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So now we learn some term because we are going
to use this term frequently throughout this class So first among these are phase velocities
and group velocities So when one electromagnetic wave is propagating through free space or
to on medium or it can also propagating propagate through array umm guiding structures we are
associated with two different types of velocities The first one we call phase velocity it represents
the velocity at which the phase of the signal propagates and its a function of omega and
beta we will see and another one is group velocity So its sometimes we simply call it
a signal velocity so if any given envelope is there represents the velocity of that envelope
through the medium or through the guiding structure So lets see the first phase velocity so as
I said phase velocity of a wave is the rate at which the phase of the wave propagates
It can be given by omega by beta So where beta is equal to square root of K square by
Kc square so Kc it represents the cut of wave number so some of for some wave guiding structure
we will see that it has some finite Kc value and for some other wave guiding structure
it does not have any Kc value Kc equal to zero so beta approximately this
equal to two pie by lambda g Where lambda g it represent the guided wavelength so that
means in free space the wave length is lambda not at sixty gigahertz lets say in free space
the wavelength is five millimetre but when the signal is propagates through any medium
or through any wave guiding structure its wavelength changes So if it propagates through any dielectric
medium of dielectric constant epsilon r then approximately lambda g this is equal to lambda
not by square root of epsilon r So that means its become a function of a dielectric constant
and not only that if I plot beta versus frequency so beta is twice pie by lambda g so as I increase
frequency then lambda g it decreases so then beta it should varies with frequency linearly So if beta does not have any linear variation
so that means the phase fellow city it will be different at different frequencies Next
one is the group velocity So the group velocity of a wave is the velocity with which the overall
shape of the waves or wave’s amplitude or we can call the envelope of the wave it propagates And umm it can be thought og as the signal
velocity of the waveform but if there is no loss only in that case and vg this is del
del beta of omega so group velocity again it should be constant and with a frequency
so we can define one term we call it group delay So it can be define for any given channel
or it can be define for any two port network So lets say we have a two port network we
are transmitting any signal from port one to port two so my receiver side is placed
at port two then I am measuring the time delay taken by different frequency component to
reach my port two So if I plot these time delay versus frequency ideally it should be
constant This is the desired thing But practically for any channel or for any component if I
plot group delay versus frequency it becomes actually a variable quantity it varies Then we face problem due to the group delay
variation with frequency what is that problem if I send a pulse A pulse it will have many
frequency components Now this defined frequency components will travel with difference velocities
and it will take different times to reach my port two So then at the receiving side
if I plot the pulse again the shape will change so these effect called as dispersion and for
wide band system its a problem So we can avoid dispersion by avoiding group
delay variation but for any given components we can actually avoid this group delay variation
So what we can do we can minimize the group delay variation by choosing a proper designing
architecture or if this variation is too much in that case we have to use some equalization
technique which will minimize this group delay variation and minimize theeffect as a Now if I measure How to measure this group
delay This is related to the angle of transmission so you know scattering parameters so the angle
of umm transmission that means is to one if I umm plot the variation of this angle with
respect to omega that will give you the group delay So mathematically group delay tau d this is
equal to minus del del omega of angle of S twenty one So by using any pectro network
analyser or by any means If we can measure the total angle from port one to port two
so then we can easily calculate the group delaySo this is a typical group delay plot
versus frequency for passive components so usually for passive components So usually for passive components we face
a concave type group delay profile so at mid band frequency the group delay is minimum
and at band edge left hand side and right hand side group delay is maximum so all the
components it has a finite bandwidth it can operate over all the millimetre wave frequency
or all over the electromagnetic spectrum so it will be higher at the left and right band
edge of any given two port network Then the second term slow and fast waves So
how we define slow wave if VP the phase velocity is less than C that means the speed of light
in free space we call it slow wave So at fixed frequency that means for slow wave lambda
g should be smaller than lambda not so VP then only VP decreases and beta increases
so we represent actually another important at one TT beta by K not so K not this is two
pie by lambda not increased space and beta this is two pie by lambda g any for any given
medium or for any wave guiding structure So then if we plot beta by K not umm it should
be actually it should not vary with frequency so if any wave guiding structure its supports
slow wave then usually its non radiating mode so it will guide that transmission mode through
the structure and its radiates only at discontinuities So whenever we are going to design any wave
guiding structure like micro strip line or CPW line what we will expect We expect that there should be some slow wave
inside so that their wont be any radiation from the structure So we have to keep this
thing in mind also when we are going to design any components at millimetre wave frequency
as well as at microwave and RF frequencies So just opposite to this is the first wave
where the phase velocity is more than that umm of light in free space the disadvantage
of fast wave is that it radiates continuously along its length if its a semi open structure So we will see later that rectangular wave
guide it supports fast wave but since rectangular wave guide its a close structure it does not
radiate along its length But if we have some semi open structure like micro strip line
