 Today we will be having the seventh lecture of the series in the last lecture on passive electronic devices for analog signal processing. We had been introduced to one port and two port devices that can perform signal processing actions and we had seen attenuators and amplifiers introduced in terms of network theory ideal amplifiers. Then we had apart from attenuation and amplification which we have touched upon in detail in terms of network topologies. We have other signal processing activities which will be discussed later filtering. Filing is the process of solution of differential equations integration differentiation etc can be done with the help of inductors and capacitors and making use of these you build filter which is equivalent to building second order or first order differential equation or nth order differential equation. Then other signal processing activities the mathematical operations that are involved amplitude modulation and demodulation which is a process of multiplication frequency modulation and demodulation designing oscillators whose frequency is linearly related to voltage or current. Filing is another operation of multiplying two variables voltages or currents this can perform modulation and demodulation D to A conversion is another way of multiplying analog signal by a digital word or something like that so it can also be treated as a multiplier A to D conversion is basically done by a comparator for example comparator is a 1 bit A to D converter analog at the input gets changed to high or low at the output corresponding to 1 or 0 of the digital world automatic gain control this is the front end of most of the receivers that is television or radio receiver or cell phone it is required to maintain a constant received strength in order to have volume which is not varying or feeding power amplification ultimately to actuate the controller or display show it in the display or video display or audio output power has to be amplified ultimately power supply management now in order to bias all these devices properly in the active region you need power supply and it has to be managed well in order that one signal does not interact with another signal processing at circuit signal generation which is normally needed for test purposes like sign view triangular view square view and saw tooth such waveforms are needed okay so for test purposes and also for acting as a synchronizing clock in most of the digital processor and some of the analog process mixed board circuits mathematical operations involved are basically so these are the signal processing activities which are of primary concern in analog the multiplication of a variable by a constant which is normally termed as amplification or attenuation multiplication of two variables so this is the mixing operation that I had already explained then comparison so compare the input voltage with a reference and get an indication as to when it crosses the reference as high or low at the output of the comparator so these are the important operations which will be involved in most of the systems that we discuss so then the devices capable of power amplification these are called active devices devices that cannot provide power amplification are termed therefore as passive devices that is characterized by we had seen in two port network it is characterized by for the passive device the forward transmission and reverse transmission factors are the same in Z and Y matrix and opposite in sign in H and G matrix representation of the two port active devices on the other hand do not follow this road devices that perform the core mathematical operations the passive devices are resistors inductors capacitors crystals and diodes these are the ones which are going to be discussed today introduced as components what are the practical devices available today okay and active devices like op-amps comparators multipliers FETs and BJTs so we will be getting introduced to the practical devices in today's lecture let us consider first the resistor R is equal to rho L by A this is well known equation for resistor in terms of resistivity rho is the resistivity in ohms centimeter if it is L is in length of the resistor in centimeters and A is the area in square centimeters now obviously you see that whether it is very high resistor or low resistor it takes huge amount of area this is the reason presently the out of the passive component the resistor which is primarily needed to convert voltage to current or current back to voltage these are to be used as FIVA number as possible in any system because it takes up area it is not compatible with the main device active device transistor for example MOS transistor for example that is used in today's electronics so and the commercially available resistors which are available in different with different materials like carbon via metal film and semiconductor semiconductors are the material is normally not suited very well for resistor because of the sensitivity with respect to temperature however in ICs one has to go for this so precision resistors are not really available in IC form available only in thin film or thick film components now these are available because of restriction on area in terms of fractions of ohms to several mega ohms only the tolerance available in discrete circuits can range from 0.1 to go up to 20% or 30% in integrated circuit components we have these available with different power ratings available