Lecture 53 - Oscillator Design - III

[Music] [Music] uh welcome to Arab transer design course we are discussing the oscillated design so in oscillator design in last class we have seen the one port oscillator design and we already have seen one approach that we used in rfic uh one port view of oscillator same can also be used and uh it is widely considered in the microwave oscillator design also and this methodology can also be used if we are going for very high frequency oscillator design like a sub terz frequencies so in that what we considered we will consider the input impedance Z in so here there is a J in input input impedance which is a input impedance when you see at the Active device side so this is a we can consider any transistor and that transistor is terminated uh with some lossless component and when it is a terminated uh it is added with terminating Network this gamma in we we need to ensure that it is providing the negative resistance and this input matching network if we design a combination of input reflection coefficient and Gamma L such that we will get the oscillation so in order to get the oscillation what we need is if we consider this Z in which is J in equal to R in a into Omega because your input impedance when you see at the Active device it has a different amplitude it is a function of amplitude and frequency that is a Omega plus JX in a comma Omega so this we can write it a input impedance that we see on the active side so uh and we can write a JL which is equal to RL Omega plus J xn Omega so here this resistance is only function of frequency because we are going to design the input matching Network or any matching Network either by using RLC circuits or by using some uh micro strip resonators or micro strip circuits any uh JL that we design is going to become a function of Omega so if you want to satisfy this condition when J JL is connected with Jin then your Jin plus ZL J in a comma Omega plus JL Omega should be equal to Z so this is a condition that uh we should satisfy in order to have the oscillation so if we consider that uh gamma L if we try to represent the same thing in terms of the reflection coefficient gamma l which is given by ZL minus Z by ZL + z equal to uh we will write ZL uh Omega is equal to minus Z in so we can write minus Z in minus J kn/ by minus J in + J which eventually uh we can write J in plus Z if we take a negative by J in minus Z KN which is equal to 1 by GMA in so finally the condition for oscillation is gamma uh gamma in gamma L is equal to 1 so this is how we uh we can consider gamma in gamma L is equal to 1 that is a condition to have a one port oscillation now this uh uh this is a this Jin and JL are it is a frequency dependent so that in order to have a stable uh oscillation condition there is a one kurova identity that needs to be satisfied what is says Kuroko identity that your Del Del R in by D A means which is changing the amplitude with the where a is equal to a multiply with the D XL Omega / D Omega at Omega is equal to Omega minus d xine e Pi d a for a equal to a Del RL by D Omega for Omega is equal to Omega this should be greater than zero so this is a kuroka condition of stable oscillation and mostly this RL Del RL by D Omega is zero so this this condition this Del Rin by d a function of amplitude at a equal to a d XL Omega by D Omega Omega is equal to Omega not should be greater than zero and we'll try to see that how we can uh design the oscillator by using this method so when we try to find out the design or uh Power uh we we need to make sure that our series resonator means this is here A series resonator and this is r in a now if we consider this input impedance which is a uh negative resistance and changing with the amplitude that resistance is changing with the amplitude we can Define it as a one function and that function that at amplitude zero it has a negative resistance R and as amplitude is increasing your negative resistance is decreasing so we want to have a maximum power so in order to establish that what we need is condition for maximum oscillator power the operating point a not and Omega not we need to find the amplitude a and Omega not it should be stable so it should satisfy the kurakova identity and the condition for starting the oscillation so now we will try to find out based on this function what is what are the different values so for condition for maximum power if we write R in this function R in which is equal to minus r it is a decreasing means resistance is increasing minus r a 1 - A by A M so when a is equal to a m your input resistance R in is equal to 0 and for when a is z it is a minus r so we need to find out the values value of amplitude and the resistance for the maximum imum power so power can be given by 1/ 12 real part of V1 I conjugate we can write a half i1 s into i s r i1 square into R in a where R in can be replaced with here so it can be given by 1 12 a² r 1 - A by A M less than Z as power is generated now here if we try to find out that DP by da differentiate with the A and equate equal to zero we can get both the amplitude and the resistance are in for getting the maximum power so if we equate it equal to zero maximum power occurs for a not Max which is 2/3 a so at this point you will get maximum power if you put a 2/3 am in this equation we will get R in a Max which is equal to minus r by 3 so this is a model that we have considered for the input uh resistance and then based on this model this anot Max that is required is 2/3 of am and R in which is equal to minus r by 3 now we will see all the uh we will find out all the values input impedance uh R in X in and is it stable or not so what are the operating point and stability so operating point is RL + r in a equal to Z so RL which is given by R because uh that is minus so here r 1 - A by m so RL which is equal to R by 3 XL Omega plus x a