So next topic is filter at micro wave and

millimetre wave frequencies we use couple theory to design filters and the method we

use it is called insertion loss method So in this method what we do power loss ratio

we define it by using a umm a polynomials so here power loss ratio it is equal to power

available from source divided by power deliver to load So that is equal to P incident divided

by P load If I consider normalised power values in that case we can consider P incident that

is equal to one and divided by this is one minus magnitude of gama omega square So gama it is the reflection coefficient and

if I consider one minus gama square it will represent the power deliver to load now what

we do in this method we actually change the input impedance or input impedance it becomes

a function of frequency and that will be vary umm according to our requirement For example

if we have to design one bandpass filter then in band or in pass band it will have good

impedance matching with the source and load And out of band the impedance mismatch it

will be as high as possible and it is done by changing theparameter of this component

whatever we used for filter design So then above the S parameters transmission

parameter is twenty one which is a function of frequency we can write down the transfer

function in terms of this S twenty one as magnitude of S twenty one square this is equal

to one by one plus epsilon square into FN square here capital omega It is a frequency

variable so for now you can consider capital omega is equivalent to small omega and epsilon

it is repel constant now we define different function for this Fn accordingly we name the

filter in different names like butter hot filtersfilter electric filter It can be shown that for a linear time in

variant network the transfer function it twenty one it also can be represted by ratio two

polynomials N function of P divided by D function of p so where this NP and DP they are polynomials

in a complex frequency variable P is equal to sigma plus J capital omega Then we can

define transmission loss response of the filter if we have a S twenty one from that we can

calculate what is the loss Loss its a positive quantity so this is equal

to ten log of one by magnitude of S twenty one square in decibel so once we have transmission

loss which represents the loss power loss from port one to port two for a two port network Then we can easily calculate reflection loss

for a loss less network so LR reflection loss this is equal to ten log of one minus S twenty

one square so for loss less network is eleven square plus S twenty one square that is equal

to one so from using that relationship we are calculating reflection loss here Phase

response of the filter phi from port one to two that is equal to the angle value of S

twenty one again its a function of frequency and the phase response is very important in

particularly white band applications Why Because the delay of phase represents sorry

the slope of phase represents group delay it can be represented as tou D this is equal

to minus del del omega of Phi twenty one Now this is the time taken by your signal to reach

port two its a function of frequency that means this time it varies with frequency so

if it varies more in that case we will be facing pulse distortion and the medium it

becomes dispersive so there should be a minimum allowable value for this time delay variation

over the band to keep that distortion minimum Now we define different types of filter by

choosing different types of polynomials They have their own properties the most popular

one are butterworth or maximally flat response Second one is chebyshev response then elliptic

function response and Gaussian or flat group delay response So let me discuss about one of this so whatever

response for this one you can see S twenty one square transfer function it is represented

by one by one plus capital omega to the power twice N And if I plot loss which was defined

by this relationship LA this is equal to ten log of one by S twenty one square So if I

plot that we will be having a low pass type response you can see here so at capital omega

equal to zero loss is zero and after that it is increasing It has the property that maximum number of

twice N minus zero derivatives at gama at omega equal to zero So that means here for

a loss less system we have magnitude of S twenty one equal to one And from this point

if I increase capital omega then loss its slowly increases and after omega C let us

call it the cut off frequency omega isis defined where we have a specified loss value L AR

after this loss increases rapidly so this is for a low pass filter we can design band

pass filter by using low pass to band pass transformation Next another example this is for cheby chebyshev

response you can look at the plot we have ripple in passband and in stop band we don’t

have any ripple So it exhibits equal ripple passband and maximally flat stop band And

the transfer function is defined as one by one plus epsilon square into TN square where

epsilon this is called the ripple constant it can be given by square root of ten to the

power LAR by ten minus one LAR this is the specified loss value for example

you can define your cut off frequency omega C by three DB in that case LAR that is equal

to three DB you can define your cut off frequency by lets say point one DB in that case LAR

that is equal to point one DB so TN this is cosinocydal function or cos hyperbolic function

depending on the omega value Now if I compare this chebyshev press forms with the previous

one Butterworth lowpass filter response so you

see in pass band for butterworth we almost don’t have any variation where as for the

chebyshev we have ripples in the passband and for the same order we will discuss later

