18. Oscillators (Intentional)

all right much of our study of feedback systems has really revolved around trying to prevent oscillations in feedback systems or even preventing getting very close to oscillation when we consider the relative stability of many of our systems today I'd like to look at the the converse of that problem in particular let's consider what happens when we wanted to want to design an oscillator and in fact a good quality sinusoidal oscillator an oscillator that provides very high purity sinusoidal signals as an example of describing function analysis we had looked at one function generator Lu which combined two Schmitt trigger and an integrator and we found out that that made a very convenient laboratory generator in that we were able to get simultaneously square waves and triangle waves and then by shaping the triangle wave very frequently manufacturers use that technique to build a oscillator or a function generator that can also provide sinusoidal signals the problem is that it's very very difficult to get sinusoidal signals with high spectral purity via that technique you can get ones that have harmonic distortion possibly as much as 50 or 60 DB below the fundamental but it's hard to get anything better than that by shaping a triangle wave the typically very high purity oscillators use other techniques and it really attempts to make a system that looks very much like a linear feedback system with a complex conjugate pair our poles on the imaginary axis one of the popular topologies is the Wien bridge oscillator and I show that here there is the bridge network that at least in this implementation consists of a series and a parallel branch using equal RC values and then again in in this implementation we have an operational amplifier and one way to view this thing is that the operational amplifier takes the output of the Wien bridge and notice the amplifier is connected for a non-inverting gain of +3 we have a feedback resistor of 2 r1 and an input resistor equal to r1 so from here to here we get a gain of 3 plus 3 through the operational amplifier and so what we have is an operational amplifier connected for a gain of plus 3 and then the Wien bridge taking the output of the operational amplifier and feeding it back into the positive input terminal of the operational amplifier this is actually a positive feedback system as we'll see in a moment we can write the transfer function for the Wien bridge itself this is vo as shown in the view graph where vo is the output of the operational amplifier if we assume that the input impedance to the operational amplifier is sufficiently high so that the Wien bridge is not loaded by the input to the operational amplifier then VA which is the voltage coming out of the Wien bridge in fact the voltage applied to the non-inverting input terminal of the operational amplifier that transfer function through the Wien bridge is simply equal to RC s over R squared C squared s squared plus 3 RC s plus 1 if we look at this transfer function we find out that at low frequencies the phase shift approaches plus 90 degrees at low frequencies the s terms and the s squared term in the denominator are small compared to 1 the numerator simply gives us plus 90 degrees a phase shift at sufficiently high frequencies the squared term dominates the denominator we get something falling off as one over R CS at sufficiently high frequencies and so we get minus ninety degrees a phase shift at the tune frequency of the Wien bridge in other words at s equals J Omega where Omega is 1 over RC the S squared term cancels the plus 1 in the denominator at that point the transfer function gives us a gain of RCS over 3 RCS or one third so the Wien bridge then gives us positive phase shift at low frequencies negative phase shift at high frequencies and out of frequency Omega equals 1 over RC the Wien bridge gives us no phase shift and an amplitude or a magnitude of one third from the input of the Wien bridge to its output consequently if we surround the Wien bridge with an amplifier with a gain of precisely 3 we can form a positive feedback loop where the loop transmission magnitude is precisely one at only one frequency in particular at Omega equals 1 over RC at that frequency the Wien bridge provides an attenuation of one third the operational amplifier as I showed the view graph provides a gain of one third or I'm sorry a gain of three and so we have a loop transmission whose magnitude is precisely plus one we can see the effect of this by drawing a root locus diagram let me stress the fact that this is a positive feedback system I've also normalized things to RC equals one second if we factor the denominator of the Wien bridge transfer function of course all the frequency dependent elements are concentrated in the Wien bridge i've assumed the operational amplifier has negligible dynamics we find that by factoring that denominator we get poles at s equals minus 1.5 plus radical 5 over 2 minus 1.5 minus radical 5 over 2 two for the two poles associated with the Wien bridge transfer function and there is of course a zero at the origin reflecting the numerator now we normally have drill drawn root locus diagrams for negative feedback systems but there's certainly no reason that we can't extend the technique to use it for positive feedback systems and let's see what modifications we have to make to our rules we develop most of our root locus rules by arguing that that in order to be at a point on the root locus when we evaluated the