 Today our 22nd lecture will be on the topic of passive filters before we go into the discussion of passive filters let us just review what we had done earlier so we had discussed about the mathematical theory of filter approximation the mathematical functions that represents closely the box like approximation so we started off with structure this blue color corresponding to 1 by 1 plus X to power 4 the maximally flat function of second order it is a sort of one of the most common maximally flat functions to be used in a variety of applications filter is one of them you can also design trans conductors with signal level okay plotted this way and becoming maximally flat okay so is an important mathematical function then we went over to variation of that which is called inverse championship function where again we have no X function in the numerator only the denominator has that 1 minus X squared plus 2 X to power 4 that means it is an even function of X okay still remains an even function but here this minus X squared indicates the its influence close to X equal to 0 in enhancing the amplitude that means actually now you can cause it to peak within the pass band so that it can cause a ripple in the pass band so the extent to which this dominates tells by what extent this has gone above that is as you increase the coefficient of this it increases faster and faster so the magnitude of this is controlled by the coefficient of X squared okay compared to the coefficient of X power 4 so that is the ripple in the pass band structure then if you sort of increase the coefficient of X squared you can increase the ripple you can see now another way of doing it is instead of increasing the coefficient here you can have a coefficient in the numerator a polynomial having numerator function of X squared which is 1 minus 0.25 X squared divided by 1 minus 1.25 that means in a sense it is equivalent to having 1 minus X squared this 0.25 gets cancelled with 0.25 when X is close to 0 okay so we have in a sense this being imitated by this okay and this is going to be the ahh function which is going to have ripple in the pass band so then you can have this as positive and this as negative which is again going to cause increase negative function okay it is equivalent to having 1 minus let us say 2 X squared in the denominator so this will be increasing it further okay and this particular function 1 minus 0.25 instead of 1 plus X squared here okay is going to facilitate introduction of a 0 in the stop band beyond X equal to 1 if the coefficient of this is less than 1 you can introduce a 0 outside the pass band. So when it is positive and this is negative effectively you can cause a ripple in the ahh pass band okay with no 0s coming into picture if the denominator and numerator both have negative then you can cause a ripple in the pass band and a 0 in the stop band okay that kind of thing function is called elliptic function. So we have this as let us say elliptic that is ripple in the pass band ripple in the stop band okay. Then we have this function which is Butterworth maximally flat okay so it has only the coefficient highest coefficient of X squared that is X 4 here rest of the coefficients are all 0 and no numerator coefficients. This one is going to be just ripple in the pass band okay which is called Chebyshev okay this one this one this is similar to the Chebyshev function okay this one okay is called inverse Chebyshev. So the characters features of inverse Chebyshev is these two coefficients are the same so that it is maximally flat okay in the pass band and there is ripple in the stop band there is a 0 here. So maximally flat in the pass band okay in the pass band and ripple in the stop band that is the characteristic feature of inverse Chebyshev okay so it is this green function okay. So this one is Chebyshev function okay so there is ripple in the pass band and there is no 0 coming into picture this is called elliptic ripple in the pass band and ripple in the stop band ripple both in the pass band and stop band Butterworth no ripple maximally flat. So these are the mathematical function is easily understood and setting the coefficient can be done experimentally I suggest that you take any graph plotter which is freely available in the internet and try all these manipulations of the function and see these plots and understand the design of filter functions right. Now this is another ahh part of filter design that we can discuss higher order wide band filter with stagger tuned narrow band filters of lower order how do you bring about this let us say we are starting with this function which is 1 by 1 plus x square that is the Butterworth function of first order that is plotted here around x equal to 0 this is going to give you the characteristic this way. Now how to get a higher order Butterworth that is the exercise that means how do I get this as 1 by 1 plus x to power 4 by using 2 such things right it understood that I have to multiply this okay by another of this type. So what is the way to