Chapter 2 Fully-Analytical Multi-band Coupling Matrix Synthesis
2.7 Conclusion
In this chapter, the novel analytical method to synthesize a dual-band or multi-band filtering function has been successfully developed. Based on the synthesized composite filtering function, the transversal coupling matrix can be obtained. Moreover, the arbitrarily located transmission zeros, arbitrary return losses within each passband, and various bandwidth of each passband are available in this method. Compared with the most popular method, analytical iterative method, the proposed method in this chapter provides the equal-ripple potential to give the practical insight in the filter design.
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Chapter 3
Dual-band Filter Design Using Parallel-Coupled Line
3.1 Introduction
The coupling scheme corresponds to a physical placement of resonators in the filter design. By carefully adjusting the coupling strengths between resonators, the performance of the filter can be determined. For the single-band filter design, the coupling schemes are widely studied for the specific electrical responses. For examples, the tri-section topology is proposed for generating one transmission zero above or below the passband, and the quadruplet topology is used to generate two transmission zeros above and below the passband.
The coupling matrix provides an advantage of hardware implementation. Based on the advantage, the coupling matrix of the transversal topology needs to be transformed into some specific coupling schemes for practical realization of a filter. For the dual-band filter design, some coupling schemes, which are cul-de-sac [98], inline topology [97], and extended box topology [100], are generated via a series of similarity transformations, matrix rotation or optimization [98], [105]. Some of those topologies, however, are difficult to be realized by microstrip lines.
Considering the dual-band microstrip filter, dual-mode and frequency-separated coupling scheme is proposed to realized the dual-band filter characteristics. The phenomenon in placing transmission zeros related to corresponding coupling topology, however, is not clear yet such that the mechanism in transmission zeros generation of the dual-band filter is still unobservable under proper coupling scheme.
49
In this chapter, single-path and dual-path topologies are provided for the microstrip dual-band filter design. The single-path coupling scheme has rare physical insight and needs run-and-try process to check if the coupling scheme is valid for the requested specifications. For the dual-path coupling scheme, it is able to illustrate the dual-path characteristics via frequency-separated path. Furthermore, the mechanism of transmission zeros and the separation between two adjacent passbands, furthermore, will be studied via the coupling matrix.
3.2 Single-path Coupling Scheme
The single-path coupling scheme for a dual-band filter has a direct coupling path with some cross-coupling paths. To capture the phenomenon of the prescribed transmission zeros, the well-known cross-coupling topologies, such as tri-section and quadruplet coupling schemes, are used to the run-and-try process. To apply the trisection and quadruplet coupling schemes to the dual-band filter design in the single-path coupling scheme, finite transmission zeros will be placed to separate two passbands. Following is an example. A dual-band filter is designed to have two second-order filtering functions; one has a normalized transmission zero at 0 rad/s and the normalized central frequency at 0.75 rad/s with the multi-band lowpass domain bandwidth δ1 of 0.5 rad/s, while the other one has a normalized transmission zero at 0 rad/s and the normalized central frequency at -0.75 rad/s with the multi-band lowpass domain bandwidth δ2 of 0.5 rad/s. The return loss in each passband is 20 dB. The synthesized filtering functions and corresponding S-parameters are shown in Figure 3-1.
The transversal coupling matrix is shown in Table 3.1. Due to two transmission zeros from each passband are used to contribute the separation of two passbands, the quadruplet
50
topology is used. In order to convert the transversal coupling matrix for the single-path coupling scheme with quadruplet coupling scheme, the optimization proposed in [94] is used. The coupling scheme is shown in Figure 3-2 and the corresponding coupling matrix is listed in Table 3.2.
Figure 3-1 (a) Filtering functions for two single-band filters of degree 2 (CN1 has the transmission zero at 0 rad/s and the central frequency at -0.75 ras/s with the multi-band lowpass domain bandwidth δ1 of 0.5 rad/s, and CN2 has the transmission zero at 0 rad/s and the central frequency at 0.75 rad/s with the multi-band lowpass domain bandwidth δ2 of 0.5 rad/s), and the composite dual-band filter (CN1 // CN2). The in-band return loss is 20 dB in each case. (b) The corresponding S11 and S21 for the dual-band filter.
