Welcome to this first lecture on the NPTEL course on basic building blocks of microwave engineering.

Now the theme of this first week lecture will be the mathematical model of microwave transmission.

You know that from the source, the electromagnetic energy propagates. Now it propagates through

various channels and then reaches its destination or receiver or sink. Now we will model this first

part of the transmission and when that theme the first lecture today that will address this concept

of mode in microwave transmission. Myself Amitabha Bhattacharya here in E & ECE department of IIT

Kharagpur. You can reach me in my email here. Now let us see if we look at the universe,

when we are, if we look beyond our territory that is earth, you see so many stars,

so many galaxies etcetera. Now do you know that all of them, they radiate either light that's

why we are seeing them or they are also radiating electromagnetic radiation and there is one Nobel

Laureate Michael Booth who measured and found that in earth everywhere we have something like

2 degree Kelvin of microwave radiation always that the background microwave radiation. So always we

are getting bombarded with microwave radiation, we are living with it. Also, You know Sun emits solar

flare that's why when the solar flare reaches earth, it's not always but sometimes when it

reaches solar flare is different from the normal light, so solar flare when it reaches all the

activities in communication etc. They get stopped, so ISRO has a calendar they keep track of when the

solar flare reaches us and that's why they notify and that time although our satellites etc. they

stopped working because solar flare creates magnetosphere and that creates huge magnetic,

electromagnetic field. All of you are familiar with this site in rainy season you will see

lightning. Now you see a lightning striking the earth. It is a huge source of electromagnetic

emission due to the high potential, high static charge potential there is air gets breakdown and

that's why you see that due to that breakdown the light gets produced, the sound get produced. Now

sound is not electromagnetic energy but this light is and that reaches earth. It you know

if you try to receive this energy will get killed but that energy comes here when it is

not striking neither a tree or anything but in the whole atmosphere you are pervaded with these

terrestrial lightning. Then you see satellites, natural satellites in the right side. In the left

side the man-made satellites huge number of them. All of them are communicating with earth, by what,

by electromagnetic signal. So you are always getting signals from them. Then you see, let us

see a typical family how we are surviving today, there are Wi Fis, that is emitting electromagnetic

environment. You see satellite that is doing that, you see TV that is doing that, you see mobile

phone nothing but electromagnetic radiation, you see a ground station satellite, you see your tab

or pager something you see mobile phone you see electric towers, you see so many devices. So we

are leaving with it. Now we need to understand this environment, if we want to master this

technology, so let us see that electromagnetic spectrum. You see starting from a soccer field

that large size of wavelength to what our molecule etc. all our electromagnetic environment but we

won't talk with this whole spectrum our microwave technology means. You see we have noted that radio

waves you can see that up to the soccer field to roughly the baseball size of wavelength,

baseball diameter wavelength that is radio waves, but after that from baseball to let us say this

a point dot that wavelength type of the radiation for which wavelength is that long that is denoted

as microwave. As you can see that this is the zone of microwave. So here will mainly talk with

that and just beyond that you see. If we go still further in wavelength that means still higher you

know frequency you can appropriately see from here we are reaching terahertz radiation. Now microwave

radiation as the technology is progressing day by day, it is going towards terahertz radiation

and similarly this terahertz radiation also coming day by day beyond. So almost these two

technologies are getting merged or I think when you people will be finishing your carrier that

time it will be completely merged. So we need to understand this technology and for that this

following lectures will help you to understand the basics of that. Now you see from all these field

if I look at any environment, actually we are doing in a particular place we were trying to map

the electromagnetic field distribution and it came something like. This is a color plot , so you see

the red ones they are the maximum things, maximum radiation level, electromagnetic radiation,

then it is coming down it's actually the radiation density, electric field radiation density at that

point and you see that gradually it is going down. Obviously this blue etc. those are very low but

you see there are various types of these field distributions are possible. Now the question is,

if we want to understand and microwave engineering then we need a model that is

the basic task of any engineering, that when we try to analyze or we try to understand something,