in different packages the resistor can have on idealities in terms of parasitic capacitance series inductance thermal noise is a source of voltage source that is generating noise because of its inherent nature of resistance. Parasitics become important in high frequency and high precision analog signal processing circuits that is the equivalent circuit of resistor because of series inductor and shunt capacitor it almost becomes a resonant circuit if the shunt capacitance comes into play first the effect is to reduce the impedance to values less than R as frequency increases and the bandwidth of this resistance is limited to 1 over R into CP when the inductive effect comes into picture it might resonate in certain cases may be with free high Q so 1 over root LP CP is the resonant frequency and net effect is the impedance increases with frequency effect of parasitics therefore limits its use to a certain bandwidth 50 kilohertz and carbon type about 1 megahertz foil resistors can cope up to 100 megahertz. Now coming to the capacitor so since resistor is only converting voltage to current or current to voltage we need not change this variable in signal processing can retain either that as voltage or current in signal processing and use only once a precision resistor outside an integrated circuit okay for converting voltage to current if it is a reference voltage it can be converted to a reference current by using a precision resistor and once this is done the signal processing can be carried out either only in terms of voltages or in terms of currents. Capacitors are generally made with dielectric materials and which between two conductive electrodes C is equal to EA by D epsilon is the permittivity in terms of Farage per centimeter of the insulating material separating the two electrodes with area A and D is the distance between the electrodes in centimeters. Now capacitor is an important component fortunately for us the integrated circuit capacitor almost looks like the discrete circuit counterpart however the tolerance is pretty poor in integrated circuits. The discrete circuit capacitors the dielectric materials used are ceramic tantalum, polyester, polystyrene, polypropylene, polycarbonate and metallized paper, teflonic air etc. Electrode material is primarily made out of aluminium. Energy stored in a capacitor in electrostatic form is half CV square so it can be used as a storage element. Polarized capacitors have pre specified polarity and of a large value of capacitors. However in integrated circuit and systems it is necessary to minimize the use of discrete capacitors because they are going to be costly and out of proportion compared to the size of the system that we are emitting in electronics today. So we have three picofarads to several hundreds of micro farad of these discrete circuit capacitors available. But in integrated circuits are limited to only few picofarads or less. Voltage ratings with different voltage ratings these capacitors are available and with different mountings okay and different formats. Again capacitors have a leakage resistance equivalent series resistance which is very important ranging from 0.01 to several ohms. So ideal capacitor is a short circuit at high frequencies whereas in fact because of the series resistance it is no longer considered as a short circuit the series ESR limits its ability to short. And lead inductance further makes it non-ideal and instead of acting as a short it may act as an open circuit. The effects of these parasitics become important in again high frequency signal processing. As the frequency increases the net impedance decreases that is the normal ah capacitor ideal capacitor. When the inductive effect comes into picture the resonant frequency makes it ah that is beyond 1 over root LC LP into C the net impedance becomes open circuit instead of a short circuit. Electroletic capacitors have inductors beyond a few megahertz which is why ah small ceramic capacitors are normally put in parallel with them in order to make it appear as a true short circuit at high frequencies. Ah Aluminium and tantalum electrolytic capacitors with non-solid electrolyte have high ESR values up to several ohms. So this has to be borne in mind while using electrolytic capacitors. Inductors are actually we have seen that inductor does the same signal processing activity like integration and differentiation that the capacitor does. So inductors are not compatible with present day ah IC technology ah because of their coil and ah nature of being available in the form of a coil. So ah they are ah not preferred in electronics today ah well except at very high frequencies like gigahertz range the size of the inductor becomes small enough to be considered for ah signal processing activity. Inductors are coils on a substrate or coils bound around magnetic cores unlike resistors and capacitors inductors are not so easily made available commercially they are made to odd. L is equal to mu n squared A by L again area place error that means ah large valued inductors are unthinkable in present day electronics. So they are not used at all in ah base band signal processing activity. Inductor has a series resistance RS associated with it making it have what is called a quality factor. The quality of the inductor is defined as ratio of inductive impedance divided by the series resistance. So RS ideally speaking should be 0 so that 3 of the ideal inductor is infinity whereas in