Table 3.1 The Transversal Coupling Matrix for the Dual-band Filter in Figure 3-1.
LOWPASS PROTOTYPE FREQUENCY (rad/sec)
S 0.0 0.5315 -0.3046 0.3046 -0.5315 0.0 1 0.5315 1.2338 0.0 0.0 0.0 0.5315 2 -0.3046 0.0 0.4053 0.0 0.0 0.3046 3 0.3046 0.0 0.0 -0.4053 0.0 0.3046 4 -0.5315 0.0 0.0 0.0 -1.2338 0.5315 L 0.0 0.5315 0.3046 0.3046 0.5315 0.0
51
Table 3.2 The Coupling Matrix for the Dual-band Filter in Figure 3-1 with Coupling Scheme in Figure 3-2.
Figure 3-2 The single-path coupling scheme for the dual-band filter in Figure 3-1.
The single-path coupling scheme, however, is not always can be analyzed systematically using the information of transmission zeros. In most cases, the cross-coupling path needs to be run-and-try, and it even has no proper topology while the positions of transmission zeros are asymmetric to the passbands.
3.3 Dual-path Coupling Scheme
For the dual-band filter design, the single-path coupling scheme do not have an obvious relationship with the dual-band filter characteristics. To relate each passband with the coupling topology, the dual-path coupling scheme is considered. For example, there are two filtering functions; one is the 3rd order filtering function, which has the normalized transmission zero at -1.8 rad/s and the normalized central frequency at -0.8 rad/s with the
S 1 2 3 4 L
S 0.0 0.5624 0.0 0.6590 0.0 0.0 1 0.5624 0.0 0.4590 0.0 0.0 0.0 2 0.0 0.4590 0.0 0.5379 0.0 0.0 3 0.6590 0.0 0.5379 0.0 1.0893 0.0 4 0.0 0.0 0.0 1.0893 0.0 0.8663 L 0.0 0.0 0.0 0.0 0.8663 0.0
S 2
3 L
1
Source/Load Resonator
4
52
multi-band lowpass domain bandwidth δ1 of 0.4 rad/s, while the other is the 3rd order filtering function, which has the normalized transmission zeros at 1.8 rad/s and the normalized central frequency at 0.8 rad/s with the multi-band lowpass domain bandwidth δ2 of 0.4 rad/s. The synthesized filtering functions and the corresponding S-parameters are shown in Figure 3-3. The transversal coupling matrix in this example is listed in Table 3.3.
The transmission zero for the separation of two passbands is created as demonstrated in the discussion in Section 2.4.1. In this case there are three finite transmission zeros within the entire normalized low-pass domain there are -1.8, 0, and 1.8 rad/s. To illustrate the dual-band characteristic and let each path govern one passband, the trisection portion of each path is used to provide one transmission zero on the stopband. Figure 3-4 shows the coupling scheme, and the corresponding coupling matrix is rotated by following steps [4].
The values of diagonal elements of the transversal matrix are categorized into two groups, which are positive values and negative values, and then the original matrix can be separated into two parts with values shown in Table 3.4. Based on these two sub-matrices, the rotation sequence in Table 3.5 are applied and then the matrix for the coupling scheme shown in Figure 3-4 are extracted with values listed in Table 3.6.
Figure 3-3 (a) Filtering functions for two single-band filters of degree 3 (CN1 has the transmission zero at -1.8 rad/s and the central frequency at -0.8 rad/s with the multi-band lowpass domain bandwidth δ1 of 0.4 rad/s, and CN2 has the transmission zero at 1.8 rad/s
LOWPASS PROTOTYPE FREQUENCY (rad/sec)
53
and the central frequency at 0.8 rad/s with the multi-band lowpass domain bandwidth δ2 of 0.4 rad/s), and the composite filter (CN1 // CN2). The in-band return loss is 20 dB in each case. (b) The corresponding S11 and S21 for the dual-band filter.
