Filter Design Examples and Results

The proposed filters are built on a 20-mil-thick Rogers RO4003 substrate with εr

= 3.38, tanδ = 0.0021. The commercial EM simulation software Sonnet 9.0 [100] is used to perform the actual computation as described above. The specifications of the filter in Fig. 3.1(a) are described in previous section. Here, the dimensions of the resonator 1 are the same as the resonator 4 and the resonator 2 are the same as the resonator 3. The values of Z1 and Z2 for resonator 1 and 4 are chosen as 121 and 41.78 Ω, respectively. The values of Z1 and Z2 for resonator 2 and 3 are chosen as 121 and 24.93 Ω, respectively. Thus, from (3.3)-(3.6), the values of RZ and θ1 for resonator 1 are 0.206 and 24.41 degree, respectively, and the values of RZ and θ1 for resonator 2 are 0.3453 and 30.44 degree, respectively.

The initial dimensions are then fine tuned to achieve the original specifications.

The dimensions (mm) of the filter shown in Fig. 3.1(a) are W1=0.15, W2=3.10, W3=1.52, L1=4.27, L2=4.82, L3=5.28, L4=5.87, S1=0.13, S2=0.38, S3=1.29, S4=0.15, L =2.54, L =10.82. The diameter of the via-hole is 0.81 mm. The second filter shown

(a)

(b)

Fig. 3.5. The constructed filters. (a) The quadruplet filter. (b) The quadruplet filter with source/load coupling.

in Fig. 3.1(b) is a canonical form filter that two cross couplings one between resonator 1 and 4 the other one between source and load are included. This would create two extra finite transmission zeros to improve selectivity. Fortunately, one can add the source-load coupling as a perturbation that other portion of the filter could keep unchanged. Therefore, all of the dimensions of the filter in Fig. 3.1(b) are the same as Fig. 3.1(a) except for the source-load coupling enhancement line. The dimensions (mm) of the second filter are W4=0.2, S5=0.25, L7=1.17, L8=0.61. Figs. 3.5(a) and 3.5(b) show the pictures of the two constructed filters.

Fig. 3.6(a) shows the measured in-band performance of the first constructed filter.

Good agreement between the simulated and measured results is observed. The midband insertion loss is 2.7 dB and the return loss is greater than 14 dB. The simulated and measured wide-band performances of the filter from 1 GHz up to 14

(a)

(b)

Fig. 3.6. Measured and simulated performances of the quadruplet filter of Fig. 4.5(a).

(a) In a narrow-band. (b) In a wide band.

GHz are illustrated in Fig. 3.6(b). The measured transmission performance of the filter is with a rejection level better than 30 dB up to 12.5 GHz.

The in-band performance of the second constructed filter is shown in Fig. 3.7(a).

The measured positions of the two additional transmission zeros contributed by source-load coupling are just a little drift compared with the simulated results. The measured insertion loss is 2.75 dB and the return loss is greater than 17 dB. The measured upper stopband performance of the filter shown in Fig. 3.7(b) follows the simulation and achieves an attenuation level exceeding 27 dB up to 12.7 GHz.

(a)

(b)

Fig. 3.7. Measured and simulated performances of the quadruplet filter with source/load coupling in Fig. 3.5(b). (a) In a narrow-band. (b) In a wide band.

Chapter 4 Microstrip Parallel-Coupled Filters with Cascade Trisection and Quadruplet Responses

Microstrip parallel-coupled filters with generalized Chebyshev responses are presented. The basic structure of the proposed filter is a conventional parallel-coupled filter which the physical dimensions can be easily obtained by the well-known analytical method. With the aid of the equivalent circuit corresponding to a conventional parallel-coupled filter, the relative insertion phase from source or load to each open-end of resonators can be easily obtained by observing the two-port admittance matrix. Applying the cross coupling from source or load to a proper nonadjacent resonator, a trisection or a quadruplet coupling scheme can be realized with prescribed transmission zeros. More importantly, the proposed trisection can be designed to have a transmission zero on the lower or upper stopband by just adjusting the length of the cross coupling strip. Using the proposed structure, the conventional time-consuming adjusting procedure to obtain initial physical dimensions of filters is no longer required. In this chapter, a fourth-order parallel-coupled filter is used as the basic structure to demonstrate various combinations of transmission zeros. Simulated and measured results are well matched.

4.1 Introduction

High performance microwave filters are essential circuits in many microwave systems where they serve to pass the wanted signals and suppress unwanted ones in frequency domain [26]. Cross-coupled filters are attractive since they exhibit highly selective responses which are required in modern communication system. Among these cross-coupled filters, the cascade trisections (CT) and cascade quadruplets (CQ) [6], [71], [73], [81], [82], [95] are two of the most commonly used coupling schemes.

