Organization

This dissertation is organized as follows. Chapter 1 describes the relative researches and difficulty in the microwave dual-band and multi-band filter design. In Chapter 2, the proposed analytical coupling matrix synthesis is described in detail. The different order, different return loss, arbitrary transmission zeros, and multi-band characteristic are available in this synthesis technique. Moreover, the behavior of generating the intrinsic isolation between two adjacent passbands is also discussed. The synthesized polynomials and coupling matrices are listed to give the reader more information in checking the proposed technique.

Chapter 3 describes the single-path and dual-path topologies for the dual-band filter

10

design. The frequency-separation property can be observed in the dual-path topology, and it is useful in frequency planning and filter implementation. Moreover, the mechanism of introducing transmission zeros is similar to that in the single-band filter design, and it is also discussed the in dual-band filter design. The dual-band filter with parallel-coupled lines is then synthesized based on the user-specific coupling matrix in order to extract the filter in each path. After connecting those two filters using double-diplexing configuration, the dual-band filter with dual-path configuration is then generated.

In Chapter 4, the analytical synthesis procedure in designing two-mode dual-band filter is proposed. E-shaped resonator is used to analyze for its even- and odd-mode properties. The two-mode dual-band filter with E-shaped resonators is then synthesized analytically based on the user-specific coupling matrix with dual-path coupling scheme.

The guide line in using the back-to-back E-shaped topology is also provided for the pre-defined coupling coefficient. Chapter 5 describes the tri-band and quad-band filter design using the E-shaped resonator proposed in Chapter 4. The tri-path and quad-path topologies are used to generate the corresponding coupling matrix. To generate the tri-band and quad-band filter, the two dual-band filters are connected using double-diplexing configuration.

In Chapter 6, we conclude the dissertation and draw suggestions for future works.

 

11

Chapter 2

Fully-Analytical Multi-band Coupling Matrix Synthesis

2.1 Introduction

In this chapter, the procedure of multi-band coupling matrix synthesis is discussed in detail. For single-band filter design, the coupling matrix synthesis is proposed in [90], and it has the advantage in hardware implementation. For the multi-band filter design, the most popular procedure is the analytical iterative method [102]. The iterative method, however, may not only have the convergence problem under specific requirement, but also generate some unwanted performances in multi-band filter design. The phenomenon will also be discussed latter.

2.2 Analytical Multi-band Filtering Function Synthesis

To develop a novel and fully analytical multi-band coupling matrix synthesis technique, here a modification is applied to the well-known single-band coupling matrix synthesis and generalized this procedure into the multi-band filter design. The single-band coupling matrix synthesis procedure is shown in Figure 2-1 and is described briefly in the following. For a two-port lossless filter network with N inter-coupled resonators, the transfer and reflection function can be expressed as a ratio of two N-th degree polynomials

( ) ( )

( ) ( ) ( )

( )

11 ^N , 21 ^N

N N

F P

S S

E

ε

E

Ω Ω

Ω = Ω =

Ω Ω (2-1)

12

where Ω is the real frequency variable, the related complex frequency variable s = jΩ , and ε is a normalization constant related to the prescribed return loss level; all polynomials have been normalized so that their highest degree coefficients are unity. S11(Ω) and S21(Ω) have a common denominator EN(Ω), and the transmission zeros of the transfer function are contained in the polynomial PN(Ω). Using (2-1) and the energy conservation for a lossless network, S11(Ω)² + S21(Ω)² = 1, S21(Ω) can be represented as

CN(Ω) is known as the filtering function of degree N. Here, the proposed filters have the form of the generalized Chebyshev characteristic.

In the procedure in Figure 2-1, the filtering function CN(Ω) governs the filter performance. To extend the single-band performance to multi-band one, the property of filtering function is described first. The filtering function satisfies the following conditions:

( )

Here a 3^rd order filtering function is used as an example. Figure 2-2 shows the

13

performance of the filtering function with and without a pair of transmission zeros. It can be noted that the magnitude of the filtering function is less than 1 within the passband. The magnitude of the filtering function increases, as the normalized frequency moves away from the passband. Furthermore, if the transmission zeros appear, the magnitude of filtering function grows dramatically.

Figure 2-1. The procedure of the single-band coupling matrix synthesis [90].

