The Asymmetric Dual-path Dual-band Filter

Table 3.9 lists the coupling matrix with the coupling scheme shown in Figure 3-9.

The central frequencies of the two passbands in the practical design are 2.26 and 2.7 GHz, while the fractional bandwidth is 5% in each passband. For the practical implementation, a 0.508-mm-thick Rogers RO4003 substrate, with a relative dielectric constant 3.58 and a loss tangent of 0.0021, is used to implement such a dual-band filter. The design parameters can be obtained using (3-6) and (3-7). The layout of the dual-band microstrip filter is shown in Figure 3-23 with the dimensions are listed in Table 3.17. The circuit photograph is shown in Figure 3-24 and Figure 3-25 shows the measured results, which agree well with the simulated performance.

Frequency (GHz)

75

Figure 3-23 The layout for the asymmetric dual-path dual-band filter with the coupling matrix listed in Table 3.9.

Table 3.17 Dimensions for the Layout Shown in Figure 3-23. (Unit: mm)

W1 W2 W3 W4 W5 W6 W7 W8

1.2500 1.2500 0.9250 0.2000 0.6000 0.7000 0.2000 1.2500

W9 W10 L1 L2 L3 L4 L5 L6

0.6000 1.0250 20.0500 12.1000 19.6500 15.1000 18.6500 19.5000

L7 L8 L9 L10 S1 S2 S3 S4

20.1250 17.4750 19.9500 18.0250 0.2750 0.6500 1.2500 1.0750

S5 S6 S7 S8 S9 S10 Wd D1

0.2000 0.2000 1.2250 0.7500 1.1750 0.2000 0.2000 10.2250

D2 D3 D4 D5 D6 D7 D8 G1

46.6500 16.4500 0.7750 0.6000 12.5750 57.3500 6.5000 0.2250

G2 Wa U1 U2 U3 U4 Wb B1

0.2000 0.2000 23.4500 6.3250 4.9600 23.6000 0.2250 1.6500

B2 B3 B4 Wc C1 C2 Wf F1

1.7000 1.7000 1.6500 1.67500 10.0500 10.0500 1.1000 2.5000

76

Figure 3-24 The photograph of implemented microstrip asymmetric dual-band filter with the coupling matrix listed inTable 3.9.

Figure 3-25 The responses for the asymmetric dual-path dual-band filter in Figure 3-24.

Furthermore, to observe the mechanism of the cross-coupling in each passband, the current density under two passbands is simulated. In Figure 3-26 (a), the current mainly flows through the upper path and the cross-coupled quadruplet at 2.69 GHz. At 2.263 GHz, the current flows through not only the lower path, but also the cross-coupled quadruplet within the upper path and the resonator 3, as shown in Figure 3-26 (b). It agrees with the discussion in Sec. 3.4.2.

Frequency (GHz)

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

|S11|, |S21| (dB)

-80 -70 -60 -50 -40 -30 -20 -10 0

Measurement Simulation Frequency (GHz)^{2.0 2.5 3.0}

Group Delay (ns)

0 3 6 9 12

2.0 2.5 3.0 -10-8-6-4-20

77

Figure 3-26 Simulated current density of the filter in Figure 3-24 (a) at 2.69 GHz, and (b) at 2.263 GHz.

3.7 Conclusion

In this chapter, we introduce the relations between the coupling coefficients and parameters of the parallel-coupled line. For the convenient of the implementation, the single-path and dual-path coupling schemes are proposed. Within these two topologies, the dual-path coupling scheme shows the advantage of the physical insights, and the transmission zeros can be assigned to each passband using the well-known mechanisms in the single-band filter design. By adding the additional quadruplet or trisection coupling scheme in the coupling paths, the dual-band filter with finite transmission zeros is design and implemented. The measured results have shown good agreement with simulated results.

mm

mm

0 50 100 150

0 10 20 30 40

10 20 30 40 50 60 70 80 90 100 110

mm

mm

0 50 100 150

0 10 20 30 40

10 20 30 40 50 60 70 80 90 100 110

(b) (a)

78

Based on the synthesis procedure in Chapter 2, the proposed dual-band filters have shown properties of the flexible response, good performance, and quick design procedure.

