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Hertz Dipole

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24.

Antennas

What is an antenna

Types of antennas

Reciprocity

Hertzian dipole
near field
far field: radiation zone
radiation resistance
radiation efficiency
Antennas convert currents to waves
An antenna is a device that converts a time-varying
electrical current into a propagating electromagnetic wave.
Since current has to flow in the antenna, it has to be made
of a conductive material: a metal.

And, since EM waves have to


propagate away from the antenna,
it needs to be embedded in a
transparent medium (e.g., air).

Antennas can also work in reverse: converting incoming


EM waves into an AC current. That is, they can work in
either transmit mode or receive mode.
Antennas need electrical circuits
• In order to drive an AC current in the antenna so that it
can produce an outgoing EM wave,
• Or, in order to detect the AC current created in the
antenna by an incoming wave,

…the antenna must be connected to an electrical circuit.

Often, people draw illustrations of


antennas that are simply floating in
space, unattached to anything.

Always remember that there needs to


be a wire connecting the antenna to a
circuit.
Bugs also have antennas.
But not the kind we care about.

not made of metal

The word “antenna” comes from the Italian


word for “pole”. Marconi used a long wire
hanging from a tall pole to transmit and
receive radio signals.

His use of the term popularized it.

The antenna used by Marconi


for the first trans-Atlantic radio
broadcast (1902)
Guglielmo Marconi
1874-1937
There are many different types of antennas
• Omnidirectional antennas – designed to receive or
broadcast power more or less in all directions (although,
no antenna broadcasts equally in every direction)
• Directional antennas – designed to broadcast mostly in
one direction (or receive mostly from one direction)
Examples of
omnidirectional antennas,
typically used for receiving
radio or TV broadcasts. “Yagi-Uda”
antenna
“rabbit ears” automotive
“whip” antenna

parabolic Examples of directional antennas,


dish typically used for higher
antenna frequencies, e.g., microwaves
pyramidal and millimeter waves
horn antenna
Why is there no such thing as a perfectly
omnidirectional antenna?
Suppose there was one. Consider a large sphere
centered on this hypothetical antenna.

Everywhere on the surface of this


sphere, there would need to be a non-
zero tangential electric field vector.

But this can’t happen. It is impossible


to completely cover the surface of a
antenna
sphere with a continuous tangential
vector field that has no zeros.
This is known as the Hairy Ball Theorem:
There is no nonvanishing continuous
tangent vector field on a sphere.
OR:
You can’t comb a hairy ball flat without
creating at least one cowlick.
Reminder: accelerating charges
qa  t  sin 
Accelerating charges radiate E  r, t  
electromagnetic waves. 4 0 rc 2 
a
0
330 30

This is true whether the 300 60

charges are bound inside 270 90

atoms:
240 120

210 150
180

N electrons/m3
or they are freely
flowing (like a
time-varying
area A
current in a wire):
distance vave t
Reciprocity
qa  t  sin 
E  r, t  
4 0 rc 2
Note that the radiated electric field is proportional to the
acceleration of the charge.
z Since Maxwell’s equations are linear:
a small E, B  a t 
metal
when a charge acceleration gives rise
element to a field, the same field can induce the
containing
accelerating same acceleration:
charges
4 0 rc 2
y a t   E  r, t  cause effect
q sin 

x
Antenna reciprocity: a receiving antenna’s properties
are identical to the transmitting properties of the same
antenna when it is used as a transmitter.
Antenna reciprocity
Antenna reciprocity: a receiving antenna’s properties
are identical to the transmitting properties of the same
antenna when it is used as a transmitter.
Properties: this includes
everything about the
antenna
• antenna gain
• radiation pattern
• radiation resistance e.g. this antenna transmits mostly in
• polarization one particular direction. It also receives
mostly from that same direction.
• bandwidth