multiply okay so that it becomes this that means all the other coefficients of ahh x square etc should go to 0 after multiplying. So what we do is mathematically you can see that this characteristic can be shifted on either side of x equal to 0 by certain amount let us say alpha. So what happens then this shifting function will be say k divided by 1 plus let us say x minus alpha okay whole square so I have shifted it by alpha and this one can be multiplied by another which is k divided by let us say 1 plus x plus alpha we know that it should be a symmetric function around x equal to 0 in order that it remains symmetric you have to multiply 1 that is shifted okay onto this side peaking at x equal to alpha and this is peaking at x equal to minus alpha. So you multiply this you get this as k square divided by you can see that this is 1 plus x minus alpha whole square x plus alpha whole square is going to be 2 x square plus 2 alpha square and then this into this is going to be plus x square minus alpha square whole square. So here for this function you will see that the coefficient of ahh x square can equal to 0 when alpha is equal to 1. So this is the choice for the alpha so that means actually it becomes a function which is going to be ahh 4 okay divided by okay ahh sum actually if it is 2 divided by 1 plus 1 plus x square okay this will be 1 plus okay 1 plus x square okay into 1 minus x square into 1 square minus x square whole square added to this. So that is what it is right so and it will therefore become equal to okay ahh by properly selecting this you can now make it equal to 1 by which is this 1 by 1 plus x to power 4 okay this divided by 4 ultimately. So this is how the choice of k is equal to 2 for that you can show that it becomes equal to this. So this is the mathematical procedure of making a wider band okay structure using a narrow band okay multiplying it twice okay this into this okay shifting it this way and shifting it that way and multiplying these 2 and then making it become equal to this by making alpha equal to 1. If you increase the alpha slightly then there will be ripple in the pass band. So the extent to which it deviates from this alpha equal to 1 gives more and more ripple in this as alpha is less than 1 it gets narrowed down okay it does not become wide band. So these are the techniques of designing wide band filters if you have 3 of them okay it can still become wider or it will be written by 1 plus k x to power 6 that is what we can synthesize. So try it out as a problem to solve you have 3 such peaks 1 okay at x equal to minus alpha another at x equal to plus alpha and the third one which is at x equal to 0 multiply these 3 and get a wider band. So what are the passive filters filters that use only passive components R L and C are the passive components that are used these are known as passive filters before the commercial availability of the op amps all base band filters are mainly passive in nature. So almost all telephone filters were passive filters that also in the base band they were very bulky okay. However because of reliability precision okay and low sensitivity temperature variation and aging these were preferred transformer was the other passive device which is used for impedance matching. So these were passive linear network elements which were used for filter design. So presently base band filters no longer use discrete transformers okay the base band filters presently are mainly using discrete components are never used because it is mostly done by digital filtering or active filters have replaced all these passive filters. Passive filters are still used in microwave and RF region and even AF interconnect I mean as band pass filters these are the most preferred ones okay are selective even for rejecting and selecting okay this still form very small size may be compatible with presently electronics and therefore these are still in use. Interconnect models now interconnect is an important thing now okay in most of the layout we have connections from one set of group of active devices to another group of active devices whether they are analog signal processing ICs or okay digital signal processing ICs interconnect is a common thing. So the interconnect from one node to the other can be modeled crudely as results causing some dissipation and capacitance parasitic so it is essentially acting like a low pass filter. So since interconnect become very important okay at very high frequencies their effect on both digital signal processors and analog signal processors become very important to understand and therefore this is now ahh something that one has to understand in order to understand signal processing the effect of parasitic on the other hand if at very high frequencies the inductive parasitic also comes into picture apart from the resistance then it becomes a low pass filter of second order that is the ahh approximation that you can so do for the interconnect okay between one and one node and another node okay the transmission line so that becomes