Page 319 - Maxwell House

FEED LINE BASICS 299

equivalent radius in (6.15) we obtain for the first high mode ≅ 3.142(1 + /) √ . The
⁄

rigorous analysis shows that this cut-off wavelength estimation is almost perfect when →
and increases to ≅ 3.336 √ when / = 0.25. The latter discrepancy of 6.2% is not
⁄

very significant.
The reader should beware that this result is not only academic but carries important information
for practical applications imposing a limitation on coaxial line transversal sizes. Actually, from
the above discussion surrounding the fibers, we came to conclusion that the simultaneous
propagation two and more wave modes can lead to noticeable signal distortions. We learned
that basically, each mode reaches the receiving point at the different moment of time. If so,
instead of one expected signal we have the whole chorus of echo signals that is undesirable.
Therefore, this effect must be avoided as much as possible especially in lines of length
comparable to wavelength and lengthier. So it is highly desirable to keep in any line (not only
coaxial) so-called one mode regime at all frequencies within the signal bandwidth. In other
words, the shortest signal wavelength must exceed the cut-off wavelength of the first mode or
≥ and ≤ . Then according to (6.15)

≤ ( √ (1 + /))⁄ (6.16)
Let us check how this inequality works at 300 GHz or = 1 mm for air-filled micro-coax
with / = 0.5. From (6.16) we have 2 ≤ 1.33 mm. The production of such tiny line is reality
and the art of technology [1]. The applications of said lines are mainly the ultra-wideband
miniaturized devices in millimeter-wave frequency band.
The next high wave mode with m = 2 can only exist at frequencies where the inner and outer
conductors are relatively large by comparison with the wavelength. If so, the incident and
reflected waves take opportunity to run inside the ring as shown in Figure 6.4.2c and (2) ≅

( − )/2 while ≅ 1.57( − ) √ . The correct value is ≅ 2( − ) √ . The 21%
⁄
⁄

error is not alarming and demonstrates the imperfection of our model as the wave mode pattern
becomes more complicated.
The “Achilles’ heel” of coaxial lines is the central conductor of small diameter practically
defining such basic line parameters as attenuation and RF power handling. Looking back at
Figure 6.1.1d we can see that the electrical field has the highest intensity nearby and on the
surface of the inner conductor. It can be shown that this intensity is proportional to the applied
between conductors’ voltage and the inverse function of radius, i.e. = . Therefore,
⁄

that can initiate corona effect or even reach the breakdown values for quite moderate voltages
as the radius of inner conductor reduces. In addition, we can expect the extreme dielectric
heating around the inner conductor. The intensity of magnetic field varies the same way,
i.e. = , where is the electric current in line conductors. According to the boundary
⁄

conditions we formulated in Table 2.2 of Chapter 2, the tangential component of magnetic field
(look at Figure 6.1.1d) is equal to the electric current density on the conducting surface.

Therefore, high intensity of magnetic field means the high density of this surface current and,
2
⁄
⁄
as aftermath, high conductive (Ohmic) loss proportional to ( ) . Assuming that = 3.25
(standard 50 Ohms coax with = 2 ) we can come to conclusion that 91% (!) of Ohmic loss

falls on the central conductor. The main coaxial cable advantage is low production cost, low
weigh, flexibility and excellent shielding.