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1 De antenne
For overruling of the DCF77-signal with a hand-held device it is important to have an
efficient and small antenna. To understand the choices in the design process well, the
chapter starts with some theoretical background. After that the technical specifications
of the realized antenna will be discussed and finally the test results of the entire
antenna system will be presented.
1.1 Theory of the antenna system
1.1.1 Power Transfer
To create an electromagnetic wave with a frequency of 77.5 KHz an antenna is
needed. The system of a transmitter is roughly divisible in a power supply and an
antenna in series. The impedance of the latter one is given by:
Z A  RA  jX A
(3-1)
With RA the real and XA the imaginary part. The reactive part is divisible in two parts:
(3-2)
RA  Rr  RL
RL is the loss resistance and Rr the radiation resistance. The loss resistance is the part
which converts electrical energy into heat, the radiation resistance is the part which
represents an equivalent part to the energy converted to electromagnetic radiation.
The generator also has internal impedance:
Z g  R g  jX g
generator
antenne
Vg
RL
Ig
Rg
Rr
Xg
XA
Figure3.1 Model of a transmitter
To calculate the power converted to electromagnetic radiation, it is necessary to
calculate the current through the circuit first:
(3-3)
Ig 
Vg
( Rr  RL  Rg ) 2  ( X A  X g ) 2
( A) [fill me]
(3-4)
With Vg the modulus of the amplitude delivered by the generator. The calculated
current can be used to determine the radiated power:

2
2
Rr
Pr  I g Rr  V g 
(W ) [fill me]
2
2 
 ( Rr  RL  Rg )  ( X A  X g ) 
(3-5)
The aim is to deliver power to the antenna as efficient as possible, for that the internal
impedance of the generator should be as small as possible.
1.1.2 Radiation zones
There are three radiation zones in the surroundings of an antenna. The first zone is the
reactive near field, the border of this zone is at a distance r of:
r  0,62 D 3 /  [fill me]
(3-6)
D is the largest dimension of the antenna. However when the antenna is much smaller
than the wavelength, this zone will reach a distance of λ/2π. The field in this zone is
derived from the theory [fill me] of circular apertures. The energy (E) is given by:
sin( x)
E
[fill me]
(3-7)
x
However this formula is for an infinitesimal small aperture and a realistic antenna
hasn’t got negligible dimensions. Therefore a certain distribution should be added to
the formula. The field would then be like in the figure below. Furthermore, a field
with a periodic function which isn’t dependant of the wavelength doesn’t appear to be
plausible. A deep research to this phenomenon was out of the scope of this project, so
this is just the doubtful theory.
Figure 3.2 Energy in the field
It is clear that the field falls off rapidly initially, and in the region behind 20 meters it
is almost steady. This is also the area where the antenna will be used.
The area surrounding the reactive near field is the radiating near field also known as
the Fresnelzone [fill me]. In this zone the energy in the field drops with a fourth
power. When the antenna is very small compared to the wavelength, this zone
wouldn’t be present. For 77.5 kHz with a handheld device, this zone would not exist.
For conventional antenna’s the borders of this zone will be:
0,62 D 3 /   r  2D 2 /  [fill me]
(3-8)
The last zone is the far field also known as the Fraunhoferzone[fill me]. In there the
power of the field drops quadratic. This zone is important in the research of the signal
strength of the real DCF transmitter. The border is given by:
r  2 D 2 /  [fill me]
(3-9)
1.1.3 Intensity, directivity and efficiency
The beam intensity (the Poyntingvector) of the electromagnetic wave in the far field is
given by the cross product of the electric field and the magnetic field:
S
1
0
E  B  Watt/m^2[fill me]
(3-10)
The radiation intensity of a transmitter, U , can be calculated from the beam intensity
multiplied by the distance from the source squared:
U  r 2 S Watt [fill me]
(3-11)
Directivity is a relative quantity which compares the antenna to an isotropic antenna
in a particular direction:
D
U
4U
[fill me]

