Transmission Lines and E.M. Waves Lec 26

8/20/2019 Transmission Lines and E.M. Waves Lec 26

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Transmission Lines and E.M. Waves

Prof R.K. Shevgaonkar

Department of Electrical Engineering

Indian Institte of Technolog! "om#a!

Lectre$%&

Welcome, in the last lecture we saw the wave propagation in a medium which is not ideal

but a good dielectric, now we investigate the wave propagation in a medium which is a

good conductor that means in this medium the conduction current is much larger

compared to the displacement current. So let us take a medium which we call as the good

conductor as we characterize earlier a good conductor is characterized as σ is much larger

compared to ω ε0 εr that means the conduction current density is much larger compared to

the displacement current density then we call that medium as the good conductor.

s we did in the previous case again we !ind out what is the propagation constant !or this

medium, we separate the real and imaginary parts and !rom there we get the attenuation

and phase constants. s we have seen the propagation constant " !or this medium will be

( )0 # # r ωµ σ ωε ε + .

$%e!er Slide &ime' 0('() min*


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+ow i! substitute this condition σ --- ω ε0 εr then this approimately can be written as

#ωµ σ because now this /uantity is negligible compared to this /uantity. So i! have a

conductor essentially the displacement current contribution is neglected now and because

o! that we have the propagation constant which essentially is given by that.

+ow i! you separate the real and the imaginary part o! this then the #ωµ σ

is straight

!orward we want the #

so #

can be written as #

(eπ

which again e/ual #

)eπ

that is

e/ual to

#

( (

+

.

$%e!er Slide &ime' 01')( min*


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So i! substitute this into that will get this /uantity propagation constant " will be e/ual

to

#

( (ωµσ

+

.


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$%e!er Slide &ime' 0)'02 min*

s you have done earlier we call this real part o! this /uantity as the attenuation constant

and we call the imaginary part as the phase constant. 3or this medium we have the

attenuation constant 4 5 (

ωµσ

and we have phase constants 6 5 (

ωµσ

that is !or this

medium the attenuation constant 4 and the phase constant 6 are almost e/ual so we have

4 is almost e/ual to 6.


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$%e!er Slide &ime' 07'7 min*

+ow i! you go back to our understanding o! &ransmission 8ine we have characterized the

&ransmission 8ine as a lossy line whenever we have attenuation constant comparable to

the phase constant. n this situation the attenuation constant is e/ual to the phase constant

means this medium i! at all we want to visualize similar to the &ransmission 8ine this is a

etremely 8ossy &ransmission 8ine.

So !or a good conductor since the attenuation constant is e/ual to the phase constant this

medium is like a very lossy line case. &hat means when the wave tries to propagate in this

medium its amplitude dies down very rapidly in the direction o! propagation. So that

means when the wave tries to enter the conducting medium its amplitude reduces very

rapidly because it is having the attenuation constant which is e/ual to the phase constant.

So a good medium can be visualized like a etremely 8ossy &ransmission 8ine.

9owever there is a small di!!erence between a 8ossy &ransmission 8ine and the

conducting medium. n a 8ossy &ransmission 8ine the power was getting lost into the

resistance o! the line or the conductance o! the medium separating the two conductors o!


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the line so there was a loss o! power when the wave propagated along the &ransmission

8ine. 9owever, in this case when the wave is attenuating not necessarily the power is

getting loss into the conductor. n !act we will see little later when we have the case σ 5 :

so 4 becomes e/ual to in!inity, that means over a zero distance the wave will die down as

you try to penetrate in this medium but in the small thickness there is no power loss. So

we will see that when we talk about the conductors the wave cannot penetrate in this

medium because the 4 is very large there is a lot o! attenuation, that does not mean that

the power is getting loss into the heating o! the medium something else hammers to the

power and that we will investigate it later.

9owever, at this point we can say that i! a good conductor can be characterized like a

8ossy &ransmission 8ine and there is the reason we can have this /uantity 4 5 6 and then

the wave amplitude will vary as e ;4 where is the direction o! the wave propagation.

