# Non-Sinusoidal Oscillators and VCO (FM &amp; FSK Generators) today our thirty fifth lecture is on non sinusoidal oscillators we had earlier in the thirty fourth lecture discussed about the schmitt trigger circuit and we had seen how this schmitt trigger or regenerative comparator can be used for a variety of application as a mixed mode circuit pulse width modulation that is the basic schmitt trigger is having regenerative positive feedback we saw this is represented as an inverting type of schmitt trigger with the hysteresis inside so actual hysteresis of this is going to be looking like this this is v naught this is vi and the amount of hysteresis can be controlled by r1 by r1 plus r2 which we will call as beta so this when it is this can be only at plus vs or minus vs not any intermediate point whatever be the input so this particular this is going to be plus or minus beta v uh vs plus or minus beta vs so this was the schmitt trigger that was used as inverting another one topology which is called non inverting schmitt trigger that is the input and the ground for example they get interchanged this earlier input is grounded now so the feedback returns is regenerative action only the input now is fed here output is taken here so this is what the inverting uh the non inverting schmitt trigger represented by this kind of symbol without this inversion ok and this characteristic is uh the something like this this is v naught and this is vi so how these uh schmitt triggers can be used for fm generation is another basic principle that we have this comparator regenerative comparator one feeds here a triangular wave form and this here a reference wave form then you get here ok based on what the v reference is a parallel or a rectangular wave rectangular wave has a duty cycle which can be uh made to be proportional to the uh reference so the duty cycle generator or fm or fsk generator has this kind of characteristic it is half into one minus vr into one minus beta by vp vp is the peek of the square wave input so this is beta r1 by r1 plus r2 so using this one can uh design a class d power amplifier dc to dc converter etc based on this then it can also be used for fm generation ok uh when uh this uh uh vr is varied according to uh audio signal that is fm if it is uh digital data ones and zeros dc un changes from one value to another then it is the fsk generation it can be used for that now astable and multi vibrator function generation so this is another important use of the schmitt trigger so we are now taking this uh inverting type of schmitt trigger here let beta equal to r1 by r1 plus r2 that is the amount of voltage fed back and this is either at plus vs or minus vs so this is changing state at this point when vi corresponds to this plus or minus beta vs so what happens let us assume that initially at plus vs so initially at plus vs so the capacitor is going to start charging from zero so it is going towards plus vs if this connection is made so the input now sees a voltage which is exponentially increasing because of the charging of the capacitor with the time constant equal to r into c so this exponentially increasing voltage is trying to reach plus vs but as soon as it becomes equal to beta vs same as this beta vs change of state has to occur from plus to minus so now the capacitor is seeing minus vs and it is discharging at the transient time constant rc so it will be discharging towards minus vs as soon as it reaches minus beta vs this is at plus beta vs again it changes back to plus vs so this will keep on happening as this toggles from plus vs to minus vs this will toggle from beta vs to minus beta vs so the capacitor will be charging and discharging continuously so it is going to acquire this beta vs again so these two time durations are the same because the capacitor is going to be uh having a voltage of beta vs to minus beta vs that is twice beta vs in a time let us call this half time period because these two times are same this is the time period so this is half time period so charging and discharging take the same amount of time because they are the same time constant rc and acquiring the same voltage of two beta vs within that time t by two so you can consider this portion for example a capacitor is trying to charge up to plus beta vs it would have gone on like this so the voltage applied is minus beta vs plus vs which is vs into one plus beta that is the voltage applied and t by two it is acquiring a voltage of two beta vs so this equation is the charging of capacitor when a voltage is applied across it this is the total voltage applied this is voltage it acquires within that time t by two so this case t equal to uh two rc log one plus beta by one minus beta and f is nothing by one over t so this is a uh function generator or astable multi vibrator which gives a uh uh exponential increase and exponential decrease in waveform if the time interval is very short this can be considered linear and it is a triangular waveform here you get a square waveform with frequency equal to one by t both of the same frequency so this is simulated for r1 equal to r2 beta equal to half so beta equal to half in this case so uh two rc log one plus beta one plus r by one minus r is log three so that is the time period so t t equal to two into r is one k c is one microfarad log three this comes out to be roughly equal to t uh two rc log three right which is equal to two point two milliseconds one over t is four hundred and fifty four hertz so the simulation exactly tallies with the expected results so