U0
Prad
(3-12)
When the angle with respect to the antenna isn’t specified, the directivity in the
direction of maximum intensity is given. In a diagram the directivity in dB’s is
usually shown as a finction of the angle.
The efficiency of an antenna is a number between 0 and 1, and is defined as follows:

e0  er ecd  ecd 1  
2

 Rr   Z in  Z 0

 1 
 RL  Rr   Z in  Z 0
2

 [fill me]

(3-13)
With er the efficiency of the transfer of the transmission line to the antenna and Z0 is
the impedance of the transmission line.
Important properties of an antenna are the efficiency and the direction of the power.
Relevant for those properties are the radiation resistance and the directivity. This
should be mentioned in the design process.
1.1.4 Isotropic antennas
An isotropic antenna is an imaginary, ideal, lossless, point shaped antenna. This not
realizable in practice, but it can be used as a reference for other antennas. The
transmitted power of the antenna is equally distributed over a sphere and decays with
the square of the distance:
 P 
U 0   rad2  W / m 2
 4r 
U moet S zijn[fill me]
(3-14)
1.1.5 Coil antennas
An electromagnetic wave doesn’t consist of an electric field only, but it also has a
magnetic field. Initiation of a wave can be done by means of creating a dynamic
electric field or a magnetic one. The energy stored in the magnetic field is a factor c
smaller than in the electric field.
One way to transmit an electromagnetic wave is using a coil. This is also called a
magnetic dipole. Below you can see the azimuth diagram on the left and on the right
the directivity in the vertical direction, the orientation of coil is shown in the
diagrams. In the most favorable direction the directivity is 4.78 dB with respect to an
isotropic antenna.
Figure 3.3 Directivity of the antenna
The radiation resistance of a coil antenna is given by the following formula:
RR  2  197  e 
n2 r 4
[fill me]
4
(3-15)
With μe the permeability of the material in the coil, n the number of turns in the coil
and r the radius of the ferrite rod. Higher radiation resistances are obtained with this
antenna than with a conventional wire antenna at a wavelength of 3868 meters. If the
coil is for example wound in a circle of 10 cm diameter with 1000 windings and a
ferrite with μe of 1000 in the core the resistance will be 8.6 mΩ.
This antenna will be such connected that the circuit resonates at the desired frequency.
The losses of this antenna are predominantly in the parasitic resistance of the
transmitting coil, the series resistance of the capacitance necessary to let the circuit
resonate and in the ferrite, with resistance:
R f  2fe
A
''
0n2
[fill me]
'
L
f
(3-16)
f
With  ’’ the imaginary part (the lossy part) of the permeability and  ’ the real part.
Af and Lf are the area and length of the ferrite.
The vectorpotential in a coil caused by an AC-current I (t )  I 0 cos(t ) can be
calculated as follows using spherical coordinates.
 I 0 bn2
A(r, t ) 
4
2

0
cos  (t  r / c)
r
cos  'd ' yˆ
(3-17)
With r the distance to the center of the coil and b the radius (see also figure 2.4).
After evaluation of this integral, the result in spherical coordinates is:
A( r ,  , t ) 
b 2 I 0 n 2  sin(  ) 1
4


r


 cos( (t  r / c))  sin(  (t  r / c))̂
c

 r
(3-18)
Figure 3.4 Coordinates of a coil antenna
With this vector potential the electric and magnetic fields can be calculated using the
c
equations of Maxwell. In the far field ( r  ) the first term of the vector potential

can be neglected:
A b 2 I 0 2 n 2  sin(  ) 


 sin  (t  r / c)̂
t
4c
 r 
b 2 I 0 2 n 2  sin(  ) 
B   A  

 sin  (t  r / c)̂
4c 2
 r 
E
(3-19)
(3-20)
The Poynting vector van be calculated with this result:
S
1
0
E  B 
  b 2 I 0 n 2 2  sin(  ) 

c
4c


r
2

 cos (t  r / c) rˆ


(3-21)
The beam intensity will then be:
 b 4 I 02 n 2 4  sin 2 
1
 2 r̂ [fill me]
W  cE02  
3
 r
2
32
c