R L and C. So if you have actually large number of nodes then from here you will have again the same model with different value of R depending upon the length and the capacitance okay this will be different again this will be modeled as another set of value R. So all these are interconnect passive okay filter like approximation. So filters are also components which can be simulating any higher order linear system okay for understanding even mechanical and aeronautic systems we can use the model ahh electrical model for this. So most of the filter topics are discussed in a course on networks again the systems analog systems with feedback we have discussed the effect of the same transfer function on the performance of the ahh system. So all these things can be modeled in terms of passive filters. Let us see how it can be done first order RC network we have already discussed all these transfer functions when we discussed the ahh two port ahh passive networks earlier. Let us revisit that V naught by VI is equal to 1 by 1 plus SCR that is a low pass transfer function for sinusoidal excitation S is replaced by j omega. So V naught by VI is 1 by 1 plus j omega CR V naught by VI magnitude square is nothing but this H of j omega H star of j omega okay T of j omega and T star of j omega okay. So that is nothing but the magnitude square okay which is equal to 1 by 1 plus omega CR square 1 by 1 plus capital omega square where capital omega is called the normalized frequency. So let us understand these terminologies which are often used in filters normalized frequency. So this is the way all standard filters are built in terms of normalized frequency 1 by 1 plus capital omega square. Similar to what we wrote earlier in terms of X 1 by 1 plus X square which is a Butterworth function okay as far as the magnitude ahh power magnitude is concerned. So it is equal to power magnitude V naught over VI square will be 1 by 1 plus omega square V naught over VI therefore magnitude is square root of this. It is similar to maximally flat magnitude Butterworth function of first order. So this is maximally flat magnitude function of the first order. So that is why it is 1 by 1 plus omega square so it is a natural Butterworth filter first order is always a natural Butterworth filter function. Response is similar to that of low pass filter as far as the phase response is concerned okay Phi is equal to angle of V naught over VI tan inverse omega CR minus tan inverse omega that is a phase lag okay. So minus d phi by d omega change of phase with frequency is called delay function. This delay we have already discussed in ahh signal okay processing have already indicated if this is the signal and it is going through some system which is not effecting the magnitude okay but only causing okay delay then the signal will be shifted in time by this delay function. So that delay is defined as change of phase with respect to frequency okay delta phi by minus delta phi by delta omega is all tau the delay. So if you have the phase function here 1 over 1 plus d omega CR is the transfer function so the phase is really equal to okay tan inverse omega CR okay minus so this is the phase phi of this low pass okay. So if you now take delta phi by delta omega of this minus of that that becomes again nothing but CR divided by 1 plus omega CR square okay or it is normalized okay where the omega CR is capital omega square so it is actually equal to omega naught into tau okay that is the delay function okay tau is the delay function. So this square magnitude and the delay are frequency dependent in the past. So this also is similar to what we saw earlier that is both magnitude function square and delay function are of the same nature 1 by 1 plus omega square okay. So these are plotted here V naught over VI magnitude 1 by 1 plus square root of 1 plus omega square and delay tau by tau naught equal to t these are plotted here and they are maximally flat. So this corresponds to magnitude function which is 1 by square root of 1 plus omega square this corresponds to delay function. So let us this is 1 by 1 plus omega square root okay whereas this is 1 by 1 plus omega square okay as far as the bandwidth is concerned omega okay in the power half power point okay it becomes half okay and the square root if you take it is 1 by root 2 okay or 3 dB point it is called. So filters with maximally flat magnitude functions are called whatever filters filters with maximally flat delay characteristics are called Bessel's okay and Thomson filters okay rate of attenuation at the edge of the path band okay omega equal to 1 is minus 0.5 first order low pass filter RL filter. So we have the RC filter you just put L in series and R in shunt in the RC filter it was R in series and C in shunt at the output. So it just is changed over and then it becomes okay low pass RL filter here earlier 1 over RC is the cut off frequency omega naught okay or the what is that this is the normalizing frequency and here it is R by L that is the normalizing frequency that means this