Review of electromagnetic principles, we will
continue with that chapter. Previously we
have seen details of Maxwell’s equation
module 1A and 1B. In this module that is module
number 2, we will look into uniform plane
waves in different media.
First we will define what is meant by uniform
plane waves and what is meant by transverse
electromagnetic waves or TEM waves. We will
derive the wave equation, then after that
we will look into wave propagation in pure
dielectric that is lossless media, then in
a lossy media which has got a finite conductivity,
we will also introduce the concept of skin
depth which will be used several times in
the remaining chapters.
What is meant by plane waves? For this look
at the sketch over here in Cartesian coordinates.
Imagine that you have a wave that is travelling
in the set direction for convenience. Then
a plane perpendicular to this direction lies
in the XY plane, so XY plane so imagine a
sheet lying along XY plane and if all your
electric field vectors and magnetic field
vectors are lying in this plane then we call
it as a plane wave. So in a plane wave, electric
and magnetic field vectors are in a plane
perpendicular to the direction of propagation
so in this case in the XY plane.
Now what is meant by a uniform plane wave?
Imagine that the E vector and the H vector
has the same value independent of the position
in its XY plane that is everywhere along the
XY plane, then we call it as uniform plane
wave so it is uniform all along this plane.
Now what is meant by transverse electromagnetic
waves? So uniform plane wave is a special
case of trance waves called transverse electromagnetic
waves, it only means that E vector and H vector
lies in a plane that is perpendicular to the
direction of propagation. Now another example
of transverse electromagnetic waves is a transmission
line that is two wires two parallel wires
or one wire above ground plane, we will consider
transmission lines in later modules.
Now just from this figure we can derive some
properties for this electric and magnetic
field vectors say for example, E and H vector
do not have any component in the direction
of propagation and they are denoted with respect
to X and Y because it is uniform. Now we write
these properties in terms of equation in the
next slide.
Say from the above figure we can say that
rate of change of E with respect to X is 0
and E is aligned to the X direction by choice
without any loss of generality and H is aligned
to Y direction without any loss of generality,
so the other components are all 0. Now, let
us take the curl of the vector E and applying
the Faraday’s law that we have seen before,
so we take the differential form or the point
form of the Faraday’s law. If you now perform
your vector algebra, the curl of E is defined
by this matrix, here there are the unit vectors
with differential symbols corresponding to
X, Y and Z and these are the components of
E vector.
So since by our choice of coordinate system
and without any loss of generality, E is aligned
with X direction, E Y and E Z are 0. Similarly,
rate of change of E vector with respect to
X and with respect to Y are also 0, so this
curve can be simplified by this equation over
here that is curl of E equals rate of change
of magnetic flux density with respect to time
is simplified like this, over here the special
variation is with respect to centre, so S
wave is moving forward in the set direction,
vector E can change.
Now let us consider the time harmonic variation,
so this is the fraction of space and time
so if the time variation is in the harmonic
manner in the sinusoidal manners which is
frequency domain is represented by E to the
power J Omega t, where Omega is the angular
frequency and F with tilde on top this is
the phaser and that phaser can be written
as magnitude of the phasor and this is angle
with respect to your reference so we can write
it like this also F 0 angle Theta. Now if
you substitute this into this equation and
similarly same equation can be written for
magnetic field also in phaser form then E
to the power J Omega t on both directions
in both sides will cancel each other and what
is remaining is this equation, rate of change
of E with respect to Z equal to - J Omega,
J Omega is coming from the differentiation
of this, J Omega Mu H so this is one of the
equations.
Now similarly, from the differential form
of ampere’s law states that curl of magnetic
field intensity vector is equal to the sum
of conduction current and displacement current.
So this can be simplified taking the curl
here, instead of E it will be H in this form
and in phaser notation we can simplify it
in this way, where d by dt of E vector will
yield J Omega here. Now, from this equation
and this equation combined that is differentiating
with respect to Z and substituting in each
other we get the wave equation total like
this, so 2nd derivative of E with aspect to
Z equal J Omega Mu Sigma + J Omega Epsilon
multiplied by E.
