The document presents information on loop antennas. It discusses that a loop antenna consists of a loop of wire that carries RF current. It describes loop antennas as either electrically small or large based on circumference size. For a small circular loop antenna, it derives expressions for the electric and magnetic fields radiated using cylindrical coordinates. It finds that the power radiated from a small loop is proportional to the fourth power of its radius. It also determines the directivity and maximum effective area of a circular loop antenna.

1. LOOP ANTENNA
Presentation on Electric
field of small loop antenna
and directivity of circular
loop antenna with uniform
current
SUBMITTED TO: SUBMITTED BY:
Mr.Sudhir Mishra Priya Hada
Amity Institute of Engineering and Technology
Priya Hada Title 1/19

2. LOOP ANTENNA
LOOP ANTENNA
• A loop antenna is a radio antenna consisting of a loop of
wire with its ends connected to a balanced transmission
line
• It is a single turn coil carrying RF current through it.
• The dimensions of coil are smaller than the wavelength
hence current flowing through the coil has same phase.
• Small loops have a poor efficiency and are mainly used as
receiving antennas at low frequencies. Except for car
radios, almost every AM broadcast receiver sold has such
an antenna built inside of it or directly attached to it.
Priya Hada Title 2/19

3. LOOP ANTENNA
LOOP ANTENNA
• A technically small loop, also known as a magnetic loop,
should have a circumference of one tenth of a wavelength
or less. This is necessary to ensure a constant current
distribution round the loop.
• As the frequency or the size are increased, a standing
wave starts to develop in the current, and the antenna
starts to have some of the characteristics of a folded dipole
antenna or a self-resonant loop.
• Self-resonant loop antennas are larger. They are typically
used at higher frequencies, especially VHF and UHF,
where their size is manageable. They can be viewed as a
form of folded dipole and have somewhat similar
characteristics. The radiation efficiency is also high and
similar to that of a dipole.
Priya Hada Title 3/19

4. LOOP ANTENNA
LOOP ANTENNA
• Loop antennas come in a variety of shapes (circular,
rectangular, elliptical, etc.)but the fundamental
characteristics of the loop antenna radiation pattern(far
field) are largely independent of the loop shape.
• Just as the electrical length of the dipoles and monopoles
effect the efficiency of these antennas, the electrical size of
the loop (circumference) determines the efficiency of the
loop antenna.
• Loop antennas are usually classified as either electrically
small or electrically large based on the circumference of
the loop.
• electrically small loop :circumference λ
10
• electrically large loop :circumference λ
Priya Hada Title 4/19

5. LOOP ANTENNA
SMALL CIRCULAR LOOP
For the field analysis of a loop antenna, position the antenna
symmetrically on the x-y plane, at z = 0.The wire is assumed to
be very thin and the current spatial distribution is given by
Iφ = I0 where I0 is a constant. Although this type of current
distribution is accurate only for a loop antenna with a very small
circumference,a more complex distribution makes the
mathematical formulation quite cumbersome.
Priya Hada Title 5/19

6. LOOP ANTENNA
RADIATED FIELDS
To find the fields radiated by the loop, the same procedure is
followed as for the linear dipole.The potential function A given
by :
A(x, y, z) =
µ
4π c
Ie(x , y , z )
e−jkR
R
dl (1)
where R is the distance from any point on the loop to the
observation point and dl⣙is an infinitesimal section of the
loop antenna. In general, the current spatial distribution
Ie(x , y , z ) can be written as:
Ie(x , y , z ) = axIx(x , y , z ) + (ay)Iy(x , y , z ) + azIz(x , y , z )
(2)
Priya Hada Title 6/19

7. LOOP ANTENNA
RADIATED FIELDS
The form is more convenient for linear geometries. For the
circular-loop antenna whose current is directed along a circular
path, it would be more convenient to write the rectangular
current components of (2) in terms of the cylindrical
components using the transformation


Ix
Iy
Iz

 =


cos(φ ) −sin(φ ) 0
sin(φ ) cos(φ ) 0
0 0 1




Iρ
Iφ
Iz


Ix = Iρcosφ − Iφsinφ (3)
Iy = Iρsinφ + Iφcosφ (4)
Iz = Iz (5)
Priya Hada Title 7/19

8. LOOP ANTENNA
RADIATED FIELDS
Since the radiated fields are usually determined in spherical
components, the rectangular unit vectors of (2) are transformed
to spherical unit vectors.
ax = arsinθcosφ + aθcosθcosφ + aφsinφ (6)
ay = arsinθsinφ + aθcosθsinφ + aφcosφ (7)
az = arcosθ − aθsinφ (8)
Substituting (3,4,5,6,7,8) in (2) reduces it to:
Ie = ar[Iρsinθcos(φ − φ ) + Iφsinθsin(φ − φ ) + Izcosθ] +
aθ[Iρcosθcos(φ − φ ) + Iφcosθsin(φ − φ ) − Izsinθ] +
aφ[−Iρsin(φ − φ ) + Iφcos(φ − φ )]
Priya Hada Title 8/19

9. LOOP ANTENNA
RADIATED FIELDS
For the circular loop, the current is flowing in the φ direction (Iφ)
so that above equation reduces to:
Ie = ar[Iφsinθsin(φ−φ )]+aθ[Iφcosθsin(φ−φ )]+aφ[Iφcos(φ−φ )]
(9)
The distance R, from any point on the loop to the observation
point, can be written as:
R = (x − x )2 + (y − y )2 + (z − z )2 (10)
x = rsinθcosφ
y = rsinθsinφ
z = rcosθ
x2
+ y2
+ z2
= r (11)
x = acosφ
Priya Hada Title 9/19