this is beta vs so in this case beta is half and vs is uh ten volts even though vs is ten volts the op amp used is tl zero tl zero eight two ok so uh when supply two volts supply is used it is going only up to about eight volts ok minus eight volts so that is why the square wave amplitude is limited by the uh ability of the op amp to give the highest output as eight and minus eight when ten volt supply is used and correspondingly this voltage is eight by two which is four volts so the amplitude gets fixed the frequency gets fixed and please note that the frequency is independent of the uh amplitude uh of uh supply voltage magnitude of supply voltage vs so that is the advantage of the simple circuit this has no need for the sophisticated amplitude stabilization schemes that become necessary in the case of sinusoidal oscillators so very simple thing what is now done is in simulation two right the uh this voltage is shifted to a reference value this particular thing is sort of uh grounded so grounded the plus supply only minus supply is there so this is single supply utility of the same circuit so this is made zero this is minus twenty and the v reference is minus ten so beta is still half alright it is again plotted you can see here the same frequency and the same swing ok for the uh square wave as well as the uh exponential wave occurs ok so the value of this remains the same that is two rc log three so frequency is same as before what has now happened is there is an offset ok around which these things are toggling that means minus ten volts is the offset around which this is toggling as because of four plus four ok minus ten minus ten minus four ok so uh the this the uh minus ten uh plus eight ok minus ten minus eight this will be minus ten plus four minus ten minus four so the frequency remains the same so even with single supply we can get the same kind of uh waveform get for uh the square wave and the exponential wave and ap a direct application of this is an on off temperature controller so again this schmitt trigger is right there r1 and r2 to control the hysteresis beta that controls the extent of hysteresis around vr into one minus beta so let us consider this that uh this is a heater coil which is switched on or off depending upon whether the temperature had reached what is the called the set point temperature this particular point is called set point temperature it is going to be in terms of the voltage at this point uh so this diode is a temperature sensor which we had explained earlier in the old applications so it is forward biased ok so minus vs is supplied to r through it so it is forward biased this voltage is typically there minus point seven volts so as the temperature increases the characteristic we have told that delta vbe or delta v diode by delta t is negative and it is almost uh constant at minus two point five millivolts by degrees centigrade that is till diode forward drop temperature coefficient so this voltage v gamma actually are the continual pitch of the diode this keeps decreasing for the same uh current ok the forward voltage decreases at this rate two point five millivolts per degrees centigrade so it is a negative temperature controller that means v gamma becomes less in magnitude so since we are biasing it in this manner the voltage v gamma magnitude when it is reducing it is becoming less negative that is it is going towards positive so when the temperature is at room temperature it is the highest magnitude ok and as temperature increases it becomes less and less in magnitude ok so the voltage ok is uh let us say uh becoming less negative as temperature increases ok now that characteristic is obtained ok uh for this astable multi vibrator action the kind of hysteresis it has because they are using negative voltages and negative reference just like the example that we had shown earlier in this case so it is all in one content negative voltages ok for both x axis and y axis so we use a mosfet as a switch here right so actually the uh thing is located somewhere here initially ok so that it is on it heats up the coil and this voltage is going to become uh less negative in magnitude so it is moving in this direction so at some point it is going to change state ok and it is coming over to this so that means it is going to be switched off it becomes less than voltage here it is greater than voltage so this is an on off control which is adopting the same principle of charging the capacitor however this is nothing but the thermal time constant for it to reach the set uh above the set point temperature ok and then it is switched off so it starts cooling and then the uh it is coming towards ok uh a value which is ok below the uh set point temperature right if this is the room temperature this is the set point temperature right so it is coming below the set point temperature like that it will go on so this is nothing but an astable multi vibrator designed to incorporate it to a temperature controller so these are the deviations around the set point so we have this deviating by delta t on either side of set point temperature so this is delta t on this side and delta t on this side that is determined by the hysteresis so this is the hysteresis bringing about uh fluctuation around the set point so depending upon the accuracy with which we want this to be set right this particular uh delta t can be varied and delta t is directly proportional to beta ok so now monostable multi vibrator action if we just connect a diode across the capacitor then what happens is this particular