79

Chapter 4

Two-mode Dual-band Filter Design Using E-shaped Resonators

4.1 Introduction

Two-mode resonators are attractive for its advantage of the size reduction in the filter design. Based on the property of two-mode, the interaction between these two modes of the resonator can be completely separated for the physical consideration. Many articles have been provided to design the dual-band filter with two-mode resonators. However, the fully analytical design has not been proposed yet.

E-shaped resonator is validated in dual-mode single-band filter design [122]-[124]

and is a good candidate in dual-band filter design [125]. The even- and odd-mode analysis of the E-shaped resonator is proposed in [122] and corresponding coupling scheme is proposed in [124]. In this chapter, the analytical approach for two-mode dual-band filter synthesis using E-shaped resonators is proposed. Based on the dual-band coupling matrix synthesis proposed in Chapter 2, the odd-mode of the E-shaped resonators is firstly analyzed to determine the dimensions corresponding to the odd-mode portion of the filter.

Then, the central open-stub of the E-shaped resonator can be used to adjust the slope parameter of the even-mode to fit the requirement of the even-mode filter parameters. In addition, the out-of-phase property of the coupled edge of the E-shaped resonator is also discussed and used to improve the separation of two adjacent passbands. By properly arranging the filter layout, the filter order can be increased and the requested transmission zeros are available. The details are discussed in the following sections.

80

4.2 E-shaped Resonator

Figure 4-1 (a) shows the layout of the E-shaped resonator. Due to the electrically symmetric of the layout, the even- and odd-mode analysis can be applied, as shown in Figure 4-1 (b) and (c). Considering the boundary conditions of the electric field, that is, magnitude of electric field is maximum at the open end and is zero at the short end, the electric field distributions for even and odd modes of the E-shaped resonator are illustrated in Figure 4-2. In the figure, Lb = Lb1 + Lb2, so that the λo is larger than λe. Consequently, the self-resonant frequency of the even mode is higher than that of the odd mode. Based on the property, the self-resonant frequency of the even mode can be adjusted by tuning the length of the central stub of the E-shaped resonator(i.e., Lc) while the self-resonant frequency of the odd-mode is kept unchanged.

Due to the property of the separately adjusting the resonant frequency of these two modes, the dual-band filter is now analyzed using even-odd mode analysis and the analytical synthesized procedure is provided based on the synthesized coupling matrix.

Figure 4-1 The schematic of the E-shaped resonator. (a). Layout. (b) Odd-mode. (c) Even-mode.

L_a L_a

L_b L_b L_c

(a) (b) (c)

81

Figure 4-2 The E-field distribution for the (a) odd-mode and (b) even mode of the E-shaped resonator.

4.3 Analytical Approach in Two-mode Dual-band Filter Design Using E-shaped Resonators

Due to the even- and odd-mode of the resonators are at different resonant frequency, they are independent of the effects between each other at their operated frequency. Using these two modes to design a dual-band filter, each passband will be governed by its corresponding mode. Therefore, the proposed two-mode dual-band filter can be described by the coupling scheme shown in Figure 4-3. The two-mode mechanism is controlled by even- and odd-modes. The contributions from even- and odd-modes can be separated by two different coupling paths, and the dual-band performance of the two modes are then extracted using the coupling matrix synthesis. Each mode governs the performances of one passband. Here the back-to-back E-shaped resonator is used to construct the two-mode dual-band filter. Figure 4-4 (a) shows the layout for the proposed two-mode dual-band filter. Removing the central open stubs from the layout, as shown in Figure 4-4 (b), the odd-mode response can be analytically synthesized from the corresponding odd-mode portion of the coupling matrix. And then, the effects from the even-mode will be

L_a

L_b L_c L_b2 L_b1 L_a

L_a+ L_b= λ_o/4 L_a+ L_b1= L_b2+ L_c=λ_e/4

|E| |E|

(a) (b)

82

introduced by adding the open stubs, which dimensions are also determined in an analytical approach. In the following, we will introduce these analytical procedures to synthesize the two-mode dual-band filters.