So i! plot the wave amplitude as a !unction o! i! plot this will get essentially a

eponential curve which is this law. So i! travel a distance 5

α then the wave

amplitude would die down to one over e o! its initial value so i! take this value to be

zero and have a certain wave amplitude at 5 0 here, over a distance o!

α the wave

amplitude will die to one over e o! its initial value.


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$%e!er Slide &ime' 0


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so no matter what is the thickness or the depth o! this conductor only this thickness where

the !ield die down to one over e o! its initial value they are the one which simply matter

in the propagation o! electromagnetic wave, beyond that whether the path conducting

medium is there it doesn=t really a!!ected.

So in practice when ever we are having the conducting boundaries essentially this region

which is very close to the sur!ace o! the conducting boundary will decide the propagation

characteristic o! the electromagnetic wave. What is deeper in the medium is it is o! no

relevance because the !ields essentially die down very rapidly as we go inside the

conducting medium.

So i! write this 4 over this depth what ever this /uantity is let us call that /uantity as δ .

So here the e!!ective depth δ is

α and i! substitute the value o! 4 !rom here then that

will be

(

ωµσ .

! substitute !or ω which is two by !re/uency so that will be e/ual to

(

( f π µσ where (

will cancel so !rom here essentially can get

f π µσ .

$%e!er Slide &ime' ('7> min*


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So have an e!!ective depth o! penetration o! the electromagnetic wave inside a good

conductor that is inversely related to the s/uare root o! the !re/uency it is inversely

related to the s/uare root o! the conductivity. So higher the conductivity or higher the

!re/uency less will be the depth o! penetration, not only that when the conductivity is

in!inite that means i! take the ideal conductor in that case this depth will be zero that

means the wave will not penetrate the medium at all. So i! you take a ideal conducting

medium there is no penetration o! electromagnetic energy in the ideal conductor.

Similarly i! go to very arbitrarily high !re/uencies the depth o! penetration becomes

etremely small.

8et me take some typical numbers and #ust to get the !eel !or what depth we will have i!

you operate at the radio !re/uencies. So let us say take some arbitrary medium !or

which the conductivity is approimately 0? @ m J , it is a good conductor and let us say

take a !re/uency ! 5 00 A9z a typical !re/uency which is used !or the &B transmission

or !or the %adio propagation.

$%e!er Slide &ime' )'1 min*


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+ow !rom here i! substitute !or this δ get this δ is e/ual to s/uare root o! one upon C

into !re/uency which is 00 mega hertz which is 02 9z so this is into 02 into D and let

us assume the permeability o! this medium is the same as the !ree space which is !our

power 0;? so let me assume the permeability o! the medium D 5 D 0 which is e/ual to )C

0;? 9enry per meter so can write here )C 0 ;? multiplied by σ which is 0?. !

simpli!y, this will be)

( 0π × meters or can multiply this by 0> to make it micro

meters so this will be approimately about !i!teen micrometers.

$%e!er Slide &ime' >') min*


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! take a good conductor which is having the conductivity o! order o! 0? and let me

remind you that a good conductor like copper have a conductivity o! 7.> × 0? the

conductivity even higher than this. ! take a good conductor material in practice like

copper or silver and i! take the !re/uency let us say hundred mega hertz then the depth

o! penetration o! the electromagnetic wave is only !i!teen microns that means the wave

propagation is #ust !i!teen microns !rom the sur!ace o! this good material which is having

conductivity 0? and we saw as the !re/uency increases the skin depth becomes smaller.

So, i! you go to !re/uency o! Eiga hertz the !re/uency which is used !or Satellite

communication or the microwave !re/uency or the millimeter wave !re/uencies this

number will become even smaller.

So what we !ind is e!!ectively !or a good conductor i! you take the !re/uencies which are

typical radio !re/uencies the depth o! penetration typically is o! order o! !ew microns or

!ew tens o! microns. Fasically this phenomenon is taking place only on the skin o! the

material which is o! this order o! !ew microns to !ew tens o! microns. &hat is the reason

why we call this e!!ective depth as the SG+ HIJ&9 o! the material. So we call the delta

an e!!ective width as the SG+ HIJ&9 o! the material.

