Review of electromagnetic principles continuation.
In this chapter we will inspect electric and
magnetic fields from small dipoles; these
are the electric dipoles and the magnetic
dipoles. Magnetic dipole is basically a small
loop; we will introduce the concept of wave
impedance and also find the expression for
maximum possible radiated field.
Now take this picture on the left over here,
if we want to find the radiation field from
a piece of wire, this piece of wire can be
component leads or it can be a track in the
printed circuit board or it can be just a
connecting wire between two circuits. So if
we want to find the radiation field from such
a wire then you can divide that into very
small dipoles, these dipoles are shown here.
Then if you know the expression for electric
and magnetic fields from a small dipole like
this then you can add up all these dipole
fields from the various dipoles to find the
total field, so that is the principle involved.
So the basic expression that we want to know
is that from the small electric dipole, these
are called electric dipoles.
Now you can imagine another scenario in which
you have a wire that is closed like a loop
like that. So here we have 2 possibilities;
one possibility is that you can divide this
wire as we did before into very small dipoles,
find the electrical and magnetic fields from
each of these dipoles at a point where we
are interested in finding the fields then
sum it up. Another way is that, this area
of the loop that you can divide into very
small loops so that it covers more or less
the complete area, then these are called the
magnetic loops or magnetic dipoles. Then from
there you can find the total magnetic field,
so there are 2 possibilities in finding the
fields.
Now first consider the electric dipole, so
this is represented by a small piece like
this which has a dipole moment, this dipole
suppose this is part of a wire a small part
of a wire of length small l and carry a current
I then the electric dipole moment is defined
as
(Me = I × l) M subscript e = I into L,
so the current is directed in the Z directions
so this is Cartesian coordinate X, Y, Z what
is shown, but we will find expressions in
spherical coordinate because that is more
convenient for us, so this spherical coordinate
is defined like this. So if this is the any
point is space p then from the origin to this
point the distance is called R that is one
of the coordinates.
Then from the Z direction an angle (Theta)Theta
so this is the second of the coordinate then
the rotation Phi(Phi) from the X axis, this
orthogonal system R Theta and Phi, so Phi
is expressed in terms of R Theta and Phi so
the orthogonal spherical coordinate system.
Now if that is a case, we can find expression
for the R component of E field, the Theta
component of E field and the Phi component
of H field. So H will have only one component
that is Phi because if you have current in
this way then the fields are around this in
the Phi direction so we have only Phi components
for the H field from the symmetry. Then for
the E field we have only R component and theta
component and you do not have any Phi component
for the E field.
Now if you look at the expression for the
electric field and the magnetic field, Z 0
is the free space impedance 377, we will not
go into the details of this equation because
we are interested in only one term of this
equation usually. Now what you can notice
is that this is varying with respect to the
distance, now it can vary as R square or R
cube or if it is Theta component it can vary
as inversely proportional to R or inversely
proportional to R square or R cube, so here
you can see that it can vary as 1 over R or
1 over R square. So as you are moving far
away from this dipole, by the way it is assumed
that the length L of the dipole is so small
compared to the distance that you are interested
in the field as well as wavelength involved,
then only this expressions are true, this
is a very small dipole compared to the wavelength
as well as the distance R.
Now very far away from the dipole, 1 over
R cube times and 1 over R square times fall
off very fast, and basically the terms that
are significant in value are 1 over R terms.
So where are the 1 over R terms? So in this
you do not have because this is either 1 over
R square or 1 over R cube so here you have
one term that is varying as 1 over R and here
also you have 1 term varying as 1 over R,
so these phase are more dominant far away
from the dipole. So let us look at the expressions
for those fields so E theta and H Phi and
they are orthogonal to each other, these 2
terms and far away from the dipole this is
almost like transverse electromagnetic waves,
so the pointing vector or the energy flow
is in the R direction faraway.