Figure 4-3 The coupling scheme for the two-mode dual-band filter design.

Figure 4-4 (a) The proposed two-mode dual-band filter. (b) The layout for the odd-mode portion of the two-mode dual-band filter. The number shows the resonator index.

IN

OUT for odd-order

OUT for even-order

1 2

3 4

N-1

N

L L

S

IN

OUT for odd-order

OUT for even-order S

1 2

3 4

N-1 N

L L

(a) (b)

83

4.3.1 Analytical Approach for the Odd-mode Analysis

For the odd-mode analysis, the central open-stub in each E-shaped resonator has no effect. Therefore, it can be removed when designing the odd-mode filter. Figure 4-5 shows a generalized representation of a bandpass filter with resonators and J inverters. To evaluate the values of Js, (3-6) is applied. Once the values of the J inverters are obtained, the even- and odd-mode characteristic impedance for parallel-coupled line can be calculated by (3-7). For the anti-parallel coupled line [141], the formulation is shown in (4-1). Note that the electrical length if the anti-parallel coupled line should not equal 90^o.

( )

Figure 4-5 A generalized band-pass filter circuit using admittance inverters.

4.3.2 Analytical Approach for the Even-mode Analysis

Now adding the central open-stub to the E-shaped resonator as shown in Figure 4-6 to design the even-mode-filter, note that the only adjustable parameters are the impedance and the electrical length of the central open-stub. These parameters must be adjusted to match the even-mode resonant frequency and the slope parameter to fit the requirement of J^ej-1,j. Based on the extracted Zoe, j-1,j and Zoo,j-1,j for the coupled lines, the J^ej-1,j is obtained

84

by (3-7). By the known J^ej-1,j and M^ej-1,j, the slope parameter can be obtained by (3-6).

To calculate the slope parameter from the equivalent circuit, Figure 4-7 shows the corresponding half circuit for even-mode analysis. In Figure 4-7, the slope parameter for even-mode half circuit can be obtained by (4-2)

'

where ωe is the even-mode resonant frequency.

In Figure 4-7, the even-mode resonant condition is

0 2

Figure 4-6 The circuit schematic of the E-shaped resonator.

85

Figure 4-7 (a) The even-mode analysis for the E-shaped resonator in Figure 4-6. (b) The equivalent circuit for the even-mode analysis.

By rearranging (4-2) we have

2 2 '

0 2 2 0 1 2

' 2 '

0 1 1

sec 2 tan tan 4 2 sec .

e

S e

Z Z b

Z b Z

θ θ θ θ

θ θ

= +

− (4-4)

To solve (4-3) and (4-4), the variables ZS and θ2 can be determined. Because the Zoe

and Zoo based on M^oj,j+1 are already fixed when designing the odd-mode filter, once the ZS

and θ2 are obtained, the corresponding coupling coefficient for even-mode M^ej,j+1 is fixed accordingly. This phenomenon restricts the achievable filter parameters for the even-mode.

(a)

(b)

86

To identify the available dual-band characteristics, the M^ej,j+1 of the extracted layout needs to be estimated. Figure 4-8 shows the circuit that is used to identify the coupling coefficient between two E-shaped resonators operating at even-mode.

Figure 4-8 The circuit is proposed to identify the coupling coefficient between two E-shaped resonators operating at even-mode.

The S21 can be derived based on the ABCD matrix of the circuit shown in Figure 4-8.

The poles of S21 are dominated by element C’ of the sub-ABCD matrix shown in Figure 4-8 due to the weak coupling (i.e., a very small value of capacitor CK). To find the roots of C’, the details is derived as follows. Firstly, the well-known equation for determining the coupling coefficient is

2 2

1,2 ^H2 ^L2.

H L

f f

k M f f

= Δ = −

+ (4-5) Let

in in

in in

A B

C D

⎡ ⎤

⎢ ⎥

⎣ ⎦

' '

' '

A B C D

⎡ ⎤

⎢ ⎥

⎣ ⎦

87 2 .

e fH fL

f = + (4-6)

The above equations can be represented as

2 1

Based on the coupling coefficient, two resonant frequencies can be obtained.