$%e!er Slide &ime' 2') min*


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So skin depth is a parameter which tells you over which the electromagnetic energy is

going to eist !rom the sur!ace o! the good conductor. Feyond this depth again as

mentioned the properties o! the material do not really matter. So i! you have a hal!

!re/uency system a component since the energy is going to lie or the !ields are going to

lie only within a distance o! !ew tens o! microns we have to make that layer very good

because any perturbation in the properties o! that layer is going to a!!ect the propagation

o! the electromagnetic wave.

8et us say have a solid conductor and want to send !rom high !re/uency signal using

this conductor the !ields are essentially is going to lie only on the sur!ace o! that

conductor within the skin depth. So i! take a conductor let us say something like that

and you have to say the electromagnetic energy is going to propagate along this

conductor. &he current which are going to be ecited because o! these !ields are not going

to lie in the entire cross section o! the conductor they essentially be con!ined to the skin

depth o! the conductor only this is the region where the !ields are going to eist and

because o! that the currents are going to !low.

$%e!er Slide &ime'


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So this region though you are having conductor here is not really contributing to the !low

o! current because the current is #ust con!ined to this where this thickness is δ which is

typically o! the order o! !ew tens o! microns.


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$%e!er Slide &ime' (0'0) min*

So even i! you make this conductor hollow the propagation o! the energy along this

conductor will not get a!!ected because this layer still remains intact. So two things

happen here !irstly since the phenomenon is only taking place on the sur!ace this material

does not play role in the propagation o! electromagnetic energy, secondly i! you wanted

to have a good propagation the properties o! this small thin layer should be very good

because any perturbation in the conductivity o! this layer would a!!ect the properties o!

electromagnetic waves signi!icant. So that is the reason when we go to the microwave

component where the !re/uency is very high and the skin depth becomes only a micron

or something like that the sur!ace has to be etremely good and per!ect because any small

perturbation on their sur!ace where most o! the energy is con!ined to will a!!ect the

propagation o! electromagnetic wave.

Knce we understand this then we !ind something interesting now and that is let us say we

have a !inite conductivity o! this medium through which we were sending the signal

around this conductor and we have certain area o! cross section or radius !or this you

have etremely low !re/uencies the current will be uni!ormly distributed over this so

can !ind out the area o! the cross section and !rom the area o! cross section can !ind out


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what is the resistance o! this wire per unit length because know the resistivity o! the

medium.

9owever as we increase the !re/uency and have the skin depth starts coming into the

picture the current is now getting more and more con!ined to the sur!ace that means the

area o! cross section over which the current is !lowing is reducing now. s a result the

resistance o! this conductor starts increasing as you go to higher !re/uency. So because o!

the skin depth the resistance is now becomes a !unction o! !re/uency. 3or a given

conductivity as the !re/uency increases the skin depth reduces, as the skin depth reduces

the area o! cross section over which the currents are !lowing have reduces and because o!

that the resistance o! the conductivity increases.

So now what we have !ound is the resistance which we treated more like the property o!

the wire is not so when we go to high !re/uencies and when the skin depth starts playing

a role. s we go to higher and higher !re/uency the same conductor start showing more

and more resistance and the wave propagation essentially is con!ined to the sur!ace o! the

conductor.

nother aspect o! the high conductivity o! the medium is i! we look at the Lharacteristic

mpedance o! the medium or the ntrinsic mpedance o! the medium cη

is we saw earlier

is

#

#

ωµ

σ ωε + and now we are having σ much larger compared to ωε so this is

approimately

#ωµ

σ .


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gain i! you separate the real and imaginary parts then the intrinsic impedance o! the

medium will be #

( (

ωµ ωµ

σ σ +

or i! you write in terms o! the magnitude and phase this

/uantity will be)7

ωµ

σ ∠ o

.