Table 3.3 The Transversal Coupling Matrix for the Dual-band Filter in Figure 3-3
Figure 3-4 The dual-path coupling scheme for the dual-band filter in Figure 3-3.
Table 3.4 (a) The Transversal Coupling Matrix for the Upper Path (M1).
(b) The Transversal Coupling Matrix for the Lower Path (M2).
S 1 2 3 4 5 6 L
S 0.0 0.2295 -0.3419 0.2534 0.2534 -0.3419 0.2295 0.0 1 0.2295 1.0975 0.0 0.0 0.0 0.0 0.0 0.2295 L 0.0 0.2295 0.3419 0.2534 0.2534 0.3419 0.2295 0.0
S 2 3
S 0.0 0.2295 -0.3419 0.2534 0.0 1 0.2295 1.0975 0.0 0.0 0.2295 2 -0.3419 0.0 0.8653 0.0 0.3419 3 0.2534 0.0 0.0 0.5272 0.2534 L 0.0 0.2295 0.3419 0.2534 0.0
S 4 5 6 L
S 0.0 0.2534 -0.3419 0.2295 0.0 4 0.2534 -0.5272 0.0 0.0 0.2534 5 -0.3419 0.0 -0.8653 0.0 0.3419 6 0.2295 0.0 0.0 -1.0975 0.2295 L 0.0 0.2534 0.3419 0.2295 0.0
(a) (b)
54
Table 3.5 Rotation Sequence for Reduction of the Transversal Matrix to the Requested Matrix with Topology in Figure 3-4
θr = -tan-1(cMkl/Mmn)
Table 3.6 The Coupling Matrix for the Dual-band Filter in Figure 3-3 with the Dual-path Coupling Scheme shown in Figure 3-4.
It can be noted that the values of diagonal elements in the extracted matrix are also categorized into two groups, which are positive values and negative values, and corresponds to the resonant frequency of each resonator. Hence, the upper path governs the lower passband, and the trisection portion of the upper path provides a transmission zero on the lower stopband (i.e., -1.8 rad/s). Similarly, the lower path governs the upper
S 1 2 3 4 5 6 L
S 0.0 0.4739 -0.0958 0.0 0.4835 0.0 0..0 0.0 1 0.4739 0.9033 0.1880 0.0 0.0 0.0 0.0 0.0 2 -0.0958 0.1880 0.8653 0.2047 0.0 0.0 0.0 0.0 3 0.0 0.0 0.2047 0.8247 0.0 0.0 0.0 0.4835 4 0.4835 0.0 0.0 0.0 -0.8247 0.2047 0.0 0.0 5 0.0 0.0 0.0 0.0 0.2047 -0.7619 0.1880 0.0958 6 0.0 0.0 0.0 0.0 0.0 0.1880 -0.9033 0.4739 L 0.0 0.0 0.0 0.4835 0.0 0.0958 0.4739 0.0
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passband and the trisection portion of the lower path generates a transmission zero on the upper stopband (i.e., 1.8 rad/s). By using such a dual-path coupling scheme, the transmission zeros on the upper and lower stopband can be generated by the trisection portion, while the additional transmission zero used to separate two passbands is generated by out-of-phase property of CN1 and CN2.
3.4 Transmission Zeros Determination in Single-path and Dual-path Coupling Schemes
The coupling matrix is related to the responses of reflection and transfer function S11
and S21 via the following equations [46]:
[ ] [ ] [ ]
1 1
11 1,1 21 2,1
0
1 2 , 2 ,
,
S j S j N
j
− −
⎡ ⎤ ⎡ ⎤ +
= + ⎣ ⎦ = − ⎣ ⎦
= Ω + −
A A
A U M R (3-1)
where N is the order of filter, Ω is the angular frequency in the low-pass domain, [U0] is similar to the (N+2)-by-(N+2) identity matrix except [U0]1,1 = [U0]N+2, N+2 = 0, [M] is the (N+2)-by-(N+2) coupling matrix, and [R] is the diagonal matrix with [R] = diag{1,0,…,0, 1}.