So we don’t need to keep saying “transmit or receive”. We can


talk about just one mode, since the other one is the same in
reverse.
In an antenna, changing currents give rise to
radiated electromagnetic waves
Obviously, the many different
possible antenna geometries
give rise to many different
radiated fields, due to the
different spatial distributions of
changing currents.
electrical connection for
driving current in the
We need to treat a simple case antenna
first:
length of the
the “Hertzian dipole” antenna

V(t)
The Hertzian dipole antenna
The Hertzian dipole is a linear d << 
antenna which is much shorter than
the free-space wavelength: V(t)
For our purposes, we can treat this as a
wire of infinitesimal length d, carrying a
current I(t) = I0 cos(t) → I0 ejt.

In 1887, Hertz used a sub-wavelength antenna


to generate (transmit) and detect (receive)
radio waves – the first wireless broadcast. It is
considered the first experimental proof of
Maxwell’s equations. Heinrich Hertz
1857 - 1894

Asked about the ramifications of his discoveries, Hertz replied,


"Nothing, I guess."
Hertzian dipole – coordinate system
To treat this problem, we use spherical coordinates.

The small antenna is located at the origin, and


oriented along the z axis.
z Spherical coordinates: this
point is both (x,y,z) and (r,,)

r cos 

 This shows 2 of the 3 unit vectors for


ˆ spherical coordinates, pointing along the
radial (r) and azimuthal () directions. (The
r

d
y  unit vector is not shown in this diagram.)

s
co

 The field must obey Maxwell’s



in
rs

r sin  sin  equations in empty space


x
everywhere outside of the wire.
The field of a Hertzian dipole antenna
The resulting electric and magnetic fields are:
 0 2 I 0 d   1 j 
E  r , , t    2 cos     rˆ 
4 c0   k r  k r  
2 3
  0 0 
 j 1 j  ˆ   jk0 r jt
sin       e e
 k 0 r  k r  2  k r 3  
 0 0  

 0 2 I 0 d  j 1  ˆ  jk0 r jt


B  r , , t   sin     e e
4 c0 2
 k r  k r 
2

 0 0 

Complicated expressions! But we can gain physical insight


by looking at limiting cases.
 
Note: even without any simplification, it is obvious that E  B.
The near field of a Hertzian dipole antenna
The properties of these fields are quite different if you are
near the antenna or far from it.

First limiting case: very near the antenna. So: k0r << 1

In this case, the 1/r3 terms are the only ones that
matter in the E-field expression. Also, exp(jk0r) ~ 1.
 I0d
E  r , , t    2 cos  ˆ
r  sin  ˆ  e jt

j 0  4 r 3

To interpret the meaning of this, recall that the charge on


a conductor is related to the current, according to:
dQ
I t   For our assumed current,
dt this implies that I0 = jQ.
The near field of a Hertzian dipole antenna
 Qd
E  r , , t    2 cos  ˆ
r  sin  ˆ  e jt

 0  4 r 3
This is exactly the same result as one finds for the field of
a simple static dipole (two charges of opposite sign
separated by distance d), except for the time dependence.
For the magnetic field, with the
same approximations, we find:
 0 I 0 d
B  r , , t   ˆ sin  e jt

4 r 2
which is the same result one
finds for the static magnetic field
of a current element I0d (again
except for the time dependence).
The near field is the quasi-static regime
Very close to the antenna, the electric and magnetic fields
are the same as what one finds in electrostatics, except
that they oscillate in magnitude according to ejt (just as the
current in the antenna does).

Thus the near fields are called “quasi-static” fields.

Exploiting near-field interactions


is important in proximity sensors,
such as the touch screen on your
phone or tablet.

An illustration of the electric


and magnetic near fields of
a dipole antenna
The near field is not radiation
 Qd
E  r , , t    2 cos  ˆ
r  sin  ˆ  e jt

 0  4 r 3

Note: this has a radially


r̂ directed field component.