is also written as 1 by 1 plus S by omega naught. So omega naught is equal to R by L this is equal to 1 by 1 plus capital S right. So now consider the second order whatever passive low pass filter. So we have now a combination of L and C we had the low pass filter with R and C and the low pass filter with L and R now we have R and L coming in series with C in shunt forming what is called second order low pass filter. So output is equal to input for DC so 1 by 1 plus SCR okay plus S squared LC I mean how this is to be written is very simply this is the methodology which we have been adopting 1 by 1 plus okay the admittance in shunt that is SC and impedance in series which is R plus SL so we get this. So this is therefore written as 1 by 1 plus S by omega naught Q plus S squared by omega naught square. So that is the normalizing frequency omega naught equal to 1 by root LC. So this is the series resonant circuit coming into picture so we have omega naught equal to 1 by root LC that is the resonant frequency and that is the normalizing frequency then S by omega naught Q so Q is something that we are going to identify as root L by C divided by R okay by comparison of coefficients okay this is S by omega naught Q. So omega naught Q is equal to S by omega naught okay and omega naught CR is 1 by Q so from that you get this substituting this. So that is what omega naught is equal to 1 by root LC okay which is the normalizing frequency of S squared and at S equal to J omega we can now introduce this as 1 by 1 plus J omega CR minus omega squared LC okay this becomes that. So again replacing it with normalizing frequency and putting omega by omega naught as capital omega okay you get this equation as the magnitude function as 1 by 1 minus omega squared whole squared the real part plus omega squared by Q squared which can be written as 1 by 1 plus 1 by Q squared minus 2 omega squared plus omega to the power 4. So this is the square of the magnitude function so these are the definitions Q is equal to root LC by R is known as a quality factor and if Q is made equal to that is the choice okay 1 over Q squared is made equal to 2 or Q is equal to 1 over root 2 will make this go to 0 and then it becomes Butterworth 1 by 1 plus omega to the power 4 it becomes second order Butterworth filter. So a choice of omega naught and Q decide the function of the filter uniquely and Q equal to 1 over root 2 its frequency response is maximally flat flatter than corresponding to the first order. So as you keep on increasing the order okay you can make it flatter and flatter over a wider range of pass band frequencies and attenuation will be more rapid in the stop band it is going to a closely approximated box like characteristic that we want. So now consider the phase these are the two factors that are very important so the same transfer function this is the real part and this is the imaginary part so phase of V naught over VI5 is equal to minus tan inverse the imaginary part omega by Q divided by the real part 1 minus omega square. So T is defined as delay is defined as minus T phi by D omega. So if you actually differentiate this okay then T becomes equal to 1 over Q the numerator it is 1 plus omega square divided by the same denominator that we had earlier obtained 1 plus minus 2 plus 1 over Q square omega square plus omega to the power 4 this is the same as the denominator of the magnitude function. So however this has a numerator polynomial so if you actually differentiate this and maximize this you can show that the maximum occurs at omega equal to 1 okay or at the resonant frequency T max is equal to 2Q value of this at omega equal to 1 is 2Q okay. So this is very important it is directly proportional to the slope it is directly proportional to Q that means the entire change of phase from 0 to 180 degree because change of phase occurs from 0 to 180 degree okay as omega keeps shifting right towards infinity from 0 the phase changes from 0 to 180 degrees and that rapidity of phase change occurs okay or keeps on increasing okay as Q increases that we will plot and see for actual low pass filters built by us. So we have here for different Q values okay this is for a Q of 10 okay and this is now this is for a Q of 10 and this is going to be okay higher Q than this. So if this is for a Q of 10 this will be 2.5 higher than that because R comes in the denominator. So you can see this is for the lower Q and this is for the higher Q the plot of magnitude it peaks okay. This we had noted in the system design also higher the Q more is the peak around the resonance. Now what happens to the phase characteristic this also we had noted okay higher Q okay the phase changes more rapidly okay and still higher if you consider okay the perhaps the Q is going to make it change very rapidly okay. So this is going to be the fact that the Q change that is for higher the Qs the phase change is going to be concentrated all around the resonant frequency which is about 16 kilo hertz for this that we have chosen okay. So now let us