(3-22)
The total power transmitted will be the integral of the area of the Poyntingvector:
P
1
0
b
 E  B  da 
I 02 n 2 4
12c 3
4
(3-23)
The directivity can be calculated from this value:
4r 2W 3
D
  cos 2 ( )
P
2
(3-24)
These results can be used for calculations in the designing process.
1.1.6 Coil resistance
For an efficient transmitter system the antenna itself should be suitable, but the
parasitic resistance of the coil should be minimal too. This effect causes an extra
resistance of the total system where a considerable part of the power is dissipated.
The coil has significant parasitic resistance, especially at large inductances. Therefore
attention must be paid to that effect, which is also dependent on the frequency. At
higher frequencies due to skin effect only the outer part of the wire it will be exploited
for conducting the current, whereby the resistance will increase. The width of that part
has been specified as:
d
K
(maak f het volgende: f * 1 e -3 en zet voor de wortel een factor
f
1000)[6]
(3-25)
d is in meters and f in Hz. K (in meters per [wortel hertz] ) is a material dependant
constant, for copper this is 2.08 and for aluminum 2.77.
Figure 3.5 The skin-effect
To prevent obstruction is experienced of this effect, Litz wire can be used. This is a
wire existing from several thin wires insulated from each other. Because those wires
are thinner, the unused part is relatively smaller and by volume more wire is
conducting.
Furthermore several windings interfere with eachother inside the coil, so that as a
result extra resistance is caused. To minimize that effect the distance between
windings must be equal to their diameter.
1.1.7 Capacitor resistance
An ideal capacitor doesn’t exist; especially when the circuit should be as efficient as
possible it is important to minimize the parasitic series resistance.
Different types of capacitors have different loss angles, this loss angle is defined in
the complex plane as follows:
Figure 3.6 Loss angle
The ideal capacitor should only have an imaginary part, however in practice there
always is a real loss part in the impedance. This part is usually specified as tan δ. With
that value the series resistance of the capacitor can be calculated as follows:
R
1
 tan 
 C
(3-26)
1.1.8 Strength of the DCF77 signal
For overruling the DCF77 signal it is necessary to know the strength of the signal
originating from Germany. The transmitter in Mainflingen transmits 25 kW. The
signal isn’t transmitted in a sphere, but in the same shape as figure 3.3. But the power
will decay quadratically anyway. With respect to an isotropic antenna it has a gain of
4.78 dB.
This can be converted to a gain of 3. Enschede is situated approximately 300 km away
from Mainflingen, therefore the intensity of the power should be approximately:
25kW
* 3  8,33  10 7 W / m 2
2
(300km)
(3-27)
The beam intensity is dependant of the electric field strength of the electromagnetic
wave as follows:
1
2
I  c 0 E 0 [4]
E0, 0’letje weg….
(3-28)
2
From this the electric field strength can be calculated, this is approximately:
E0 
I
0,5c 0
 0,025V / m Weer de 0 weg bij E
(3-29)
In an antenna the voltage induces an electromagnetic wave. For this a closed loop
antenna can be used. The induced voltage in the antenna depends on the electric field
according to [fill me]:
U
2E0 A

[fill me]
(3-30)
The wavelength of the DCF77 signal is:

c 3  108

 3868m
f 77500
(3-31)
When a closed loop antenna with an area of one square meter is used to receive the
signal, a voltage is induced with the following RMS-value:
U
2  0,025  1
 40,6V
3870
(komma’s weghalen)
(3-32)
The power delivered is equal to:
P
U 2 (40,6  10 6 ) 2

 32,9 pW
R
50
(3-33)
After measuring with both a closed-loop antenna and an original coil antenna from a
DCF77-klok no peek appeared on the spectrum analyzer at 77.5 kHz, the measured
value did not end up above the noise level. The maximum value of this noise was
approximately -80 dBm.
This logarithmic value is calculated with the received power, with the following
formula:
 Psignaal 

