Circuit Modeling

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 prototype corresponding to the circuit model in [43].

In this dissertation, a doublet as shown in Figure 5-2 (b) is used to model the

circuit in Figure 5-2 (a). In Figure 5-2(b), the resonator 1 represents the odd-mode resonance, where the center plane of the E-shaped resonator is an electric wall (E-plane). On the other hand, the resonator 2 represents even-mode resonance, where the center plane of the E-shaped resonator is a magnetic wall (H-plane). With the notation shown in Figure 5-2(b), the corresponding coupling matrix M can be written down as

(5.1)

There are some interesting properties of the doublet filter in Figure 5-2(a). First, since the E-shaped resonator exhibits symmetry, the relationship MS1=-M1L and MS2=M2L holds. Second, |MS1|>|MS2| is always true for this structure since the coupling strength between the odd mode and external feeding network is always larger than that of the even mode.

To get more insight of how to control a transmission zero of a doublet filter in this configuration, an explicit expression relating the coupling elements and the transmission zero Ω is provided in a low-pass domain as follows

Ω=(M₁₁M_S²₂−M₂₂M_S²₁)/(M_S²₁−M_S²₂) (5.2)

Note that the mapping between normalized frequency ω′ and actual frequency f isω′=(f / f₀ − f₀/ f)Δf / f₀, where f₀ and Δ are center frequency and bandwidth f of a filter, respectively.

Based on the Eq (5.2), observations are summarized in the following:

1. The transmission zero is always located at finite frequency since M_S1≠M_S2. In

other words, the structure exhibits finite transmission zero inherently.

2. The transmission zero can be moved from upper stopband to the lower stopband, or vice versa, by changing the sign of M11 and M22 simultaneously. This property makes it possible to generate upper stopband or lower stopband finite transmission zero with similar structure.

3. If M₁₁ >0 and M₂₂ <0, Ω would be greater than zero. In a more explicit expression, M₁₁ and M₂₂ can be related to the resonant frequencies of odd mode, f_odd, and even mode, f_even, respectively by the following equations.

) 1 2

(

0 0 11

f f f M

f_odd = − ×Δ (5.3) )

1 2 (

0 0 22

f f f M

f_even = − ×Δ (5.4)

where f₀ and Δ are center frequency and bandwidth of a filter, respectively. That f is if f_odd < f₀ and f_even > , the transmission zero would be on the upper stopband. f₀ 4. If M₁₁ <0 and M₂₂ >0, Ω would be smaller than zero. That is if f_odd > f₀

and f_even < , the transmission zero would be on the lower stopband. f₀

To get the related electrical parameters indicated in Figure 5-2(a), one can take the following procedures. First, synthesize a coupling matrix M corresponding to the prescribed response. Then, consider parameters concerning the odd mode only by removing the open stub on the center plane. Once the open stub is removed, the circuit becomes a first-order hairpin filter. The first-order hairpin filter can be synthesized by the conventional method [44] with the center frequency set to be the resonant frequency of odd mode, which can be expressed as

) 2 / 1

( _{11 0}

0 M f f

f

f_odd = − ×Δ . At this step, one can specify the values of ϑ_C,Z₁ and ϑ1 and obtain the values of Z_oe,Z_oo by analytical method [44]. Second, put the open stub back. The two parameters of the open stub,Z₂ and ϑ₂, can be adjusted to

achieve the desired resonant frequency and the needed external coupling strength of the even mode. Here, the resonant frequency of the even mode is

) 2 / 1

( _{22 0}

0 M f f

f

f_even = − ×Δ .

To illustrate the procedure, an example is taken of a second order generalized Chebyshev filter with a passband return-loss of 20-dB and a single transmission zero at a normalized frequency Ω=3. The corresponding coupling coefficients are MS1=1.1110, MS2=0.6170, M11=1.4545, and M22=-1.6260. For filter with center frequency f0=2.4 GHz and fractional bandwidth FBW=0.05, the ideal response is depicted in Fig. 3 as solid lines. After getting the coupling matrix, ϑ_C could first be specified. Here, we set ϑ_C =60^o , Z₁=50 ohm, and ϑ₁ =60^o and obtain

2552 .

=75

Zoe ohm,Z_oo =38.1022 ohm for a uniform impedance resonator with characteristic impedance Z₀ =50ohm at frequency f_odd =2.3127GHz. Next, put the open stub back and adjust the values of Z₂ and ϑ₂ by the optimization method to let the response of the circuit match with the ideal response calculated from the M matrix. The optimized values of Z₂ and ϑ₂ are 62 ohm and 86.8^o, respectively, at frequency 4976f_even =2. GHz. According to the obtained electrical parameters in Figure 5-2 (a), the corresponding response is shown in Figure 5-3 as circled lines. The frequency response contributed only by the odd mode is also depicted in Figure 5-3 as dashed lines to let us understand the procedures clearer.

