Phase Relationships and Generation of Finite Transmission Zeros

coupling path from source or load to resonators and to apply suitable phase shifts of the cross coupling paths to generate finite frequency transmission zeros on either upper stopband, or lower stopband, or both of the stopbands. Let us take a fourth order filter as an example.

Fig. 4.2. The equivalent lumped-element circuit of a fourth-order parallel coupled filter.

In the beginning, the initial design of the proposed filters is based on the conventional parallel-coupled filter presented by Cohn [97]. Fig. 4.2 shows the lumped-element equivalent circuit of a fourth-order parallel-coupled filter shown in Fig. 4.1(a). Cross couplings are not introduced at this moment. Here, it should be pointed out that the lumped-element equivalent circuit should include the phase-reversing transformer in every resonator. Although the phase-reversing transformer is often omitted in a conventional parallel-coupled filter due to no effect on the magnitude of filter response, it is, however, very important in the proposed cross-coupled filters. Let us now check phase relationships from source or load to resonators. In order to observe the relative phase conveniently, we sequentially number the corresponding nodes of Figs. 4.1(a) and 4.2 as A-J from source to load.

Therefore, the relative phases in the lumped-element circuit model of Fig. 4.2 can be determined, and all of the insertion phases between node A and nodes B-J in Fig.

4.1(a) are obtained easily. Consider each box in Fig. 4.2 which represents an ideal admittance inverter having constant image admittance and constant phase shift of － 90^o for all frequencies. Let nodes A and B to be the input and output ports of the admittance inverter J . The matrix element Y of the two-port admittance matrix can

then be determined. Thus, the phase of YBA is －90^o over both the frequency ranges f

＜ f0 and f ＞ f0 where f0 is the center frequency of the filter. Also, the phase shift of the phase-reversing transformer is －180^o for all frequencies. Consequently, the phase of YCA is －270^o over both the frequency ranges f ＜ f0 and f ＞ f0. Next, consider YDA that the shunt inductor/capacitor pair as shown in Fig. 4.2 is a resonator.

The phase shift of a resonator at off-resonance frequencies is dependent on whether the frequency is above or below resonance. As f ＜ f0 the admittance of the resonator is inductive and the phase shift should be －90^o. Similarly, as f ＞ f0, the admittance of the resonator is capacitive and the phase shift should be ＋90^o. As a result, the phase shift of YDA is －90^o when f ＜ f0 (－90^o－90^o－180^o－90^o ＝－450^o ＝－

90^o) and ＋90^o when f ＞ f0 (－90^o＋90^o －180^o－90^o ＝ －270^o ＝＋90^o).

Following similar analyzing procedures described above, one can observe every relative phase shift between node A and nodes B-J. The phase relationships from source or load to resonators can be easily observed by using any commercial circuit simulator. Table 4.1 summarizes the phase relationships between node A and nodes B-J as f ＜ f0 and f ＞ f0. The method is applicable to any order of a parallel-coupled filter. As a result, the relative phase shifts between node A and nodes B-J in Fig. 4.1(a) are identical to those of the lumped-element filter in Fig. 4.2

Next, the cross coupling paths will be studied. When a cross coupling path is applied to node A and another node in the nonadjacent resonator and its phase delay is 180^o out of phase with the main path, a transmission zero appears.

The trisection coupling scheme in this filter can be formed by adding a cross coupling path from source to the second resonator. The two ends of the second resonator corresponds to node D and node E. Assume the cross coupling path is applied from source to node D. Because the phase of the main coupling path YDA is

＋90^o as f ＞ f , the phase of the cross coupling path from source to node D should

be －90^o as f ＞ f0 in order to have a upper stopband transmission zero. On the other hand, if the cross coupling path is applied between source and node E, the filter can also have a upper stopband transmission zero when the phase of cross coupling path YEA is ＋90^o as f ＞ f0 due to －90^o of phase in the main path YEA as f ＞ f0. In contrast to a upper stopband transmission zero as discussed above, a lower stopband transmission zero could also be possible by applying－90^o phase shifts of the cross coupling path YEA as f ＜ f0 where it is 180^o out of phase with that of the main path YEA. In spite of the source, the load can also be cross-coupled to the third resonator in Fig. 4.1(a) to form another trisection.

Another popular coupling scheme is so called quadruplet where two transmission zeros, one on upper stopband and the other on lower stopband, are generated by applying just one cross coupling path. The quadruplet cross coupling could be from source to the third resonator or from load to the second resonator. Utilizing similar phase analysis method as trisection, the phase relationship of the quadruplet coupling scheme can be easily obtained. Let us use the cross coupling path from source to node G as an example. The phase shift of the main path YGA is ＋90^o as f ＜ f0 and f ＞ f0. As mentioned above, as long as the phase shift of the cross coupling path YGA is － 90^o, two transmission zeros on both lower and upper stopband should appear.

Similarly, the quadruplet cross coupling path could also be formed from source to node F.

Table 4.1 summarizes the phase relationships that may help a designer to judge the relative phase of the main coupling from source or load to each node and to apply proper phase of the cross coupling to create desired transmission zeros. After the positions of the transmission zeros being qualitatively determined, the proper strength of the cross coupling should be quantitatively determined for a desired specification.

Full discussion will be presented in section 4.3.