consider that this circuit is going to be having a characteristic as we saw the magnitude characteristic this is a second order filter the magnitude characteristic can peak around resonance and come down like this. So if you really find out the peak that depends upon Q okay that is the ripple that you can promise. So it is equivalent to a Chebyshev filter okay peaking in the pass band with ripple if you make Q equal to 1 over root 2 that it becomes Butterworth. So it just becomes a Butterworth filter right. So by changing the Q you can go over from Butterworth to Chebyshev right as far as this topology is concerned these are the ones which are plotted here this is close to Butterworth and this is the Chebyshev as both of them are somewhat Chebyshev because Q is much greater than 1 over root 2. So if you want to make the it truly Butterworth maximally flat you can do it by making the Q go to 1 over root 2 that means selecting the resistance value appropriately. Now if you change this we had earlier this network let us say 80 whichever resistance it was 40 you would sort of increase the I mean decrease the Q increase the resistance decrease the Q to half its value and you replace this using the same inductor but a capacitor to resonate at a certain frequency okay and this was the earlier capacitor you use an appropriate inductor to resonate at the same frequency right omega naught let us say then you can shift the entire characteristic from 0 to any value what omega naught. So the same characteristic which was earlier coming as this got shifted to from 0 let us say it got if this is 0 from 0 it got shifted to okay any center frequency you want that means by putting appropriate resonant circuits which resonate at the same value this is a series resonant and this is parallel resonance. So this effect of band pass can be shifted to okay occurring around 0 frequency can be shifted to any frequency omega naught this we will later on discuss as frequency transformation that means most of the filters are designed for low pass prototype and appropriate shifting of this occurs to any other frequency by replacing the in capacitor by SC by SC plus 1 over SL where omega naught is 1 over root LC that is the resonant circuit okay and as far as the inductor in series is concerned it is replaced by SL is replaced by an impedance SL plus 1 over SC again resonating at omega naught equal to 1 over root LC this mathematically we will explain as shift from 0 to omega naught please remember this the filter now which was originally second order filter now becomes a fourth order filter function it has the same magnitude function centered around omega naught now this is the simplicity of design in case you want a band stop of this type okay which permits all the signals and stops signals here and again permits all signal beyond this then it is necessary to replace this inductor by a parallel resonance with this L coming here and C coming in shunt that means the circuit will be this L 1 M and 0.04 micro farads and this will be okay same capacitor 0.1 micro with 0.4 milli coming in series so this is the network for band stop function okay something goes like that so you can see that it is pretty simple as far as a designing filter is concerned and for delay equalization you can see that it is only necessary to make here this maximally flat how do you make this maximally flat you can see that this coefficient is 1 omega square so 1 is equal to coefficient here minus 2 plus 1 over Q square so this becomes Q squared is equal to 1 by 3 or Q is equal to 1 by 3 that is called a Thompson filter of the second order or Bessel's filter so this is what has been designed you can see that it is going to be for the further it is compared as 1 by 1 plus 0.5 X square okay and the other one is 1 plus X square by 1 plus X square plus X square 4 so you are making this Q equal to 1 over root 3 then this is the maximally flat delay characteristic or Thompson or Bessel filter so we have now seen Q equal to 1 over root 3 for the second order makes it a Bessel's filter maximally flat delay characteristic so variety of such filters can be designed Chebyshev filter or equal filter can be designed now Q has to be greater than 1 over root 2 so it is of this type as far as magnitude is concerned now K1 being controlled by the factor 2 minus 1 over Q squared for this network so if K1 is 0 it is Butterworth that is for Q equal to 1 over root 2 so this is the thing K1 is 0 for Butterworth Q has to be 1 over root 2 so K1 has to be positive okay for Chebyshev and it is K1 that determines the extent to which ripple exists in the pass pad and for that you can actually maximize this function okay and find out the frequency at which the maximum occurs so this peak is okay root of K1 by 2 okay after differentiating and this point K1 okay corresponds to okay 2 minus 1 by Q squared the extent sorry the peak at which omega I mean peak at which this occurs corresponds to root K1 by 2 and K1 is equal to 2 minus 1 over Q squared and then this peak value okay