practice RS is limited for very high frequencies it may be of the order of ah 3 to 5 ok gigahertz range that is the kind of inductor that is available today in integrated circuit compatible device. Inductor store energy as half Li squared in electromagnetic form. So this also can store energy ah in ah electromagnetic form. So parasitic associated with again in all these things you will see that the ah device ultimately becomes a resonant circuit either series or parallel resonance equivalent. An inductance has a series resonance ah resistance RS as the series resistance parallel capacitance of CP. So inductors have resistance inherent in metal conductor of the order of your ohms. This can further have hysteresis and eddy current loss added to the ah DC resistance that we have already considered. Crystal this is an important component which is ah still in irreplaceable in IC form right. It cannot exist in IC form because of the very nature. The crystal is a vibrating mechanical resonant system with an equivalent electrical resonant circuit shown. It is mainly a series resonant circuit ah with a very high Q value ranging from 10 to power 4 to 10 to power 6. You can see the ah series resonance found by the inductor coming in series with ah finite resistance and a series capacitance. So this is forming a series resonance circuit. Ultimately when resonance occurs at 1 over 2 pi root L 1 C 1 series resonant frequency this is the limiting resistance R 1. Across which we have the parasitic capacitance that is coming C naught and the parallel resonance that occurs is at a frequency 1 over 2 pi L 1 into C 1 C naught by C 1 plus C naught. So that series resonance and parallel resonance ah occurring very close to one another that is the beauty of the crystal that it can change from series to parallel around the same frequency okay and therefore realizing a huge range of impedances inductive or capacitive around the same ah frequency that is the resonant frequency. So this resonant frequency of crystals can range from hundreds of kilohertz to about few megahertz tens of megahertz is represented as shown mainly used for generation of precision frequency clock signals. The impedance function of the crystal is given by if you define Q S as omega L divided by RS and omega S L by RS omega P L by R S is Q P then the impedance 1 over SC is having both zeros and poles governed by this equation. So it has parallel resonance where this goes to zero and series resonance where this goes to zero. So that is the ah nature of a crystal series resonant frequency is 1 over root L 1 C 1 and parallel resonance is omega P is equal to root of 1 by L 1 C 1 C naught by C 1 plus C naught. So it is close to the series resonant frequency differing by only this factor since C 1 by C naught is normally much less than 1 okay this factor goes to zero close to zero compared to 1. So omega P is close to omega S the quality factors Q S and Q P are defined as this as this is the case omega P is very close to omega S. Now let us consider the only ah two terminal element which is passive but ah very useful non-linear device this is the ideal diode this we have already explained to you the ideal diode characteristic which is characterized by this kind of relationship that for I greater than zero it is called forward biased the V voltage V is zero uniquely for V less than or equal to zero I is equal to zero this is the other region reverse biased region that means it can be ah switched on by biasing by a current it can be switched off by applying a voltage in the reverse direction this is an important thing it is a switch which is a control switch we will see how it can be used in a variety of applications later. Now coming to the practical semiconductor diode which is very important in its application a semiconductor diode is formulated by a P type material and N type material okay forming a junction PN junction is important. So once you have a PN junction like this there is okay ah motion of okay holds from here to here where there is abundance of electrons okay electrons move from here to here causing a built in potential here that this N region becomes positive and P region becomes negative and there is a built in potential between the junction. So this particular thing is going to oppose current flow okay when it is reverse biased this potential will increases okay and therefore there is no flow of current there is only flow of leakage current which is called IS okay. Then when the built in potential is decreased by cause biasing gate okay so that current flows okay in the forward direction then the potential decreases and the current flow increases okay as the current flow increases okay it conducts more and more. So it is a short circuit almost equivalent to okay in the forward direction and the open circuit in the reverse direction short circuit means you can take any current in that direction. This is explained in terms of the equation that I the current flow of a diode semiconductor diode is governed by the this is the reverse saturation current IS into exponent V is the forward bias voltage by VT KT over Q this is the thermal voltage minus 1 is the famous equation that governs the diode current flow. This is the forward characteristic and this is the reverse characteristic where it goes to saturation ultimately it breaks down this breakdown can be either a Zener breakdown dominated phenomena or avalanche breakdown okay at higher voltages