we have higher attenuation slope for chebyshev response So if attenuation criteria are more

prominent than the passband criteria in that case we can choose chebyshev lowpass prototype

response But for butterworth polynomial we have one

advantage if we plot the group delay variation it will have lower variation compare to chebyshev

one So for wide band applications then butterworth should be preferred if we want to avoid any

dispersion so in thisyou can use Gaussian response which will have a flat group delay Theoretically no group delay variation over

the whole bandwidth but its attenuation responses poorest among all this four and this third

one umm electric function response it shows ripples both in passband as well as in stopband

and it is having maximum group delay variation inside the bandwidth So next how to realize this functions So we

start with lowpass prototype so considering you see if I asked to design a lowpass filter

usinglike capacitor or inductors how will be connecting so capacitor usually we use

in shunt configuration so high frequency component will pass through capacitor and inductors

will be using in series configuration so it will attenuate frequency components so the

basic lowpass filter configuration it can be a pi type or a T type configuration so

in this case we are showing both of this two different types In the first type where we are using a basic

pi type you can see two capacitors separated by one series inductor now this elements usually

they are represented by their normalised values we call simply G parameters Gone G two G three

if N is given then last element will be one inductor and if N is odd in that case the

last element will be one capacitor And you can see G nought and G N plus one

they represents shorts impedance and the load impedance represented by two resistors we

have develop this in this case in place of pi we are using a T network two inductor one

G three and one shunt capacitor given by G two Now if N is even in that case last element

is a capacitor and if N is odd in that case last element is one inductor So we have actually different tables available

for this G values for different LAR specified LAR for different types of polynomials here

for example we are showing the element values for a chebyshev lowpass filter where it consider

that LAR equal to point zero one Db Okay let me discuss one point how to define

this LAR better let me draw it So lets say we have a two port network this

port one this port P two and it is behaving as a lowpass filter now if I plot S twenty

one versus frequency so along X axis I am representing frequency in gigahertz and along

Y I am representing magnitude of S twenty one in DB Now if we have perfect transmission

from port one to port two so that means S twenty one in Db it is zero DB and since we

are considering lowpass filter as the frequency increases in that case attenuation increases

or Seventy one in DB it becomes negative goes to minus infinity Now how to define the cut off frequency lets

say I am using this lowpass filter to separate the IF and my IF requirement is hundred megahertz

so IF bandwidth minimum we need hundred megahertz so whatever signal we have it is in this hundred

megahertz we don’t want any attenuation for this hundred megahertz signal so we can

define LAR lets say point one DB or point zero one Db and it should have at least a

minimum value FC of hundred megahertz so LAR then we define LAR this is equal to lets say

point one DB Now in filter applications many times we simply

define three DB cut off frequencies but not here if I define my three DB cut off frequency

at FC equal to hundred megahertz that means already we I am assuming that my fifty percent

power near the cut off frequency it is being attenuated by the lowpass filter which I don’t

want so when I define LAR for this cut off frequency it should be much lower Now to have some specified attenuation at

some given frequency you can define one more cut off frequency lets say for this application

we need at least thirty DB attenuation above FC equal to one gigahertz So in that case

you can define another cut off frequency But you remember whenever we are going to use

this charts this table we first consider what is the signal bandwidth and accordingly you

can choose LAR equal to lets say point one DB or point zero one DB or point zero one

zero zero one DB anything Now after fabricating this filter so whatever

bandwidth I am defining lets say point one or point zero one it can be measure Because

of fabrication tolerance and also umm accuracy problem of VNA this Seventy one it will vary

inside the fast band and this variation will be more than lets say point zero one DB So

thats why it is really difficult to measure that point one DB or point zero one DB bandwidth

of a filter and practically we measure three DB cut off frequencies it is easier to measure So now lets go back now you see in this lowpass

prototype filters if I use one single pi section obviously attenuation slope attenuation will

vary slowly so if I want to improve this curve selectivity that means the attenuation roll

of In that case we have to increase number of section or we have to increase the order

of the filter So we have to use more number of elements This table it shows till G equal to ten G

nought it is they are all normalised values G nought this is equal to one and omega C

its considered as one Then for example lets say if I consider N equal to two we have Gone

equal to point four four eight nine G two equal to point four zero seven eight and N

plus one that load point one one zero zero eight similarly depending on your attenuation

requirement you can choose the order of the filter for example lets say you decided your

order of the filter at least it should be N equal to 7 then you can choose this G parameters This particular row now we have to realize

this G parameters or L and C values so similar types of table available for different LAR

values and also for different polynomials Now practical filter design so already lets

say we have the required G parameter we have to now find out then what is the actual L

and C values Because the G parameters those are normalised values given in the chart So

for that we have to use some sort of impedance transformation Next step we have used omega C equal to one

that is again normalised value so we have to use one transformation for our required