So this value J Omega Mu times Sigma + J Omega
Epsilon, so this is the function of material
property in terms of conductivity and electric
permittivity and the frequency Omega, so this
factor is called as propagation constant square
so Gamma square, so similarly with the H field
also we can write a wave equation.
So our propagation constant gamma is square
root of J Omega Mu Sigma + J Omega Epsilon
that is, it is the property of frequency and
the property of the material through which
it is propagating. Now this can be separated
into real and imaginary parts, and let us
say Alpha is real part and Beta is imaginary
part. Now the real part is called that attenuation
constant and this imaginary part Beta is called
the phase constant. Now both Alpha and Beta
has unit per meter however, just to distinguish
between imaginary and real part we call Alpha
that is at attenuation part as neper per meter
and Beta as radians per meter. Now a general
solution for wave equation in terms of travelling
waves in forward direction is given by E as
a function of Z and t equals some magnitude
value E to the power - Gamma Z E to the power
J Omega t.
A general solution for wave equation in terms
of travelling waves in forward direction,
E_x (z,t)=E_x0 e^(-Gammaz) e^jOmegat=E_x0
e^(-Alphaz) e^(j(Omegat-Betaz));
Also H_y (z,t)=H_y0 e^(-Gammaz) e^jOmegat
So Gamma can be expressed in terms of Alpha
and Beta so if you expand it, this part is
the attenuation part E raisee to - Alpha Z
then you have a phase part also similarly,
H can also be written in this manner. Now
since it has got the real part and the imaginary
part, if you try to measure the field with
an instrument, what we measure is the real
part of the field that is the physical real
part of the field and that part is given by
E x e to the power - Alpha Z Cos of Omega
t – Beta Z so that is the real part of the
field. So you can see that amplitude is varying,
not only that there is a periodicity in the
amplitude, not only that wave is going through
the medium, it is decreasing exponentially
by this factor.
Now, the wave has a velocity into the medium
and that velocity we can derive by tracking
the points where Omega t - Beta C becomes
constant, or you can say that okay you pick
up a fix point along the time varying and
attenuating wave, so that condition is rate
of change of Omega t - Beta C with respect
to time is equal to 0 and from this we get
d Z by dt if you differentiate both terms
with respect to time, so that is the phase
velocity or V and that is from this we get
it as Omega by Beta, Beta is the phase constant.
So this may not be speed of light because
it depends upon the property of the medium,
and in vacuum of course will be the speed
of light.
Now we can define another property of the
medium, intrinsic impedance of the medium.
We have seen previously that E is given by
this expression and similarly H is given by
this expression and in general we know that
impedance is expressed as E x by H y, but
even without using this you can manipulate
this equation and find that H is given by
E x by Z m and Z m equal to J Omega Mu by
Gamma propagation constant, so that is equal
to square root of J Omega Mu divided by Sigma
+ J Omega Epsilon. So impedance of the medium
is given by this frequency as well as in properties
of the medium there are 3 basic properties
for a medium; one is the conductivity Sigma,
the electric permittivity Epsilon and the
magnetic permeability Mu, so from this we
get the impedance of the medium.
E_x (z,t)=E_x0 e^(-Gammaz) e^jOmegat
H_y (z,t)=H_y0 e^(-Gammaz) e^jOmegat=E_x0/Z_m
e^(-Gammaz) e^jOmegat
Z_m= jOmegaMu/Gamma= Square root (jOmegaMu/(Sigma+jOmegaEpsilon))
Now let us go to the lossless media that is
a good dielectric, where conductivity is equal
to 0. Previously we have seen proposition
constant Gamma which can be expressed as Alpha
+ J Beta, now we can solve for from this equation
we can solve for Alpha and Beta, you know
you can remove this square root by squaring
both sides and then comparing all the real
terms with the imaginary terms, you will get
for Alpha this particular expression that
is the attenuation constant and for Beta you
will get this expression that is propagation
constant, so you can try it at home and see
that this is correct. Now Alpha and Beta differs
in this term over here, here it is - and here
it is +.