To determine the locations of transmission zeros from the coupling matrix in arbitrary coupling topology, it is equivalent to find the roots of S21. In order to find the roots of [A-1]N+2,1, the technique in [136] is used,
56
( )
1, 21 2,1
cofactor( ) Det .
N N
− +
⎡ ⎤ + =
⎣ ⎦ A
A A (3-2)
Hence the transmission zeros are consequently the zeros of the cofactor of [A]1,N+2. Now we present three examples of dual-band filters with single-path and dual-path coupling schemes, and show the mechanism in introducing transmission zeros under signle-path and dual-path topology.
3.4.1 Single-path Dual-band Filter Characteristic
In the first example, the dual-band filter consists of two 4th order filters with central frequency at -0.75 and 0.75 rad/s and transmission zeros at ±0.1118, ±1.3126, and
±j12.2234 rad/s, which j is the imaginary number. The multi-band lowpass domain bandwidth δ for both passband is 0.5 rad/s. The corresponding coupling matrix is listed in Table 3.7 and the performance is shown in Figure 3-5.
Table 3.7 The Coupling Matrix of the Single-path Dual-band Filter
S 1 2 3 4 5 6 7 8 L
S 0.0 0.6849 0.0 -0.2442 0.0 0.0 0.0 0.0 0.0 0.0
1 0.6849 0.0 0.9474 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2 0.0 0.9474 0.0 0.3120 0.0 0.0 0.0 0.0 0.0 0.0
3 -0.2442 0.0 0.3120 0.0 0.6952 0.0 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 0.6952 0.0 0.3670 0.0 0.0 0.0 0.0
5 0.0 0.0 0.0 0.0 0.3670 0.0 0.8023 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0 0.0 0.8023 0.0 0.3811 0.0 0.4198
7 0.0 0.0 0.0 0.0 0.0 0.0 0.3811 0.0 0.5614 0.0
8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5614 0.0 0.5937
L 0.0 0.0 0.0 0.0 0.0 0.0 0.4198 0.0 0.5937 0.0
57
Figure 3-5 The performance of the single-path dual-band filter and the corresponding dual-path coupling scheme based on the coupling matrix inTable 3.7.
To understand the mechanism of the transmission zeros. Here the cross coupling paths are removed to see and estimate the transmission zeros. In Figure 3-6 (a), the cross coupling path S-3 is removed, then the outer transmission zeros are disappeared; while removing the path 6-L, the inner transmission zeros will be disappeared, as shown in Figure 3-6 (b). Applying (3-2) to evaluate the transmission zeros, the formulation is as follows:
Frequency (rad/sec)
-2 -1 0 1 2
|S11|, |S21| (dB)
-80 -60 -40 -20 0
S11 S S21
2
3 4 5 6
7 L
1 8
Source/Load Resonator
58
From the above solutions, if the matrix is symmetric, then the transmission zeros are symmetric to 0 rad/s. While removing path S-3 (i.e., MS,3 = 0), the transmission zeros of
±1.3126 rad/s are disappeared, and transmission zeros ±0.1118 rad/s will be disappeared when the path 6-L is removed.
Figure 3-6 The performances for the single-path dual-band filter with removing of cross couplings (a) removing path S-3 and (b) removing path 6-L.
3.4.2 Symmetric Dual-path Dual-band Filter Characteristic
Frequency (rad/sec)
59
In the second example, the dual-band filter consists of two 4th order filters with central frequencies at -0.75 and 0.75 rad/s in two passbands, and transmission zeros at ±0.1078,
±1.4657, and ±j0.9613 rad/s. The corresponding coupling matrix is listed in Table 3.8.