we make a model. So that's why we say that if you want to understand, you want to design, you

want to analyze any practical system, you want a mathematical model. So for microwave transmission

system also we want model of this EM signal as I shown you just before that, if I want to have

what is happening I need a model, mathematical model of EM signal. Now EM signal again I have

listed for your convenience from that spectrum graph that you know x-ray, light ray, infrared,

microwave, TV, radio, radar, mobile telephone, landline telephone everything comes under EM

signal. Out of that I can say that microwave TV then radar then somewhat mobile telephone these

are in our microwave zone. So the question is do we need for each of these type of signals because

each of signal is different that's why they are quiet listening at any place all these signals are

present. Now that means if I want to understand do I need a model for each of the signal. Then my job

is just remembering all those models it will be a huge task, so fortunately it's not so. So now all

the same signals they will be Maxwell's laws, all EM signals light you know Sir J. C. Bose,

he proved that okay light is indeed a form of electromagnetic radiation. So let us see light

and mobile phone signal now what can be light can be seen what well phone signal cannot be seen but

still both of them obeys Maxwell situation. There are various solutions of Maxwell's laws,

light signal is one such solution mobile phone signal is another such solution radar signal is

another such solution etcetera etcetera. So do I need to understand all these, so that means

let us make a set of all EM signals, so one can make a set of all such EM signals. Are all these

signals independent? If it was then we had to understand all these signals but no, answer is no,

fortunate for us there are a minimal number of independent solutions such that all EM signals

can be expressed as linear combinations of them. So you see that though there are various solutions

but they are not independent there is a minimal list of independent solutions and all solutions

are linear combinations of them. So that we are fortunate and these minimal number of independent

solutions are called modes. So you see that people generally say that modes, what is a mode, modes

are solutions of Maxwell's equation. I say it is partly to the answer because any electromagnetic

signal as I said that light signal or a microwave signal or a radar signal that is also solution of

electromagnetic signal but all signals are not called modes. Modes are the number of

independent solutions minimal number in which we can express. Now those who are familiar with set

theory they understand that basically this we can call as a basis set. You see if I have a set of

signals then there is a minimal number of sets my minimal number of elements of that set in terms of

which I can express all the elements of that set. That set it all the Macey said like for electric

electrical signals pre-approved the famous furrier by his period theorem we proved that okay you can

take a exponential basis 8 to express all sorts of electrical signal that's why today we take

either cause our sign or sinusoidal signal we analyze that and say okay we know all the

signals because all the signals can be linearly expressed like that. So Fourier took a basis set

okay there are other basis sets also possible. Now modes, it's precise definition means modes

are mutually independent solutions of Maxwell's equation such that every possible electromagnetic

field configuration can be expressed as a linear combination of the modes. so for our learning

purpose if I understand the field distribution in a mode then I can construct any signal,

so my model requires that what will be the this modes field distribution. So let us come that

obviously when Em signal is produced there needs to be a source, that source may be near the signal

because electromagnetic signal propagates as a wave. So when I am looking at the signal the

source may be near may be far away but to start with one kind of source is a point source. Now

point source is actually an ideal source like the concept of point in geometry which you know

that it's an abstraction similarly ideal so point source it's an ideal source it's an abstraction no

real source is a point source. All distance stars as you showing in the first slide that you see so

many stars but they appear to be us as a somewhat not point something more but if we are really far

away then all distance starts appearance point to our eyes but if we go closer to the star, what we

see we see the star it's a huge size there are a variety of contours there are mountains here,

there are rivers here, there are caves here, craters here etcetera etcetera. So any faraway

distributed source of EM field is considered as a point source and if the source is nearby

then we know what is its contour so we can say that it is summation of all those point source.

