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