Figure 5-3. Responses generated from the coupling matrix and from electrical network shown in Fig. 2(a) with synthesis parameters.

B. Extended-Doublet Filters

Based on the doublet filters developed in previous section, the emphasis is put on how to extend the design to extended-doublet filters in this section. There are two possible arrangements suitable to form extended-doublet filters. One possible arrangement is indicated in Figure 5-4 where the extended doublet filter consists of a doublet filter plus a grown resonator. The grown resonator is a half-wavelength resonator with both ends open. In this case, the grown resonator would mainly couple to the odd mode of the E-shaped resonator. And for the even mode of the E-shaped resonator, it acts as a non-resonant element, which slightly perturbs the resonant frequency of the even mode. Another possible design is shown in Figure 5-5 where both ends of the grown resonator are shorted to ground. In this case, the grown resonator mainly couples to the even mode of the E-shaped resonator and acts as a non-resonant element to the odd mode of the E-shaped resonator. To clarify the

coupling relationship between each resonator, the coupling routes are accompanied with layouts in Figure 5-4 and Figure 5-5.

Figure 5-4. A layout of extended-doublet filter and its corresponding coupling scheme.

The design is for flat group delay response

Figure 5-5. A layout of extended-doublet filter and corresponding coupling scheme.

The design is for skirt selectivity response

The extended-doublet filter has a pair of finite transmission zeros. For the design in Figure 5-4, the pair of transmission zeros is on the imaginary-frequency axis. On the other hand, to generate a pair of real-frequency transmission zeros, the design in Figure 5-5 must be adopted. The difference between the two designs can be

understood from the governing equation of finite transmission zeros. Since the proposed extended doublet filters are symmetric structures, the relations |MS1|=|M1L| and |MS2|=|M2L| always hold. Thus, the governing equation of finite transmission zero can be expressed as

₂

2 2

1 2 23 2 2 1

S S

S

M M

M M

= −

Ω (5-5) As discussed in the design of doublet, the coupling coefficient of source to odd mode is stronger than that of source to even mode. Thus, for the design in Figure 5-4,

|MS2|>|MS1|, which leads to Ω² <0. On the contrary, for the design in Figure 5-5,

|MS2|<|MS1|, which results in Ω² >0. In conclusion, the design in Figure 5-4 can be used to generate delay-flattening transmission zeros while the design in Figure 5-5 can be used to generate a pair of attenuation poles.

To illustrate the procedure of the design, a generalized Chebyshev filter with passband return loss of 20-dB and a pair of transmission zeros at Ω=±2 is taken as an example. The design of an extended doublet starts from the synthesis of coupling matrix, which can be done using the technique in [9]. The synthesized coupling matrix is shown in Figure 5-6(a). Using the information of MS1 and MS2, one can construct the doublet by the method provided in section 5-2A. Excluding the M23 and M32 in the coupling matrix, one can calculate the response contributed from the doublet only. For instance, if the center frequency and fractional bandwidth of the designed filter are 2.4GHz and 5% respectively, the responses of the doublet are shown as dotted lines in Figure 5-6(b). After getting the initial design of doublet, add the grown resonator. Since Ω² >0 in this case, the layout in Figure 5-5 must be adopted. Ideally, the response of the extended-doublet would be the solid lines shown in Figure 5-6. The physical implementation of this design will be presented in section 5-3 to confirm the validity.

(a)

(b)

Figure 5-6. The extended-doublet filter with in-band return loss RL=20dB, normalized transmission zeros at Ω=±2. (a) its coupling matrix (b) Responses of extended doublet filter and responses contributed by doublet only.

C. Box-section Filters

The fourth order filter in the “box-section” configuration was first proposed in [38] and realized by coaxial resonators. With the zero-shifting property, it is possible to use the similar filter structure to realize the finite transmission zero either on the upper stopband or on the lower stopband. The box-section filter is suitable for the complementary filters of a transmit /receive duplexer [42] since it has asymmetric response with high selectivity on one side of the passband. The microstrip box-section filter was first reported in [41] with open square loop resonators. Because the box-section coupling diagram is symmetric where MS1=M4L, M12=-M24, M13=M34, and M11=M44 should be held in the coupling route shown in Fig. 5-7(a). Therefore, it is preferable to layout the filter symmetrically because a symmetrical-layout filter can inherently obtain symmetrical coupling coefficients. The asymmetrical layout causes the filters in [41] to be difficult to keep the coupling coefficients to be symmetric.