$%e!er Slide &ime' ()'12 min*

! you recall the electric and magnetic !ields in a medium or related to this /uantity called

the intrinsic impedance o! the medium we have seen !or dielectric medium or !or a

medium where the magnitude o! the electric !ield and the magnitude o! the electric !ield

are e/ual to η . So in this case the ratio o! this will be e/ual to the Lharacteristic

mpedance or the ntrinsic mpedance o! the medium and now this /uantity is e/ual to

)7ωµ

σ ∠ o

.


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$%e!er Slide &ime' (7'(7 min*

So what we !ind is the electric and magnetic !ields are perpendiculars to each other the

trans electromagnetic wave nature is still there but now there is a time lag between the

magnetic !ield and the electric !ield. So i! you take the ratio o! these two the ratio is now

dependent on !re/uency also not only on a medium parameter that is used to be in the

case o! dielectric medium and also the magnetic !ield now lacks the electric !ield by)7

o

.

So when we talk about a good conducting medium important things happen where when

the wave tries to penetrate in the conducting medium it penetrates over a very short

distance called a skin depth and also the electric and magnetic !ields are not in time phase

any more but there is a phase di!!erence in time !or the electric and the magnetic !ield and

which is !or a good conductor is approimately )7o

.

So what we !ind !rom this discussion is a wave propagation in a dielectric medium and in

a conducting medium are completely di!!erent n a dielectric medium the wave can really

go deeper into the medium the electric and magnetic !ields are in time phase the wave

amplitude reduction is very small whereas i! go to the conducting medium the wave


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attenuates rapidly it cannot go really deeper into the medium, also the electric and the

magnetic !ields have a phase di!!erence in time with each other.

+ow i! you take a general medium which is a combination o! the Hielectric and the

Lonductor. &hen o! course all these relationships are going to be comple we cannot

make any approimations. So !or those media !or which the conduction current is

comparable to the displacement current but in the two etreme cases !or a good dielectric

and !or a good conductor these approimations can be made easily and one can

understand the wave propagation in a more comprehensible manner than in a general

media.

With this understanding now we go with another aspect o! the wave propagation and that

is what the velocity o! the wave when it tries to move in this medium.

+ow !irst let us take a medium which is again a good dielectric and without loosing

generality let us say the losses in dielectric are very small so can treat this dielectric

almost like a lossless dielectric. &hen how the wave is going to propagate in this

mediumM s we have seen i! the medium was lossless then the propagation constant is

purely imaginary and then any !ield I can be written as some 0I

e;#6 and i! want to put

eplicitly the time !unction e #ωt.


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$%e!er Slide &ime' (


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+ow i! stand on a particular location in the space that means i! is constant and look at

the wave which is moving in the space the phase o! this wave will be varying with

respect to time as ωt that means the phase is linearly increasing as a !unction o! time. !

instantaneously look at the wave in the space again we see the phase is varying as N6

because again it is varying linearly at a !unction o! distance in the space.

So i! we become an observer and stand in the space and look at the wave which is

moving then at any particular point o! the wave i! you monitor its phase will vary linearly

the !unction o! time.

9owever suppose hold on to a particular phase point on the wave let us say this is the

wave i! take any particular point and hold on to this point so the observer is standing

and watching the wave moving but let us say we are holding on to a particular phase

point on the wave so this is the point want to hold on to. +ow since the wave is moving

i! have to hold on to a particular phase point must move along the wave then only

will hold on to this point, otherwise the wave will simply move and the phase will

change. So an observer has to move along with this wave with a velocity with which the

wave is traveling i! he does not want to loose this particular point to which he is holding

on to or in other words now i! the observer is not losing this point i! you look at the phase


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o! the wave the phase is not changing anymore !or him because he is holding on to his

point we are looking at the wave and !or him the phase is always same because he is

moving with the wave.


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$%e!er Slide &ime' 1('0< min*

So i! we consider an observer such that he holds on to a particular point and thus what

ever is needed not to loose this point then !or that observer the phase appears to be

constant at the !unction o! time. So what we do is we take this /uantity the total phase o!

the electromagnetic wave and i! make this /uantity constant as a !unction o! time i!

make this stationary at a !unction o! time what ever speed the observer has to move to

make this stationary is the velocity o! this wave. So what we say is OAake the phase

stationary as a !unction o! time to get the velocity o! the wave.