FC so the example which I was showing FC equal to hundred megahertz So then the practice steps you first change

capital omega to practical frequency small omega Then the change terminating impedance

to port impedance Z nought previously it was shown by G nought and G N plus one It requires

obviously impedance scaling So how we define impedance scaling factor gama nought this

is equal to Z nought by G nought when G nought is a resistance and it is equal to G nought

by Y nought when its a conductance So typical Z nought values for any millimetre

wave system or micro wave system fifty ohm Next we have to find out the real LC values

The prototype elements are transformed as L this is gama nought equal into small L so

it is actually that G parameter then capital C equal to small C by gama nought again this

right hand side C this ih the normalised G parameters Similarly we can find out the real

capital R and capital G all the right side they are the G parameters Let us consider then how to design a band

pass filter directly so for band pass filter design we use resonator coupled resonator

systems so obviously it will be a two port network port one and port two they are connected

to my millimetre wave systems which is asystem so in simulation we can represent then port

one and port two by the equivalent resistance fifty ohm and then power should be coupled

from port one to resonator so we have to umm study this coupling mechanism in details and

how the filter design steps it can be represented in terms of this coupling So what we do The coupling from port one to

fast resonator or umm port two to last resonator we represent it by quality factors We call

it external quality factor so let me show you in diagram So here you can see the input port input feed

line and the output feed line we are considering a four resonator systems Also these are conceptual

diagram then the power transfer from feed line to resonator one it is being represented

by external quality factor QE one and for theresonator to feed line it is being represented

by QEN and then the coupling between two resonators It is represented by coupling coefficient I will discuss later what is coupling coefficient

and how we can extract the values by using any full wave simulator This coupling it can

be electric coupling due to the electric field it can be magnetic coupling or even it can

mixed coupling So these are complex situations and the relationship with the prototype LPF

element I am showing you directly the values QE one we can relate it to that G parameter

this equal to G nought Gone divided by FBW FBW represents the fractional bandwidth so

if you have the bandwidth then simply bandwidth divided by this mid band frequency it will

give you fractional bandwidth QEN this is the external quality factor for the second

port this GN into GN plus one divided by FBW And the coupling between I and I plus one

resonator that is a function of fractional bandwidth and the G parameters so FBW divided

by square root of GI multiplied by GI plus one where I it varies from one to N minus

one So you see then if we need higher fractional

bandwidth or for white band design we have to obtain higher coupling So for any given specification then how to

implement any filter So this is we call synthesis tapes first we have to choose the type of

polynomials depending on the requirements fastband lets say group delay variation Fastband

is two hundred and twenty one variation then stop band attenuation requirement curve selectivity

depending on all this criteria we can choose the type of polynomial it can Be anything

butterworth chebyshev electric or a short group delay or Gaussians response So if nothing is specified you can select

chebyshev response most probably Because chebyshev it is somewhat in between butterworth and

electric Then determined the low pass prototype elements from that tabular data that means

the G parameters then calculate the coupling coefficient so you have to determine number

of resonators from umm order of the filters then the required coupling coefficients that

means the coupling matrix basically we have to synthesis so M I And I plus oneit is already defined we discussed

previously then determine the external coupling parameters choose the type of resonators so

it can be of different shapes and implementation also it can be in micro strip line it can

be in lets say image guide it can be in rectangular wave guide anything So depending on again

power handling capability capacity depending on loss requirement we have to choose the

proper umm implementation scheme Next determine filter dimensions so this filter

dimensions we have to use some sort of full wave simulator to realize this filter and

than the optimisation for final design So its a lengthy procedure and it starts with

the QE and K calculation So here what we will do We will assume that we have the required

G parameters and from that already we have calculated the external quality factors and

the coupling matrix So next we will discuss then how to implement this filter after a

short break Thank you