Filter Design Examples

In this section, we will develop two novel quadruplet filters with source-load coupling and utilize the CAD tool introduced in previous section to do diagnosis of proposed filters. The design procedures are summarized as following. Follows the synthesis method described in [9], one would get the ideal coupling matrix with the topology shown in Figure 3-2(a). The corresponding spacing between every resonator is determined through the characterization of the couplings as described in chapter 8 of [12]. After EM simulation, the values of unwanted cross couplings are extracted.

Fixing the values of unwanted couplings, the optimization technique is then applied to determining the required frequency shifts of resonators and the change of other coupling elements to compensate the distortion of |S11|[17]. Two examples are given to show the design procedures. The first filter, shown in Figure 3-4, is designed to

have two pairs of real frequency transmission zeros at normalized frequency

6 , 2±

±

=

Ω for skirt selectivity. The second filter, shown in Figure 3-7, is intended to have one pair of real frequency transmission zero at normalized frequency Ω=±4.5 for selectivity and another pair at Ω=±j1.55 for in-band flap group delay. The center frequency, the fractional bandwidth, and the maximum in-band return loss of both filters are 2.4GHz, 3.75% and 20dB respectively. The filters are built on a 20-mil-thick Rogers RO4003 substrate with ε_r =3.38 , tanδ =0.0021. The commercial EM simulation software Sonnet 9.0 [32] is used to perform the simulation.

(a)

(b)

Figure 3-4. (a) quadruplet filter with the capacitive S/L coupling controlled by the controlling line (b) photograph of the fabricated filter with dimension (in mils) S1=4, S2=8, S3=41, E1=90, E2=20, W1=64, W2=30, h1=310, h2=250, g1=42, g2=26, Line=160

A. Quadruplet filter with two pair of real frequency transmission zeros

In order to see the effect of the controlling line, we exclude the controlling line at first and adjust the quadruplet filter with the previously mentioned procedures. After extracting the unwanted diagonal cross couplings of the quadruplet filter and compensate them, we would get the EM simulated response shown as circles in Figure 3-5(a). Using the CAD tool developed in section 3.3 together with the cost function defined in Eq. (3-7), the extracted coupling matrix M1 (with the value of cost function U =10⁻⁷) is obtained as following.

The corresponding response of M1 is also shown in Figure 3-5(a) as solid line for comparison.

After adding the controlling line of source-load coupling, the EM simulated response is shown in Figure 3-5(b) as circles. The corresponding extracted coupling matrix M2 (with the value of cost function U =10⁻⁷) is

The corresponding response of coupling matrix M2 is also shown in Fig. 5(b) as solid line.

(a)

(b)

Figure 3-5. (a) response of quadruplet filter (b) response of quadruplet filter with controlling line of source-load coupling. Circle: EM simulated results; solid line:

circuit model.

Comparing M1 and M2, it can be easily observed that the introduction of controlling line is only a small perturbation to the original quadruplet. In other words, the controlling line has negligible contribution to the passband response. Besides, the existence of the tiny unwanted diagonal cross couplings MS4 and ML1 in matrix M2

explain why the response is asymmetric because the response becomes symmetric as the MS4 and ML1 are excluded from M2. Taking matrix M2 into equation (3) and (4), and setting unloaded quality factor Q_u =150, the results are shown in Figure 3-6 as dashed lines. The measured responses are also shown in Figure 3-6 as solid lines.

Comparing the circuit model responses with measured responses an excellent fit can be observed except some frequency drift toward lower frequency.

Figure 3-6. Experimental and circuit model results. Solid line: experimental results, dashed line: circuit model including loss term.

(a)

(b)

Figure 3-7. (a) quadruplet filter with the inductive S/L coupling controlled by the controlling line (b) photograph of the fabricated filter with dimension (in mils) d=20, Line=800, s=4, L3=575, L1=940, L2=770, L3=575, h1=340, h2=304.

B. Quadruplet filter for flap group delay and skirt selectivity

As mentioned in section 3.2, the unwanted cross couplings M13 and M24 would destroy the in-band group delay flatness. To reduce the strength of unwanted couplings, we use the L-shaped resonator and arrange the resonators in square to maximize the distance between diagonal resonators as shown in Figure 3-7. The coupled lines with length L1, L2, and L3 control the strength of coupling between L-shape resonators respectively. The inductive source-load coupling is effectively controlled by changing the length of controlling line with both ends connected to ground. Resonant frequencies of resonators can be tuned by adjusting the length h1 and h2. Following similar procedures in the previous design, we can get the extracted coupling matrix M3 as

The corresponding response of M3 fit well with the EM simulated results as shown in Figure 3-8. Taking M3 into equation (3-3) and (3-4) and setting unloaded quality factor 150Q_u = , we have the filter responses shown in Figure 3-9 as doted lines. The experimental results are also shown in Figure 3-9 as solid lines that they are similar to the circuit model results except similar frequency drift as the former example. The frequency drift might come from the discrepancy of the substrate dielectric const. In other words, the dielectric const ε might be greater than data sheets’ value 3.38. _r

(a)

(b)

Figure 3-8. Response of quadruplet filter with controlling line of source-load coupling.