structure has now one stable state that is why it is called monostable stable state what is it let us assume that it is plus vs so when plus vs is here this capacitor tries to charge up to plus vs with rc time constant but as soon as it reaches v gamma this diode conducts and never allows it to charge above v gamma so at that point of time this is plus into plus vs and this particular thing is at plus beta vs so the voltage affectively across the uh uh op amp or the comparator is positive here more positive here so this state is a stable state we have assumed it to be at plus vs because it is at a large positive voltage so now what is done is that a trigger pulse comes momentarily it comes and makes this voltage immediately go to a negative value more negative than this so the moment that goes to a negative value or a value less than v gamma this immediately goes turns to minus vs so this is the trigger point so the capacitor now is seeing a voltage which is minus beta times vs so it will uh charge with the time constant rc up to minus vs but as soon as it reaches this voltage reaches minus beta vs it will come back to the stable state of plus vs so it has changed the state to minus vs at this point and after a time duration t let us call it the delay it has come back to plus vs again it will go on like this remaining until the trigger pulse is applied so let us say it is applied arbitrarily at this point then it will again generate a pulse width of t and then come back to plus vs so this can be used as a timer application where any process actually starts at this point and ends at ok after time t so one such timer i see which is very popular is lm triple five costs about point four eight but that contains two comparators here just using one comparator one can also build this uh sort of timer so this particular thing is again this time duration t can be evaluated this way the voltage applied is let us say this is v gamma and this is going on up to minus vs so effective voltage is vs plus v gamma one minus e to power minus t by rc that is the time the voltage across by the uh across the capacitor at that time is ok uh this uh twice that is beta times vs ok plus v gamma that is the voltage acquired by the capacitor so t is equal to t is equal to rc log one by roughly one plus one minus beta we can ignore v gamma compared to beta vs that is what is simulated beta is again kept as half and t uh is nothing but rc log two so that comes out to be point six nine three milliseconds so this has been tried with uh the trigger pulse coming at a rate of hundred hertz so this is the point where the trigger pulse has appeared so it has gone from v gamma to minus beta vs and that is t again after uh this it has to discharge and it will keep on going up to this but at this point the diode comes into picture and holds the voltage at v gamma across the capacitor so the next trigger pulse has to come only after this has reached this this state that is the limitation of the frequency of triggering ok so the time t is equal to rc log two is exactly fixed so industrial timers make use of this principle for uh actually generating a fixed time duration function generator so we have here the inverting uh non inverting type of uh non inverting type of uh schmitt trigger we had seen the inverting type of schmitt trigger being used for function generation suppose we replace it with the uh non inverting type of schmitt trigger with a which we had discussed earlier in the last class with an integrator which is again near ideal we had used earlier rnc which is a rough approximation to an integration operation so we are using this as an ideal integrator now op amp so what happens to this astable multi vibrator let us see so once again output of this can be either at plus vs or minus vs so it is pumping in a current of vs by r into this and this voltage across this is going to change linearly so this voltage is going to be zero here and therefore it is minus vs by rc into t ok so that means for plus vs we have a linearly de decreasing voltage here and as soon as this voltage reaches a value such that for plus vs here this should become equal to zero that means it should take on uh minus vs ok so the voltage at this point is continuously changing then it is plus vs here it is vs into r1 ok plus v naught prime into r2 by r1 plus r2 that is the voltage at that point that when it becomes equal to zero this changes state from plus to minus so what is that voltage v naught prime can be from this equal to minus vs into r1 by r2 so as soon as it reaches minus vs into r1 by r2 this changes state from plus to minus so this will now change slope from minus vs by rc into t to plus vs by rc into t so this will linearly increase this way so as soon as it reaches plus vs into r1 by r2 again this minus will change over to plus this way it will go on this time duration is t by two this also is t by two so within a time t by two it is changing at a rate of vs by rc into t by two it is acquiring a voltage of twice vs into r1 by r2 so again the time period t is independent of vs because vs vs get cancelled and t becomes equal to four uh four r1 by r2 ok into rc so here we get a triangular wave output here you get a square wave output of this frequency f equal to one by t so this is what is derived here you can see that this is changing from minus r1 by r2 into vs to plus r1 by r2 into vs at a rate vs by rc again discharge is at the same rate but negative vs by rc so frequency is r2 by four rc into r1 so this is what is shown here in simulation r