Furthermore, the resonant frequencies can be also derived from the circuit in Figure 4-8.

To use the cascade ABCD matrix, the ABCD matrix of the whole circuit can be represented as

Based on the ABCD matrix, the S21 can be represented as

( )

The resonances are the roots of the denominator of S21. For the weak coupling test, here the

88

capacitance of CK is chosen as a very small value, say 0.0001 pF, Hence the 1/(ωCK)² has relatively larger value than other terms, so the roots of C’ is close to the roots of the denominator of S21.

To find the expression C’, the analysis procedure in [141] can be applied to obtained the Ain, Bin, Cin, and Din by terminating two terminals of the coupled line with transmission lines, which have θS length and characteristic impedance 2ZS. Finally, C’ can be represented as

From the above equation, it can be noted that C’ is a real coefficient equations and it is a second order equation with variable χ = -2ZScotθS. Hence the roots of C’ can be

89

obtained using the quadratic formula.

Based on (4-10), the roots of the denominator of S21 can be obtained by root finding for C’ under the weak coupling test. The exact value of the roots can be also obtained by (4-7). Due to the solutions from (4-7) are equal to the roots of C’, the derived equation is obtained as follows:

Solving the equation, two resonant frequencies, which are fH and fL, can be extracted, and they are corresponding to two peaks of S21. Then the coupling coefficient is

2 2

90

4.3.3 Impact of the Constrained Even-Mode k

^e

on the Filter Performance

Because the even-mode coupling coefficient of the back-to-back E-shaped resonator is fixed during designing the odd-mode filter, the available even-mode filter parameters have limitations and will be discussed in the following. Here two cases are analyzed. The first one is to change the bandwidth ratio with frequency ratio as a parameter but the odd-mode bandwidth is fixed as 10%. The second condition is with a fixed frequency ratio but the bandwidth ratio is varied. Using (4-12), the impact on the return loss of the even-mode passband due to the relative error of coupling coefficient (Δk^e) can be observed through following examples. Here three topologies are used, as shown in Figure 4-9.

Figure 4-10 shows the theoretical results of the two-mode dual-band filter with different central frequency ratio but the odd-mode bandwidth is fixed to be 10% and the bandwidth ratio is also fixed to be 1.3. Figure 4-11 shows the calculated Δk^e of the proposed analytical design of back-to-back E-shaped resonator versus bandwidth ratio with frequency ratio as a parameter where the odd-mode bandwidth keeps being 10%. On the other hand, Figure 4-12 shows the theoretical return loss of the same coupling scheme. In this case, however, the central frequency ratio is fixed to be 0.75 with different fractional bandwidth. Again, Figure 4-13 shows the Δk^e versus bandwidth ratio.

91

Figure 4-9 Three topologies used for estimating k^e. (a) The coupling scheme for the coupling matrix in Table 4.1. (b) The coupling scheme for the Example 1 in the following section. (c) The coupling scheme for the Example 2 in the following section.

Figure 4-10 S11 with various even-mode frequencies. (a) Coupling scheme shown in Figure 4-9 (a) with 15 dB return loss. (b) Coupling scheme shown in Figure 4-9 (b) with 20 dB return loss. (c) Coupling scheme shown in Figure 4-9 (c) with 15 dB return loss. (Circle: fo

= 0.8fe. Triangle: fo = 0.75fe, X: fo = 0.7fe). All cases are under the 10% fractional

92

Figure 4-11 The difference between the exact and the estimated coupling coefficients with various fractional bandwidth on even-mode and different frequency ratios of two passbands. (Triangle: Coupling scheme shown in Figure 4-9 (a) with 15 dB return loss.

Circle: Coupling scheme shown in Figure 4-9 (b) with 20 dB return loss. Square: Coupling scheme shown in Figure 4-9 (c) with 15 dB return loss). All cases are under the 10%

fractional bandwidth on odd-mode.