it will radiate continuously along its length So we cant use it as a wave guiding structure
then But another application it has that we can
design antenna actually there is a category of antenna which utilizes this continuous
radiation along its length we call it the leaky wave antenna and for this leaky wave
antenna the beam direction its a function of beta or frequency So if I change frequency
then beam direction will change so continuous scanning is possible by frequency swipe And in that case the attenuation constant
alpha of that wave it will determine the beam with of the signal So we can control the beam
with of such antenna as well as the propagation direction or beam angle of the antenna So
in this case so lambda g should be higher since VP is higher and beta is lower than
K not Another important term is skin depth So this
term actually explain why at millimetre wave frequencies we cant use metals for wave propagation
So how we define skin depth This is the depth below the surface of given conductor at which
the current density has fallen to one by e times of JS where JS is the surface current
density value So now look at this top right picture so we
are sending some electromagnetic signal through some metallic wire now we are plotting the
current density over the cross section of this wire So you as you can see this current
density is highest on the surface of the wire and if we go inside further inside the current
density decreases it decrease exponentially given by this relationship J equal to JS into
e to the power minus d by delta So if we so what is this delta This is the
skin depth its given by square root of twice rho divided by omega Mu R into Mu not Rho
is the resistivity of this material so this is approximate relationship and it holds good
for metals or lower resistivity you can replace rho by sigma then it will be twice by omega
sigma Mu R Mu not So if I look at this expression one important thing we observed that its function
of omega If I increase frequency skin depth will decrease
So that means at millimetre wave frequency this current will be mostly surface current
component it will flow through a th thin layer of metal just situated on the surface of this
wire So since its utilizing a very thin layer surface resistance will be very high at millimetre
wave frequency the surface resistance is so high the metal it will be very lossy In fact at optical wavelength the frequency
is so high we cant use any metal at all thats why we use in optical fibre always the dielectric
material now lets calculate the skin depth value for at some give at some frequencies
lets say for copper so if I calculate skin depth at fifty hertz for copper its eight
point fivw millimetre and at ten kilohertz it decreases to six hundred and sixty micrometer
at te gigahertz its point sixty six micrometer if I further increase the frequency to hundred
gigahertz it is just point twenty one micrometer So inside the metal we don’t have any current
component in other way we can say that the metal thickness we need at millimetre wave
frequencies it very small The thumb rule is that you just take the thickness five times
than the skin depth value So for example at hundred gigahertz if I take
a metal thickness of one micro meter so it is sufficient to attenuate or to support your
current density We don’t have anything beyond one one micrometer inside the wire So this
is the general expression of skin depth about for metal we use this simplified one Next boundary conditions so lets first consider
a dielectric dielectric boundary The first medium it has a dielectric constant of epsilon
R one and the second medium it has a dielectric constant of epsilon R two and n cap it represent
the surface vector and interface then from the boundary conditions We know that if we assume that no charge or
surface current density is there on the interference then the normal component of displacement
factor is continuous across the boundary and the normal component of B is continuous across
the boundary so that means whatever we have due one inside epsilon R two so if I take
the perpendicular component it is equal to umm D two sorry D two whatever we have in
epsilon R two it is equal to D one in epsilon R one if I consider just the normal component So similarly it can be shown that the tangential
component of electric field it is continuous across the boundary so the tangential component
ET one that is equal to ET two so just inside medium one and just inside medium two is parallel
electric field component they are equal Similarly for the magnetic field H one and H two tangential
component they are continuous Now if we have a dielectric metal boundary
so second example let us consider a PEC so how we define perfectly electrical conductor
so for that sigma is infinite if sigma is infinite then we don’t umm we don’t have
any charge inside this PEC so in that case all the charge it will appear only on the
metal surface or PEC surface So then the normal component of D it is discontinues
by the charge density rho is normal component of B is zero normal and the tangential component
of electric field is zero so this is one important observation for PEC or sometimes we call it
electric valve tangential electric field component is zero So if there is any electric field
on PEC it might be then perpendicular and the tangential component of H is discontinuous
by the surface current density JS Similarly we may have magnetic wall interface
so sometimes we call it umm open circuit condition So for magnetic wall interface we have tangential
magnetic field zero and the relationships are given here normal component of D is zero
normal component of B is zero and tangential component of electric field it is discontinuous
by an imaginary magnetic surface current density and the tangential component of H is also
zero So two important observations from this last
two electric wall and magnetic wall interface for the electric wall then we have only the
normal component of electric field or tangential electric field component zero and for the
magnetic wall we have tangential umm magnetic field zero so we have only the normal component
of magnetic field so this first one electric wall sometimes we call the short circuit condition
and the second one magnetic wall sometimes we call the open circuit condition So now with this terms so lets see what are
the different challenges we face at millimetre wave frequency and then we learn how to overcome
them so whenever we are going to design any millimetre wave systems or any millimetre
wave component we have to deal with this challenges So lets start with this first one Simulation
how we design any components We use different types of EM silver we call it electromagnetic