angle of the AF product it had a that angle how to be an odd multiple of 180 degrees in other words the real part of the AF product had to be minus one and the or the real part had to be negative with no imaginary part in order to be at a point on the root locus diagram that was necessary to cancel the inversion that in our standard topology we assumed occurs at the summing point if we don't have that inversion if we have a positive feedback system why the difference is that the rules that evolved from that hundred and eighty degree condition we have to we have to modify those rules to say that the angle associated with the AF product now has to be a multiple an integral multiple of 360 degrees well one of the results of that is that branches now lie on the real axis where there are an even number of loop transmission poles and zeros to the right so in that in this case we have branches on the real axis here where we have two poles one pole and one zero to the right and let's see they they also exist on the entire positive real axis where we have no poles or zeros to the right again an even number and again we can make appropriate modifications to the other rules we find out that the if we evaluate the root locus diagram for this positive feedback system these two brand should simply come together they break off the axis and if we keep track of of that there's a rule that tells us where the breakaway point occurs in particular factoring the characteristic equation or the loop transmission with respect to s and setting that equal to zero we find out that the breakaway occurs at s equals minus 1 there's a reentry point at s equals plus 1 we've seen enough of this sort of thing to realize that we'll get a circle something like so the cross over into the right half-plane of course occurs for our normalized system with RC equal to one this is a circle breakaway as of minus one reentry is a plus one so this is simply J my note this is plus J and down here we have minus J the if we evaluate the gain necessary to get a pair of poles right on the imaginary axis that's the oscillator condition of course we want to use this thing as an oscillator and have it provide sinusoidal amplitude Y what we want is to have a complex conjugate pair of poles on the imaginary axis we find out that the a not F not necessary and our positive feedback system to bring us to this point is simply three we had argued that before on physical grounds recognizing that at the frequency where we get no phase shift through the Wien bridge network we had an attenuation of one third through the network we make up for that by surrounding the network than amplifier that provides a gain of three well when we go to do this we have to worry somehow about stabilizing the system stabilizing the amplitude of the oscillation if we simply build up our linear system of course we'd never get things precisely right the poles would either be in the left half plane or they'd be on the right half plane we would never be able to get them precisely on the imaginary axis so we have to somehow take care of that fact and at least one possibility is to modify our amplifier a little bit as shown our amplifier provides a gain of a plus three however we could make the feedback resistor for example nonlinear such that for small signals we got a gain of three and for larger signals we got two somewhat reduced and in fact we have to do a little bit better than that and the problem is the component tolerances tell us that that for any real physical wing bridge network will probably need a gain greater than three in the amplifier in order to ensure that the poles are in the right half-plane are actually get to get the poles on the imaginary axis the extent to which the gain has to exceed three is a function of component tolerance if the values are precisely as shown we need a gain of exactly three in the operational amplifier closed-loop gain of exactly three however any deviation from that symmetry means that we need to gain somewhat greater than three and so we can as a function of the kinds of components we anticipate determine how much larger than three the gain of the amplifier must be to ensure that we will always be able to get poles at least on the imaginary axis we might do that by making one resistor that determines the gain of the operational amplifier r1 and the second resistor equal to 2 r1 plus three times Delta r1 if we evaluate the closed-loop gain of the amplifier will then find out that the gain from here to here is simply equal to 3 times 1 plus Delta where Delta is presumably a small number so that's fine we can choose Delta's sufficiently large so that we guarantee under small signal conditions with the poles of the system are in the right half plane consequently when we start the system up we'll get an exponentially growing oscillation at approximately the desired frequency 1 over RC radians per second however we have to do something now to limit the amplitude of the oscillation and so now we could view the operational amplifier in a describing function sense let's design a system that has a gain for small signals that's 3 times 1 plus Delta but then let's put some sort of a limiter possibly across this resistor I've shown it again as a pair of back-to-back Zener diodes we saw on in the last session that that was an effective way of providing a limiter in this case in the non-inverting amplifier configuration when signals become large enough to short out these diodes the incremental gain from this