And Theta component and Phi components are
lying in the plane perpendicular to the direction
of propagation on the fields. So this is the
expression for the Theta component and this
is the expression for the Phi component, they
are proportional to the current and also it
varies as a function of Sin Theta so it is
angular dependent. So when Sin Theta is 90
degree, the field will be maximum, and when
Sin Theta is equal to 0, in this direction
the field is 0 so the radiation pattern is
more like this as I am drawing here, so this
will be the radiation loss faraway.
Now consider the case of magnetic loop, so
here again uh from the symmetry there is a
loop here in the XY plane, against spherical
coordinate is defined and this has locked
a dipole moment I times A, where A is the
area of the loop. So the shape of the loop
is not very important even though for convenience
but it shown as round, it can be any shape
as long as this loop is very small compared
to the wavelength and the distance where we
are interested in finding the field, so the
shape does not matter really so this is the
dipole moment.
Now here also, from the symmetrical configuration
you can see that okay any small voltage can
dry the current easily around it so it will
create magnetic field as well, it can create
an electric field drop around the loop, so
you can see that electric field will be in
Phi direction because it can easily create
an electric drop. Then magnetic field will
be in both R direction as well as Theta direction
so you have 3 components only, other components
are 0. So here also these components can vary
as 1 over R cube or 1 over R square or 1 over
R and far away when Beta are far less than
1, Beta is to Pi by Lambda.
So at far distances we are interested only
in terms that is varying as 1 over R because
those terms will be the dominant one and others
will be approaching 0 so those terms are written
over here and you can see that Phi and H are
orthogonal to each other and they are also
orthogonal to the direction of propagation,
so far away you can assume that this produces
something like a TEM wave. Now E field and
H field both are proportional to the dipole
moment I 0 times area of the loop and also
Sin Theta, so here also the radiation component
is I mean radiation is maximum when theta
equal to 90 degree in this direction the plane
of the loop and minimum or 0 perpendicular
to the loop in the Z direction far away from
the field.
Wave impedance is defined as the ratio of
electric field to magnetic fields but those
components that are perpendicular to the direction
of propagation. So if direction of propagation
is R away from the origin, then the electric
and magnetic fields in a plane orthogonal
to that direction is taken to find the wave
impedance. So for electric dipole, it becomes
ratio of Theta component of the E field to
the Phi component of the H field, and for
the magnetic dipole it becomes the Phi component
of the E field and Theta component of the
H field. So this wave impedance is a function
of distance or electrical distance from the
dipole and it is also dependent upon Beta
r, now let us find the wave impedance of the
E dipole.
So if you take uh the expression for the Theta
component of E and Phi component of H that
you have seen in the previous graph these
expressions.
Then you can see that it can be simplified
by this expression, Beta = 2 Pi by Lambda
and Beta r = 2 Pi R by Lambda. Now if we have
Beta r far greater than 1 or far from the
dipole, the far field condition then this
becomes very small and basically what is in
the square root is just 1, so you can see
that wave impedance for electric dipole far
from the dipole is nothing but free space
impedance Z 0 that is 377 Ohms, but situation
is different when Beta r is far less than
1 or near to the dipole. When wave impedance,
you can see that now Beta r is small so this
becomes very big i.e one by Beta r .so simplifying
you get it as Z 0 by beta R and wave impedance
is far greater than free space impedance,
Z 0 that is square root of Mu by Epsilon or
C = 1 by square root of Mu by Epsilon.
Now the expressions for wave impedance, simplifying
you get as 1 by 2 Pi F Epsilon R, where Epsilon
is the electric permittivity. So you can see
that very close to the dipole wave impedance
becomes quite big and far from the dipole
it should be this, but of course this expression
is valid only very near to the dipole Beta
r far greater than 1, beyond that it is not
valid.
When far from the dipole the wave impedance
is equal to free-space impedance.
When
Near to the electric dipole, wave impedance
is far greater than free space impedance
Now we can find wave impedance for H dipole,
so from the expressions for the E field and
H field and the ratio of that, it can be shown
that as wave impedance equals Z 0 multiplied
by this expression, where Beta equal to 2
pi by Lambda, Betar=2Pir/Lamda (Beta r = 2
Pi r b y Lambda). Now here we can take 2 conditions;
one is when Beta r is far greater than 1 so
under that condition you will see that wave
impedance is nothing but the free space impedance.