Circle: EM simulated results; solid line: circuit model.

From above two examples, we can conclude that the controlling line of source-load coupling can effectively adjusting the position of finite transmission zeros with negligible perturbation to the passband. It is suggested that one can design the symmetric folded coupled-resonator filter at first and then adds the controlling line to control the source-load coupling without fine-tuning other portion of the filter. The design method may apply to higher order symmetric folded coupled-resonator filter.

(a)

(b)

Figure 3-9. Experimental and circuit model results (a) return loss and insertion loss (b) group delay. Solid line: experimental results, dashed Line: circuit model including loss term.

Chapter 4 Modified Parallel-Coupled Filter with Two Independently Controllable Upper Stopband Transmission Zeros

In this chapter, a microstrip cross-coupled filter with two independently controllable transmission zeros on upper stopband is presented. The initial filter structure is a conventional Chebyshev-response parallel-coupled filter that can be easily realized by the analytical method. The newly proposed coupling/shielding lines can effectively control the cross and main couplings without changing the original filter layout. This approach allows designer to eliminate tedious segmentation method, which is usually used to establish the relation between coupling coefficient and physical distance between resonators. A 3-order filter is designed and fabricated for demonstration.

4.1 Motivation

The cross-coupled microstrip filters have been extensively studied in recent years.

Research efforts are focused mainly on two aspects. One is finding new shape of resonator. Another is developing novel synthesis methods, which enable designer to arrange resonators in different ways to achieve advanced response such as generalized Chebyshev response. Resonators with different shape, such as loop [11], hairpin [33], and patch [34], have been arranged in specific topologies for improving the selectivity or in-band group delay of filters. Some widely applied topologies are cascade quadruplet (CQ) and cascade trisection (CT). Besides CQ and CT, novel synthesis methods have leaded to novel topologies containing couplings of source/load to multi-resonator [23, 35]. In a word, novel physical structures accompanied with advanced synthesis methods have enriched the possibilities of a microstrip filter.

However, the designs of cross-coupled filters are not as straightforward as conventional ones such as parallel-coupled filter, end-coupled filter, etc. In the design

of cross-coupled filter, there are no explicit expressions to relate synthesized electrical parameters to physical dimensions of a filter. Therefore, when designing a cross-coupled filter, the first step is to synthesize a coupling matrix. Then, use segmentation method to relate coupling strength to physical distance between resonators [12]. The drawback of the design procedures is that once the size of resonator changes, designers must redo the segmentation method to find physical dimensions of filters. Moreover, since segmentation method can provide only approximated dimensions of filter, fine tunings are always needed.

To skip the tedious designing routine of segmentation method, we propose an easy designing procedure to realize a filter with two upper stopband transmission zeros. The basic structure of proposed filter utilizes the conventional microstrip parallel-coupled filter [36], as shown in Figure 4-1(a), to serve as the initial design.

Then, vertically flip feeding lines of the source and the load as shown in Figure 4-1(b).

As described by Chang and Itoh in [37] that the physical dimensions keep the same during flipping. Next, adding the proposed coupling/shielding lines at the ends of input and output feed lines as depicted in Figure 4-1(b). Figure 4-1(c) shows the coupling diagram of Figure 4-1(b) and the coupling elements can be optimized and fine-tuned by the method given in Chapter 2. The proposed layout of the filter is somewhat similar to those of [37]. Nevertheless, the strengths of couplings MS2 and ML,n-1 in the filters described in [37] are extremely weak and not taken them into account during filter design procedures. In this paper, we introduce these coupling/shielding lines to control the strength of MS2 and ML,n-1, which makes it possible to independently control two transmission zeros in upper stopband.

(a)

(b)

(c)

Figure 4-1. (a) The conventional parallel-coupled filter. (b) The modified filter. (c) The coupling route of the modified filter.

4.2 Circuit Description and Design Feasibility

The design procedures are started with the conventional microstrip parallel-coupled filter. Following easy design procedures, dimensions of a Chebyshev-response parallel-coupled filter as shown in Figure 4-1(a) can be obtained.

Then, vertically flip feeding lines of the source and the load with respect to the resonator “1” and resonator “n” respectively, which is shown in Figure 4-1(b). Note that the gap spacing Si in Figure 4-1(a) is identical to that in Figure 4-1(b). During practical layout, designers may shorten resonators in advance to prevent the feed lines from directly connecting to the resonator “2” or resonator “n-1” if needed.