point to this point does not go to 0 but rather to unity because we have then effectively a direct incremental connection between the output and this point we have the non-inverting follower the gain of one amplifier so if we plot the VA verses V out transfer characteristics or V out as a function of V a the slope will be 1 plus Delta until we get up to a voltage equal to the breakdown voltage of the Zener diodes beyond that the slope is 1 we can look at the describing function for that element it starts out of course with a value of 3 times 1 plus a when the test signal into our nonlinear element in particular the amplitude of the signal here becomes equal to the Zener break down voltage the describing function magnitude begins to drop off it eventually asymptotically approaches 1 and of course we'd anticipate that our system would oscillate assuming perfect components symmetrical components in the Wien bridge when the amplitude into our operational amplifier had become large enough so that the gain of the amplifier from here to here in a describing function sense lowered to 3 that's this point so we'd anticipate that the amplitude of the fundamental component of the signal applied to the operational amplifier would be this value that's the value that squeezes down the gain to 2 precisely 3 the difficulty with doing that is that we get harmonic distortion certainly we would typically in this situation like to use the signal V out as the output voltage of our our system because it's buffered it comes from an operational amplifier that be a nice point to use as a an output for the overall oscillator but that signal has harmonic distortion and in fact the harmonic distortion is unattenuated by the frequency selective Network in this case because we apply the input to the amplifier here the non-linearity occurs in the amplifier so it really squeezes down the top and bottom of the sine waves we get some harmonic martien out of that we can reduce the harmonic distortion by making Delta small in that case we just sort of edge into the limiting characteristics of the Zener diodes get very little flattening very small harmonic distortion however if we want to build a design a design that's relatively tolerant of inaccurate components we have to make Delta large enough to accommodate the spectrum or spread of anticipated component values and again if we are not awfully careful in our choice of components Delta may have to be fairly large to guarantee that every member of the population that we build up if we build a bunch of these oscillators will in fact start oscillating we could of course tweak things up but then there's temperature dependencies and that sort of thing that that again lead us in the direction of having a conservative value for Delta so that we're sure that all of the oscillators we build oscillate all of the time and the effect of making Delta larger of course is to increase harmonic distortion because for sort of a typical member of the population we have to go fairly far out in the curve get a measurable amount of flattening in the output sinusoid another strategy would be something as follows we could for example consider taking our oscillator and maybe have a potentiometer that allowed us to adjust aim and we could look at the output of the amplifier or the oscillator on an oscilloscope for example and if we found that the amplitude were shrinking or were smaller than we want to wanted it to be we might up the gain of the operational amplifier that would presumably move the poles the system either into or toward the right half-plane if we move them into the right half plane the amplitude would begin to grow we could let it grow to the amplitude we wished and then back off a little bit on the potentiometer and and hopefully by us playing servomechanism we could keep a constant amplitude oscillation that of course is a little bit impractical but it happens that some of the very high quality linear quote linear oscillators one this ones that give us very harmonically pure sorts of sinusoidal outputs basically use this a pro what they do is build up a feedback system that looks in some sense at the amplitude of the oscillation and they then adjust some parameter in a loop and on a very long time constant they basically make a measurement over many many cycles of oscillation and with a very long time constant adjust some parameter in the loop with the hope of keeping a pair of poles precisely on the imaginary axis I understand although I haven't verified this myself but I understand that one of the first products made by hewlett-packard was an audio oscillator that basically use this technique they used a wing bridge kind of a circuit as the variable element they used an incandescent lamp an incandescent lamp has temperature temperature dependent resistance and by applying the outdoor the signal in the Wien bridge the lamp formed part of the Wien bridge as the signal got bigger or smaller the resistance of the lamp changed because of heating in the lamp that occurred with a very long time constant at least compared to the period of oscillation and the resistance of the lamp changing would automatically act to stabilize the amplitude of the oscillation it's kind of an interesting passive if you will feedback system I also understand but I'm not sure how true that is that that oscillator was originally developed for some early Walt Disney movies but I haven't verified any of that personally what I'd like to do is look at an example of an oscillator