INPUT: (1) Filter order(2) Return loss (3) Prescribed transmission zeros

From the recursive technique，the fractional forms of S₁₁and S₂₁are:

S₁₁(Ω) = F(Ω)/E(Ω), S₂₁(Ω) = P(Ω)/(ε E(Ω)), The filtering function is defined as

C_N(Ω) = F(Ω)/ P(Ω)

Transfer S-parameters into Y-parameters S₁₁(Ω) = F(Ω)/E(Ω), S₂₁(Ω) = P(Ω)/(ε E(Ω)),

Y₁₁(Ω) = m(Ω)/n(Ω), Y₂₁(Ω) = P(Ω)/ n(Ω),

Y-parameters derived from the transversal coupling matrix are Y₁₁^Mand Y₂₁^M. Let

Y₁₁= Y₁₁^M，Y₂₁= Y₂₁^M

OUTPUT

Single-Band Transversal Coupling Matrix

14

Figure 2-2. The response of the 3^rd order filtering function. (a) The linear scale. (b) The log scale.

To combine two filtering functions into a composite filtering function and the resulting composite filtering function has to satisfy (2-4), the parallel addition of two filtering functions is applied. The parallel addition of two filtering functions is defined as the reciprocal of the composite filtering function equals to the sum of the reciprocal of two filtering functions just like the total resistance of two parallel connected resistors. After the operation, the filtering function with small value dominants the performance of the composite filtering function. For the filtering function, it has small values within the passband, and very large values while |Ω| > 1, which follows the property of Chebyshev characteristic. Based on above property, while applying parallel addition of two filtering functions with different central frequencies, the dual-band filtering function can be obtained. In Figure 2-3, the solid line represents a dual-band filtering function, which comes from summing up the reciprocal of two 2^nd order filtering functions with different central frequencies. In this figure, it is clear to show that the small value of two filtering functions will dominate the value of the composite filtering function. Hence the operation

Ω = 1 Ω Ω = -1

| C_N |

| CN | = 1

Without transmission zeros With transmission zeros

Ω log(| C_N|)

Ω = -1 Ω = 1

(a) ^-¶ (b) ^-¶

15

of summing up the reciprocal of each filtering function is useful in the dual-band filtering function synthesis.

Figure 2-3. The performance of the 4^th order dual-band filtering function.

Based on above description, the composite filtering function can be obtained as follows:

where CN1(Ω) and CN2(Ω) are single-band filtering functions with shifted central frequencies. The principal advantage of this technique is that the individual filtering function CN1(Ω) and CN2(Ω) can be obtained analytically by the efficient recursive technique and frequency shift. Hence the polynomial of the composite filtering function can be derived as

16 original filtering function through

( ) ( )

are all generated by recursive technology analytically [93]. In addition, the transmission zeros can be generated using those CN1(Ω) and CN2(Ω) corresponding to their central frequency at each passband via (2-6).

In case of two passband with different bandwidth, these polynomials should be modified. For the i-th filtering function with frequency shift Ωsi and the multi-band lowpass domain bandwidth δi, the polynomial can be represented as

17 (2-8) will be explained in the next section.

The case can be extended to multi-band situation that suppose there are m passbands for a multi-band filter and corresponding filtering function for each passband is CN1, CN2, …, and CNm so that the composite filtering function can be obtained as

( )

individual filter order and the number of transmission zeros. By carefully placing the transmission zeros, the requested frequency response can be obtained under desired specifications.

To update the numerators FN(Ω), PN(Ω), and ε and evaluate the denominator EN(Ω) of

18

In (2-10), the passivity of the rational function representations of the S-parameters is not guaranteed. To enforce the passivity, those roots of EN(Ω) with positive real part in the s domain (s = j Ω) are modified by changing the sign of the real parts, as shown in Figure 2-4. Finally, the transversal coupling matrix based on the generated polynomials is obtained using the method in [93].

Figure 2-4. The passivity enforcement for polynomial EN(Ω).

To transfer the response to the bandpass domain, the following equation is used:

Non Passive root Passive root jΩ

σ

19

where Ω is the frequency in the multi-band low-pass domain, f is the frequency in the bandpass domain, fC, fH, and fL are the central frequency, the upper edge of the highest passband, and the lower edge of the lowest passband in the bandpass domain, respectively, and Δ is the fractional bandwidth.