It is worthy to be noted that the diagonal terms of the coupling matrix are classified into two groups to present the central frequencies of two passbands, i.e., resonators numbered from 1 to 4 (passband with central frequency 0.75 rad/s), and resonators numbered from 5 to 8 (passband with central frequency -0.75 rad/s). The corresponding performance is shown in Figure 3-7, and the dual-path coupling scheme is shown in the subplot. For the dual-path topology, each path has a cross-coupled quadruplet, so that there are two transmission zeros symmetric to the central frequency of each passband. Once the cross couplings of these quadruplets are removed, the transmission zeros vanish, and it can be observed by the dashed line in Figure 3-7.
Table 3.8 The Coupling Matrix of the Symmetric Dual-path Dual-band Filter
Due to the frequency-separated property, the mechanism of generation of corresponding transmission zeros can be observed in each passband. In Figure 3-8 (a), the quadruplet cross-coupling within the upper path introduces two transmission zeros
S 1 2 3 4 5 6 7 8 L
S 0.0 0.5110 0.0 -0.0433 0.0 0.5128 0.0 0..0 0.0 0.0
1 0.5110 -0.7914 0.2328 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2 0.0 0.2328 -0.7405 0.1622 0.0 0.0 0.0 0.0 0.0 0.0
3 -0.0433 0.0 0.1622 -0.7464 0.2191 0.0 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 0.2191 -0.7911 0.0 0.0 0.0 0.0 0.5128
5 0.5128 0.0 0.0 0.0 0.0 0.7911 0.2191 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0 0.0 0.2191 0.7464 0.1622 0.0 -0.0433
7 0.0 0.0 0.0 0.0 0.0 0.0 0.1622 0.7405 0.2328 0.0
8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2328 0.7914 0.5110
L 0.0 0.0 0.0 0.0 0.5128 0.0 -0.0433 0.0 0.5110 0.0
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symmetric to the central frequency at 0.75 rad/s, which is the resonant frequency of resonators numbered from 1 to 4. Similarly the cross-coupled quadruplet within the lower path provided two zeros symmetrical to -0.75 rad/s, as shown in Figure 3-8 (b). Once the quadruplet cross coupling is removed from each path, the transmission zeros vanish.
Furthermore, to identify the locations of transmission zeros in each path, (3-2) is applied to the following matrices [A]up and [A]low:
Consequently, the transmission zeros are 1.4897 and 0.0536 rad/s of the upper path, and they are -1.4897 and -0.0536 rad/s of the lower path. Moreover, it is noted that the transmission zeros can be obtained similarly in the dual-path topology without cross-coupled quadruplets, and they are ±j1.8360 and ±j0.2883 rad/s. It means even no cross-coupled quadruplet, dual-path topology also provides complex transmission zeros, so that the inherent separation between two adjacent passbands exists, as shown in Figure 3-7.
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Figure 3-7 The performance of the symmetric dual-band filter and the corresponding dual-path coupling scheme based on the coupling matrix in Table 3.8.
Figure 3-8 The performances of the (a) upper path and (b) lower path in the symmetric dual-band filter.
3.4.2 Asymmetric Dual-path Dual-band Characteristic
In the third example, the asymmetric dual-path dual-band filter consists of a 3rd and a 4th order filters with central frequencies at 0.7778 and -0.7778 rad/s in the lowpass domain with the multi-band lowpass domain bandwidth δ of 0.4444 rad/s for both passbands. It
Frequency (rad/sec)
62
contains six transmission zeros, which are -1.4305, -0.2078, 0.0948, 1.4802, and -0.7663 ± j0.5029 rad/s. The corresponding coupling matrix is listed in Table 3.9.
In Figure 3-9, one group of resonators, namely resonators 1 to 3, governs the central frequency at 0.7778 rad/s, while the other group of resonators 4 to 7 governs the central frequency at -0.7778 rad/s. The value of entry M5,L, however is small enough to be ignored without influence on the dual-band characteristic.
Table 3.9 The Coupling Matrix of the Asymmetric Dual-band Filter.