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$%e!er Slide &ime' 11'(0 min*

So i! say that this /uantity has a total phase which is a combination o! space and time i!

make that constant say ωt N 6 is constant !or that observer which is moving with the

velocity o! the wave. &hen !rom here i! take a derivative o! this with respect to time

will get ω N 6

d.

dt 5 0 will give me this /uantity

d.

dt which is nothing but velocity that is

e/ual to

ω

β .


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$%e!er Slide &ime' 1)'2 min*

+ow since this velocity we have calculated !rom the stationarit! o! the phase we call

this velocity as the phase velocity o! the wave which means the !aster phase point o! the

wave is moving with this velocity. 8ater on we will see this concept is essential because

there is another velocity with which actually the energy travels. n this case when we are

talking about unbound medium you have chosen a coordinate system such that the wave

is traveling along direction, this appears very trivial.

9owever in general when we talking about a wave moving about an arbitrary direction

this concept o! saying that the phase is moving with this velocity but the energy might

move with some other velocity will become more apparent. n this situation when you

said that the phase is moving with this velocity we call this velocity o! the wave as the

phase velocity. So we have to eplicitly saying that this velocity is the phase velocity,

normally it is denoted by v p and this /uantity is nothing but

ω

β .


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So the /uantity v p is not the /uantity which people know a priory !or any medium !irst !or

a given !re/uency we !ind out what is the phase constant then

ω

β gives me a velocity

which is velocity and we call that velocity as the phase velocity. So essentially

knowledge o! propagation constant or phase constant is important because that tells me o!

what is the velocity o! the phase o! the electromagnetic wave when it travels into the

medium. n !act, !or a traveling wave this is very straight !orward so get a relation

which is like this.

9owever the concept o! phase velocity can be etended to any arbitrary wave

propagation. 3or eample suppose have two waves which are traveling in opposite

direction !orm the standing wave can still de!ine the total phase which is a combination

o! space and time and by making that /uantity stationary as the !unction o! time may get

the epression !or the phase velocity o! the wave. So the concept is very general though

!or traveling wave it is very simple and then we get a phase velocity o! a traveling wave

which is e/ual to

ω

β . So we will see that this relation that we will use very o!ten when we

go to the propagation o! electromagnetic wave.

$%e!er Slide &ime' 1?'0> min*


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+ow as you have seen earlier i! take a Jure Hielectric with dielectric constant ε r then !or

this medium the phase constant 6 is 0 r ω µε ε

. Since the medium is dielectric medium

permeability o! the medium permeability o! the !ree space so the phase constant is

0 0 r ω µ ε ε .

+ow let us consider a medium which is !ree space that means !or which epsilon εr is . So

this is !or this dielectric !or !ree space where ε 5 ε0 and 6 5 0 0ω µ ε

.

$%e!er Slide &ime' 12'(2 min*


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So the velocity !or the !ree space v p 5 0 0

ω

ω µ ε which is again e/ual to 0 0

µ ε .

! substitute !or 0 0 µ ε

in this then this will be

? <

) 0 0

1>π

π

− −× × × and that will be

e/ual to

2

1 0 m

s× .

So this is the velocity o! the wave when the wave is traveling in the !ree space and that is

21 0 m s

×. t will immediately strike to us this is the velocity o! the light in !ree space.

So since light is a transverse electromagnetic wave it is traveling with a velocity which is

given by this /uantity which is the phase velocity and that is the reason why velocity o!

light is

21 0 m s

×.

$%e!er Slide &ime' )0'00 min*


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So not only light but any wave which is transverse electromagnetic wave and it is

traveling in the !ree space will have a velocity which is e/ual to

21 0 m s

×.

So what we can do is we can substitute !rom this one into this. So 0 0

µ ε is the velocity

o! light in vacuum and normally is denoted by the parameter c.


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$%e!er Slide &ime' )0'10 min*

So essentially the velocity o! wave in a dielectric medium v p will be 0 0 r

ω

ω µ ε ε which is

some 0 0

r µ ε ε

. nd this /uantity 0 0

µ ε is nothing but velocity o! light in vacuum

which is c so that is e/ual to r

c

ε .