equal to r1 equal to one k r2 equal to two k the reason why we have chosen r2 to be now different is because in the case of this kind of variation r1 by r2 if it is less than only this scheme will work if r1 by r2 is greater than one the voltage here has to go above the maximum limit possible for the schmitt trigger output ok which is not going to make it toggle at all so r1 by r2 in this case has to be less than one in order to make it toggle and continue with this kind of generation ok so this is something that has to be remembered so we made r2 safely equal to two point two k so this particular thing has now resulted in this is uh uh vs uh into r1 r1 is one k uh by two point two time the maximum which is eight volts ok and uh this is minus eight by two point two this is plus eight and this is minus eight so the time period is one point eight two uh milliseconds f naught is five hundred and fifty hertz now let us say we have put here an asymmetrical voltage that this is v asymmetry what does it change in terms of symmetry earlier the current of charging and current of discharging of the capacitor c was the same in magnitude it was changing direction from vs by r to minus ss by r the moment you put a va here this is changing between plus or minus vs because it is the schmitt trigger output so the current here while charging is vs minus va by r this is charging that means the voltage uh at that point here is changing as vs minus va by rc into t so it is negative so it is minus so this is the slope vs minus va by rc and in the other direction this is minus vs this is va so the total current in this direction is vs plus va by r so that means actually it is increased ok comp that means it will take a less time to come to this which is because its slope is higher vs plus va by rc into t plus so if there is an asymmetric causing voltage here which is positive then this slope is lower than this slope so we get a saw tooth at this point of course this will become a rectangular wave so this will be at plus vs all the time and at minus vs for less time so rectangular wave here and a saw tooth here is what you get because of asymmetry taken an asymmetry voltage of five volts or actually i think so for ten volt supply we have taken an asymmetry of five volts so i have shown you the result of asymmetry so we have seen here vs minus va by rc into t1 is the same voltage from here to here the voltage changes remaining the same as before two r1 by r2 into vs so t1 changes however because the rate of charging and discharging are different now so t1 and t2 are different t1 plus t2 is now t so t1 is this and t2 is this from these equations and t1 plus t2 is this so we a saw tooth generation here so asymmetry is five volts as i told you the charging is occurring at uh uh faster uh rate so it takes less time discharging occurs at a slower rate takes more time and uh uh this is the saw tooth with va changing over to minus five you have now charging taking more time discharge takes less time and the saw tooth is of this type so by nearly changing the asymmetry voltage va one can actually get the uh different types of saw tooth waveforms from this function generator so this function generator essentially can be used for triangular waveform at this point and square waveform at this point and this triangular waveform is converted ok using a diode function generator which we had discussed earlier it can convert a triangle to a sine wave triangular to sine wave converter we have discussed complete design on this in the earlier lectures so here you get a sine wave of the same frequency f equal to one over t so these are the important functions necessary for testing ok in a test lab this is a test oscillator ok which is uh one of the cheapest oscillators available in laboratories today all over the world so ics are available which use these principles ok i will come to that later at the end so now we will convert this into uh ok function generator with offset if you now put instead of bringing about asymmetry we are putting an offset voltage here v offset so this ground is lifted and offset voltage is put here then what happens is as far as the triangular waveform is concerned it remains the same as before the charging and discharging remain the same ok because this is grounded now so with this offset here what can only happen is this output waveform of the triangle is offset by certain amount ok so that is uh what is being shown here now so with an offset ok this triangular waveform which was uh swinging around zero has got shifted to a positive voltage ok so because of an offset of one voltage introduced there ok so this offset can create uh problem sometimes because if it is too much this particular thing will be getting distorted because it cannot go above eight volts here so the offset should be set that the triangular waveform is produced only within this swing that is possible for the schmitt trigger output now this function generator can be simply converted to a vco which is an important block again in communication systems vco is nothing by fm generator or fsk generator so this particular uh schmitt trigger is followed by now not an ordinary integrator but an integrator preceded by a multiplier this kind of conversion we had done also in the case of voltage control filter converting a double integrator loop into a voltage control filter or oscillator was done by replacing the integrator by a multiplier followed by an integrator so this multiplier followed by an integrator is uh going to make us a current here