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MORE COMPLICATED ELEMENTS OF FEED LINES 425

17
Figure 8.4.8 illustrates two most favorite planar layouts of such filters . In Figure 8.4.8b the
half-wavelength resonators are folded into “hairpin” configuration to reduce the total filter
length. Looking back at Section 8.2.3 of this chapter, we see that each of such filters is the
cascaded directional couplers shown in Figure 8.2.2a. To convert the directional coupler into a
filter element, port 2 and 3 of the coupler are not terminated (open-circuit) or grounded through
the vias (short-circuit). Thereby, the injected into port1 of coupler EM wave repeatedly moving
back and forth between port2 and port3 over a quarter-wavelength coupling section in the same
manner as in any in-line resonator (see the bouncing diagram in Figure 8.1.1c). To investigate
the properties of such type of resonator, bring up the expression (8.5) assuming that port2 and
port3 are open-circuit. We know from discussion in Section 8.2.3 that EM wave of magnitude
inserted into coupler port1 is divided between port2 and port3 while port4 is completely
1
isolated, i.e. = 0 as soon as port4 is connected to dummy load. The EM wave passing in
4
port2 with magnitude reaches the open-circuit end of line and totally reflects back getting
2
− − − . This
the phase shift . It means that = . Seemingly, in the port3 =
3
2
2
3
phase shift takes into account the presence of small fringing capacitor at the end of open-
circuit line (see Table 7.1 in Chapter 7) with negligible radiation. Putting these relationships
into the expression (8.5) and turning from matrix to usual form of the linear equations we obtain
= + = ( + ) −
21 2
31 3
21 2
1
31 3
⎫
= (8.15)
2
21 1
=
3 31 1 ⎬
= + = ( + ) − ⎭
4 31 2 21 3 31 2 21 3
Excluding and we have = = ( 21 + ) − and =
2
2
⁄
3
1
31
1
2
⁄
= 2 − . Evidently, the first expression corresponds to the reflection coefficient
4
1
21 31
in port1 while the second one describes the transfer function. If so, the normalized input
impedance can be found from (3.89) of Chapter 3 as =
�1 + � �1 − � �. The results of numerical simulation are presented in Figure 8.4.8c
where the red line corresponds to = ℜ� � and the reactive part = ℑ( ) is shown

by blue dotted line. As expected, the both parameters corresponds to a parallel resonator
connected in shunt across the line. Evidently, the resonator external Q-factor increases and the
bandwidth constricts accordingly as the coupling reduces. In the same manner, we can get the
parameters of distributed resonators while port2 and
port3 are shorted. In this case, 2,3 = − and =
2,3
0. The admittance =
�1 + � �1 − � �, i.e. both admittance and
impedance are formally described by the same
expression. If so, the plots in Figure 8.4.8c illustrates
not only the normalized impedance but also the
normalized input admittance = + that
corresponds to a series resonator connected across the
Figure 8.4.9a Combline filter
17 Public Domain Image, source:
http://wikivisually.com/wiki/Distributed_element_filter/wiki_ph_id_45,
http://www.lpkf.cn/applications/rapid-pcb-prototyping/design-articles/microstrip-filter-and-coupler-
circuits.htm