avalanche breakdown dominates at lower voltages Zener dominates at our 3 volts okay both are of the same order okay. So the diode current therefore is exponentially related to voltage in the forward direction and in the reverse direction it saturates to minus IS. So this is below breakdown so this reverse current is heavily dependent upon temperature and it is roughly doubling for every 10 degree rising temperature. Another thing that we should know is the value of VT that is considered here it is an important thing so VT is KT over Q it is equal to 25 millivolts at T of about 300 degree Kelvin. So this is the important parameter associated with the diode the thermal voltage okay is T by 11600 and becomes about 25 millivolts at room temperature. Again the same equation can be rewritten as V equal to VT log I is the forward current by IS reverse saturation current. V is therefore a complex function of temperature the temperature coefficient of V is very important it is used okay for sensing temperature so doing it by DT is roughly of the order of minus 1.5 millivolts per degree centigrade rising temperature to minus 2.5 millivolts per degree. So it is very useful in today's electronics as a temperature sensor. Again this is the macro model which is a higher level macro model than what we have considered for the ideal diode. So for a semiconductor diode the conduction you can say is almost non-existence up to a cutting voltage this is called a cutting voltage of about 0.5 to 0.7 volts that is put as a battery in series with a slope inverse of the slope of this which is RF and ideal diode. So we have a variety of diodes available to us rectifier signal rectifier diodes this is for rectifying signal frequency components photo diodes light emitting diodes opto couplers light emitting diode with okay sort of light detector like photo diode is called an opto coupler or opto isolator this is also called opto isolator. So opto coupler or opto isolator get formulated with night emitting diodes in couple with in sort of photo diode or photo transistor. Varector diode is a variable capacitor short key barrier diode electro static discharge diodes protection purposes of protection of the circuit okay particularly get oxide of the MOS input okay is protected in normally if it is made available to the general public for use by this diodes which are coming in shunt with the input of the MOS RF diodes pin diode tunnel diode is a negative resistance device okay n type of negative resistance for example that we had discussed earlier DIAC okay this is the S type of negative resistance that ahh device people give as reference or indication n type and S type solar cells backward diodes that breakdown occurs right at 0 volts so it is used for as a rectifier diode for low voltage signals around this before the breakdown occurs breakdown is considered at 0.6 volts. So if the voltage is less than 0.6 then you can use it for rectification of such signals large current rectifier diodes or power diodes. So signal rectifier diodes are ahh less than less used in present the electronic circuit okay the power diodes okay may require heat sinking arrangement or sort of dissipating the heat generated. Now consider the Zener diode the Zener diode model is that is these are diodes with the specific breakdown voltage okay which are manufactured these are called Zener diodes and the equivalent circuit is the Zener voltage in series with the Zener resistance okay and the ideal diode that is what is called any current beyond which only you can consider the voltage is fairly constant at VZ. So power dissipation of the Zener corresponds to the operating current into VZ so there is a maximum dissipation which it can handle carrying a certain current. Temperature coefficient of this normally if it is avalanche breakdown that is beyond 3 volts it is going to have ahh positive temperature coefficient and less than 3 volts it has negative temperature coefficient. So typically you can see this is a commercial ahh Zener 12 volts Zener it has a ahh leakage current knee current of about 10 microamperes RZ is of 30 ohms and the temperature coefficient of Zener as about 13 millivolts per degree centigrade Toshiba CMZB12. Now this can be very well used in sensor diodes applications if you have two perfectly matched diodes two diodes in a package then they would have the same leakage current okay if they are identical in geometry then V1 the forward voltage of diode is KT over Q log I1 by IS V2 is equal to KT over Q log I2 by IS IS being the same for both because they are matched Q over K is 11600 millivolts per or old millivolts per degree centigrade the diodes are matched the differential voltage V1 – V2 becomes KT over Q log I1 by I2 KT over Q we know already is 25 millivolts at 300 degree Kelvin so it will be 25 millivolts by 300 degree centigrade it is the value of K over Q. So it is a very well defined constant and such a constant is very rarely possible as a temperature coefficient for any voltage so accurate so precise it is enough if you make log I1 by I2 as a design factor and even if I1 by I2 varies by a small amount due to reasons unknown then logarithm of that varies very little with the resultant effect this is used as a reference in most of the electronic applications of present day systems so it is least sensitive to variation in ratio of the current right. So this is about 11 millivolts per degree centigrade this is the commercial pair