2.3 Frequency Transformation

The previous procedure to synthesize the multi-band filtering function begins with the filter information in the lowpass domain. The practical specifications, however, are almost described in the bandpass domain. To relate with the information between two domains, here the frequency transformation is provided. Figure 2-5 shows the variables in the bandpass domain and the lowpass domain. In this figure, the relations between variables of the i-th passband in the bandpass domain are shown below:

,

Based on (2-11) and (2-12), the following relations can be derived:

20

The requested specifications are return loss (RL), the central frequencies and fractional bandwidths of both passbands in the bandpass domain. The so-called multi-band lowpass domain bandwidth δ1 and δ2 in the Figure 2-5 can be obtained as

i iH iL,

δ = Ω − Ω (2-13)

where i is the index of the i-th passband.

Figure 2-5. The requested design variables in the bandpass domain and the lowpass domain.

RL is the prescribed return loss.

f

21

The corresponding transmission zeros in lowpass domain can also be obtained using (2-12). When inserting the transmission zeros in the single-band coupling matrix synthesis procedure [90], the transmission zeros in the single-band lowpass domain should be modified as

( )

, , , ,

2

TZ i j TZ i j ci

i

P ⁼

δ

^{Ω − Ω} (2-14)

where δi is the multi-band lowpass domain bandwidth of the i-th passband, ΩTZ,i,j is the j-th transmission zero of the i-th passband in the lowpass domain, and PTZ,i,j is the j-th transmission zero of the i-th passband for inserting into the synthesis procedure in [90].

2.4 Computational Examples

In this section, three examples are used to demonstrate the validation of the proposed synthesis procedure in dual-band filter design. Moreover, prescribed transmission zeros are discussed to be properly described in single-band filtering function.

2.4.1 Example 1: Symmetric Dual-band Bandpass Filter

In this example, the specifications of symmetrical dual-band bandpass filter are provided. Two passbands both are with the filter has filter order 3 and return loss 20 dB.

The first passband has a central frequency at 2.32 GHz and 5% fractional bandwidth. The second passband has a central frequency at 2.695 GHz and 5% fractional bandwidth. The transmission zeros are 2.151 GHz, 2.5 GHz, and 2.905 GHz.

22

To apply (2-5) to design the dual-band filter, these specifications of frequency are firstly transferred into the lowpass domain using (2-11), (2-12) and (2-13), and then the corresponding polynomials are shifted and shrank using (2-8). The requested variables in the synthesis procedure are listed in Table 2.1.

Table 2.1 shows the variables in the lowpass domain used in the proposed synthesis procedure. The passband #1 has the central frequency at -0.75 rad/s and the transmission zero at -1.5 rad/s, while the passband #2 has the central frequency at 0.75 rad/s and the transmission zero at 1.5 rad/s. Applying the synthesis procedure in [93], all central frequencies in each passband need to be shifted to 0 rad/s. Hence the frequency domain in passband #1 needs to be shifted by +0.75 rad/s, that is, the central frequency is 0 rad/s (-0.75 + 0.75) with transmission zero at –0.75 rad/s (-1.5 + 0.75), and the Ωsi is +0.75 rad/s in (2-7). For the frequency domain in passband #2, it needs to be shifted by -0.75 rad/s, so that the central frequency is 0 rad/s (0.75 – 0.75) with transmission zero at 0.75 rad/s (1.5 – 0.75), and the Ωsi is 0.75 rad/s in (2-7). The transmission zero ΩTZ2 listed in Table 2.1 is the intrinsic transmission zero from the proposed synthesis procedure, which will be discussed latter. The settings in the synthesis procedure are listed in Table 2.2 and the resulting coupling matrix is listed in Table 2.3.

Table 2.1 The Requested Frequency Variables in Example 1. 

Specifications Bandpass Domain Lowpass Domain f1 2.32 GHz fC 2.5 GHz Ω1 (2-12) -0.5 rad/s f2 2.695 GHz f1L 2.26 GHz Ω 2 (2-12) 0.5 rad/s Δ1 5 % f1H 2.38 GHz ΩC1 (2-12) -0.75 rad/s Δ2 5 % f2L 2.63 GHz ΩC2 (2-12) 0.75 rad/s ΤΖ1 2.151 GHz f2H 2.76 GHz ΩTZ1 (2-12) -1.5 rad/s

ΤΖ2 2.5 GHz Δ 25 % ΩTZ2 (2-12) 0 rad/s

23

ΤΖ3 2.905 GHz fC 2.5 GHz ΩTZ3 (2-12) 1.5 rad/s

Table 2.2 The Requested Setting Variables in Synthesis Procedure in Example 1.