Figure 4-12 S11 with various fractional bandwidths on odd-mode. (a) Coupling scheme shown in Figure 4-9 (a) with 15 dB return loss. (b) Coupling scheme shown in Figure 4-9 (b) with 20 dB return loss. (c) Coupling scheme shown in Figure 4-9 (c) with 15 dB return

Δ^odd/Δ^even

93

loss. (Circle: Δ^odd = 5%. Triangle: Δ^odd = 10%). All cases are under fo = 0.75fe and Δ^odd / Δ^even = 1.3.

Figure 4-13 The difference between the exact and estimated coupling coefficients with various fractional bandwidth on both odd- and even-modes. (Triangle: Coupling scheme shown in Figure 4-9 (a) with 15 dB return loss. Circle: Coupling scheme shown in Figure 4-9 (b) with 20 dB return loss. Square: Coupling scheme shown in Figure 4-9 (c) with 15 dB return loss). All cases are under fo = 0.75fe.

Figure 4-11 gives designers a guideline to choose the frequency ratio under the specific coupling scheme. Here Δ^odd /Δ^even = 1.3 is used as an example to explain how to choose the frequency ratio. For the case which has the coupling scheme in Figure 4-9 (b) with 20 dB return loss (circle symbol), it shows Δk^e is about +11% under fo = 0.7fe (solid line), Δk^e is about +0.5% under fo = 0.75fe (dashed line), and Δk^e is about -7% under fo = 0.8fe (dotted line). Compared with the results shown in Figure 4-10 (b), the S-parameters under fo = 0.7fe with +11% relative error is close to the dotted line with X symbols, the S-parameters under fo = 0.75fe with +5% relative error is close to the solid line with triangle symbols, and the S-parameters under fo = 0.8fe with -7% relative error is close to the dashed line with circle symbols.

From the discussion above, while the coupling scheme in Figure 4-9 (b) with 20 dB Δ^odd/Δ^even

1.1 1.2 1.3 1.4 1.5 1.6 1.7

Δke =(ke exact-ke )/ke exact(%) -10

-5 0 5 10

Δ^odd=5%

Δ^odd=10%

94

return loss chosen with Δ^odd /Δ^even = 1.3, the frequency ratio fo = 0.75fe is the better choice for the small relative error. For the example shown in Figure 4-18, the Δ^odd /Δ^even = 1.3 and fo = 0.718fe , such that the relative error of coupling coefficient is within 0~10% as shown in Figure 4-11. The same analysis can be applied to Figure 4-13. For other specific coupling scheme, the figure of relative errors has to be firstly analyzed to provide the guideline in two-mode dual-band filter design.

4.3.4 Analytical Calculation Example: Fourth-order Two-mode Dual-band Bandpass Filter

To demonstrate the proposed analytical approach for two-mode dual-band bandpass filter design, a forth-order two-mode dual-band filter is used as an example. The filter parameters for the dual-band synthesis are as follows: The first passband central frequency is firstly shifted from 0 to -0.764 rad/s at the lowpass domain and the multi-band lowpass domain bandwidth is 0.4587 rad/s. Similarly, the second passband central frequency is shifted from 0 to 0.8211 rad/s and the multi-band lowpass domain bandwidth is 0.3489 rad/s. Both filters are the second-order filters with return loss of 15 dB. After parallel addition of two filtering functions, the corresponding coupling matrix is obtained in Table 4.1 with coupling scheme shown in Figure 4-9. The four-pole dual-band filter is then transformed to the bandpass domain with central frequency at 2 GHz and the fractional bandwidth bandwidth 40%. It means the frequencies for odd- and even-mode are 1.707 and 2.376 GHz, with odd-mode bandwidth 1.3 times larger than that of even-mode. The circuit schematic of the two-mode dual-band filter is illustrated in Figure 4-14. To evaluate the parameters for odd-mode firstly, the central open-stubs are removed and the corresponding layout is shown in Figure 4-15. The values of the J inverters are then calculated by (3-6)

95

and Zoe and Zoo of the coupled line are obtained by (3-7) and (4-1) and shown in Table 4.2 with E1 = E2 = 2E3 = 60^o. The Zoe and Zoo for the MS,1o and M2o

,L are calculated by (3-7), and Zoe and Zoo for the M1o

,2o are calculated by (4-1). The odd-mode performances are simulated with ADS and shown in Figure 4-16, where the performances from the extracted circuits agree well with that from the coupling matrix.

