Fundamentals of Electromagnetics - RF/Microwave Interaction with Biological Tissues - page 31

sical physics, assuming that the radiator may possess all energy levels and that
transitions are by a continuous process [17].
The classical Johnson-Nyquist expression for the noise emitted by a
resistor can easily be derived from the Rayleigh-Jeans approximation. A
circuit is formed by connecting a resistor at both ends of a lossless transmis-
sion line. Both resistors are equal to the characteristic resistance of the line;
hence the line is matched and there are no reflections. The matching resistors
and the line are all at the same temperature. Hence, the circuit is a one-
dimensional blackbody in which the energy emitted by the resistor at the left
propagates on the line and is absorbed by the resistor at the right end, while
the energy emitted by the resistor at the right and propagating on the line is
absorbed by the resistor at the left. As for establishing the three-dimensional
Planck's law, one multiplies the energy per photon by the number of photons
in a given mode and by the number of modes in the given length. The
expression is reduced assuming that the Rayleigh-Jeans approximation is valid
( hf << kT ). Dividing by 2 to obtain the spectral power density for one
resistor only yields the classical expression for the noise energy emitted by a
resistor:

()=

Sf kT W Hz (1.38)

- 1

It is interesting to observe that this expression is independent of the value of
the resistor: Two resistors, with values 1 W and 1 MW, respectively, produce the
same energy. It is important to note that expression (1.38) is valid only under
the Rayleigh-Jeans approximation. The power emitted in a given bandwidth
B is of course

P = kTB W

(1.39)

From this expression one can define the noise temperature of a signal: It is the
temperature at which a pure resistor should be maintained to produce the
same power spectrum:

()

Sf
k

()=

Tf K (1.40)

In particular, this is the adequate definition for the antenna temperature , which
is the absolute temperature at which a pure resistor must be maintained to
produce the same power density as that measured at the output of the antenna.
In other words, if a pure resistor maintained at the antenna temperature
replaces the antenna, the receiver does not observe a difference. One then uses
expression (1.39) for calculating the noise power emitted by a pure resistor,
considered as the source of a generator with the resistor as the series resist-
ance. A resistor of the same value to match the source terminates this source.
It is well known that the generator then delivers the maximum possible power.
Calculating this power yields