which is if this is 1 this is epsilon 1 1 plus epsilon 1 corresponds to 1 by 1 minus K1 squared by 4 so you can fix the epsilon 1 and determine the value of K1 that is necessary for getting this kind of peak so we have designed it for epsilon 1 equal to 0.1 and K1 comes out as 0.83 and that is what is plotted here okay so 1 by square root of 1 minus 0.83 omega squared plus omega to the power 4 okay which is the designed thing for a ripple of 0.1 in the pass pad that means this goes from 1 to 1.1 okay if you select the K1 as 0.83 and corresponding Butterworth function which deviates by 0.1 is plotted here at X equal to 1 so you can compare the Butterworth with the Chibichir so this design has been complete here by suitably selecting the Q so K1 equal to 0.83 you can find out the value of Q needed so this is 0.83 2 minus 1 by Q squared you can find out the value of Q from this okay. Now if you select the same RLC network with same value of R and L and C okay as far as the denominator is concerned is reproduced exactly now that L I am going to split it as L1 plus L2 that means I make this L equal to L1 plus L2 I can split L1 so that the this acts as a short at very high frequencies okay this ultimately the high frequencies these two impedances dominate so the attenuation at very high frequencies will be L1 by L1 plus L2 that can be fixed at any value you want okay then actually this introduces a 0 at 1 over 2 pi root L1 into C okay so that it will therefore be a function which can actually peak just as we had designed it for a high peak okay because the denominator still remains 1 minus may be 0.83 okay X squared plus X to power 4 okay so it peaks at the same point but there has been a 0 that has been introduced outside the band okay and that 0 corresponds to a frequency of 1 over 2 pi root L1 into C that can be conveniently located at any point we want right so that is nothing but an elliptic filter okay if it is peaking and there is a 0 then it is called an elliptic filter if it is maximally flat okay by suitably selecting the value of Q you can make it maximally flat and that there is a 0 at this point then it is called inverse inhibition filter. So this is the elliptic filter that has been designed if this Q is chosen such that it is maximally flat then it becomes inverse inhibition filter so these are the values of components that are used okay for the same network okay L2 plus L1 is the same value 1 milli Henry's C is the same value 0.1 micro Farad so everything else remains the same but there is a 0 that is introduced outside the pass by so this is the way we have this resonant at omega equal to 1 over root L1 C okay and the denominator remains the same as before okay and Q is the same as before okay and only the 0 frequency is now having introduced the 0 appropriately okay in the numerator we have 1 minus N1 omega square okay and N1 is decided by the ratio of omega not by omega Z okay whole square. So that can be fixed appropriately by knowing where exactly we want the 0 to be introduced so this is the point that we 0 is introduced so inverse inhibition filter okay we can fix all these things appropriately okay so we have actually introduced a condition where it is maximally flat so 2 N1 is same as K1 that is for the inverse inhibition in the case of elliptic okay we have to make K1 greater than 2 N1 so that there is a peak K1 is made greater than 2 N1 for elliptic these are the functions elliptic low pass filter okay ripple in the pass band and ripple in the stop band and we have the BataWatt filter second order okay and then inverse TBCA filter right inverse TBCA filter is not there okay so we have only the elliptic filter with different peaks okay and then the BataWatt filter for comparison so N1 is equal to 0.25 K1 is equal to 0.7 so that is one elliptic filter that we have chosen. So in conclusion we have seen how to design the basic filters in first order and second order if you know how to design basic filters in first order and second order you can just cascade these filters okay and get any higher other filter okay that also has been demonstrated mathematically at least. So important thing is we have maximally flat magnitude filter or BataWatt filter that has been designed first order and second order and then TBCA filter with ripple in the pass band peaking in the pass band and the extent of ripple can be decided by appropriately selecting the Q okay BataWatt also Q selection is the crux of the design and then inverse TBCA where it is maximally flat in the pass band and ripple in the stop band 0 is introduced in the stop band peaking in the pass band as well as ripple in the stop band is called elliptic okay and this is the one that has the highest rate of attenuation in the stop bandage for the same order. We had also discussed the maximally flat delay characteristic or Thomson filters and Bessel Smith. In the next class we will be discussing further about the frequency transformation starting from the low pass prototype how to go over to pass band stop high pass etc.