Bassband RL Filter Order ΩC ΩTZ δ (2−13) PTZ (2-14)

#1 20 3 -0.75 rad/s -1.5 rad/s 0.5 rad/s -3 rad/s

#2 20 3 0.75 rad/s 1.5 rad/s 0.5 rad/s 3 rad/s

After applying the procedure in [90], the corresponding F'N1(Ω), P'N1(Ω), F'N2(Ω), and P'N2(Ω) are obtained in Table 2.4. Applying (2-8), the synthesized polynomials are then shifted and shank, and then PN(Ω) and FN(Ω) of the dual-band filter are obtained. Finally, applying (2-10) to update PN(Ω) and FN(Ω) and obtain the EN(Ω) and ε. Finally, the representation of dual-band filter in (2-1) can be completed. Table 2.4 shows the calculated polynomials. To check the passivity, the synthesized EN(Ω) has roots as shown in Table 2.5. To enforce the passivity, the negative imaginary parts of those roots need to be changed as positive.

The magnitude and phase of the composite filtering function and the S-parameters are shown in Figure 2-6. The transmission zeros of the composite filtering function are slightly shifted (2.121 GHz and 2.945 GHz in Figure 2-6(b)), and this can be noted in Figure 2-6(a).

The frequency shift comes from the combination of two filtering functions and can be eliminated by careful designing these two filtering functions. Figure 2-6(b) shows the corresponding S-parameters. In this figure an additional transmission zero 0 rad/s is introduced. This is because the phase of CN1 and CN2 is 180 degree out-of-phase around 0 rad/s. Furthermore, the transversal coupling matrix is obtained based on the derived polynomials and is shown in Table 2.3.

24

Table 2.3 Transversal Coupling Matrix for the Example 1.

Table 2.4 The Polynomials in (2-8) in the Example 1.

Passband #1 ε1 = 18.7449

Before Freq. Shift F'N1(Ω) Ω ³ + 0.0429 Ω ² - 0.0464 Ω - 0.0013

P'N1(Ω) Ω + 0.75

After Freq. Shift FN1(Ω) Ω ³ + 2.2929ω² + 1.7054 Ω + 0.4098

PN1(Ω) Ω + 1.5

Passband #2 ε2 = -18.7449

Before Freq. Shift F'N2(Ω) Ω ³ - 0.0429 Ω ² - 0.0464 Ω + 0.0013

P'N2(Ω) Ω - 0.75

After Freq. Shift FN2(Ω) Ω ³ - 2.2929ω² + 1.7054 Ω - 0.4098

PN2(Ω) Ω - 1.5

Dual-band (2-8) & (2-10)

After (2-8)

FN(Ω)

FN(Ω)= FN1(Ω) FN2(Ω)

Ω ⁶ – 1.8465Ω ⁴ + 1.0290 Ω ² – 0.1680

PN(Ω)

PN(Ω)= PN1(Ω) ε2 FN2(Ω) + PN2(Ω) ε1 FN1(Ω) 29.7255 Ω ³ – 80.5389Ω

S 1 2 3 4 5 6 L

S 0.0 0.2432 -0.3811 0.2933 0.2933 -0.3811 0.2432 0.0 1 0.2432 1.1169 0.0 0.0 0.0 0.0 0.0 0.2432 2 -0.3811 0.0 0.8712 0.0 0.0 0.0 0.0 0.3811 3 0.2933 0.0 0.0 0.4212 0.0 0.0 0.0 0.2933 4 0.2933 0.0 0.0 0.0 -0.4212 0.0 0.0 0.2933 5 -0.3811 0.0 0.0 0.0 0.0 -0.8712 0.0 0.3811 6 0.2432 0.0 0.0 0.0 0.0 0.0 -1.1169 0.2432 L 0.0 0.2432 0.3811 0.2933 0.2933 0.3811 0.2432 0.0

25

ε =ε1ε2 -351.372

After (2-10)