So any distributed source nearby we call it is summation of point sources. Now point source it

radiates equally in all direction, this point it does not have a preference of radiation to

any particular direction. Again I am reminding that no real source is a point source, every

real source has a preferred way of radiating, a preferred direction in which it radiates more than

others but when we idealize or when we learn thing point source we define as which radiates equally

in all directions. So this that means that does origin there is a point source so in all possible

directions you are giving a wave can be seen equally bright in all directions you have not seen

anything till now which does like that but ideally you can say because if you observe a distance

star, now in the art if you move reasonable distance suppose in a region etcetera you don't

think that there is any change in that, that means that is as a to you it is appearing as a point

source but if you really travel far then you are traveling much so something suppose from northern

hemisphere to southern hemisphere if you come then you would see that okay that is changing.

So we can say that energy is coming equally in all direction from point source and people have

found out the solutions for that it has been seen that image spherical waves in all directions. So

if you are a center of a sphere then the energy is equally going along larger and larger spheres and

coming to you. At a particular distance from the source if we connect all the equal amplitude and

equal phase points that means whose any field electric field or magnetic field it's equal

amplitude and equal phase points if you connect then the locus becomes a spherical surface. so

it's phase front is called spherical so that means when elect any point source is electromagnetic

signal then at a particular distance if we connect all the equal amplitude and equal phase that means

equal signals where they are then if we connect that that becomes a spherical surface. That's why

we say that point source emits spherical waves in all directions. Now at a large distance from

the source you know that sphere if you go on constructing larger and larger spheres now very

large sphere at a particular place you will see that the spherical surface is almost becoming a

planar surface. So that's why we are saying that at a large distance from the source this spherical

surface becomes approximately a plane. Then these waves are called plane waves. Now we'll see plane

waves and or you have already seen plane waves in EM theory plane waves means in that plane there

are electric field vector magnetic field vector they don't have any variation they are constant

over the plane that is called uniform plane wave. First plane wave that all equal amplitude

and equal phase points they are plane and then if you have uniform plane which is a variety

of plane wave where they will have the there is no variation in that plane of the amplitude or

phase of that thing. So that plane waves you have seen already its solution we have seen in anything

it is that Cos Omega t minus KZ type of thing or e to the power minus kz did that thing. So

for plane waves, you have also seen that electric field vector magnetic field vector and direction

of in EM energy propagation of EM energy direction that form a right-handed orthogonal triplet. This

you have already seen. So for plane waves this is two. Now we see the second type of or an another

type of source instead of point source suppose we have all these are obstructions but it will

help you to understand what is a source and how it radiates. Suppose we have an infinite sheet of

surface current source that means a whole plane where we have a one directed surface current.

It is called sheet of surface current source, so here we have taken that in the Z is equal

to 0 plane the sheet of surface current source is like. So what we have written that not only point

sources some other sources also can produce waves whose electric and magnetic field vectors both are

orthogonal to the wave propagation direction. So consider an infinite sheet of conducting surface

current density let us call that surface current density Js and consider obviously since we are

talking of electromagnetic field that means this is not a DC current it is an AC current that I

am writing consider time varying GS the surface current density is time varying it has a time

function it is dependent function of T and also for simplicity I have assumed that it is in the XY

plane it is X directed it can lie in any direction but we can orient our XY thing so that for a

simplicity it becomes X directed thing. Now since you see this source, this source it is an infinite

source though my drawing does not be present but in XY plane it has an infinite variation actually

there is an X here you'll see this is X though I don't think it isn't mixed with the contour so in

XY plane it is infinite. So since it is infinite and you know that if there is an electric there

is an conduction current that means in that direction there are the electric field also. So

if J s is in X Direction obviously the electric field results from in the X direction but since

there is no variation in the whole infinite plane in either X or Y, please remember that it may be

X directed the current is X directed electric field is also X detected but they don't have any

variation because the whole source doesn't have any variation in the neither X or Y direction but

it definitely has a variation in the Z direction because suddenly it is here just at Z is equal to

zero - it is not there at Z is equal to zero plus it is not there. So it has a variation only in the