It is worthy to be noted that one quadruplet cross coupling introduces four transmission zeros. To make clear the mechanism in generating of transmission zeros, the analysis is applied as followings. Figure 3-10 (a) shows the performance in the upper path, and it is obvious that the transmission zeros come from the quadruplet cross coupling, To identify the transmission zeros of the lower path, the cross-coupled path is retained. This extra path consists of the quadruplet cross coupling within the upper path and the resonator 3, which is shown in the subplot of Figure 3-10 (b). Moreover, the transmission zeros can be determined via (3-2), and they are 0.1133 and 1.4806 rad/s of the upper path, while -0.2613, -1.4191, and -0.7546 ± j0.4837 rad/s of the lower path. Furthermore, even no cross coupling, the dual-path topology inherently maintains the isolation between two
S 1 2 3 4 5 6 7 L
S 0.0 0.5005 0.0 -0.0597 0.4902 0.0 0.0 0..0 0.0
1 0.5005 -0.8187 0.2353 0.0 0.0 0.0 0.0 0.0 0.0
2 0.0 0.2353 -0.7752 0.2086 0.0 0.0 0.0 0.0 0.0
3 -0.0597 0.0 0.2086 -0.8179 0.0 0.0 0.0 0.0 0.5033
4 0.4902 0.0 0.0 0.0 0.8164 0.2002 0.0 0.0 0.0
5 0.0 0.0 0.0 0.0 0.2002 0.7782 0.1572 0.0 0.0
6 0.0 0.0 0.0 0.0 0.0 0.1572 0.7786 0.1983 0.0
7 0.0 0.0 0.0 0.0 0.0 0.0 0.1983 0.8165 0.4876
L 0.0 0.0 0.0 0.5033 0.0 0.0 0.0 0.4876 0.0
63
passbands, as shown in Figure 3-9.
Figure 3-9 The performance of the asymmetric dual-band filter and the corresponding dual-path coupling scheme based on the coupling matrix in Table 3.9.
Figure 3-10 The performances of the (a) upper path and (b) lower path in the asymmetric dual-band filter.
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3.5 Analytical Approach for Dual-band Filter Design Using Parallel-Coupled Lines
The previous section shows the specific coupling scheme for the microstrip implementation. Here the relations between the coupling coefficients and the parameters of the microstrip lines are discussed. To introduce the slope parameter of the resonator, the value of Js of the parallel-coupled line model can be obtained using the following equations [137]:
where Δ is the fractional bandwidth. Once the values of J inverters are obtained, the even- and odd-mode characteristic impedance for non-quarter-wavelength parallel-coupled line [138] can be calculated by
Based on (3-6) and (3-7), the parameters of the microstrip parallel-coupled lines can be obtained. Here an example for the dual-band filter is provided to show the validation.
The filter design for the dual-band GPS at 1227 MHz and 1575 MHz is used to show the closely adjacent passbands. In the design, each passband has 10% fractional bandwidth.
65
Use (2-13), the central frequency of the dual-band filter is 1390 MHz with fractional bandwidth 35%. To transfer the information into the lowpass domain, the passband #1 has the normalized central frequency at -0.713 rad/s and the passband # 2 has the normalized central frequency at 0.713 rad/s. Both filter orders are 3 and the return loss are 15 dB. The dual-path coupling scheme is used in this example, as shown in Figure 3-11. The corresponding performance is shown in Figure 3-11 and the coupling scheme is listed in Table 3.10.
Figure 3-11 The performance and coupling scheme of the dual-band filter design for GPS system.
Table 3.10 The Coupling Matrix of the Dual-band Filter for GPS System Design.
Frequency (rad/s)
66
Based on the coupling matrix, now the parameters of the parallel-coupled lines can be obtained in (3-6) and (3-7). Let f1 = 1575 MHz and f2 = 1227 MHz. For the microstrip implementation of the upper path of the dual-path coupling scheme, the schematic is shown in Figure 3-12. For the passband at f1, the corresponding path is S-1-2-3-L.
