Page 354 - Maxwell House

334 Chapter 6

desired specifications. Some of them probably are unrealizable due to extraordinary impedance
value, negative section length, or an excessive number of sections.
Looking back through Chapter 5, we can realize that (6.45) is likewise the expression (5.86)
that we successfully used for linear array synthesis. Evidently, we can proceed the same path if
we will treat the reflection coefficients in (6.45) as the excitation magnitude, i.e. = , of

linear array elements in (5.86). To finish the conversion of (6.45) into Fourier series and avoid
too general discussion, let us limit the class of transitions assuming that a) all sections are the
same length, i.e. = = ⋯ = = and b) the dominant mode in all line sections is TEM,

2
1
i.e. = = 2 ( √ )⁄ and is the overall dielectric constant. Then

4(/ 0 ) 4(/ 0 )
2 = =
⁄
( 0 ) √ √ 0 ⎫
⎪
∑ = (6.47)

=1
−2 ∑ = ⎬

=1
2 ⎪
= −2 , =
0 0√ ⎭
Finally, (6.45) and (5.86) formally coincides since
= ∑ (6.48)

=0

The expression (6.48) should be a Fourier series while 0 ≤ − ≤ 2. Therefore, according to
the last expression in (6.47) our approach to the transition synthesis is limited in frequency
⁄
⁄
domain as 0 ≤ ≤ . It is worthwhile to point out that the transition where = =
0
0
1
= ⋯ = is a one of rare cases when the combine reflection coefficient can be analyzed
2

analytically without numerical assessment. Eventually, the series in (6.48) is a geometric
progression with common ratio and can be modified to

(+1) sin (/2)
1−
| | = | |�∑ � = | | � � = | | � � (6.49)
0 =0 0 0
⁄
1− sin ( 2)
The graph of this function is depicted in Figure 6.7.3b for N = 10 and | | = 0.05 meaning
2
0
that only 5% of energy is reflected from the first step. According to this graph | | ≤ | | at

0
all frequencies within the relative bandpass 77% while 95% of input power from line #0
transfers to line #11. Typically, the electrical length of line sections is the quarter of wavelength
in line, i.e. (/ ) √ = 1/4 in (6.47). Evidently, their lengths must be corrected to
⁄

0
compensate the parasitic step-capacitance the same way as we have done before.
Using the similar ideas that were outlined in Section 5.4.4 of Chapter 5, we can develop a wide
variety of different transitions. In particular, the Dolph-Chebyshev technique lets synthesize the
transition satisfying some specifications and having the minimal number of steps, i.e. the
shortest one. Just note that the described Fourier series method can be applied not only to coax-
coax line transitions but with some minor modifications to coax-waveguide, waveguide-
waveguide, waveguide-stripline and many other transitions.
Looking back at the plot in Figure 6.7.3b, we can detect some bizarre outcome: at some
frequencies | | > 1, i.e. the reflected power for some reason excesses the power delivered to

the transition. Clearly, this result is meaningless because the transition is a passive devices and
could not generate extra power. We know the reason for such physically inconsistent data. That
is the approximation we have made suggesting that the amplitude of each partial reflection wave