT divided by C or if applied voltage is the one that is of concern then I naught becomes C DVI by DT. So similarly the inductor acts as either an integrator of voltage or differentiator of current. Now what is the actual signal processing activity integration and differentiation if it is a sign function that you are feeding the sign gets converted to cosine function right. So a phase shift of 90 degree gets generated right and therefore it is called quadrature function that is a sign and cosine are quadrature functions right phase shift of 90 degree exist. Now if you put a combination of a source with capacitor ideal not ideal source non-ideal source with capacitor then let us see what it does it is nothing but a filter function II is the independent variable V naught is the dependent variable and therefore we have here a classic case of a filter a filter is nothing but differential equation it may be first order in this case order of the filter depends upon the number of independent reactive elements like capacitor or inductor that come in the network. So in this case there is only one capacitor so it is a first order network which simulates a first order differential equation. We see this the output voltage V naught by R is the current through the resistor and CDV naught by DT is the current through the capacitor so total current IS source current is going to be dependent upon this plus this right DV naught by DT therefore is equal to IS by C – V naught by RC this is a first order differential equation if you solve this you will get V naught by IS as a transfer function which is R by 1 plus SCR it is an impedance function first order impedance function. So this is nothing but a low pass filter right that means basically if you plot this function right as a function magnitude of this function as a function of omega S equal to j omega then you will see that this is at low frequency is equal to R and beyond a certain frequency that it gets decreased at 20 decibels per decade. So this is what it is R divided by 1 plus j omega CR magnitude which is R by square root of 1 plus omega CR square all these things are known to you right. So this is a low pass characteristic or it is a low pass filter function. Now let us do some modification to this to understand what exactly is going to happen if it is made active the previous network was passive network purely now by shunting it by means of a negative resistance of minus R1 negative resistance of R1 or resistance of minus R1 what happens to the first order differential equation we get V naught by R minus V naught by R1 plus CDV naught by DT equal to IS. So it is still a first order differential equation what can now happen is earlier all the coefficients were remaining positive now this coefficient of V naught can become negative by proper selection of R1 in relation to R. So this is what is going to be done next so the parallel combination of R with R1 is a single resistance R dash which is R R1 by R1 minus R so it can become positive negative R infinity. So if R is less than R1 okay then what happens is this becomes negative this no this is R is less than R1 this is still positive and therefore it is a low pass filter if R is greater than R1 on the other hand the transfer function as negative real power and the impulse response of this independent variable now what happens is earlier the solution was solution to this is going to be E to power minus T by R dash C into some a magnitude this is the solution to this. So whatever energy is put here decays exponentially this is a decay if R dash is positive if R dash is negative this whatever energy is put in this in terms of a voltage or current that keeps on exponentially increasing exponential decay means this is the decay from the initial value at T equal to 0 exponential growth is this that if R dash is negative it is exponentially growing it explores right. So the system becomes unstable such systems are not to be considered this is what we had already enunciated earlier in terms of admittance function having negative real part. So any admittance function that has negative real part causes the energy that is put inside that to explode okay become infinity. So it is unstable so integrator itself is unstable because at omega equal to zero this becomes infinity okay. So whether it is equal to zero or less than zero it is unstable as long as the negative real part is not existing or it is always positive real part the system has stable response the energy that is put inside the system always decays finally after the energy decays fully right it is again ready for its function right this is the transient response of such systems. So this is what happens in a first order network with active device incorporated in it okay. Let us now go to the most important one input network that we have to consider that is made up of resistor the passive element inductor okay capacitor and negative resistance. So this is these three are the passive components and we have the active device here. So just like the previous case this is another differential equation however this differential equation that is written with respect to V naught as the node voltage V naught by R is the current in this okay and 1 over L integral V naught dt is the current in the inductor CDV naught by dt is the current in the capacitor and the current in the negative resistance is this minus V naught by R1. So this is indicating a second order differential equation okay and it can be solved and V naught by IS now becomes the transfer function which is the impedance of this okay given input current input current into impedance gives the output voltage okay. So V naught by IS is the transfer function here which is an impedance function which itself is represented as SL by S squared LC plus SL by R dash plus 1 and this can be written in what is called normalised form okay this is 1 okay this is S by omega naught Q this SL by R dash is written as S by omega naught Q and this is S squared by omega naught squared. S by omega naught is called the normalised frequency S by omega naught is dimensionless and it is called capital S and that is called the normalised frequency the way okay this is the way it is normally written. So omega naught is called the resonant frequency 1 over root LC this is really called a tank circuit which can store energy okay depending upon the relative magnitude of R and R1 will let us see omega naught equal to 1 over root LC is the resonant frequency okay the capacity reactance gets cancelled with inductive reactance okay at this frequency Q is a parameter okay which characterises the second order system this Q is normally written as in control systems 1 over 2 zeta zeta is the damping factor right that by comparing the coefficients we can say Q is equal to R dash by omega naught L it is the relative magnitude of the resistance effective resistance R dash is a parallel combination of R and R1 with the reactive impedance omega naught L that is Q. So R R dash is into root of C by L it is let us see its response okay. Now this is going to be having 3 cases to be discussed if for example R is equal to R1 what happens is Q becomes actually infinity because R dash goes to infinity so Q becomes infinity Q is infinity so at omega equal to omega naught at resonance this whole function goes to infinity so that is the case of what is called oscillator this becomes a harmonic oscillator equation the coefficient of dv naught by dt goes to 0 so this is a second order differential equation with dv naught by dt coefficient