that basically uses this technique we won't use a Wien bridge I'd like to introduce another topology for an oscillator and the oscillator that I'd like to look at is called a quadrature oscillator and it consists of two integrators here we have a familiar non-inverting integrator I'm sorry inverting integrator simply R and C so the transfer function from here to here is minus 1 over RCS the other configuration is is a little less common it consists of a low-pass RC section and if from we assume that Delta is equal to zero we then have a CR from the output back to the inverting input of the operational amplifier and if you write the transfer function from here to here for Delta equals zero you find out that transfer function is plus 1 over RC s so if Delta 0 if the RCS are matched we have a minus 1 over RCS from here to here we have a plus 1 over RCS from here to here we have a system whose loop transmission is minus 1 over RCS quantity squared and it's quite easy to show that that gives us a pair of poles on the imaginary axis again we have the problem of stabilizing the amplitude of such an oscillator well let's see let's consider what happens if we allow Delta as shown in the view graph to be variable let's let that be our pop the thing that we're going to adjust to control the amplitude of the oscillation we can write the loop transmission or maybe the negative of the loop transmission as simply one over RCS that's the transfer function of the first integrator or actually a negative of it the first integrator gives us an inversion and we'll take the negative of that to get minus L of s so we get 1 over RCS we then have the low-pass characteristics of the RC that precedes the second operational amplifier we get 1 over our RCS plus 1 and then we write the transfer function that links the output of that that gives us the gain from the non-inverting input terminal of the second operational amplifier to its output and if we write that exactly we find out that that quantity is 1 plus Delta R plus 1 over C s over 1 plus Delta times R if we collect terms we find out that we get a 1 over 1 plus Delta a 1 over RCS quantity squared and then we get a pole 0 doublet if Delta is small the doublet of course is closely spaced a positive value of Delta in other words if the resistor going to ground is larger than the series resistor in the input RC network that in fact the zero lies at a lower frequency than the pole we can analyze this system in a variety of ways one fairly convenient way is to use a root locus approach and if we draw a root locus diagram for the system we find that we have a pair of poles at the origin associated with loop transmission that's the one over our CS quantity squared and as I've shown things we have a pole at s equals minus 1 over RC here we have a zero located at a slightly lower frequency a slightly longer time constant in fact we have a zero that's at minus 1 over 1 plus Delta times RC and if the L is small compared to 1 we can use a series expansion for 1 over 1 plus Delta and approximate that series by its first term and under those conditions we get this at approximately minus 1 plus Delta divided by RC that's the location of the 0 in the loop transmission well the root locus diagram for this setup of course is simply that this branch moves toward the 0 to counter that this is a system where the average distance remains a constant because we have three poles and only one zero these branches spring out at right angles or spring out along the imaginary axis but they have to come back somewhat to counter the fact there's a branch moving to the right in the S plane asymptotically for very large values of a naught F naught product let's see this moved to the right an amount Delta each of these branches would have to move back an amount Delta over two we can get that as a say average distance argument or by calculating asymptotes for these branches and either case we'll find out that they move back in a mount Delta over two times RC for the a naught F naught that corresponds to this expression we find out that the closed-loop poles of the system lie here and of course the complex conjugate to that point and they have an imaginary part equal to simply one over RC or J over RC and these displacement from the real axis this quantity is an amount Delta divided by four RC it's fairly easy to show that by directly factoring the characteristic equation that corresponds to this expression for the negative of the loop transmission well now we see our strategy as we change Delta we move the location of this zero a positive value for Delta gets us a situation shown which results in a stable system all the closed-loop poles lie on the left half plane if we make Delta negative make the resistor going to ground smaller than the series resistor going into the operational amplifier make that time constant a little bit shorter than the input time constant then this zeros on the other side of the pole under those conditions the closed-loop poles are in the right half of the S plane and so now our strategy parallels that that I suggested for the Wien bridge we can look at the amplitude of the oscillation being generated by our system and we effectively have a pot that changes Delta what we would do is increase Delta make Delta positive if we found that the amplitude of our oscillation was too large because that would move the closed-loop poles to the left half-plane the amplitude would begin to shrink when it had shrunk back to the right size we