To introduce two independently controllable transmission zeros on the upper stopband, the design procedures could be started with the Chebyshev-response coupling matrix and perturb it by introducing cross couplings MS, 2 and ML, n-1 to form two trisection blocks as shown in Figure 4-1(c). During this procedure, it is found that in order to keep equal ripple in-band response, the strength of M1, 2 and Mn-1, n must be decreased and the frequencies of resonators need to be adjusted. Therefore, a suitable manner to simultaneously introduce the couplings, M_S2 and ML, n-1, and decrease the strength of M1, 2, and Mn-1, n is needed. The coupling/shielding lines shown in Figure 4-1(b) seem to be a perfect candidate. The coupling/shielding lines can introduce couplings, MS2 and ML, n-1, and decrease the strength of M12 and Mn-1, n by shielding part of the coupling gaps of them. Practically, length, width, and vertical position of the coupling/shielding line can be adjusted. Here, we fix the line width and adjust the line length and vertical position. The vertical position of the coupling/shielding line has little effect on shielding but has strong influence on cross-coupling. In Chebyshev-response parallel-coupled filters, the relations S1<S2 and Sn+1<Sn always hold, which makes it possible to add coupling/shielding line at the end of feed lines.

Another merit of the parallel-coupled filter structure is that when adjusting the length

of resonator to align the resonant frequencies, the coupling between resonators is nearly unchanged. The feasibility of nearly independently tuning the coupling and frequencies makes it easy to implement the asynchronous tuned filter as that in Figure 4-1(c).

Figure 4-2. The layout of the fabricated filter (unit: mil). L1=354, L2=354, L3=354, L4=354, S1=11, S2=35, W1=19, W2=21, K1=19, K2=20, T1=87, T2=39. The line with of coupling/shielding lines is 8mil.

4.3 Design Example and Experiment

To show the feasibility of the proposed structure, an example is given below. The center frequency, in-band return loss, and fractional bandwidth of the filter are chosen to be 5GHz, 20dB, and 5% respectively. The filter is built on a Rogers RO4003 substrate withε_r =3.58, thickness=20mil, andtanδ =0.0021. The initial dimensions of the parallel-coupled filter are obtained by analytical method described in [7]. The coupling/shielding lines with length T1 and T2 are added at the ends of feed lines as shown in Figure 4-2, to introduce two transmission zeros separately. Two prescribed transmission zeros are located at 5.35GHz, and 5.7GHz respectively. The initial value of T1 and T2 can be arbitrarily set, say, T1=50 mil, T2=30 mil. The S-parameters of the filter is then obtained with the help of the commercial EM simulator Sonnet [32].

Next, the method described in chapter 2 can be used to extract the coupling matrix from the simulated S-parameters. The physical dimensions of filter are then adjusted

according to the extracted coupling matrix to match the prescribed response. After totally five of EM-simulation, matrix extraction, and physical parameters adjusting loops, one can get the simulated results as shown in Figure 4-3. The measured results are also shown in Figure 4-3 for comparison. The corresponding physical sizes are shown in Figure 4-2. And the corresponding coupling matrix M is extracted as follows

⎥ ⎥

Figure 4-3. Simulated and measured responses. Solid line: measured results. Dashed lines: EM simulated results.

It should be emphasized that filter shown in Figure 4-2 has exactly the same layout as the initial design except two coupling/shielding lines. So, designers can

easily realize this filter even by trial and error method without using of matrix extracting procedure.

Figure 4-4. EM simulated results of three different cases. The dimensions of the simulated filter are the same as these shown in Figure 4-2 except T2 is set to different values.

It is mentioned in section 4-2 that the introduction of the coupling/shielding lines in this way can effectively adjust the transmission zeros with slight perturbation of the passband return loss. To demonstrate the merits of easy tuning of the proposed structure, three EM simulations are taken in which the length of the coupling/shielding line, T2, are set to 14mil, 24mil, and 39mil respectively while the other dimensions are the same as these given in Figure 4-2. From the EM simulated results shown in Figure 4-4, it is obvious that the transmission zero can be tuned over a wide range with negligible change in the passband return loss.

Chapter 5 Microstrip Realization of Generalized Chebyshev Filters with Box-Like Coupling Schemes

In this chapter, generalized Chebyshev microstrip filters with box-like coupling schemes are presented. The box-like coupling schemes taken in this chapter include doublet, extended doublet, and fourth-order box-section. The box-like portion of the coupling schemes is implemented by an E-shaped resonator. Synthesis and realization procedures are described in detail. The example filters show an excellent match to the theoretical responses.