Table 4.1 Coupling Matrix for the Two-mode Dual-band Filter

Figure 4-14 Two proposed layouts of back-to-back E-shaped resonators.

S 1^o 2^o 1^e 2^e L

S 0.0 0.4973 0.0 0.4404 0.0 0.0 1^o 0.4973 0.8043 0.2872 0.0 0.0 0.0 2^o 0.0 0.2872 0.8043 0.0 0.0 -0.4973 1^e 0.4404 0.0 0.0 -0.8614 0.2202 0.0 2^e 0.0 0.0 0.0 0.2202 -0.8614 0.4404

L 0.0 0.0 -0.4973 0.0 0.4404 0.0

96

Figure 4-15 The corresponding layout for the odd-mode part of the filter

Table 4.2 The Calculated Impedances for the Odd-mode Analysis MS,1o M1o

,2o M2o

,L

J 0.0079 0.0018 0.0079

Zoe 84.9270 62.3482 84.9270

Zoo 36.9231 40.9896 36.9231

Figure 4-16 The performances for the odd-mode part of the filter in Figure 4-15.

Furthermore, it is worth to point out that the two output ports in circuit A and circuit Frequency (GHz)

1.6 1.8 2.0 2.2 2.4

|S11|,|S21| (dB) -60 -50 -40 -30 -20 -10 0

Ideal Circuit A Circuit B S21

S11

97

B receive signals with the same amplitude but 180 degree out-of-phase, as shown in Figure 4-16 and Figure 4- 17. It can be illustrated using the odd-mode analysis. Due to the odd-mode, the electrical fields ate two ends of the resonator have same magnitude but they are 180-degree out-of-phase. This phase inversion has no influence on a single mode filter but can provide an extra transmission zero for two-mode filter. Now, the odd mode filter design is completed.

Figure 4- 17 The 180-degree out-of-phase between two output ports in Figure 4-15.

To introduce the even-mode, we add the central open-stubs. Use (4-3) and (4-4), ZS,4

= ZS,5 = 20.3602 Ω and E4 = E5 = 49.025^o. Using (4-11) and (4-12), the estimated k^e1,2 is 0.0829, such that the approximated M1e,2e is 0.2072. (Here the ideal M1e,2e is 0.2202.) The simulation results are obtained using ADS and shown in Figure 4-18.

Frequency (GHz)

1.6 1.8 2.0 2.2 2.4 2.6

Phase Difference (degree)

179.0 179.5 180.0 180.5 181.0

98

Figure 4-18 The performance of the two-mode dual-band filter of the circuit A in Figure 4-14

It should be pointed out that as the sign of elements M2o

,L and ML, 2o in Table 4.1 is changed, i.e., -0.4973, a transmission zero appears between two passbands and provides a good rejection of these two passbands. As we have mentioned, different circuit prototypes of back-to-back E-shaped resonators provide 180 degree phase difference, and it corresponds to the sign change in the coupling coefficient. Hence when choosing the circuit B in Figure 4-14 without changing the layout dimensions, the new response is shown in Figure 4-19. It provides an easy physical mechanism to change the sign of the coupling coefficient. In this case, with moving the output port location, the stopband rejection can be enhanced.

In this example, the Δ^odd/Δ^even = 1.3 and f^o = 0.718f^e, such that the relative error of the coupling coefficient is within 0~10% as shown in Figure 4-11. Hence in Figure 4-10 we predict the return loss level of the second passband will be larger than 20 dB, and it agrees with the result shown in Figure 4-19.

Frequency (GHz)

1.6 1.8 2.0 2.2 2.4 2.6

|S11|,|S21| (dB) -60 -50 -40 -30 -20 -10 0

Coupling Matrix

Coupling Matrix

在文檔中 多頻濾波器耦合矩陣之合成及其實現於微帶線平行耦合濾波器結構之研究 (頁 92-0)