which was earlier simply vi by r or plus minus this vi was only changing between plus or minus vs it is either plus vs or minus vs so what happens here when you multiply with vc this will become plus minus vs divided by ten which is the reference voltage for the multiplier into vc so the current in this which was earlier vs by r in this direction or this direction is now going to change to vs by ten r ok into vc plus or minus ok so this is going to change in the same direction but you can think of this modifying the original r as ten r by vc that means all those uh frequency and time period get modified simply by this factor that is earlier we had this as r2 by r1 into four rc here as f so now this r is going to be replaced by r into ten by vc which results in vc by forty rc into r2 by r1 so it becomes a linear vco and if you now find out the sensitivity of the vco which is defined as delta f by delta vc this becomes simply since it is linear becomes constant equal to r2 by forty rc into r1 or it can be also written as f divided by vc hertz per volt this is an important parameter associated with a linear eco if it is non linear you have to find out at the constant of frequency of oscillation the slope so vco is an important building block in a phase locked loop which we will discuss which we have already discussed earlier this is strictly speaking not really a phase it is a frequency locked loop we will henceforth call this phase locked loop we have discussed it as something that we had used in the filter for self tuning of the filter for locking onto for example pi by two phase shaped here this block vco is an important building block in a feedback system that is called frequency locked loop so the vc voltage control filter vcf is replaced by vco there becomes an independent system generating a frequency of oscillation depending upon the control voltage function generator is an oscillator which produces square wave triangular wave rectangular wave with specific duty cycle and saw ok also it can produce a uh saw tooth ok it can also output sine wave by converting the triangular wave to sine wave using diode function generator the vco circuit that was discussed earlier ok which is uh available as an ic xr two two two zero six or lm five six six but this function generator is very popular as test oscillator chip which is used in all laboratory function generators ok for test purposes now fsk generation how to do fsk generation so we can actually use this vco and modulate it that modulation simply means that uh we are now connecting at this point vc uh to the uh positive voltages one indicating one digital one and the other indicating zero digital zero then it becomes a fsk generator so this is high this is low both have to be positive right so this gives high frequency this gives low frequency this is what is used in modems ok for uh modulating the frequency next instead of the square wave applied as vc one applies a dc with a sine wave superimposed so that it gets now modulated continuously the frequency is slowly becoming highest and slowly becoming lowest like that it goes on this is nothing but fm generation right so at this point ok at this particular voltage it is to carrier frequency ok and it more than the carrier frequency it is less than the carrier frequency here so that is depicted for the triangle also that way for the sine wave also it same uh kind of characteristics exist and therefore you can take the output of the sine wave output and it will be fm generation here these are the characteristic features of this uh ic xr two two zero six and uh it is continued here you can see the sensitivity sweep range two thousand to one so five six six is the tiic which is capable of being used as a uh vco now the limitations due to the op amp rise uh or the comparator rise time and fall time uh will not permit it to be used uh for very high frequencies it will be uh more so with uh op amps being used as comparator here the limitation is due to slew rate right so if uh slew rate is known right you know that see uh from plus vs to minus vs the op amp is going to rise at slew rate ok and therefore the voltage twice vs divided by roe if roe is the slew rate or the op amp that is the time taken for as rise time in the case of op amp which is going to make it too slow right so if it is uh for example one volt per mic uh microsecond ok uh and uh vs is let us say uh ten volts so this will be twenty divided by one ok so uh that is the rise time ok twenty volts per microsec uh that is uh twenty microseconds it is going to take which is uh going to result in the rise time being too large in the case of actual comparators the rise time is going to be pretty small and therefore it can be used for very high frequencies ok so the accuracy of the frequency is better with comparators ok so it can uh typically be used up to megahertz very easily with full output swing possible whereas this is limited to few kilohertz ok as uh plus minus ten volts amplitude because of this fundamental limitation so we have seen how uh op amps should not be used as comparators comparators are used in mostly open loop or positive feedback and they do not have any frequency compensating capacitor put the ones used in popular general purpose op amps with frequency compensation that fundamentally limits the frequency of operation it makes it very slow so uh comparator uh like three eleven etc are fast comparators there are no uh capacitors added there is no need for them because it is never used in negative feedback thank you very much