FN(Ω) Ω ⁶ – 1.8465Ω ⁴ + 1.0290 Ω ² – 0.1680 PN(Ω) Ω ³ - 2.7094 Ω

ε -11.8206

EN(Ω) (non-passive)

Ω ⁶ - 1.8465 Ω ⁴ - j0.0846 Ω ³ + 1.029 Ω ² + j0.2292 Ω - 0.168

EN(Ω) (passive)

Ω ⁶ - j1.1616 Ω ⁵ - 2.52121 Ω ⁴ + j1.7398 Ω ³ + 1.57656 Ω ² - j0.4863 Ω

- 0.168

Table 2.5 Roots of Non-passive and Passive EN(Ω).

Roots of EN(Ω) (non-passive) Roots of EN(Ω) (passive) 1.0702 - j0.1037 1.0702 + j0.1037 -1.0702 - j0.1037 -1.0702 + j0.1037 0.8214 + j0.2904 0.8214 + j0.2904 -0.8214 + j0.2904 -0.8214 + j0.2904

0.3957 - j0.1867 0.3957 + j0.1867 -0.3957 - j0.1867 -0.3957 + j0.1867

26

Figure 2-6. (a) Filtering functions for two single-band filters of same order 3 (CN1 has the transmission zeros at -1.5 and central frequency -0.75 rad/s and CN2 has the transmission zeros at 1.5 and central frequency 0.75 rad/s), and the composite dual-band filtering function (CN1 // CN2). The in-band return loss level is 20 dB in each case. (b) Corresponding S11 and S21 for the symmetric dual-band filter in lowpass and bandpass domains.

To modify the transmission zeros in each bassband to achieve the specifications, that is, transmission zeros are 2.151 GHz, 2.5 GHz, and 2.905 GHz.

2.4.2 Example 2: Asymmetric Dual-band Bandpass Filters

For an asymmetrical dual-band bandpass filter, the frequency response is not symmetric about the central frequency. Two filtering functions used to illustrate the dual-band characteristic have following specifications. Passband #1 is the third-order

LOWPASS PROTOTYPE FREQUENCY (rad/sec)

27

function, which has central frequency at 1.8 GHz with transmission zeros at 1.613 GHz and 2 GHz, and the fractional bandwidth is 9.1%. Passband #2 is the fifth-order function, and it has the central frequency at 2.24 GHz with transmission zeros at 2 GHz and 2.495 GHz, and the fractional bandwidth is 7.28%. Return loss is 20 dB for both passbands. The requested frequency variables can be obtained using the proposed procedure and are listed in Table 2.6. Table 2.7 shows the requested setting variables, and the synthesized polynomials are listed in Table 2.8. The corresponding responses are shown in Figure 2-7 and the transversal coupling matrix is listed in Table 2.9.

Figure 2-7. (a) Filtering functions for two single-band filters of different order 3 (CN1 has the transmission zeros at -1.433 rad/s and 0.004 rad/s and central frequency -0.696 rad/s, and its multi-band lowpass domain bandwidth δ1 is 0.607 rad/s.) and order 5 (CN2 has the transmission zeros at 0.004 rad/s and 1.482 rad/s and central frequency 0.756 rad/s, and its multi-band lowpass domain bandwidth δ2 is 0.486 rad/s), and the composite dual-band filtering function (CN1 // CN2). The in-band return loss level is 20 dB in each case. (b) Corresponding S11 and S21 for the symmetric dual-band filter in lowpass domains.

Table 2.6 The Requested Frequency Variables in Example 2.

Specifications Bandpass Domain Lowpass Domain f1 1.8 GHz fC 2 GHz Ω1 (2-12) -0.393 rad/s

28

Δ1 9.1 % f1H 1.88 GHz ΩC1 (2-12) -0.696 rad/s Δ2 7.28 % f2L 2.16 GHz ΩC2 (2-12) 0.756 rad/s ΤΖ1 1.613 GHz f2H 2.32 GHz ΩTZ1 (2-12) -1.433 rad/s

ΤΖ2 2 GHz Δ 30.2 % ΩTZ2 (2-12) 0.004 rad/s

ΤΖ3 2.495 GHz fC 2 GHz ΩTZ3 (2-12) 1.482 rad/s

Table 2.7 The Requested Setting Variables in Synthesis Procedure in Example 2.