Choosing θS1 = θ12 = θ23 = θ3L = θ1 = θ2 = θ3 = 60o (i.e., the length of each resonator is 180o, so that the slope parameter b = π/2/Z0), the calculated parameters of the parallel-coupled lines are listed in Table 3.11. The similar procedure is applied to the lower path, and the schematic is shown in Figure 3-13 and parameters are listed in Table 3.12. In Table 3.10, the asymmetric property of the passband comes from the different diagonal terms, so that the length of the coupled line should be slightly tuned.
Figure 3-12 The schematic of the parallel-coupled line for the upper path of the dual-path coupling scheme.
Table 3.11 The Calculated Parameters for the Parallel-Coupled Line of the Upper Path.
f1 = 1575 MHz
MS1 M12 M23 M3L
J 0.0075 0.0014 0.0014 0.0075
Zoe (Ω) 82.5198 57.8927 57.8927 82.5198 Zoo (Ω) 37.1236 40.0080 40.0080 37.1236
67
Figure 3-13 The schematic of the parallel-coupled line for the upper path of the dual-path coupling scheme.
Table 3.12 The Calculated Parameters for the Parallel-Coupled Line of the Lower Path.
f2 = 1227 MHz
MS4 M45 M56 M6L
J 0.0075 0.0014 0.0014 0.0075
Zoe (Ω) 82.5108 57.8963 57.8963 82.5108 Zoo (Ω) 37.1245 40.0060 40.0060 37.1245
Figure 3-14 The performances for the parallel-coupled line model and coupling matrix of lower and upper paths.
Table 3.13 The Tuned Lengths of the Parallel-Coupled Line in Figure 3-12 and Figure 3‐13
68
θ12 65o θ45 65o
θ23 65o θ56 65o
θ3L 55o θ6L 57o
θ1 58o θ4 59o
θ2 49o θ5 50o
θ3 58o θ6 59o
After tuning the lengths, the performances for the upper and lower path are shown in Figure 3-14 with the tuned lengths listed in Table 3.13. To combine two filters into a dual-band filter, the double-diplexing configuration is used, as shown in Figure 3-15. The f0 in the figure is the central frequency of the dual-band filter. In [132], the imaginary part of the input admittance for the path S-1-2-3-L (which is passband at f1) is zero at f2, while the imaginary part of the input admittance for the path S-4-5-6-L is zero at f1. Applying this procedure to the dual-band filter for the GPS system and fine tuning the lengths of each filter, the final lengths are listed in Table 3.14. The performances are shown in Figure 3-16.
Figure 3-15 The double-diplexing configuration proposed by [132]
69
Table 3.14 The Final Lengths of the Dual-band Filter for GPS System.
f1 = 1575 MHz f2 = 1227 MHz f0 = 1390 MHz
θS1 58.5o θS4 57.5o θ1 150o
θ12 68o θ45 64o θ2 97o
θ23 68o θ56 64o
θ3L 58.5o θ6L 57.5o
θ1 54o θ4 57.5o
θ2 45o θ5 50o
θ3 54o θ6 57.5o
Figure 3-16 The bandpass performances of the dual-band filter for the GPS system.
3.6 Microstrip Implementation for Single-path and Dual-path Dual-band filters
The synthesis procedure has been illustrated completely. In this section, the microstrip Frequency (GHz)
1.0 1.2 1.4 1.6 1.8
|S11|, |S21| (dB)
-80 -60 -40 -20 0
TL Model
Coupling Matrix S
2 3 5 6 7
L 1
Source/Load Resonator
70
implementations for the filters described in the previous section will be used to demonstrate the validation of the proposed filter synthesis.
3.6.1 The Single-path Dual-band Filter
Table 3.2 lists the coupling matrix with the coupling scheme shown in Figure 3-2.
The central frequencies of the two passbands in the practical design are 2.3 and 2.7 GHz,
The central frequencies of the two passbands in the practical design are 2.3 and 2.7 GHz,