Page 442 - Maxwell House

422 Chapter 8

Here ℒ ̂ = / , ̂ = ℒ and = 1 . Therefore, the shunt resonance circuit is converted
2
2
⁄
1
1
0
0
into series resonance circuit depicted in Figure 8.4.5b. The resonance frequencies of both
circuits are automatically synchronized since = 1/�ℒ ̂ ̂ = 1/�ℒ . The Q-factor of
1 1
0
loaded series circuit is = ℒ ̂ / while the shunt one is ℎ = / ℒ . If so, for
0
0 1
0
0
2
the capacitive J-invertor in Figure 8.4.4d the ratio / ℎ = 1/( ) . Therefore,
0
0
the value expresses the coupling coefficient defining the external quality and passband of
0
the series resonator relative to the shunt one. Note that the negative ℒ and shown in Figure
8.4.4 should not be confusing because they are included as the part of adjacent lumped elements
ℒ and of filter diminishing accordingly their values. Special treatment should be given to
1
1
the first and last inverter.
Note that for the sake of simplicity resonator was assumed lossless. Similarly, we can check the
performance of all other inverters to make sure that they transform series-connected element to
shunt-connected, and vice versa. It means, for example, that band-pass filter in Figure 8.4.1
with series and shunt resonators can be converted into the filter with only shunt resonators as
shown in Figure 8.4.5c.
8.4.5 Quarter-Wavelength Section of Feed Line as Inverter
Before discussing the practical realization of inventors in Figure 8.4.4, let us consider the
fundamental property of any line section named the impedance transformation. Looking back
at the expressions (3.87) and (3.89) in Chapter 3, assume that the lossless single mode line with
characteristic impedance is terminated by the load ≠ . As we demonstrated in

Chapter 3, the line reaction is entirely predictable. In response, the voltage and current change
−
= (8.11)
+

For example, if = 0 (the line is shorten meaning that + = 0) = −1

(total reflection). In case of = ∞ (the line is open meaning that + = 0),
= +1 (total reflection again). Putting the reference plane at the beginning of the line
where the matched generator is connected, we can estimate the reflection wave phasor at any
plane as (check (3.87))
= −2 (8.12)

Here = 2/Λ is the propagation constant of the dominant mode and Λ is its wavelength.
Recall that for TEM-modes Λ = λ = / as the mode propagates in free space. The phase shift
in (8.12) is doubled because the incident wave before reaching the load picks up the phase
shift (−) and passes this phase shift to the reflected wave. Then the reflected wave gets its
own phase shift running back the same distance l. Putting (8.12) into (3.87) we obtain the input
impedance of feed line section of length l

−2
1+ 1+
= =

1− 1− −2
− −2
1+
+ + tan ()
= − −2 = (8.13)
1− + tan ()
+