And when Beta r is far less than 1, you will
see that it is given by free space impedance
multiplied by 2 Pi by r by Lambda, so near
to the magnetic dipole, wave impedance is
smaller than free space impedance and it is
2 Pi f Mu r, Mu is magnetic permeability.
When   Far from the dipole the wave impedance
is equal to free-space impedance.
When
Near to the magnetic dipole, wave impedance
is far smaller than free space impedance
So we have 2 different kinds of expressions
near to the dipole for electric dipole and
magnetic dipole, whereas far from the field
far from the dipole both gives free space
impedance, so let us plot this out. So in
this, wave impedance is plotted with distance
from the dipole so to normalise for the frequency
or wavelength, we take Beta 2 Pi r by Lambda
as the X axis, so Beta r = 1 will be 2 pi
r = Lambda, when Lambda equal to 2 pi r, then
it becomes 1 then this is the wave impedance
and this blue line is for electric dipole
and red line is for magnetic dipole.
You can see that for the magnetic dipole is
transformed a very long value, the impedance
then as distances increased, it reaches 377,
for the E dipole starts from a high impedance
when you are very close to, then it comes
down and reaches 377. So often we talk of
magnetic fields, when we have a problem of
magnetic fields we talk of low impedance magnetic
fields and we often talk of high impedance
electric fields when we are closed to the
source, so the reasons for those expressions
are coming from this graph. High impedance
electric field is close to electric dipole,
low impedance magnetic field close to magnetic
dipoles, far from the dipole wave impedance
is free space impedance for air.
Wave impedance concept we will be using quite
extensively in the analysis of electronic
shielding in later chapters, so that we will
talk about what is ECL to shield against high
impedance electric fields, whereas it is very
difficult to shield against low impedance
magnetic field or we will say that shielding
is more difficult when it is low frequency
magnetic field.
Now, where will be the maximum radiated field?
So this is a dipole, first we look at the
far field, maximum electric field we have
seen before when Theta = 90 degree, so in
this direction expression is given by this,
so maximum radiated field is proportional
to the current as well as electrical length
of the dipole L by Lambda, and inversely proportional
to the distance from the dipole so this is
perpendicular, maximum radiation field is
on a perpendicular plane to the dipole. Now
here, maximum radiation is in the plane of
loop in the far field and A is area of the
loop, so maximum field in the Phi direction
is proportional to the current inversely proportional
to the distance and ratio of the area of the
loop divided by Lambda square wavelength square.
Now in the near field maximum couplings are
along the dipole length and perpendicular
to the loop. So this is opposite to that in
the far field so in the near field maximum
coupling is along the dipole for the electric
field and perpendicular to the loop for the
magnetic loop.
Now this shows example calculation for the
maximum radiated field at a distance of 10
meter and 10cm dipole versus 3 by 2 square
centimetre loop, again shape of the loop can
be anything. So in about the standards it
has been specified that you know electric
field should not exceed 32 micro volts per
meter at 30 megahertz, so you will already
exceed this limit if you have a small dipole
carrying a current of 170 microampere, and
if it is for this small local 450 microampere
will be the limit, above that you already
exceed this value permissible value by the
Gamma regulations.
Similarly, at 230 megahertz to reach this
value you need only 22 microamperes and for
a loop you need only 8 microamperes because
for the loop 8 divided by Lambda square that
is why the changes are much faster here. Now
sometimes we can find the radiated field in
time domain other than the frequency domain,
then you can see it is proportional to rate
of change of current d I by dt and the length
of the dipole and inversely proportional to
the distance and Mu 0 4 Pi these are all constant,
C is also constant you can write it in this
way 10 to the power – 7 l by r d I by dt
volts per meter. And B Phi flux density is
10 raise to - 7 divided by C r d I by dt Weber
per meter square, so you can see that ratio
of E and B it is speed of light, radiation
field is proportional to time derivative of
the current.
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