5.4 EM TRANSDUCER: A LINEAR ANTENNA
5.4.1 Basic principles
One way to produce EM radiation is to create a directional ﬂow of accelerating/decelerating electrons,
e.g., by inducing alternate electrical current in a linear metal wire – an antenna, as shown in Figure 5.3.
The radiation properties of such an antenna strongly depend on the antenna length. The effects of the
antenna size are due to the propagation properties of the wave along the antenna wire. When an oscil
lating voltage source is attached to one end of the antenna, the electric ﬁeld will push the ﬁrst electron
(Fig. 5.3), which will in turn push the second one, then the third one, etc., thus creating a train of electron
displacements. The speed of the train is close to the speed of light, c. These displacements will propagate
until the other end of the wire is reached, at which point the last electron hits a barrier and is reﬂected
back, changing the direction of its movement to an opposite (in this idealistic picture, possible energy
losses are neglected). Now there are two trains, moving in opposite directions and if they ‘collide’, wave
propagation is annihilated. Thus, efﬁcient radiation by a short antenna is possible only during the time
interval,
s
, before the wave propagates distance L
ant
to the end of the antenna:
s ¼
L
ant
c
¼
L
ant
l
Q (5.1)
For longer antennas, conditions can be found for nondestruct ive wave propagation along the antenna.
Destructive ‘collisions’ of waves are avoided whe n the direct and the reﬂected waves are synchro
nized. This means that the wave must arrive to its point of origin after traveling distance 2L
ant
in the
same phase, i.e. the time to travel is equal to the period of the oscillation:
2L
ant
c
¼ nQ (5.2a)
User’s
information
source
Oscillator (T1)
Antenna (T2)
Power
amplifier (T3)
Antenna (R1)
Tuner (R2)
Amplifier (R3)
Demodulator(R4)
Received
information
FIGURE 5.2
Block diagram of a radio communication channel
5.4 EM transducer: A linear antenna 127
or
2L
ant
Q
l
¼ nQ (5.2b)
where n is an integer (n ¼ 1, 2, .). Thus an ‘optimum’ antenna length is
L
opt
¼ n
l
2
(5.2c)
Common practical antennas are the halfwave antenna (n ¼ 1, L
ant
¼
l
/2) and the fullwave antenna
(n ¼ 2, L
ant
¼
l
).
5.4.2 Short antennas
While short antennas (L
ant
<<
l
) are much less efﬁcient, they are often used when the system
dimensions are strictly constrained. Below an intuitive orderofmagnitude analysis of the radiation by
a short antenna is offered. A short straight wire has a capacitance C
wire
:
C
wire
w3
0
L
ant
(5.3)
The corresponding energy stored in the wire is
E ¼
CV
2
2
(5.4)
For simplicity, the alternating (e.g. sine form) voltage V(t) applied to the wire will be approximated by
triangle wave, as shown in Figure 5.3:
V
D
w4V
0
t
Q
¼ 4V
0
L
ant
l
(5.5)
22
antant
L
c
L
t
t=0
antant
L
c
L
t
antant
L
c
L
t
L
2
e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

FIGURE 5.3
An intuitive interpretation of wave propagation and energy loss in short antennas
128 CHAPTER 5 Nanomorphic cell communication unit
The average value of voltage in the interval from 0 to L
ant
(L
ant
<<
l
)is
hV
D
i¼2V
0
L
ant
l
(5.6)
Putting (5.6) into (5.4) obtain:
E ¼
ChV
D
i
2
2
w
1
2
3
0
L
ant
,4V
2
0
L
ant
l
2
¼ 23
0
L
ant
V
2
0
L
ant
l
2
(5.7)
Next, the maximum power which can be radiated during the time interval,
s
, of nondestructive wave
propagation (5.1) is
P
rad
¼
E
s
¼ E,
c
L
ant
(5.8)
Substituting (5.7) into (5.8), obtain the maximum power radiated by a short antenna:
P
rad
w23
0
L
ant
V
2
0
L
ant
l
2
,
c
L
ant
¼ 2,V
2
0
ð3
0
cÞ,
L
ant
l
2
w
V
2
0
Z
0
L
ant
l
2
wI
2
0
Z
0
L
ant
l
2
(5.9)
where Z
0
¼
1
3
0
c
z 377
U
is the impedance of free space.
Note that Z
0
L
ant
l
2
in (5.9) has dimensions of resistance and is calle d the radiation resistance of
the antenna:
R
rad
wZ
0
L
ant
l
2
w377
L
ant
l
2
(5.10a)
More rigorous derivations of the short antenna radiation resistance can be found in the antenna theory
literature [2,3]. The resulting expressions can differ depending on the input assumptions. For example,
Blake [2] obtains (assuming uniform current distribution):
R
rad
z 790
L
ant
l
2
(5.10b)
while Jackson [3] gives (for nonuniform current distribution):
R
rad
z 197
L
ant
l
2
(5.10c)
The formulas (5.10a–10c) were derived for short antenna approximations assuming L
ant
<<
l
and,
generally speaking, they are not applicable to the longer antennas with L
ant
w
l
. For example, an exact
solution for an ideal halfwave antenna is [2]
R
l=2
¼ 73:1U (5.10d)
It is interesting to note that application of (5.10a) to a halfwave antenna yields an estimate of 94
U
in radiation resistance, which is remarkably close to the value given in (5.10d). This suggests
5.4 EM transducer: A linear antenna 129
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