Another microstrip box-section filter was proposed in [42]. Although the layout of the filters in [42] is symmetric, it suffers from spurious response in the filter’s lower stopband due to one of filter’s resonators to be a higher order mode resonator. In this paper, the layout depicted in Figure 5-7(b) solves the problems mentioned. The E-shaped resonator is symmetric and is free from lower stopband spurious resonances.

Due to the symmetry, only half of the electrical parameters are shown in Fig. 5-7(b).

As explained in doublet filter, the circuit layout in Fig. 5-7(b) satisfies the required sign of couplings.

(a)

(b)

Figure 5-7. A Box-section filter. (a) filter’s coupling scheme. (b) the proposed layout.

To illustrate how to obtain the corresponding electrical parameters in Figure 5-7(b) from a prescribed response, examples are taken as follows. The first example is a fourth order generalized Chebyshev filter with a passband return-loss of 20-dB, a single transmission zero at Ω=−2.57 which gives a lobe level of -48 dB on the lower side of the passband. The corresponding coupling matrix M is shown in Figure 5-8(a). After the lowpass-to-bandpass transformation, the ideal bandpass response of this filter with center frequency of 2.4 GHz and fractional bandwidth of 5% is shown in Figure 5-8(b).

The design procedures are described as follows. First, remove the open stub in the E-shaped resonator in Figure 5-7(b), which is equivalent to discarding the even mode (resonator 3 in the Figure 5-7(a)) of the E-shaped resonator. After removing the open stub, the circuit becomes a third-order hairpin-like filter. The coupling matrix M1

of this hairpin-like filter is identical to the coupling matrix M in Figure 5-8(a) except M3i and Mi3 being zero. The ideal response of this hairpin-like filter can be calculated from M1 matrix as circled lines in Figure 5-8(b). To get the electrical parameters associated with the asynchronously tuned third-order hairpin-like filter, a synchronous tuned third-order hairpin filter provides the initial design and is synthesized at first.

The synchronous tuned hairpin filter has coupling matrix M2 which is identical to M1

except Mii=0. When synthesizing the synchronously tuned hairpin filter, we set ϑ_C1 =90^o , ϑ_C2 =60^o , Z₁ =50 ohm, Z₂ =50 ohm at f = f₀ , and the characteristic impedance of each resonator to be 50 ohm. With these settings, the electrical parameters of the synchronously tuned hairpin filter are calculated and shown in Table. 5.1, which provides the initial values for the asynchronous-tuned hairpin-like filter. Then, an optimization routine is involved. The goal of optimization routine is to find a set of electrical parameters which can make the response match with the response of the ideal asynchronously-tuned hairpin-like filter calculated from M1 matrix. The optimized parameters are shown in Table 5.1 for comparison. Note that the optimized values of associated parameters are nearly identical to the initial values; therefore, the optimization routine can converge within a few times. Finally, put the open stub back and optimize the parameters Z and ₃ ϑ₃ to make the response matched with the response of the desired box-section filter’s response as solid lines in Figure 8(b). The optimized values of Z and ₃ ϑ₃ are given in Table 5.1 as well.

(a)

(b)

Figure 5-8. A fourth order box-section filter: (a) its coupling matrix (b) the responses of the box-section filter and ideal responses of the asynchronous tuned third-order hairpin-like filter calculated by M1 matrix.

Instead of a lowpass prototype filter with a transmission zero at Ω=−2.57in the first example, the second example locates the transmission zero at a normalized frequency Ω=2.57 and keeps all other parameters unchanged. According to the synthesis procedures in [38], the inter-resonator couplings are unchanged but self-couplings (principal diagonal matrix elements, M , M ,…etc., of the coupling

matrix in Figure 5-8(a)) must change sign. Following the same procedures in the previous design, one can get the electrical parameters given in Table 5.1. In Table 5.1, the column of design #1 corresponds to lowpass prototype filter transmission zero at

57 .

−2

=

Ω and the column of design #2 corresponds to a lowpass prototype filter transmission zero at Ω=2.57. The responses obtained from the electrical parameters listed in Table 5.1 and responses calculated from M matrix in Figure 5-8(a) are both plotted in Figure 5-9 for comparison.

Table 5.1. Electrical parameters corresponding to box-section filters shown in Fig.

7(b). Here, ϑ_C1 =90^o, ϑ_C2 =60^o, Z₁=50ohm, Z₂ =50ohm. All of the electrical lengthes are corresponding to the center frequency of the filter.

Design 1: in-band return loss RL=20dB, Ω=−2.57, and FBW=5%

Design 2: in-band return loss RL=20dB, Ω=2.57, and FBW=5%