presumably would change Delta such that the closed-loop poles lay exactly on the imaginary axis and and then from there on in keep adjusting the pot to make up for very small changes in in system quantities in consequence of temperature dependence or aging or whatever and work to keep the output amplitude constant to the extent that we were able to keep these poles close loopholes precisely on the imaginary axis by making this zero lie precisely on the poll we'd have a constant amplitude oscillation which would be harmonically pure we have a true linear system with a complex conjugate pair of poles on the imaginary axis well how might we physically accomplish that the the possibilities are to use a field effect transistor in order to vary the the resistor at least that's one possibility we notice the resistor whose value is 1 plus Delta times are associated with the right-hand operational amplifier we could make part of that resistor for example be a field effect transistor and for small signals applied from the source to the drain of the field effect transistor why we're able to get the we're able to change its resistance it behaves like a linear resistor and we can modulate its value and so that might be a mechanism for controlling amplitude what our strategy would be would be to measure V a determine the amplitude V a somehow and then build a control loop that changes Delta in an attempt to keep V sub a constant one of the interesting fringe benefits of the use of a quadrature oscillator which is of course where it gets its name is that when we have the loop running we ideally have two integrators ganged up and so we get two sinusoidal signals we've been focusing only on V a in the discussion but we do get two sinusoidal signals which happened to be ninety degrees out of phase since they represent respectively the input and outputs of integrators and that is a very useful feature in certain instrumentation applications let's see how we really control the amplitude of this oscillator I've mentioned the overall strategy but now we have to think a little bit about how we model this system and in particular one of the variables is how does V a react the signal amplitude are actually the envelope of that output of the oscillation react as we change Delta what we're really going to do here you see is build a very very slow loop that controls the amplitude of the output signal and it there's sort of two independent feedback processes going on at once one of them is the double integrator loop the cross is over at a fairly high frequency at a frequency Omega equals 1 over RC and that's the basic oscillator we're oscillating basically at a frequency 1 over RC radians per second the the other one is a much much slower loop a much lower cross over frequency that works on the amplitude of the signal and so what we have to do or works on the envelope of the signal to keep that envelope constant and amplitude so what we have to do is figure out a way to model the transfer function really that goes from changes in Delta to changes in amplitude our feedback system is going to measure amplitude feedback to control Delta and clearly part of that loop is the Delta to envelope or amplitude transfer function we have to find out how we model out or how we represent that well to do this let's assume that our system is oscillating with the actual signal V sub a being equal to some C sub a and amplitude associated with that signal times sine Omega T and Omega is equal to 1 over RC this is the situation that results with perfectly matched components delta equals zero we have a pair of complex conjugate poles on the imaginary axis we get a pure sinusoidal oscillator and and the amplitude is a function of initial conditions now let's see what happens if we make a step change in Delta so we change the value of that resistor at time T equals zero to some value Delta one we put in a step at time T equals zero of magnitude Delta one well for time greater than T equals zero then the voltage V sub a will be exponentially decaying we found out that positive values of Delta at least give us poles in the left half plane and so there may be a little bit of a phase shift phase change but that's not very important we're going to focus principally on the envelope our signal V sub a is then again a sub a that was the value had initially times sine Omega T that times e to the minus delta 1t over four RC reflecting the fact that the poles have a real part that's minus Delta over four RC so we get the minus Delta 1 over 4 RC times T as the exponent and we can look at this we can look at the actual envelope here we're going ahead the oscillation is running along in here at a high frequency in fact determined by the RC constant so this is running along like so prior to time T equals zero Delta is zero so the amplitude of the oscillation is constant and the magnitude is e sub a four times greater than 0 the amplitude decays exponentially the envelope going is e sub a e to the minus Delta 1 over 4 RC times T so we have the oscillation running back and forth constant amplitude up to time T equals zero for a positive value of Delta an exponentially decaying amplitude thereafter let's go ahead and write e sub a the total variable associated with the envelope well the envelope is our initial amplitude the envelope amplitude a sub capital a times e to the minus delta 1t over for RC and what I'll do is expand that in the series recognizing that the term the exponential is is hopefully small Delta is a very