Page 352 - Maxwell House

332 Chapter 6

connection. From classical circuit theory follows that 100% energy would pass through from
Line 1 to Line 2 without any reflections at all frequencies. Nevertheless, the results of
19
numerical analysis shown in Figure 6.7.1e contradict this statement and demonstrate relatively
small but thereby non-zero reflections at all frequencies between 6 GHz and 12 GHz. To
understand this phenomenon look at EM field distribution (logarithm scale) nearby the step
depicted in Figure 6.7.1b and c. Nothing unusual is detectable in the longitudinal H-field pattern
while E-field surges and accumulated around the sharp edges of step. Poynting’s theorem tells
us that any excess of electric energy in some small region can be interpreted as the shunt
capacitance in equivalent circuit as Figure 6.7.1d demonstrates. This example clearly reveals

how EM field analysis effortlessly helps building the equivalent circuits and explain the non-
zero reflection. Finally, looking back at Figure 6.7.1d we can answer the question how to reduce
the reflection, i.e. match the discontinuity: increase magnetic energy stored nearby the step and
thereby put the lumped inductor ℒ in equivalent circuit. The slight reduction in diameter of

inner conductor in Line 1 (see Figure 6.7.2a) increases the H-field intensity as it is clearly
demonstrated in Figure 6.7.2b. Remind that the H-field intensity is the inverse function of
conductor diameter. The complete equivalent circuit including two capacities and series
inductor is shown in Figure 6.7.2d while Figure 6.7.2e reflects the broadband matching
improvement. Note that such matching inductive section of line is commonly the part of the
coaxial line of bigger diameter thereby opening wider the space for EM wave to pass.

Figure 6.7.2 Matched double step in coaxial line: a) Line connection schematic, b)
Longitudinal H-field energy distribution, c) Longitudinal E-field energy distribution, d)
Equivalent circuit, e) Smith chart

The general case when the line impedances and cross section vary at the same time requires a
more sophisticated approach based on the idea that the multiple phase-shifted waves can
produce zero or close to zero sum. That is the same interference concept we employed in
Chapter 5 to explain a linear antenna pattern formation using phasor-vector diagrams in Figure
5.4.7. Referring to Figure 6.7.3a let us consider the (N+1)-step transition like a ladder
representing the whole reflected wave in AA-section as the superposition of partial waves
reflected from each step.

19 We defined the reflection coefficient in Section 3.3 of Chapter 3.