Microsystems for Bioelectronics

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 non-destruct 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 half-wave antenna (n ¼ 1, L

ant

¼

l

/2) and the full-wave 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 order-of-magnitude 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 non-destructive 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 non-uniform 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 half-wave antenna is [2]

R

l=2

¼ 73:1U (5.10d)

It is interesting to note that application of (5.10a) to a half-wave 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