not equal to Z so if you write Omega lus 1 by Omega C this is is a for series resonator equal to zero so Omega not which is equal to 1 by root LC same thing will apply to the parallel resonator as well so we get a condition RL and Omega not now this uh is it stable or not so in order to find the stability we will apply the kuroka uh stability condition Del R in a by d a d XL Omega not by D Omega minus this equal to as I told this can be cancelled out resistance RL is will not change with Omega then only that is a real part uh of the uh impedance so that is not changing so if we consider this as a zero then this condition Del Rin by n not by d a d XL Omega by Del Omega should be greater than Z so we one by one we'll find out this should be greater than Z and this should be greater than Zer so Del R in a by D A which is equal to r r by a which is greater than zero and D XL Omega by D Omega which Omega equal to l + 1 by Omega Omega squ C and based on this if we put this equation it is a 2 L that is also greater than zero so hence the operating Point whatever we have defined which is is RL which is equal to R by 3 omega which is equal 1 byot LC that is a operating point and the amplitude that we have find out that is a operating here this operating point is a stable point now we will put down all the steps and try to find out that how we can design a oscillator so this are the different design methodology you can use uh different steps various steps of the design methodology that you can use for Designing the oscillator so you have a transistor available and we which uh terminating Network that you need to design so you need to know that gamma T now you need to choose this gamma t such that your reflection coefficient means your input impedance on this side is a negative jine or it is considered as a unstable because while we have used the same method while designing the amplifier also but here when you are designing oscillator uh instead of the stable region you need to consider the unstable region so first is use a potentially unstable transistor so your transistor should be potentially unstable at the frequency of oscillation and that you can ensure by designing a various topology that we already have seen or you can uh choose certain terminating Network which will provide you the uh potential unstable transistor at your desired frequency then what you need is design the termination Network to make comma in which is greater than one so we all when I discuss the SMI chart at that time I told when we study the active component or oscillator where your reflection coefficient might be outside the me chart and their reflection coefficient is greater than one so series or sh feedback can be used to increase the value of reflection coefficient gamma in then design the load Network to resonate so we need to uh satisfy that ZN plus ZL equal to Zer and we already find out the value XL Omega which is minus X and RL Omega which is a mode of par in a is equal to 0 Omega by 3 so that R by3 we already have uh calculated now we'll try to have the example based on that so here at steady state we have a reflection coefficient gamma L gamma in which is equal to 1 therefore gamma L which is 1 by gamma in you can write this uh S11 means gamma L plus S21 gamma T S12 by1 - s22 gamma T which is a terminating Network reflection coefficient so you can rep represent the gamma l in terms of the gamma T and the Delta is given by this S11 s22 minus S12 S21 so if you try to solve the gamma T we which is equal to 1 - S11 gamma L by s22 minus Delta gamma l = to 1 minus gamma out so you we can also write it down gamma L gamma out is equal to uh 1 so that we already have seen even gamma L gamma in is equal to 1 same thing gamma L gamma out should be equal to 1 which prove that input and terminating networks are equivalent and oscillating simultaneously so uh for the condition that is for input matching Network same is for the terminating Network so we'll take one example design a transistor oscillator at 4 GHz using a gum aren MP so this is one transistor gum AR MP in a common gate configuration so this is connected in common gate configuration with 5 Nan Henry inductor in series with the gate to increase the instability so here we need to have a instability when we design the amplifier we have tried to consider the stability now additionally instability has been Incorporated by connecting the fin Nano handry inductor in series with the gate we what we need to do is choose a load Network to match a 50 ohm load and an appropriate terminating Network so what we need to design is a this input matching Network and terminating Network so this can also be mentioned as a load Network because for oscillator that is a load the scattering parameter of the transistor in a common source configuration are J not 50 ohm S11 is this S12 is this S21 and this is a uh s parameter and when you try to use the inductor and use a common gate configuration that will be converted to the s parameter which is S11 is this so now S11 is changed from this to the this S12 is this S21 and s22 so what we need to do is we need to find the output stability Circle so if you based on this s parameter you will get output stability Circle CL and RL we already have studied how CL and RL can be find out so based on the S parameter you can find out the value of Cl and RL once you have find out the CL and RL you can denote the stable region and unstable so this is a what is shown here the Smith chart and there is a region is has been shown so this here mention is a stable region so when we discuss the stability at that time we discuss that if this is a SMI chart and this point that is a if this gamma L point if gamma L point is equal to zero this point is a stable if S11 is less than zero this point is stable if s is less than zero and this point is unstable so this is a for stability