Z direction but in X or Y direction it doesn't have any variation so this type of source. So

due to the discontinuity as I was saying in the Z is equal to 0 plane, there will be EM waves

propagating away from the source in plus minus Z direction. This will always remember that if

there is any source as a discontinuity that it is suddenly there and then not there then it will

radiate. So it will radiate obviously in plus minus Z direction and already we have discussed

that the electric field is x detected the movement electric the propagation of wave or propagation of

energy is in Z direction. Now let us put the you all know boundary conditions of electromagnetic

fields. So boundary condition and Z is equal to 0 if we apply we know that what this is saying you

know n is a normal vector so n cross e 2 minus e 1 that means what tangential component should be

continuous tangential component of the electric field should be continuous that is a boundary

condition 1 of electromagnetic field. Similarly the tangential component of magnetic field n cross

H 2 minus H 1 that is discontinuous and the amount of discontinuity is the surface current density it

is a second boundary condition of electromagnetic field. In our this particular case n is the

outward normal, so n is positive az directed and e 1 H 1 we are calling it is the electric

and magnetic field in region 1. Region 1 we are defining as Z less than 0 that means below the

current sheet and Z greater than zero is above the current sheet where the fields are E2 H 2 that's

why we have written this boundary condition. Now to satisfy this boundary condition n is a Z so a

Z cross e 2 minus e 1 we know that e 2 minus e 1 will come then the H 2 minus H 1, so az it cross

this is equal to some X component. So obviously it says that H 2 and H Y they must have Y component,

simple mathematics. So we can write the fields as e1 phasor that will be equal to in ax directed and

then we assume some amplitude a first we assumed since H 1 is y directed, so H 1 let us assume in

the lower one you see to satisfy that equation in lower one it will be minus so minus a Y,

a is some amplitude then we know that the field is propagating in Z direction means it's variation

will be e to the power JK not z and you want an h1 we know that always they are related by the

wave impedance in free space it is propagating so eta not the intrinsic impedance of free space, so

there e1 will be like this similarly at Z greater than 0 we can write h2 any to some other constant

B amplitude and some other constant B here. So A B needs to be found out. So to satisfy boundary

condition 1 we get A is equal to B to satisfy boundary condition 2 we get this so solving we

get that A and B their values we get. So by that we can find out that there is an electromagnetic

field, so we see that it also produce surface current infinite surface current that produces

an electric field so that you can have the propagation in the plus minus Z direction. Now

so in this case also you see that we have electric and magnetic field vectors both are orthogonal to

the wave propagation Direction Z. So you see that E and H they are all lying in XY plane the wave is

going in Z direction the E and H field doesn't have any component in the propagation direction

this is the first mode that we introduce that if and both the electric and magnetic field they

are lying on the transverse plane that means transverse to the direction of propagation

then we call that as TEM mode or transverse electromagnetic mode. To any propagation direction

one can draw infinite number of perpendiculars that means if we have any direction to that

you can draw a perpendicular so all these are perpendicular to it you can have many because

with this I can have this as a perpendicular with that I can have other perpendiculars also. So but

all these perpendiculars they will lie in a plane that is in this transverse plane, so this plane so

there any line is perpendicular and now electric field vector can lie along any of these infinite

number of perpendiculars. Now in the transverse plane we can always find in a plane we can find

allways a couple or another perpendicular, two perpendicular couple we can always locate,

so along the perpendicular to the electric plane we can always locate another perpendicular which

is a magnetic field vector. So this TEM wave is a mode then in a guided structure like in normal

coaxial transmission line these mode propagates in unguided fields that means when we radiate

by antennas then at a far off distance from the antenna this is become say TEM wave that we have

discussed that from that starts when we are far away they becomes plane waves plane waves are an

example of TEM mode but reverse is not true apart from plane waves also there are many examples of

TEM wave field distribution, so any antenna is a distribution of point sources so at a far off

distance from the antenna the radiated field show TM mode field distribution. This is the