Bassband RL Filter

Order ΩC ΩTZ δ (2−13) PTZ (2-14)

#1 20 3 -0.696 rad/s

-1.433 rad/s

0.607 rad/s

-2.43 rad/s

0.004 rad/s 2.31 rad/s

#2 20 5 0.756 rad/s

0.004 rad/s

0.486 rad/s

-3.09 rad/s

1.482 rad/s 2.99 rad/s

Table 2.8 The Polynomials in (2-8) in the Example 2.

Passband #1 ε1 = -2.1098

Before Freq. Shift F'N1(Ω) Ω³ – 0.0038Ω² − 0.0712Ω + 0.0002 P'N1(Ω) Ω² + 0.0371Ω − 0.5155

After Freq. Shift FN1(Ω) Ω³ + 2.0842Ω² + 1.3767Ω + 0.2859 PN1(Ω) Ω² + 1.4291Ω − 0.0053

Passband #2 ε2 = -982.038

Before Freq. Shift F'N2(Ω) Ω⁵ – 0.0016Ω⁴ – 0.0746Ω³ + 0.0011Ω P'N2(Ω) Ω² + 0.0278Ω − 0.5460

After Freq. Shift FN2(Ω) Ω⁵ – 3.7848Ω⁴ + 5.6553Ω³ – 4.1679Ω² + 1.5144Ω − 0.2170

29

PN2(Ω) Ω² − 1.4855Ω + 0.0055 Dual-band (2-8) & (2-10)

After (2-8)

FN(Ω)

FN(Ω)= FN1(Ω) FN2(Ω)

Ω⁸ − 1.7006Ω⁷ − 0.8561Ω⁶ + 2.6940Ω⁵

− 0.4685Ω⁴ − 1.1816Ω³ + 0.4409Ω² + 0.1343Ω − 0.0620

PN(Ω)

PN(Ω)= PN1(Ω) ε2 FN2(Ω) + PN2(Ω) ε1 FN1(Ω)

−982.038Ω⁷ + 2313.4Ω⁶ − 243.587Ω⁵

− 3867.55Ω⁴ + 4403.23Ω³ − 1922.36Ω² + 315.313Ω − 1.1484

ε =ε1ε2 6628.44

After (2-10)

FN(Ω)

Ω⁸ − 1.7006Ω⁷ − 0.8561Ω⁶ + 2.6940Ω⁵

− 0.4685Ω⁴ − 1.1816Ω³ + 0.4409Ω² + 0.1343Ω − 0.0620

PN(Ω)

Ω⁷ − 2.3557Ω⁶ + 0.2480Ω⁵ + 3.9383Ω⁴

− 4.4837Ω³ + 1.9575Ω² − 0.3211Ω + 0.0017

ε −6.74968

EN(Ω)

Ω⁸ + (-1.7006 - j1.1821)Ω⁷ + (-1.5438 + j2.1130)Ω⁶

+ (3.9706 + j0.4535)Ω⁵ + (-0.7528 - j2.6417)Ω⁴ + (-1.8490 + j1.1961)Ω³

+ (0.8828 + j0.2649)Ω² + (0.0496 - j0.2259)Ω + (-0.0570 + j0.02439)

30

Table 2.9 Transversal Coupling Matrix for the Example 2 without Transmission Zeros Adjustment.

It is noted, however, that the transmission zero on the upper stopband of the composite filtering function is seriously influenced by the filtering function CN1. Because the filtering function CN1 has a lower order, the function value at the out-of-band is smaller than that of CN2. To compute the composite filtering function using (2-9), the filtering function with smaller value will dominate the response of the composite filtering function.

Hence, the transmission zeros on the upper stopband of the composite filtering function shifts inward with respect to the transmission zeros of CN2. This can be overcome by pre-adjusting the zero of CN2 to a higher frequency or by the method described in the following paragraph.

The alternative method to overcome the zero-shifting problem is to take advantage of the generalized Chebyshev characteristic, that is, for an N-th order filtering function, the number of transmission zeros can be smaller than or equal to N. It implies no infinite transmission zeros. Figure 2-8 shows an example, where TZs in the figure denotes the abbreviation of transmission zeros.

