Page 378 - Maxwell House

358 Chapter 7

of fields penetrates outwards developing the EM radiated wave. In full accordance with
Maxwell’s equations, the small portion of conductivity current at the line end is converted into
outward displacement current, i.e. E-fields, as seen in Figure 7.1.5a, b. Simultaneously, this
external E-field causes some electrons to move from the inner to outer surface of the coaxial
pipe as Figure 7.1.5b and 7.1.5d demonstrate. It means that now the electric current flows not
only inside but exists as the conductivity current on the outer surface. Again in accordance

with Maxwell’s equations, the two outside currents, displacement and conductivity, must be the
source of outwards EM fields (see Chapter 4 for details). It means that the coaxial line loses
some portion of energy on radiation at the open end. It helps understanding why the impedance
of the open coaxial line is not infinite anymore, the electric current at the open end is not zero,
and the reflection coefficient is close but not equal to one and becomes frequency dependable
value. Evidently, the radiation losses negligible while the transverse sizes of the coaxial line
much less than the operational wavelength. If so, they should be almost negligible at very low
frequencies and gradually increase as the frequency rises. For example, the coaxial line of 75Ω
shown in Figure 7.1.5 has 2a = 13.382 mm, 2b = 4 mm and filled with the insulator of = 2.1.

The shortest wavelength is = 30 mm at 10 GHz and the radiation loss is 7% at this frequency
(see Smith chart in Figure 7.1.5f).

The metal pipe in Figure 7.1.5b, d with current running down over the pipe surface can be

interpreted as a continuous linear array of electrical dipoles with progressive phase excitation
(check Section 5.4.7 of Chapter 5). Looking back in Figure 5.4.13, we see that such antenna
radiates mainly in the direction of current propagation. This explains that why in Figure 7.1.5e
the radiation pattern peak is directed down. Smith chart in Figure 7.1.5f reaffirms that the
reflection coefficient is only almost equal to 1 at all frequencies between 0 and 10 GHz. Such
behavior was expected due to large differences in impedances between the free space ~377 Ω
and coaxial line of = 50 Ω. Besides, there is huge mismatch in the field pattern inside and

outside the coaxial line. That is why the tip of the gradually reducing phasor moves from 1
at frequencies around 0 GHz to 0.96 at 10 GHz. If so, EM field distribution along the line (as
the superposition of incident and reflected wave) is very close to but not exact standing wave
clearly seen in Figure 7.1.5a and 7.1.5b.
From the above discussion follows that the equivalent circuit in Figure 7.1.5c must consist of
two elements: the end capacitor associated mostly with E-filed storage nearby the sharp

edges and resistor related to the EM energy leaving the line in the form of radiation. Both
Σ
values can be read from Smith chart in Figure 7.1.5f. For example, ≅ 0.15 = 0.15 ∙ 50 =

Σ
7.5Ω at 10 GHz.
7.1.4 Gap in Center Conductor
The discontinuity is depicted in Figure 7.1.6 and is much shorter than the operational
wavelength. Figure 7.1.6a demonstrates the expected accumulation of E-field energy in the gap
itself and around the sharp edges. It tells us that the equivalent circuit of this discontinuity must
include three capacitors, one is the series gap capacitor and two shunt edge capacitors

1
as shown in Figure 7.1.6c. The more accurate equivalent circuit must certainly include some
inductors reflecting the existence of H-field nearby the gap. However, according to Figure
7.1.6b, this lumped component is quite small and can be omitted. However, Smith chart in
Figure 7.1.6d indicates that the gap impedance tends to move up along the diagram and crosses
its horizontal (resistance) line at frequencies above 10 GHz as seen in Figure 7.1.6e.