going to 0 which is the harmonic oscillator equation that means it oscillates at a frequency omega naught 1 over root Clc right for R dash positive it is going to whatever energy is put in this capacitor or inductor decays in terms of oscillation if the Q is high okay it decays this is the solution to it this you had already worked out in mass right the solution of this it is a decaying oscillation right. If R dash is negative R1 okay R1 is dominating over R that is R1 is less than R then it is going to grow in magnitude oscillation is going to grow exponentially this is decaying exponentially at Q equal to infinity oscillation is sustained okay. So this is what is explained there in this slide now let us consider the example of building an amplifier using a negative resistance you have already seen the earlier application of negative resistance in the form of oscillator building or what is called a second order filter design that is a band pass filter or that is the previous one is also called a band pass filter it is the steady state solution results in a band pass action filter action okay. Now we will consider the negative resistance application for amplifier the problem given is design an amplifier using a negative resistance of a voltage gain of 10 for a voltage gain of 10 the voltage source as a source resistance of 1K and the load resistance is 2K so we have source resistance of 1K now that should be a voltage source it is a voltage source the symbol is wrongly indicated there. So we have a voltage source with source resistance of 1K having a load of 2K now we want to build an amplifier across this so we have to put a negative resistance across this minus R. So then the equivalent resistance with this 1K and 2K become 2 third K and minus R is the negative resistance so we have here minus R divided by minus R plus 2 third K which is represented as R by R minus 2 third K okay. So that is going to be multiplied by the 2 third 7 inch voltage 2 third VS okay and therefore the gain has to be made equal to 10 from which we get R equal to 5 by 7K as the answer. So if you have a negative resistance of magnitude 5 by 7K then the voltage gain is 10 that is how we have designed. So you can simulate this circuit with this kind of negative resistance and then see that it amplifies the voltage at this point across the load by a factor of 10. So we have the yeah this is what we have explained earlier that the earlier application is nothing but a resonant frequency filter which is a band pass filter with frequency of resonant as 1 over root LC and the bandwidth of omega naught by Q. So that will be represented normally that is the impedance function plotted against omega naught this is at 1 by root LC which is omega naught okay and this is the bandwidth which is omega naught by Q. So if this is the maximum you are going to 0.7 times the maximum here okay. So these are the 3 dB points and if you call this a delta omega then this is the bandwidth R 1 equal to R is a sine wave oscillator amplifier design of the negative resistance. We come to the last example that you are considering design a diode resistor 1 port network with VI characteristics as shown. This is what is called piecewise linear approximation to any non-linear characteristic that we would like to generate. So this is having a fixed slope up to 1 volt output okay for a 1 third milli ampere input okay the slope of this is equivalent to a resistance that we already seen right going in the first and the third quadrant with a constant slope through 0. So the resistance is 1 volt by 1 third milli ampere okay that means it is 3 kilo ohms. So let us try to solve this problem if you want to obtain this characteristics this 1 volt divided by 1 third milli ampere gives you 3K as the resistance to be put. So this is something that we can see we have current coming through this 1 port and across 3K okay it will develop the required voltage that we desire. So it will develop 1 third milli ampere I mean if you pump in 1 third milli ampere 1 volt will be developed across 3K that is it. Beyond that the slope has to change the changing of slope is brought about by bringing in another resistance in parallel with this 3K so that the slope gets reduced. So that the same current now passing through a parallel combination of 3K and the new resistance will create this slope okay. So the slope again can be computed this is 2 third milli ampere this is 1 third milli ampere so again 1 third milli ampere going this way okay for an addition voltage drop okay which is going to be 0.5 okay 1.5 here okay so this is 0.5 additionally so 0.5 got by 1 third milli ampere. So we can generate that kind of voltage by bringing in 3K okay this 3K is brought in parallel with this the moment the voltage goes to 1 volt this above 1 volt this diode conducts because other terminal is connected to 1 volt. So the moment the voltage goes above 1 volt the current through this is in this direction is getting forced and therefore the voltage across the diode is 0 okay. So this 3K comes in parallel with this 3K brings down the resistance to 1.5K. So we have the new slope now going okay up to the 1.5 volts at 1.5 volts we require the slope to become equal to 0 and that is facilitated the moment this voltage goes above 1.5 volts by any current flowing in this right. So that current forces the diode to conduct and maintains the potential here for all the higher currents at 1.5 volts. So this is depicted here up to this 1.5 volts okay this is new resistance that is brought in shunt with 3K manages the slope there after the slope becomes 0 the output voltage becomes constant independent of rest of the current okay because the rest of the current flows through that okay battery of 1.5 volts okay. So that is it remains constant same thing is depicted in the other direction by use of diodes in the reverse direction and voltages in the reverse direction you can see negative 1.5 volts negative 1 volt bringing in the additional 3K and this 1.5 volts okay in series with this bringing in this constant 1.5 negative voltage in this direction. So that is how we can obtain this piecewise linear okay combination of resistors okay to form a one port which can give you any non-linearity in terms of piecewise linear approximation. And this is commonly used for transforming for example a triangular wave input current okay into what we call as this kind of output voltage okay. This therefore can approximate a triangular periodic wave okay in time domain to almost a sine wave this is one of the common ways of converting a triangular wave form into a very accurate sine wave okay by piecewise linear approximations okay. So this is the last application for this lecture and after this will be discussing in the next class the two port network and its application like we did in the case of one port network and you see the power of signal processing just using the knowledge that we have gained in the previous course okay for analog signal processing application. So today's lecture we had discussed one port network elements basic network elements passive network elements like resistor, inductor, capacitor and diode and active network element negative resistance then we had seen the application of these elements their characteristics and their application in a variety of one port signal processing activities like building a net innovator building filter low pass filter band pass filter oscillator sine wave oscillator and a diode function generator to convert any periodic wave form to any shape that we desire.