In this case, these two filtering functions with same order 4 have transmission zeros at (-6 and 0) rad/s and (-6, 0, and 6) rad/s, and have the central frequency at -3 rad/s,

S 1 2 3 4 5 6 7 8 L

S 0.0 -0.2686 0.4522 -0.2492 -0.1545 0.2322 -0.2586 0.2723 -0.1829 0.0 1 -0.2686 1.1440 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2686 2 0.4522 0.0 0.7590 0.0 0.0 0.0 0.0 0.0 0.0 0.4522 3 -0.2492 0.0 0.0 0.3003 0.0 0.0 0.0 0.0 0.0 0.2492 4 -0.1545 0.0 0.0 0.0 -0.4700 0.0 0.0 0.0 0.0 0.1545 5 0.2322 0.0 0.0 0.0 0.0 -0.5761 0.0 0.0 0.0 0.2322 6 -0.2586 0.0 0.0 0.0 0.0 0.0 -0.7902 0.0 0.0 0.2586 7 0.2723 0.0 0.0 0.0 0.0 0.0 0.0 -1.0004 0.0 0.2723 8 -0.1829 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0672 0.1829 L 0.0 0.2686 0.4522 0.2492 0.1545 0.2322 0.2586 0.2723 0.1829 0.0

31

respectively. The filtering function with three transmission zeros has no infinite transmission zeros, so the stopband rejection is worse due to no infinite transmission zeros.

The stopband rejection, however, can still be kept under an acceptable level. As shown in Figure 2-8, the filtering function with three transmission zeros has the logarithm value close to 2.5 even the frequency up to 60 rad/s, which is -30 dB in S21.

Figure 2-8. (a) Two third-order filtering functions. Solid line: filtering function has two finite transmission zeros at -6 and 0 rad/s, and the central frequency is -3 rad/s on the normalized lowpass domain. Dashed line: filtering function with 3 finite transmission zeros at -6, 0 and 6 rad/s, and the central frequency is -3 rad/s on the original lowpass domain.

The in-band return loss is 20 dB in each case. (b) The corresponding S11 and S21.

Use the above property, the example for asymmetric dual-band filter is modified. One third-order filtering function CN1 has transmission zeros at -1.433, 0.0037, and 1.4134 rad/s, and it has the central frequency at -0.696 rad/s with the milti-band lowpass domain bandwidth δ1 of 0.6066 rad/s. The other one is the fifth-order filtering function CN2, and it has the central frequency at 0.7566 rad/s and the transmission zeros at 0.0037 and 1.4818 rad/s with the milti-band lowpass domain bandwidth δ2 of 0.4857 rad/s. In this case, the transmission zeros at 1.4134 rad/s of CN1 precisely locates the transmission zero on the upper stopband for the composite filtering function. Table 2.10 shows the corresponding

(a) (b)

32

coupling matrix and the performances are shown in Figure 2-9. Compared with the results in Figure 2-7, the upper stopband transmission zeros can be precisely located.

The S-parameters of the synthesized dual-band filter on the bandpass domain are shown in Figure 2-10. The transmission zeros are 1.613 GHz, 2 GHz, and 2.46 GHz, which are slightly shifted from the specifications (1.613 GHz, 2 GHz, and 2.495GHz).

Table 2.10 Transversal Coupling Matrix for the Example 2 with Transmission Zeros Adjustment.

Figure 2-9 Filtering functions for two single-band filters of different order 3 (CN1 has the transmission zeros at -1.433, 0.0037, and 1.4134 rad/s and central frequency -0.696 rad/s and itsmilti-band lowpass domain bandwidth δ1 is 0.6066 rad/s.) and order 5 (CN2 has the transmission zeros at 0.0037 rad/s and 1.4818 rad/s and central frequency 0.7566 rad/s with its multi-band lowpass domain bandwidth δ2 is 0.4857 rad/s), and the composite dual-band

Figure 2-9 Filtering functions for two single-band filters of different order 3 (CN1 has the transmission zeros at -1.433, 0.0037, and 1.4134 rad/s and central frequency -0.696 rad/s and itsmilti-band lowpass domain bandwidth δ1 is 0.6066 rad/s.) and order 5 (CN2 has the transmission zeros at 0.0037 rad/s and 1.4818 rad/s and central frequency 0.7566 rad/s with its multi-band lowpass domain bandwidth δ2 is 0.4857 rad/s), and the composite dual-band

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 27-0)