Now let us consider the lossless media where
conductivity is equal to 0 so we substitute
this value into this, immediately we see that
this is equal to 0 but this will not be equal
to 0 so this will be square root of 1 + 1,
so basically 1 and here there is a square
root. So this square root of 2 and this cancels
and you get Beta equal to Omega square root
of Mu Epsilon which is nothing but 2 Pi by
lambda. Now 
intrinsic impedance seldom becomes square
root of Mu by Epsilon that you can see by
substituting Sigma = 0 in this special form
intrinsic impedance of the medium that we
have seen before. Now phase velocity V = Omega
by Beta that is equal to substituting for
Beta, 1 by square root of Mu Epsilon which
is nothing but frequency times lambda.
These equations are strictly true for Alpha
equal to 0 but we can use these equations
with very good approximation for any dielectric
when we have Sigma Pi Omega Epsilon this part
is far less than 1. So when this part is far
less than 1, you get something like 1.002
or something like that so you can approximately
take it as 1 and this to be equal to 0, it
still these expressions can be valid. So we
can say that when conduction current is represented
by conductivity Sigma and Omega Epsilon is
displacement current. So when conduction current
is dominant over the displacement current
or Sigma by Omega Epsilon for less than 1,
these equations are true we can say like that.
Now consider lossy media where Sigma is not
equal to 0, now there is a special case of
lossy media and the good conductors like metals
like copper, iron or aluminium, where the
conduction current is far dominant over the
displacement current, displacement current
is negligible in those and conduction current
is very high so Omega Sigma by Omega Epsilon
is far greater than 1. So usually this limit
is taken as Sigma by Omega Epsilon far less
than 1 over 100.
So in that case if you go to the previous
slide this is far greater than 1, so 1 can
be neglected, so this 1 also can be neglected
so you can simplify it as Sigma by the whole
thing can be simplified as Sigma by Omega
Epsilon. So in that case you will see that
both are Alpha, Beta are equal because whatever
is in this packet is simplified as Sigma by
Omega Epsilon square then this is also coming.
So you see that Alpha equal to Beta and which
is supposed to be equal to Omega Mu Sigma
by 2. And intrinsic impedance of the medium
is given by square root of J Omega Mu by Sigma
or Omega Mu by Sigma square root of J, in
terms of phaser notation we can take it as
angle 45 degree so J represents angle 90 degree,
this is called surface impedance in the case
of very good conductors or metals because
the field is not very treating that matching
to the metal, so this is also called surface
impedance or intrinsic impedance of the medium.
And phase velocity V is equal to square root
of 2 Omega by Mu Sigma. You can see that phase
velocity is depending upon the frequency,
it changes with frequency and it depends upon
the magnetic permeability as well as conductivity
of the medium, now we can look into the consequences
of this, before that we define skin depth.
So what is skin depth? You know that waves
are penetrating into the metals but it will
be attenuating very fast, so skin depth Delta
is defined as a distance on which the amplitude
is E 0 e – Alpha Z of the wave is decreasing
by 1 by e where e is the base of the natural
logarithm. So 1 by e is about equivalent to
0.37 so this is an exponentially changing
right, suppose E 0 we take it as 1 here, now
this is exponentially decreasing so this is
the surface of the metal let us say and this
is the depth of the metal. So as we go into
the metal, when this wave this amplitude becomes
0.37 or 1 over e, we call this depth as 1
skin depth. So you can see that when you put
when you equate that you will see that skin
depth is equal to square root of 2 by Omega
Mu Sigma, you can verify it yourself and that
is equal to Omega is 2 Pi F so square root
of 1 by Pi F Mu Sigma.
So this is a function of frequency, as the
frequency is increasing what you see that
skin depth is decreasing and it also depends
upon Mu and Sigma. And similarly we can verify
that inside the metal speed of the wave V
equal to 2 Pi F Sigma. At home you can substitute
in your equation and verify all these expressions,
so wavelength lambda is nothing but 2 Pi Delta
that is skin depth. So intrinsic impedance
or surface impedance or characteristic impedance,
you can call by any of these 3 names. So intrinsic
impedance of metal is equals square root of
2 Pi F Mu by Sigma the conductivity.
Z_m=Square root (2pifMu/Sigma)
So let us do some calculation, so I have done
all this calculation and represented it in
the form of a table here and what points out
some of the properties. Let us first define
conductivity, permeability, now conductivity
of copper is very high perhaps one of the
highest in the common kind of metals that
are in use in electrical engineering, this
is 5.8 into 10 to the power 7 Siemens per
meter. So usually the materials that we use
in electrical engineering are copper, aluminium,
iron or some combinations of that so among
them this has got the highest conductivity
so we take it as a reference for easiness.
