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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

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