condition stable so gamma L is equal to 0 is unstable if S11 is greater than 0 so what is S11 in our case S11 is greater than 0 and because S11 is greater than zero this GMA L is equal to 0 is a unstable point so this is a unstable point that means this entire region is unstable and this is a stable region so what we are designing we are not designing amplifier we are designing a oscillator so what we need to use when we try to find out the reflection coefficient for load you need to keep the point in the unstable region so this is the unstable region and here the point has been chosen which is a far from the stable region we need to make sure that this gamma will not reach to the stable region for different frequency that is why we choose it at very uh far compared to the stable region otherwise we cannot have a oscillation so that gamma L we can choose value which is0 59 angle of minus one4 we can find the reflection input reflection coefficient so what we got is a gamma l so gamma L will help us to design the uh input means terminating Network and then gamma in if you have a gamma in which you have found out so S11 based on that you can find out gamma in which is z in so this is input reflection coefficient and based on that J in we have find out so condition to uh have the oscillation this uh will be minus Rin by 3 and this will be uh XL should be minus X so if we do that - 84 will be replaced with the 28 and minus J 1.9 will be replaced with plus J 1.9 at the Jedis so now we you have a gamma L and JS Point both are there you can design the input matching Network so if you try to design you can design the input matching Network by lnc or you can design the input matching Network by the uh stub matching or here the stub matching has been used so gamma L JL has been used to design the terminating Network and this is at the uh source side this is a transistor that uh is connected with the Fino hry of inductor uh then this if you if you tape out from the load side 50 ohm this uh will generate the oscillation at 4 GHz so this is how you can also design oscillator by just using a single transistor and designing input matching Network and output matching Network it it it has a many application in which if you want to design a uh specific oscillator which has a application in RF trans design that this method can be used so this is how you can design a oscillator now one of the important uh parameter of the oscillator is Phase noise so we'll try to uh discuss the phase noise so phase noise first of all what what is a phase noise if you if you see the digital circuits in digital there is a Jeter so around the zero Crossing there is a cheater and what what will happen that in reality in all our discussion what we have considered means uh when we consider the mixer and uh when we try to discuss the transer architecture every time we have consider the Delta function at F not frequency uh it is some value with a Delta at uh F not frequency there is a Delta function that we have defined now that Delta function is uh the value that we want but in reality our oscillator has a mini frequency component and that has a that is a spreaded across like this so instead of a single frequency FN there is a many other frequency which is on the side by side uh will also have if you try to draw into the frequency domain so these frequency components are nothing but not a desired frequency component and it is called as a phase noise in the oscillator and that is generated due to the noise uh in the Active Components so if uh if this is drawn as a logarithmic curve means this if we try to draw in form of a logarithmic log Delta Omega with the uh amplitude then with the power it has a three components that we can identify what are the three components because of its Behavior there is a one uh slope on the uh one side up to Delta Omega 1 by F Cube up to certain frequency Delta Omega 1 by F Cube that has a slope of three so it is a 60 DB per decade slope now if after the Delta Omega 1 by F Cube that has a another slope that is a slope of the square so this has a 40 DB per decade slope it is reducing by 40 DB per decade and then uh after Omega not by 2q this frequency it is it is going to become a constant so this uh model has been studied by leson uh it is called a lesen uh model of phase noise and in which what he has considered that there are a three frequency components that we can identify so here uh when we try to see the uh model of uh if we see the phase noise so this is a phase noise and uh it is given by this I just want to tell that what is the unit that is widely used this is a Omega not frequency or let's say we write F not because it is frequency and after certain point Delta F that is called offset frequency Offset you have a certain amplitude of in the one Herz so the bandwidth of this is a 1 Herz let's say if you have a power of zero dbm here output power at FN if you have output power at FN which is zero dbm then and if after Delta F offset at 1 Herz you have a power of minus 120 dbm so this difference is called a DBC DB dbd with respect to carrier so this is a actual carrier frequency F not and at that F not you have a power of Zer dbm and after Delta F Offset you have a power of minus 120 dbm so that difference is called a DBC and because you are considering it for one Herz of bandwidth the DBC per Herz so what is the phase noise if if for Delta F offset Delta F offset frequency the phase noise is given by 120 DBC per Hertz so this is a unit of phase noise where this Delta f is important Delta F offset so ideally what we want is this value should be as high as possible in reality in ideally it should be infinity and you should your Delta app should be as small as possible so uh then it will leads to the Delta function we'll discuss about the phase noise model lesense model in the next class thank [Music] you [Music] a