So conductivity can be represented as reference
conductivity, times conductivity of copper.
And permeability we know of a media is reference
permeability or relative permeability multiplied
by the permeability of vacuum or air, and
permeability of air is defined as 4 Pi into
10 to the power -7 henries per meter.
Free space value of V or speed of light is
3 into 10 to the power 8 meter per second,
so we can easily see that the wavelength at
1 megahertz is nothing but 300 meter that
is 3 into 10 to the power 8 meters per second
divided by frequency 10 to the power of 6,
so you get 300 meters so wavelength at 1 gigahertz
in free space or air is 0.3 meter. Similarly
we can say that free space impedance is equal
to 377 ohms. Now you remember these reference
values, so in free space you know wavelength
is of the order of meters we can say or fraction
of a meter and impedance is of the order of
100 of Ohms, 377 Ohms.
Now let us consider this in medium copper
and iron; so for copper relative conductivity
is 1 and it is a non-magnetic material so
Mu r = 1. In iron in this particular iron
let say the conductivity is already 10 percent
of copper so 0.1, so the 0.5 10 to the power
7 Siemens per meter iron is magnetic so relative
permeability let us say it is 500. So here
we have this skin depth velocity in the medium,
wavelength in the medium and the impedance
in the medium, and here we have frequency
50, 1, 1 megahertz, 1 gigahertz, same thing
for iron. Now the skin depth you can see this
is 10 to the power – 3 meter, so this is
basically in terms of millimetre, at 50Hz
the wave will penetrate into iron and within
9.35 millimetre it is reduced in value by
e that is 0.37 of the original value so that
is my skin depth and it is travelling at a
velocity of 2.936 meters per second, so you
see how different the velocity wave is in
metal.
Here in free space it is 3 into 10 to the
power 8 meters per second that is 300 billion
meters per second and here it just 2.936 meters
per second extremely slow. And what about
lambda? Lambda is 0.059 meters only, in free
space what will be lambda? At 50 hertz it
will be several thousands of kilometres, so
several thousands of kilometres versus fraction
of a meter that is 5.9 centimetres wavelength
in metal that is in copper and the intrinsic
impedance of the medium here is 2.61 milli
Ohms 10 to the power - 6 that is in free space
it is 377 Ohms. So you see that properties
of waves in metals are very different from
that in air or free space so this is very
important in EMC studies, this distinction.
Now as the frequency is increasing, we have
seen over here what happened to skin depth,
skin depth should be decreasing, impedance
is increasing then the velocity is increasing
and the wavelength is decreasing, so let us
see here kilohertz.
This is decreasing from 50 hertz, velocity
is increasing, lambda has decreased further,
the impedance has also decreased further,
so similarly you can calculate 1 megahertz
and 1 gigahertz. Now for iron with 10 percent
conductivity of copper, but 500 times more
permeability we have calculated the values
here, so you can take 1 kilo hertz, you can
see that skin depth is even smaller in here
which is more difficult to penetrate mainly
because of this permeability, we can see how
it is happening.
Skin depth, it is the property of product
of Mu and Sigma, so 0.1 and 500 so it becomes
50 here whereas for copper it is 1 so that
is why you see the difference.
So for iron it is more difficult to penetrate
because of the relative permeability, so you
get lambda but impedance of the medium is
more compared to that in copper so you can
do this calculation yourself and verify at
home as homework.
Now some of the inferences that we can take
of the table that we have seen, first of all
the characteristic impedance of metals are
extremely small compared to free space impedance,
this is one point that we can take away from
this chapter. Second, speed V and wavelength
the lambda inside metals are extremely small
compared to that in free space, so this is
other influence that we can take away and
it has got consequences when we talk about
how metals can be used to shield electromagnetic
waves. Shielding property of metals that you
see chapter 5 derives from this property.