very small quantity and if we do that we get the first two terms of this series being equal to capital e sub Capital a that's the initial amplitude that's this value minus Delta one ei over four RC times T and then there's the quadratic term and all of that if in fact this quantity the exponent is small enough we can make a linearized approximation to this transfer over to this behavior and we'd recognize capital e sub Capital a is the operating point value of the envelope and then this term is the incremental component so that's the usual linearized analysis where we consider only the first two terms in the series expansion are actually the operating point value plus the first term in the series expansion for the envelope again I I emphasize the fact that our strategy is going to be to measure the envelope hopefully do that without regard for the actual oscillation we'll try to build some sort of a detector that measures envelope do that without worrying about the exact details of the oscillation and then use envelope information measured envelope amplitude to control Delta well fine what we find out is that we make a step change in Delta and the incremental component when we draw an incremental block diagram or a linearized block diagram in response to put in making a step change in Delta we get a negative ramp for the incremental change in the envelope a sub a is something times T well let's see we have a system that we are assuming is linear by virtue of the fact that we've performed a linear is a linear as kind of analysis we're doing an operating point plus a linear change from that operating point sort of analysis so we have a linear system we put in a step and we get out of ramp we conclude that the transfer function that relates output to input for such a system must be simply an integration when you put a step into an integrator you get a ramp out consequently the fourth permutation in our hierarchy a variable and subscript a sub a of s that the transfer function that relates the transform of the incremental component of the envelope to Delta is simply the scale factor that we got from the earlier equation minus e sub cap a over four RC now times s reflecting the integration associated with our linearized analysis the fact that a step gives us a ramp change in envelope that's simply saying that if things go for a short period of time or if we look at the system for a short period of time there's a funny kind of linearization here your your linear linearizing a variable that runs continuously but but what happens is that the system works to to force itself back to equilibrium in such a way that this exponent stays pretty small the total X exponent stays pretty small it goes to zero the feedback system works in that direction well to the extent that we can use that sort of linearized analysis we're able to model this thing as simply the integration and now we can draw a block diagram we have a reference value for our envelope we have the actual measured envelope amplitude we compare those two we go through some sort of a controller and then we go through the dynamics that relate envelope to Delta so this point in the block diagram is really Delta of s we go through the transfer function we developed above capital e sub capital A over for our CS that gets us an output amplitude we compare that with a reference amplitude modified as we need to with our our compensator or loop gain or whatever and that completes the loop since there's a minus sign associated with this relationship we'll find out that that a of s actually has to be negative itself so that we end up with a negative feedback system when we go to do this there's some practical things that we have to consider one is that in order to make our analysis valid where we sort of ignore the structure associated with a high frequency oscillation the ampulla the oscillator is going providing a high frequency oscillation if we ignore that in order to ignore that structure we have to have the crossover frequency in the loop that controls the amplitude of our oscillation much much lower than one over RC as we approach 1 over RC we get into sampling kinds of problems where we're not able to get a good estimate of the envelope on an instantaneous basis and the thing that allows us to get a good estimate on a sort of instantaneous basis compared to the dynamics of our amplitude stabilization loop is to ensure that the crossover frequency in that loop is much much less than 1 over RC we also might like to have a system that will force the output amplitude to be identically equal to the reference amplitude and in general that requires some voltage for example on the control the gate of our field effect transistor and the way we can support a finite voltage with zero error is to simply use an integrator and so let's design our a of s to include an integration that will ensures to say that we can get any voltage we need to control the loop with negligible error zero error between our commanded and actual value for the envelope finally we are also we need some filtering if we think about the way we will do this we can't get a perfect detection of envelope what we might do for example is a full wave rectifier that's one possibility actually the the actual circuit that we look at is is a little bit more sophisticated than that it it does some phase shifting and then a rather neat multi phase envelope detector that gets us fairly small ripple it's a equivalent to using different phases and and just getting a