which is made out of this principle this is a CCO instrument S5813A okay sensor with sensitivity of 11 millivolts per degree centigrade so that has been multiplied by factor N so that this factor is obtained as sensitivity. So it is about output voltage of 1.94 at plus 30 degree centigrade at output voltage after amplification. Now let us consider some of the application examples this is a half wave rectifier when a sine wave is fed here this diode conducts only for positive going signals okay and therefore this removes the negative half so you get here across this a half wave rectification if this wave form is like this that you get here only a half wave this is depicted here for an input voltage of 2 volts and this frequency is 50 hertz so 20 milliseconds is time period you can see a half wave this is what is called a peak detector this is normally used in radio receivers also to detect the received AM signal. So the IF with the modulation is applied here and this gives you the sort of envelope riding over the DC right. So this is a leaky peak detector there in this case if it is just a peak detector this will try to give you the maximum voltage that is possible at any given instant of time it will hold on to that maximum if it is not leaky. So if the waveform is going on increasing like this it will keep on detecting the peak of the peaks. So this is one of the industrial applications also in order to just detect ultimately the maximum or the minimum you can change reverse the polarity and get the minimum of any time varying voltage. Now the peak detector with a leaky path here connected to a Zener regulator Zener diode forms one of the basic voltage regulated output that you can get out of the power line frequency. So this tries to charge up to VP then it is discharged through this path VP – VZ by RS is the current that will flow okay and that is the average current VP – VZ by RS okay and portion of the current goes through this after the load current whatever load current is drawn which is VZ by RS. So this is going to be so getting charged up to VP then it will discharge roughly with this current and the current drawn from here is VZ by RL all the time the rest of the current flows through the Zener which is VP – VZ by RS – VZ by RL flows through this Zener. This should be at any given instant time should be greater than I need okay and should be less than I Z max based on power dissipation allowed okay you can come up with I Z max which is nothing but power dissipation divided by I is that is power dissipation max okay VZ I Z max. So these are the equations that govern the design of these voltage regulator that is worked out here you can see that this is the input voltage applied which is VP sin omega 3 omega is 2 pi f f is 50 hertz right. So then it is only going up to the peak of this positive peak here then this charging at a rate which is VP – VZ by RC okay it almost looks linear as long as the fall is not much again it will there is a small diode drop here okay then it again discharges. So this is the charging and this is the discharging so this particular thing can be indicated this is the ripple that is occurring across the Zener. So this drop is due to the 200 ohms that RS series resistant that we have put these are the equations that have already indicated ripple peak to peak is roughly VP – VZ by R is the current of discharge that divided by C into time P rate T this is the peak to peak ripple which are if it is assumed to be linear okay if it is a good regulator we make it linear okay it is part of the exponent okay and we can approximate it to this peak to peak ripple. So we can fix the ripple at whatever value we need by selecting the proper capacitor there. So large capacitor may be needed in order to limit the ripple okay to small values and VP minimum – VZ by RS – VZ by RL minimum should be still greater than IZ the new current. So that the Zener works in the range where the voltage is constant Zener voltage remains fairly constant VP – VZ by RS VP max – VZ by RL max should be still less than PD max divided by VZ okay so that it is limited to operate below its power dissipation. Now these are the parameters connected with any voltage regulator any voltage source for that matter. Load regulation is the percentage change in output voltage for load current change from no load to full load there is a large signal parameter that you have to vary the what is that load from no load to full load and find out the percentage variation of the for example in this case Zener okay because the Zener current is changing so if the Zener voltage will change by a small amount line regulation same thing percentage change in output voltage for line voltage change from its minimum to maximum line voltage itself may be changing fluctuating. So with respect to that variation from minimum to maximum it should still work within the Zener range and there is a variation of voltage small variation. These two are large signal parameters because that depends upon the nonlinearity of the Zener regulator okay ripple rejection factor this is a small signal parameter percentage change in output voltage for a percentage change in input voltage for a given load. Now the input voltage has a ripple at 50 hertz or 100 hertz depending upon whether it is a half wave or a full wave rectifier that ripple gets reflected at the input at the output okay and what is the ability of the Zener regulator to reject that ripple. So