Besides the cross-coupled coupling schemes, other coupling topologies such as doublet, extended doublet and box-section were also found interesting [66], [86] and have been successfully implemented in microstrip form [88], [89]. In brief, all of the mentioned filters are designed to have finite transmission zeros for better selectivity.

To design a cross-coupled filter such as the filters in [6], [71], [73], [81], [95], the following procedures are usually taken. The first step is to synthesize a coupling matrix corresponding to a desired response. Secondly, decide the suitable physical layout of the resonator. Thirdly, adjust distance and orientation of two neighboring resonators two by two to get proper signs and magnitudes of the corresponding coupling coefficients. In this step, the Dishal’s method [38] is usually used. A detailed description of Dishal’s method is given in [6]. Finally, fine tune the whole circuit. The third and final steps are the most tedious and time-consuming steps because, in the third step, they need to generate design curves of coupling and external Q from an EM field solver and, in the final step, one resonator may have many neighbors that when adjusts the distance and orientation against one neighbor the coupling strength with other neighbors may change. Therefore, the iterative adjusting procedure might require. Another drawback to design the conventional cross-coupled filter is that if one coupling coefficient in the coupling matrix changes sign, the physical layout must be reconfigured. For example, in the case of cascade trisection filters of [73], completely different orientations of the resonators must be adopted for a trisection having a lower stopband transmission zero and a trisection having an upper stopband transmission zero because there is one coupling coefficient changed sign. This means that the time-consuming adjusting step described above must be done separately in two cases.

Another interesting cross-coupled filter based on a parallel-coupled filter structure was proposed by Hong and Lancaster [96]. In [96], an extra microstrip line

couples the nonadjacent resonators (resonator 1 and 4 in the paper) to produce transmission zeros. By adjusting the length of this extra coupling microstrip line and gaps of the coupling sections, the locations of the transmission zeros can be manipulated. This kind of filter has the benefit of simple layout, manageable transmission zeros, and much less time for adjusting layout than conventional cross-coupled filters. However, it has some problems. The extra coupling microstrip line has its own resonant frequencies. If the electrical length of this extra coupling microstrip line is not integer multiples of 180^o, spurious responses appear at these resonant frequencies on lower or upper stopband. The spurious resonance can seriously degrade the stopband performance of the filter. The situation becomes more severe as the extra coupling line becomes longer. If the electrical length of this extra coupling microstrip line is integer multiples of 180^o, it becomes an extra resonant node in the coupling route. This extra resonant node causes the coupling diagram more complex when synthesizing a proper coupling matrix corresponding to a desired response. One way to solve this problem is to use source or load to nonadjacent resonators cross couplings [82]. Unlike [96], the extra coupling line in [82] is directly connected to source or load so that no self-resonance of this extra coupling line will occur. The filter in [82] can largely simplify the design procedures of a CT filter due to its conventional microstrip parallel-coupled filter structure. The CT filter in [82]

introduces cross couplings of MS,2 and ML,n-1 to generate two trisections that two independently controllable transmission zeros on upper stopband are produced.

However, the realizable response of the filter in [82] is limited to be the CT filter with two upper stopband transmission zeros.

In spite of the cross-coupled schemes, the coupling schemes such as doublet, extended doublet and box-section are introduced [66], [86]. The main characteristic of these coupling schemes is the ability to shift a transmission zero from one side of

passband to the other by just adjusting the resonant frequencies of resonators in the box portion of the coupling scheme, and this is so-called zero shifting characteristic.

Recently, a fourth-order box-section filter proposed by Amari et al. [88] and filters with box-like coupling schemes proposed by Liao et al. [89] have been successfully implemented using microstrip lines. The drawback of the former is that it needs to use the Dishal’s method as described above. The latter used an E-shaped two-mode resonator, namely even- and odd-mode, to support corresponding coupling schemes such as doublet, extended doublet, and box-section. Unfortunately, when designing such a two-mode filter, the physical dimensions of the resonators are very sensitive especially the dimensions of the two-mode resonator. While tuning the filter, carefully adjusting physical parameters of the two-mode resonator is required because some dimensions of the two-mode resonator influence not only the position of the transmission zero but also the in-band return loss. It means that designer should spend much time to tune.

In this chapter, we propose new cross-coupled filters based on a conventional parallel-coupled filter, and all of the shortcomings described above can be solved.

Basically, this newly proposed filter structure takes the advantages of the Hong’s filter [96] and Liao’s filter [82]. Fig. 4.1(a) shows the schematic layout of the proposed filter with a fourth order filter as an example where the crossing coupling between source or load and nonadjacent resonators are presented by dotted lines. Its equivalent coupling diagram is shown in Fig. 4.1(b). The filter has the advantage of using the simple synthesis procedure presented in [97] to serve as the initial design. In Figs.