scalloped kind of an output waveform but there is still some harmonic content some high frequency noise associated with that measure of the envelope and what we have to do is filter that because if we think about the overall loop we're going to apply some measure of the envelope of the error to of our control element to a field effect transistor possibly if we use that as the control element and while I think the analysis is difficult probably beyond my capability modulating the control element at some high frequency can't possibly be good we'd like to have the control element have a fixed value of resistance over an entire cycle or many cycles of oscillation so what we'd like to do is include a lot of filtering so that this high frequency trash that's it's a consequence of our envelope detection gets gets filtered out in significant levels and finally we have to be very conservative in our design because there's a lot of unknowns here there's a there's a linearity problem notice that the loop transmission magnitude is a function of the commanded value for the operating point value for the envelope so if we command a large envelope we get a high loop transmission if we command a small envelope we get a small loop transmission that sort of behavior incidentally is very very or always occurs in gain control kinds of things you invariably find out that some loop transmission parameter the some loop transmission magnitude is dependent on operating point it shows up here in our amplitude stabilization loop by virtue of the operating point value of the envelope coming in as a multiplicative factor in the loop transmission there's a host of nonlinearities in this system when you use a field effect transistor as a control element I do some of that analysis in the in the text we find that that it has rather grossly nonlinear characteristics between control voltage gate voltage and and incremental resistance when we use it in this mode so we have to really our system loop transmission magnitude at low frequencies may not be what we think it is so we ought to design the system to be very very conservative design a system that has plenty of phase margin under nominal conditions so that when things are not as we thought they were we still have adequate phase margin a possible a of s might be of the following form we want an integration so we get a naught over s that gets us the integration I will will then go ahead and put in a zero because once we have the integration we have that we've intentionally added we have that integration plus a second integration that occurs at at as a consequence of the dynamics that relate envelope to Delta and so what I'd like to do is add a zero so that I get a 1 over s squared roll-off at low frequencies then a 1 over s roll-off and the vicinity of crossover let's shoot for crossover at at 10 to the minus 2nd over RC one 100th of the frequency of oscillation I mentioned earlier we we simply want to be at a small fraction of the frequency of oscillation so let's assume we're going to cross over at one 100th of 1 over RC let's put the zero a decade below that that's quite conservative we said with lag networks that we might start there but then then edge up let's put a zero so we have a 1 over s slope that extends from that starts a decade below nominal crossover and then I'll put two more poles that provide filtering a decade above nominal crossover again recognizing that we're going to shoot for a crossover in the amplitude control loop that's 1/100 of 1 over RC let me put two poles that will provide additional filtering two signals at an above 1 over RC the sort of hash on our envelope detector let's put a couple of filters that occur a decade below that and in fact are located one decade above nominal crossover frequency we can adjust a nought then so the crossover does in fact occur at 1/100 of 1 over RC we'll have a loop transmission that Falls is 1 over Omega squared at low frequencies 1 over Omega actually for a decade either side of crossover if everything occurs as we hope and then as 1 over Omega cubed beyond that the if in fact our system were precisely like this we would have about 75 degrees of phase margin a very very conservatively stable system if we if this was a kind of a loop transmission plot that we had in terms of let's focus a little bit again on on on this relationship the fact that we'd like to crossover at a frequency small compared to the oscillating frequency so that we don't get an interaction a sampling kind of a thing the only converse to that normally you see our amplitude control loop only has to make up for aging and temperature changes and that sort of thing and so a very slow amplitude loop is it's perfectly fine however it's possible in some applications that we might want to make the command changes in the amplitude of the oscillator and if we do that then we'd like to have a reasonably high crossover frequency in the amplitude control loop so that we get fairly fast response to commanded changes I haven't tried to push that at all in the system we're going to demonstrate what will we'll do is really design for a nominal crossover frequency that's one one-hundredth of the oscillating frequency and here we have the experimental system that does this there's a collection of operational amplifiers and I won't go through all the details of that but there are a number of operational amplifiers the two basic ones doing our are quadrature loop there is another operational amplifier that really mechanized