silver So what it does It basically solves Maxwell’s
equation umm over the structure so we have to first define the physical structure and
then the silver automatically it discretised that physical structure and solve Maxwell’s
equations and now umm so that discretisation number it depends on the wavelength and if
I increase the size with respect to wave length so in that case we have to use more number
of cells we call So that means the overall computational volume
it will increase so for example if we simulate any structure at very low frequency and the
same on at very high frequency lets say sixty gigahertz and above so it will consume more
computational resources so if we simulate lets say any components like a filter at six
gigahertz it can take lets say a ten to fifteen minutes umm in a three gigahertz processor
with eight gb ram But if I want to design a filter for sixty
gigahertz application in the same computer it can take a few hours So next is design
challenge we will see later there are different sources of losses and this loss is much higher
at millimetre wave frequency So how to minimize this loss when we go for any millimetre wave
system design Thats really a challenge and the second point is single mode operation So for any given umm wave guiding structure
we prefer that there will be only one type of mode present in that structure and there
is no excitation of any higher RADAR modes otherwise this high RADAR modes they will
increase the loss of the system and also dispersion so we have to avoid this high RADAR modes
then next is physical realization So we have to choose or we have to use some
materials which will give you lower loss at millimetre wave frequency and we also have
to face the fabrication challenges because we will see that many of the millimetre wave
as well as microwave components are based on transmission line theory and in transmission
line umm following this transmission line theory then this components length will be
given in terms of wavelength So for example you can consider a radiating
umm patch antenna whose length should be lambda g by two at the radiating frequency now lets
see we are designing at sixty gigahertz free space wavelength is five millimetre and in
your umm substrate it will be five by root epsilon R so already the antenna dimension
is very small now due to the fabrication tolerance if it changes by fac even a fraction of millimetre So obviously now the operating frequency will
change so you are designing some antenna for sixty gigahertz applications but due to fabrication
tolerance it can operate lets say fifty five gigahertz or at sixty five gigahertz which
is not desired at microwave frequency sense the antenna length is quiet big so we don’t
face usually this type of problems So fabrication tolerance that is another major important
issue at millimetre wave frequency range So next is system integration and packaging
so finally we have to package this millimetre wave components to protect it from different
severe weather conditions so then what type of materials we should use for packaging and
how to package this millimetre wave components without changing their characteristics so
it again another problem So we have to take into account all this issues
when we are going for any design and whenever we are going for integration for any system
then in that system as we have seen in the first picture that we don’t have only the
millimetre wave components we have also the RF and low frequency components so in the
same module how to integrate the millimetre wave system with the RF and low frequency
system that is another challenge And we have to consider all this effects so
next is testing once I design my component obviously I would like to taste the components
its working at all or not Or it is giving the desired performance or not so we need
some instruments and that millimetre wave frequency the instruments are very expensive
how expensive Lets say we want to buy one vector network
analyser which lets say will support till hundred and ten gigahertz it can cost two
crore so testing that is another important factor and its very expensive only a few labs
in our country has this millimetre wave facilities Next circuit realization so at as how we discussed
that the millimetre wave frequency the circuit size is already very small so fabrication
is a problem loss is a problem so if the circuit size become small in that case its power handling
capability will decrease So if possible use a proper architecture for a receiver so not
only for receiver for any given system there are different architectures possible so for
example lets say we are going to design a receiver So it can be homodyne receiver it can be super
heterodyne receive or it can be a six port receiver So for a six port receiver we have
many passive components parked inside the receiver For super heterodyne receiver we
have to many components to many active components to many filters so if we really want to use
it for handle device it becomes a problem so for handle device thats why homodyne receiver
is preferred or zero IF receiver is preferred so then depending
on applications we have to choose a proper architecture not only for receiver for any
other millimetre wave system So some system uses many VCO so we have to minimize the mil
millimetre wave components to minimize the cost of the circuit So if possible just use a single receiver
for your whole millimetre wave system so again losses due to minimization of interconnects
so interconnects why do we need in the system millimetre wave system will be having many
components ofstarting from antenna we have amplifier we have mixer we have other passive
components like coupler filter Now we have to use some interconnects to connect them So it can be any wave guiding structure or
it can be simple wire bounding orchip attachment but whenever we are going to use them we have
to keep in mind that already my wavelength is very small so for a guiding structure lets
say if it physical length is L so in that case total phase shipped umm from that wave
guiding structure theta it is given by beta into L So beta its a function of frequency so if
I increase the frequency for a given length L so this theta it will increase so then this
at millimetre wave frequencies the wavelength already so small that if the interconnect
even its length is one millimetre it can provide substantial phase ship fellow So we have to take into account this phase
ship as well as loss signal loss when its going through that interconnects So we have
many more challenges or I will take a short break again and then we will continue Thanking
You

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