5.1 Introduction

The microstrip filters with generalized Chebyshev response attract considerable attention due to its lightweight, easy fabrication and ability to generate finite transmission zeros for sharp skirts. In the literature, most of them are based on cross-coupled schemes such as cascade trisection and cascade quadruplet. Some representative examples of cross-coupled microstrip filters are available in the book [12].

Recently, with the progress of the synthesis technique, new coupling schemes such as “doublet”, “extended doublet”, and “box-section” are introduced [38]-[40]. As shown in Figure 5-1, these coupling schemes have a box-like center portion, so we call them box-like coupling schemes. These coupling schemes impact the filter design since they do not only provide new design possibilities but exhibit some unique and attractive properties as well. They differ from the conventional cascade trisection and cascade quadruplet mainly on two aspects. First, there are two main paths for the signal from source to load while there is only one main path in the case of cascade trisection and cascade quadruplet. Second, the configuration of doublet and box-section exhibit the zero-shifting property which make it possible to shift transmission zero from one side of the passband to the other side simply by changing

the resonant frequencies of the resonator while keeping other coupling coefficients unchanged. The zero-shifting property implies that the similar physical layout can implement a filter with transmission zero at the lower stopband or at the upper stopband, which is not feasible on the conventional trisection configuration. Besides, the third-order extended-doublet configuration, as shown in Figure 5-1(b), exhibits one pair of finite transmission zeros as that of a cross-coupled quadruplet filter. Pairs of finite transmission zeros can be used to improve the selectivity of the filter or flatten the in-band group delay. However, to the author’s knowledge, only a few studies in the literature are focused on realization of the coupling schemes shown in Figure 5-1 with microstrip line [41], [42].

(a) (b)

(c)

Figure 5-1. Basic box-like coupling schemes for generalized Chebyshev-response filters discussed in this paper. (a) doublet. (b) extended doublet (c) box-section. ( The gray area is realized by the proposed E-shaped resonator)

An important property of the schemes in Figure 5-1 is that one of the coupling coefficients on the two main paths must be negative while others are positive. The simplest way to obtain the required negative sign is to use higher-order resonance [39], [42]. Unfortunately, higher-order resonance leads to a spurious resonance in the lower stopband. Instead of using higher-order resonance, loop resonators are arranged carefully to satisfy the required sign of coupling coefficients for the box-section configuration [41]. However, a similar method can not apply to doublet or extended-doublet. To overcome these difficulties, an E-shaped resonator as shown in Figure 5-2(a) is proposed to implement the required coupling signs.

(a) (b)

Figure 5-2. A doublet filter (a) the proposed layout (gray area indicate the E-shaped resonator) (b) the corresponding coupling scheme.

The E-shaped resonator can achieve the required magnitude and sign of the coupling schemes shown in Figure 5-1. As shown in Figure 5-2(a), the E-shaped resonator comprises a hairpin resonator and an open stub on its center plane. This symmetric

structure can support two modes: even mode and odd mode. Thus, the source and the load are coupled to both modes of the E-shaped resonator. That is even though only one physical path exists between source and load, there are two electrical paths between them. Consequently, the layout in Figure 5-2(a) can be modeled by the coupling scheme, a doublet, in Figure 5-2(b). The doublet filter illustrates how an E-shaped resonator directly couples to external feeding network.

Based on the proposed E-shaped resonator, filters with extended-doublet and box-section configuration can be realized as well. The E-shaped resonator can use either its even mode or odd mode to couple an extra resonator. Thus, the extended-doublet configuration in Figure 5-1(b) is achievable. Besides, the E-shaped resonator can couple to external resonators with two of its modes simultaneously and forms the box-section configuration in Figure 5-1(c). The feasibility of realization of the basic coupling schemes in Figure 5-1 with proposed E-shaped resonator makes it possible to realize a class of coupled microstrip filters in a unified approach.

5.2 Circuit Modeling A. Filters in the doublet configuration

The E-shaped resonator filter in Figure 5-2(a) was originally reported in [43]. In [43], the E-shaped resonator was not modeled as a two-mode resonator. Instead, the circuit was modeled as two quarter-wavelength resonators with a tapped open stub in the center plane. The open stub is considered as a K-inverter between two quarter-wavelength resonators to control the coupling strength and as a quarter-wave open stub to generate a transmission zero at the desired frequency. However, the filter cannot be designed with a prescribed quasi-elliptic response since there is no suitable

The E-shaped resonator filter in Figure 5-2(a) was originally reported in [43]. In [43], the E-shaped resonator was not modeled as a two-mode resonator. Instead, the circuit was modeled as two quarter-wavelength resonators with a tapped open stub in the center plane. The open stub is considered as a K-inverter between two quarter-wavelength resonators to control the coupling strength and as a quarter-wave open stub to generate a transmission zero at the desired frequency. However, the filter cannot be designed with a prescribed quasi-elliptic response since there is no suitable