that depends upon the Zener resistance RZ this factor depends upon RZ. So smaller the RZ better it is in relation to RS it is mainly the ripple rejection is RZ divided by RZ plus RS output resistance is mainly to do with RZ actually it is RZ parallel RS. So this is kept very small and therefore it is a good factor of reduction of ripple that happens here. So this is a practical circuit with 220 volts plus minus let us see plus minus 10% voltage change so you know the minimum okay this 10% of this is 22 subtract 22 from here say 200 okay to 240 variation. So for that variation you can find out the percentage variation in the output voltage that is the line regulation factor. Then when the load changes from 1K to 10K you can find out the variation in this that determines the output resistance okay load regulation. Output resistance corresponds to the chain that is occurring when this load changes and the change in output voltage for a change in output current it is a small signal parameter. Ripple rejection when there is a 50 hertz ripple here or in this case 100 hertz because it is a full wave rectifier right. So you can find out the change in % change in ripple at this point relative to this. So this is what is called the performance factor of the linear regulator. The last application that we are considering is that of a diode multiplexer. A diode multiplexer here is nothing but a multiplier we are applying a control signal here okay such that these diodes conduct in one direction okay when this is positive this is same voltage negative that is the control signal applied. Then this these pair of diodes conduct these pairs conduct making this a short. So then the signal is connected to the load okay when this reverses this also reverses this is negative this is positive then these diodes are reverse biased these connections are off this is an open circuit this is an open circuit this is an open circuit. So the signal gets disconnected from the load. So these multiplexers are used this may be your PC this may be the temperature measuring circuit you can get it connected when you want you can disconnect it during the time other signals with similar integers connect to the PC. So this analog multiplexers are commonly used with so the PC the digital data sampling okay is done by this this is a sampling gate. So you can actually connect many similar low frequency data to the PC based on the control signals that are applied here. So it is equivalent to a multiplier when this is positive and this is negative it is connecting the signal that means it is multiplying by some alpha factor. And when this is negative and this is positive it is multiplying by zero that means nothing of the input goes to the output these are the control signals. This is when this is going positive the other one goes negative when this is going negative the other one goes positive this is where it is sampled and this is where it is not sampled this is equivalent when it is sampled okay this is the state when all the diodes are off that is the condition this is negative and that is positive. So this is a simple problem the current flowing through the control resistor RC on either side is VC – V gamma of the diode okay divided by 2 RC okay – IS max by 2 so what happens is that this control signal VC and – VC here will force a current of VC – V gamma by RC into in this and that is split into 2 to this equal short circuits okay. So and the signal current again gets divided into 2 in these 2 short circuits and therefore we get here VC – V gamma by 2 RC – IS max by 2 that should be still greater than 0 in order to keep the layers conducting and that gives us the condition that IS max should be less than this. Then for the off state to be preserved the negative voltage VN that is applied to the controls should be less than VS max of the signal from which we get IS max as VS max divided by RS plus RL parallel RC by 2. So these are the conditions for making these things work this is a specific problem given for a certain signal condition. So here you are asked to find out for a given value of RC and RS and RL okay and also for a given control signal what is the maximum VS max using the equations that are given earlier you can arrive at the fact that VS should be less than 1 volt and you can see the simulated results. So we have applied less than 1 volts about 0.8 volts as the signal and our sampling frequency is this is corresponding to so 2 pi gone here 1000 hertz and you have the sampling frequency here sampling it so many times okay. So this is the sample signal nothing flows of the signal again sample at that time you can see this peak is due to the capacitors okay which take the voltage stream okay and they are no longer considered as open circuit because capacitors still link it to the load through these peaks okay. So in conclusion we have seen how RL and C are available in practice was okay in what form in discrete components and in what form they are available in IC components. Then we had seen diode semiconductor diode its application in sensors and rectification and also as Zener diode as temperature sensors and then we had seen diode as a multiplexer. So these are the applications we have dealt with these are the important applications particularly in micro region the this is used as a mixer this diode multiplexer that we have discussed can be used with micro diode okay as a mixer so modulator it is a very important application.