TEM mode field distribution you see that it is in a coaxial line we get this type of distribution

here you see that we have so many all this is only electric field and I am showing it

is entirely in the XY plane the propagation is taking place in the Z direction. So it is the

magnitude is changing as you see from the color plot but it is streaming. So you see if we look

in the Z direction this is Z direction the in the transverse plane there is no Z component,

so this is an example of a TEM field distribution. Now let us see an infinite line source,

so this Z directed instead of a infinite sheet of current which produces TEM waves we are now seeing

an infinite line source so the source is a line infinite line. il let us call now in the Z plane

will be the transverse plane to these consider an infinitely long conducting line current source

no magnetic field can exist along this infinite line source that is bias about law or Maxwell's

law that you if you have a current source magnetic field is always perpendicular to that that means

in the along this line source there is no magnetic field component. So where is the magnetic field

vector, then it should lie along the transverse plane that means that Z is equal to 0 plane,

so magnetic field should lie there. Now electric field vector can be anything but we have already

seen TEM case. So we say that there both the electric field vector and magnetic field vector

works in the perpendicular plane. Here we say that since we are trying to see whether another mode is

possible another solution of Maxwell's equation is another independent solution is possible,

so that's why we are checking that electric field vector should not be lying entirely in this plane

because if it lies in Dell enhancement that is a TEM case. So if it is in this plane we

are going back to the TEM case, so we are trying to say another fundamental variety of mode. So

electric field vector should have a component in this transverse plane as well as some component

along the line source direction. So this mode is called transverse magnetic that means magnetic

field is entirely in the a plane and if you do the mathematics this line source gives that TEM

mode it doesn't give any TM mode the energy is propagating how energy is propagating if

you consider that considering that line current as an axis if you consider a cylinder throughout

that cylinder surface the energy is propagating these waves are called cylindrical waves. Their

field distribution is TM field distribution. This is a TM field distribution this comes in

we'll see later in circular type of wave guides not coaxial cable that means only one conductor

is here. So in a hollow metallic pipe cylindrical pipe you see electric field vector it is like this

also electric field vector we are showing in the any between thing at X is equal to zero it has a

longitudinal component. This is an example of a transverse magnetic mode. Now consider another

example that instead of that line source let us make a loop, infinite loop of current. Current

calling I, so consider an infinite loop obviously the electric field will be azimuthal you see at

every point electric field is because it should be in the direction of this conduction current

there is a conduction current here. So it is in the ePHI direction that is called azimuthal and

entirely lying in the Z 0 plane. You see here if I you all are lying in the Z is equal to 0 plane.

So the magnetic field vector should not be lying entirely in this plane, why because then we are

going back to the TEM case. Again that same logic so magnetic vector should have a component in the

transverse plane and also some component along the Z direction. These waves are called TE modes,

their field distribution is like this. This comes in will see in waveguides,

you are familiar with that, that it is an example of a T10 mode, this type of various modes but all

of them they have that you see in the transverse plane the electric field is lying, we are seeing

in a longitudinal plane you see it is basically the waveguide between that if we can see we'll see

that field distribution is like this. There is no component which is in the Z direction, so electric

field is entirely in the transverse direction and as I was saying that with mixing this type of

various modes you see three TEM type of electric field distribution if they are mixed then you

get a field distribution like this. So all these various things but all are linear combinations of

these modes so a particular field may be suppose when in a coaxial line transmission line you can

have all the modes possible TEM, TE, TM all the then can be possible. You know waveguides you

cannot have TM and so if you see the actual field distribution you will see something haphazard but

if you break it into these components you will then understand that how much of TE how much

of TM and which type of TE you also have various with numbers these two numbers they are giving two

dimensional phones so various numbers are possible so with that you get all the real-life signals. So

if you understand modes you can easily break the fields into these modes and then your analysis

will proceed. So that was our first lecture that what is the concept of mode I think I have tried

to make you understand there are three fundamental modes TM TE and T0 and wherefrom they come we have

given you examples that which type of sources can produce a pure of that variety in real life

you have a source which is mixture of all these various sources I have shown so with that they

produce various types of TM TE T0 now we need a supporting structure which can carry that so that

energy will be coming from transmitter to receiver that we will see one by one in the next lecture.