4.1(a) and (b), although the figures show multiple cross coupling routes from source to nonadjacent resonators, only one of them is chosen in the design procedures.

Similar situation occurs in the load end. Then, by observing the relative phase shifts of main and cross-coupled paths between source or load and one of the nodes of the

(a)

(b)

Fig. 4.1. The cross-coupled parallel coupled filter. (a) The schematic layout. (b) Coupling and routing scheme corresponding to (a).

interested resonator, filters with generalized Chebyshev responses can be implemented. Applying suitable cross coupling paths and phases, the proposed filter could be CT, CQ, or combination of quadruplet and trisection. It is important to note that in the trisection configuration the transmission zero can be located on either lower or upper stopband by just applying the suitable cross coupling in Fig. 4.1(a).

Therefore, the design procedures of the proposed filter are easy without using of the Dishal’s method or the method presented in [6]. Besides, it is more flexible to locate the transmission zeros.

4.2 Phase Relationships and Generation of Finite Transmission Zeros The purpose in this section is to explore the relative phase shifts of the main

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.

4.3 Cross-Coupling Schemes

Two coupling schemes, namely the CT and the mixed cascade quadruplet and trisection [65], are possible for our fourth-order examples. Fig. 4.3(a) and Fig. 4.4(a) show the CT and the mixed cascade quadruplet and trisection coupling schemes respectively. Although the CT and the mixed cascade quadruplet and trisection

The main

coupling path

The cross coupling path Response Delay line electrical

length

Frequency response predicted

f<f0 f>f0 f<f0 f>f0

YBA (YIJ) -90 -90 Not applicable Not applicable YCA (YHJ) +90 +90 Not applicable Not applicable

YDA (YGJ) -90 +90 -90 -90 Trisection with a TZ on upper stopband

0^o or 360^o YDA (YGJ) -90 +90 +90 +90 Trisection with a TZ

on lower stopband

180^o

YEA (YFJ) +90 -90 -90 -90 Trisection with a TZ on lower stopband

0^o or 360^o YEA (YFJ) +90 -90 +90 +90 Trisection with a TZ

on upper stopband

180^o

YFA (YEJ) -90 -90 +90 +90 Quadruplet 180^o

YGA (YDJ) +90 +90 -90 -90 Quadruplet 360^o

YHA (YCJ) -90 +90 *

YIA (YBJ) +90 -90 *

YJA (YAJ) -90 -90 *

* : it is a cross coupling not belongs to trisection or quadruplet and beyond the scope of this research. Table 4.1. The relative phase shifts of the main coupling path, the proper phases of the cross coupling paths to generate transmission zeros, corresponding responses, and delay line electrical length.

(a)

(b)

Fig. 4.3. The cascaded trisection filter. (a) The coupling scheme. (b) The corresponding equivalent lumped-element circuit of a fourth-order parallel coupled filter with cross couplings. Either the inverter JAD or JAE corresponds to MS2 and either the inverter JJF or JJG corresponds to M3L.

(a)

(b)

Fig. 4.4. The mixed cascaded quadruplet and trisection filter. (a) The coupling scheme. (b) The corresponding equivalent lumped-element circuit of a fourth-order parallel coupled filter with cross coupling. Either the inverter JAF or JAG corresponds to MS3 and either the inverter JJF or JJG corresponds to M3L.

coupling schemes have already proposed in literatures, the microstrip implementation using parallel-coupled structure is first proposed in this research. Here, the resonators are represented by dark dots, the source and load are empty dots, the solid lines between resonators indicate the main coupling, and the broken lines indicate the cross coupling.

The lumped-element equivalent circuit of the fourth-order parallel-coupled filter with CT cross coupling scheme is shown in Fig. 4.3(b). Either the inverter JAE or inverter JAD in Fig. 4.3(b) corresponds to the cross coupling MS2 in Fig. 4.3(a), and similar situation applies to the inverter JJF and the inverter JJG. Choosing either JAE or JAD in the source end and either JJF or JJG in the load end, different signs of MS2 and M3L can be implemented. Thus, two trisections are formed and each trisection can create a transmission zero on either lower or upper stopband. To demonstrate the mentioned properties, three CT filters are discussed as examples.

The lumped-element equivalent circuit of the fourth-order parallel-coupled filter with CT cross coupling scheme is shown in Fig. 4.3(b). Either the inverter JAE or inverter JAD in Fig. 4.3(b) corresponds to the cross coupling MS2 in Fig. 4.3(a), and similar situation applies to the inverter JJF and the inverter JJG. Choosing either JAE or JAD in the source end and either JJF or JJG in the load end, different signs of MS2 and M3L can be implemented. Thus, two trisections are formed and each trisection can create a transmission zero on either lower or upper stopband. To demonstrate the mentioned properties, three CT filters are discussed as examples.