But we know that EM radiation


ˆ
cannot have a component
pointing along the propagation
direction: 

Eradiation  k

Therefore this approximate near-field expression cannot


be describing the radiated field. To describe the radiated
field, we need to consider what is happening far away from
the antenna.
The far field of a Hertzian dipole antenna
Let’s consider the other limiting case: k0r >> 1 (far away)

In this case, the 1/r terms are the only ones that matter.
Also, exp(jk0r) is no longer equal to 1.
 0 I 0 d  e  jk0 r  jt ˆ
E  r , , t   j sin   
4  r 
The E and B fields are:
 0 I 0 d  e  jk0 r  jt ˆ
B  r , , t   j sin   
4 c0  r 

These have the form of propagating electromagnetic waves


with spherical wave fronts (no more radially directed fields).

They satisfy |E| = |B|/c, and both are perpendicular to k  k0 rˆ.
Antennas generally emit polarized radiation
Example: antennas for most
wireless networks emit vertical  0 I 0 d  e  jk0 r  jt ˆ
E  r , , t   j sin   
( ˆ ) polarization, which is 4  r 
perpendicular to the earth’s
surface.

If your cell phone had a simple linear


antenna, then it wouldn’t work if you
try to use it while lying down.
Power radiated by a Hertzian dipole antenna
  
Calculate the Poynting vector S   0 c0 E  B , and then do a
2

time average (which just gives a factor of ½):


 0 2 I 0 2 d 2  sin 2  
angle brackets S  ,      rˆ
indicate time
average
32 2 c0  r 2 
Integrate over all  and  to find the total power radiated:
2 
0 2 I 0 2 d 2
Ptotal   d  r
2
sin  S  ,   d 
0 0
12 c0

Use c0 to write this in terms of the free-space wavelength:


0 c0 I 0 2 d 2
Ptotal   2
3 
The total radiated power is proportional to the square of
the current in the antenna, and the square of its length.
Radiation resistance
Since the power radiated by the antenna varies as the square
of the current, it makes sense to draw an analogy to the power
dissipated in a resistor due to Ohmic heating, which is given
by the well-known formula:
P  I 2R

Of course, for time-varying currents, we need to time-


average, which contributes a factor of ½, making this:

1 2
Pave  I R
2

“radiation resistance” = the hypothetical resistance that would


dissipate the same time-averaged power (due to Ohmic heating)
as that radiated by the antenna.
Radiation resistance
0 c0 I 0 2 d 2
For the Hertzian dipole: Ptotal   2
3 
2 20 c0 d 2
and therefore: Rrad  2 Ptotal   2
I0 3 
1
Note, since c0  , this can also be written:
 0 0

2 0 d 2 2 d2
Rrad   2  Z0  2
3 0  3 

where Z 0  0  0  377 is the impedance of free space.

Since a Hertzian dipole antenna has d << , its


radiation resistance is much smaller than 377.
Radiation efficiency
When an antenna is driven by an AC current, it will dissipate
power due to radiation, and also due to I2ROhmic heating.
electrical power
input to the
antenna
Pinput = Prad + POhmic

The radiation efficiency is the percentage of the input power


that is converted to radiated power. Since the current is the
same for both terms, the radiation efficiency is:

Rrad

Rrad  ROhmic

For a Hertzian dipole antenna (d << ), this is usually a


small number, which means that these antennas are not
very efficient.
Radiation efficiency: example
A steel rod of length d = 1.5 meters, radius a = 1 mm is used
as an antenna for radiation at f = 1 MHz (AM radio).
2
2 d 2 2  1.5 
Rrad  Z0  2   377     0.020
3  3  300 
The resistance of a metal wire at high frequency (where the
skin depth is much smaller than the wire’s radius) is given by:
 f d
ROhmic 
 2 a
For steel,  = 0 and  = 2×106 -1 m-1. Thus Rohmic = 0.34
The radiation efficiency is therefore:
Rrad
  0.055
Rrad  ROhmic
(often this is expressed in decibels:  = 12.6 dB)

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