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$%e!er Slide &ime' )'? min*

So here !irst thing to note is when the electromagnetic wave travels in a medium it always

slows down because i! this /uantity is greater than then the phase velocity will be

always less than the velocity o! electromagnetic wave in the !ree space which is the

velocity o! light. So velocity o! any electromagnetic wave in a dielectric is always less

than the velocity in the !ree space.

We also know that the re!ractive inde o! the medium is de!ined as the ratio o! the

velocity o! light in vacuum divided by the velocity o! light in that medium. ! go by that

de!inition we know that we have the %e!ractive nde o! the medium let us say we denote

that by small n which will be e/ual to velocity o! light in vacuum divided by velocity o!

light in medium. So this /uantity is nothing but pv

c

so that is pv

c

which is nothing but

r ε

.


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So we have a very important relation that is the %e!ractive nde o! the medium a lossless

medium is e/ual to the s/uare root o! the dielectric constant o! the medium.

$%e!er Slide &ime' )1'? min*

So when ever we have a medium which is pure dielectric or a good dielectric though

%e!ractive nde and the dielectric constant are related to this relation and the

electromagnetic wave always slows down compared to the vacuum when it enters any

medium which having a dielectric constant greater than .

+ow also the same thing we can do !or the conducting medium we can take a conductor

and no matter what is more the penetration the wave has inside the conductor we can still

ask what is the velocity o! this so again !or this conductor we have 6 which is

0

(ωµ σ

so


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we can have the phase velocity v p 5

ω

β which again is e/ual to

0

(

ω

ωµ σ

so that is e/ual

to 0

(ω

µ σ .


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$%e!er Slide &ime' ))')) min*

So the velocity which we had !or a dielectric medium was r

c

ε the velocity was not a

!unction o! !re/uency so i! the dielectric constant o! the medium is not a !unction o! the

!re/uency the velocity also is not a !unction o! !re/uency that means all !re/uencies

travel with the same speed in the medium.

9owever i! take a medium which is a conductor even i! the medium properties are not

varying as a !unction o! !re/uency the phase constant to the !unction o! !re/uency and the

velocity also is the !unction o! !re/uency. &hen we call this medium as a dispersive

medium because the velocity is varying as a !unction o! !re/uency. So when v p is the

!unction o! !re/uency we call this medium as the dispersive medium.

So what we !ind is a medium which is conductor the velocity becomes a !unction o!

!re/uency that means all !re/uency does not travel with the same speed and because o!

that we have dispersion in the medium. So now we have something important

conclusions to draw. 3irst o! all the electromagnetic wave when traversed in the medium

we can de!ine a parameter what is called the phase velocity that is the parameter which


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tells you with what speed the phase or constant phase point on the wave is traveling in the

medium. &hen we !ind !or dielectric medium the phase velocity depends on the dielectric

constant o! the medium and !or dielectric constant greater than one the phase velocity is

always less than the phase velocity in vacuum that means the wave always slows down

when it travels in a medium with higher dielectric constant.

9owever a dielectric medium is not dispersive medium because the wave slows down but

all !re/uencies travel with the same speed. &hen we also !ind that the parameter called a

%e!ractive nde o! the medium which is the ratio o! velocity o! a wave in vacuum to the

ratio o! the velocity o! the wave in the medium is e/ual to the s/uare root o! the dielectric

constant o! the medium. nd then when we go to the conductor we !ind that the velocity

becomes a !unction o! !re/uency that means the medium becomes a dispersive medium.

So a pure dielectric medium is a non dispersive medium but a good conductor is a

dispersive medium.

&his essentially summarizes the propagation o! transverse electromagnetic wave in an

unbound medium. Fe!ore we proceed !urther now we investigate !irst the power !low

associated with the electromagnetic waves and then we will go to more comple analysis

o! this which is propagation o! electromagnetic wave in bound medium or in a medium

which are having at least the inter!aces dielectric or the conductive inter!aces.

&hank you.

Transmission Lines and E.M. Waves Lec 26

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