is a of s and then there's several more that do the rather sophisticated envelope detection a very low ripple type of envelope detector multi phase sort of envelope detector so that's what collection of operational amplifiers is used for there's a couple of switches a one which commands internally a fixed amplitude for the oscillation another one and the other position of that switch allows us to externally command the amplitude so we can for example look at the step response when we command to change in amplitude there's another switch that allows us to use external compensation I won't do that in this case and we have again the power supply test generator that allows us to command changes in amplitude and we observe everything on the oscilloscope what we're looking at here is the output without any commanded changes in amplitude so we simply have a constant amplitude Sonya side in this case we selected the RSC product to be ten to the minus fourth seconds we'd anticipate us elation then at ten to the fourth radians per second we're at 200 microseconds per division to again within the ability of our measurement we find out that we get an oscillating frequency that sort of ten to the fourth divided by by two pi a period of something over six hundred microseconds three divisions if you investigate this and with a spectrum analyzer you find out that the this is a very very pure sine wave the sort of distortion we get verifies the basic design principle the idea of using this is a technique for stabilizing the amplitude of the oscillator the harmonic distortion in this one is is 80 or more DB below the fundamental so the idea of putting in a slow loop to control the amplitude does a very very good job of reducing or of giving us a spectrally pure output what we're now able to do is change the the switch that connects us from an internal reference for that amplitude the amplitude now is about five volts peak or ten volts peak-to-peak to an externally commanded reference and we'll put a square wave in as the command so let's simply throw this to the other Bishan now we see the modulation we're again modulating it around about the same amplitude as we had before let's go to a situation now where we can look at the envelope and let's trigger off the signal generator and now what we what we see is exponential basically exponential changes in the envelope here we're commanding a smaller envelope here we're commanding a larger envelope let me just speed this up a little bit if we can but that's too fast again here we see the exponential changes commanding small amplitude then commanding a larger amplitude operating about five volts for this small signal things are relatively symmetrical we could estimate crossover frequencies in the control loop by looking at the time constant of this exponential since we have a system with high phase margin we design for seventy-five degrees of phase margin we get responses that look very very much first-order if we had ninety degrees of phase margin we'd anticipate responses that are indistinguishable from first-order responses this looks very very much like a first-order exponential and the time constant of the change in envelope is the order of ten milliseconds the 1/100 of one over RC that we shot for let's now look at what happens when we put in larger commanded changes so I can do that let me lower the scale on the oscilloscope a little bit and then just drive the system a little bit harder and now we begin to see how the non linearity shows up there's a number of nonlinearities in the systems I measure as I mentioned some of them have to do with nonlinearities associated with the control elements the field effect transistors for example there are other kinds of nonlinearities in the system we mentioned also that the loop transmission magnitude is dependent on actual operating condition and when we put in large changes our our assumption of linearized analysis is not operational that's well modeled as incremental changes around an operating point becomes invalid where we're making now changes that are large compared to the operating point we find out that the way this manifests itself in this particular system although there's no generality here is a very non symmetrical kind of change the increasing amplitude the dynamics associated with an increase in amplitude are much longer than the dynamics associated with a decrease in amplitude but as I say there's no real generality there that's just a peculiarity of this system again we can move up the overall amplitude there we get into saturation with me but again we notice here the fall time if you will be the change to decrease smaller amplitude is considerably faster than the time necessary to increase the amplitude so we see at least an indication of the non-linearity associated with this system or some of the nonlinearities associated with this system in conclusion I'd like to emphasize that this technique gives us the ability to build a very very low distortion oscillator and also is a very interesting example of a feedback system that has two very distinct modes of operation the basic loop that sets the frequency the 10 to the fourth radians per second or 1.6 kilohertz in this case and then the much slower loop that that controls the amplitude of that oscillation and this is this sort of a system as an example of that sort of interesting airplay in between dynamics in a system that has two distinct modes of operation this concludes our discussion of oscillators thank you