10. LOOP ANTENNA
y = asinφ
z = 0
(x)2
+ (y)2
+ (z)2
= (r)2
(11) reduces to
R = r2 + a2 − 2arsinθcos(φ − φ ) (12)
the differential element length is given by
dl = adφ (13)
Using (9), (12), and (13), the φ component of (1) can be written
as:
Aφ =
aµ
4π
2π
0
Iφcos(φ − φ )
e−jk
√
r2+a2−2arsinθcos(φ−φ )
r2 + a2 − 2arsinθcos(φ − φ )
dφ
(14)
Priya Hada Title 10/19

11. LOOP ANTENNA
Since the spatial current Iφ as given by Iφ = I0 is constant, the
field radiated by the loop will not be a function of the
observation angle . Thus any observation angle φ. can be
chosen; for simplicity φ = 0.
Aφ =
aµ.Io
4π
2π
0
cosφ
e−jk
√
r2+a2−2arsinθcos(φ−φ )
r2 + a2 − 2arsinθcos(φ − φ )
dφ (15)
For small loops, the function
f =
e−jk
√
r2+a2−2arsinθcos(φ−φ )
r2 + a2 − 2arsinθcos(φ − φ )
(16)
can be expanded in a Maclaurin series in a using:
f = f(0)+f (0)a+
1
2!
f (0)a2
+........
1
(n − 1)!
f(n−1)
(0)a(n−1)
+.......
Priya Hada Title 11/19

12. LOOP ANTENNA
where f = ∂f
∂a
a=0
, f = ∂2f
∂a2
a=0
and so on.... Taking into
account only the first two terms of (5-15a)
f(0) =
e−jkr
r
(17)
f (0) =
jk
r
+
1
r2
e−jkr
sinθcosφ (18)
f
1
r
+ a
jk
r
+
1
r2
sinθcosφ e−jkr
(19)
reduces (15) to:
Aφ
aµ.Io
4π
2π
0
cosφ
1
r
+ a
jk
r
+
1
r2
sinθcosφ e−jkr
dφ
(20)
Priya Hada Title 12/19

13. LOOP ANTENNA
Aφ
aµ.Io
4π
e−jkr jk
r
+
1
r2
sinθ (21)
In a similar manner, the r- and θ-components of (1) can be
written as:
Ar
aµ.Io
4π
sinθ
2π
0
sinφ
1
r
+ a
jk
r
+
1
r2
sinθcosφ e−jkr
dφ
(22)
Aθ −
aµ.Io
4π
cosθ
2π
0
sinφ
1
r
+ a
jk
r
+
1
r2
sinθcosφ e−jkr
dφ
(23)
which when integrated reduce to zero. Thus:-
A aφAφ = aφ
a2µ.Io
4
e−jkr jk
r
+
1
r2
sinθ
Priya Hada Title 13/19

14. LOOP ANTENNA
A = aφj
kµa2Iosinθ
4r
1 +
1
jkr
e−jkr
(24)
Substituting (24) into-
HA =
1
µ
× A
reduces the magnetic field components to:
Hr = j
ka2Iocosθ
2r2
1 +
1
jkr
e−jkr
(25)
Hθ = −
(ka)2
Iosinθ
4r
1 +
1
jkr
−
1
(kr)2
e−jkr
(26)
Hφ = 0 (27)
Priya Hada Title 14/19

15. LOOP ANTENNA
The electric field vector E is given as:
× HA = J + jω EA (28)
Putting J = 0, the corresponding electric field components can
be written as:
Er = Eθ = 0 (29)
Eφ = η
(ka)2
Iosinθ
4r
1 +
1
jkr
e−jkr
(30)
Priya Hada Title 15/19

16. LOOP ANTENNA
Power Density and Radiation
Resistance
The fields radiated by a small loop are valid everywhere except
at the origin. As the power in the region very close to the
antenna (near field, kr 1) is predominantly reactive and in
the far field (kr 1) is predominantly real. The complex power
density:
W =
1
2
(E × H∗
) =
1
2
(aφEφ) × (arHr
∗
+ aθH∗
θ )
W =
1
2
(−arEφH∗
θ + aθEφH∗
r ) (31)
is first formed. When (31) is integrated over a closed sphere,
only its radial component given by:
Wr = η
(ka)4
32
|Io|2 sin2θ
r2
1 + j
1
(kr)3
(32)
Priya Hada Title 16/19

17. LOOP ANTENNA
contributes to the complex power Pr . Thus:
Pr = W.ds = η
(ka)4
32
|Io|2
2π
0
2π
0
1 + j
1
(kr)3
sin3
dθdφ
(33)
which reduces to:
Pr = η
π
12
ka4
|Io|2
1 + j
1
(kr)3
(34)
Whose real part is equal to:
Prad = η
π
12
ka4
|Io|2
(35)
For small values of kr,(kr 1), the second term within the
brackets of (34) is dominant which makes the power mainly
reactive. In the far field (kr 1), the second term within the
brackets diminishes, which makes the power real.
Priya Hada Title 17/19

18. LOOP ANTENNA
Associated with the radiated power Prad is an average power
density Wav. It has only a radial component Wr which is related
to the radiation intensity U by:
U = r2
Wr =
η
2
k2a2
4
2
|Io|2
sin2
θ =
r2
2η
|Eφ(r, θ, φ)|2
(36)
The normalized pattern of the loop has maximum value at
θ = π
2 , and it is given by:
Umax = U|θ=π
2
=
η
2
k2a2
4
2
|Io|2
(37)
Using (37) and (35), the directivity of the loop can be written as:
Do = 4π
Umax
Prad
=
3
2
(38)
Priya Hada Title 18/19

19. LOOP ANTENNA
and its maximum effective area as:
Aem =
λ2
4π
Do =
3λ2
8π
(39)
Priya Hada Title 19/19