P[dBm ]  10  log 
P
 0 
(3-34)
From this the power of the signal could be abstracted:
80
10 10 * P0  10 11W
(3-35)
The conclusion which can be drawn from these measurements is that the signal
originating from Germany is so weak in Enschede that the induced power does not
end up above -80 dBm. This value must be exceeded to overrule the real transmitter.
1.2 The realized antenna
For the transmitting coil a piece ferrite from an ancient AM-radio is used, since a
desired piece of ferrite is not readily available. From measurements the effective
permeability of 215.6 was obtained. It would be better if the ferrite had a higher
permeability and a lower complex permeability.
For the core with an area of 6.36 * 10^-5 m2 and a length of 92 mm, 200 turns have
been wound with copper wire of 0.4mm diameter. More turns would cause quadratic
loss by the complex permeability. Thicker wires would make the antenna too large
and would produce relatively little extra profit. Double wires disturb each other too
much and produce extra loss. So this would be the best configuration for best
efficiency.
This resulted in a DC resistance of 927mΩ and an inductance of 1,84 mH. However,
at higher frequencies that resistance became higher due to the complex part of the
permeability. At 77,5KHz this was 4.4Ω extra. Larger currents through the coil also
caused a higher resistance. At the current of 28 mA, which will be eventually used,
the resistance was 18Ω
It is also of importance to consider the radiation resistance of the coil, which would be
according to (1-16) 31.3 μΩ. To bring the antenna into resonance a good capacitor
with a small loss angle was needed. For this a polypropylene condenser was chosen
with a tan δ of < 10 * 10^-4. To resonate at 77.5 KHz it had to be 2.26 nF, the series
resistance was 666 mΩ.
1.3 Test results of the antenna
To investigate the quality of the realized antenna a number of tests were carried out.
Measurements have been done using a DCF-receiver module, using a criterion of
overruling which was reached when no original bits could reach the receiver.
The criterion was already a problem itself, because the frequency of the real signal
was much more precise than the one of the function generator. Because of this two
signals with a slightly different frequency arrive at the receiver, so there will be "beat"
effects. As a result the signal had to be 20 times stronger for complete overruling than
for just disturbing.
Close to large conducting objects the antenna didn’t work very well. From
experiments it could be concluded that those objects absorbed the signal with
exponential decay. In a building with many disturbances like for example monitors
which produce noise in the same frequency region, the signal became very
unpredictable. Sometimes the original signal was such weak that overruling wasn’t a
problem at all and sometimes the disturbance was so strong that nothing caused any
effect. In a building without a large metal cage construction around it and with little
electronic equipment, the device won’t suffer from those effects
To get a more reliable measurement, a test was done outside on a lawn. The results
were as follows:
Distance Required Reciprocal
(in meters) power (in power (in
mWatts) mWatts-1)
1 0,00174
574
2
0,14
7,14
3
1,15
0,87
4
6,47
0,15
6
5,31
0,19
8
2,5
0,4
10
4,78
0,2
18
12,7
0,078
25
35
36,8
36,8
0,027
0,027
Figure 3.7 Measurement results
When the power increased, the efficiency decreased, this would make the results noncomparable to the theory. To fix this problem the required power has been normalized
to the impedance at low currents of 6Ω. The last column gives the reciprocal power,
these values are necessary because the required power is reversed proportional to the
power decay of the field. Below the graph of the last column is shown with a line
drawn through it to match with the theory.
Figure 3.8 Decay of the field
The graph has too few points to be sure whether the sineshape really continues as
indicated with the red line. The measurements of 1 and 2 meters have been omitted,
because they would make the graph too large.
These results look similar to the graph in figure 3.2, the factor of the added
distribution couldn’t be calculated exactly from the theory for the case of the coil.
Because of this influence could be different in practice, moreover there were not
enough measurement points to test the theory very well. Thus it can be concluded that
the theory is correct with the practice, as far as it was possible to test.
If the measurement results are extrapolated up to 50 meter, it can be concluded from
the theory that virtually the same transmission power is needed. To a distance of 100
meters there are still spots on the tops of the sines (see the figures 3.2 and 3.8), where
it will still work. Untill just outside the close reactive field (approximately 600
meters) receivers the signal will be disturbed. The power needed for this is 15.62
mWatts.