Proposed wideband BSF

The topology of the proposed wideband BSF illustrated in Fig. 1(a) is upgraded from traditional third-ordered BSF similar to the circuit model in [Reference Feng, Hong, Che and Xue3] by replacing two open stubs at ports with single-ended ground-coupled lines. Thus, wider bandwidth can be efficiently achieved without higher order.

Figure 1. The structure of the proposed BSFs: (a) wideband BSF and (b) dual-band BSF.

Based on the relationship between voltage and current of the coupled lines, Z parameters can be derived as in [Reference Jones16], and the expression of Z _in versus Z_{e 1}, Z_{o 1} and θ can be illustrated as

(1)\begin{equation}{Z_{{\textrm{in}}}} = j\,\left[ {{{{{\left( {{Z_e} - {Z_o}} \right)}^2}} \over {\left( {{Z_e} + {Z_o}} \right)}}\csc 2\theta - {{\left( {{Z_e} + {Z_o}} \right)} \over 2}\cot \theta } \right].\end{equation}

By imposing Z _in = 0, the three transmission zeros (TZs) are derived as follows:

(2)\begin{equation}\theta \left( {{f_0}} \right)\, = \,{\pi \over 2},\quad {\theta _1}\left( {{f_1}} \right)\, = \,{\textrm{arc}}\cos {{{Z_e} - {Z_o}} \over {{Z_e} + {Z_o}}},\quad {\theta _2}\left( {{f_2}} \right) = \pi - {\theta _1}\left( {{f_1}} \right),\end{equation}

where f ₀ represents the center frequency.

Obviously, the locations of TZs can be controlled by adjusting Z_e and Z_o of the coupled line and can further affect bandwidth as illustrated in Fig. 2(a). Trade-off exists that higher SR is attained when $k\,\left( {k = \left( {{Z_e} - {Z_o}} \right)/\left( {{Z_e} + {Z_o}} \right)} \right)$ gets lower at the expense of bandwidth. So, SR continuously rises as long as k declines, allowing flexible design for BSF of arbitrary bandwidth with wanted SR. Numerically, for the proposed wideband ABSF, 10-dB rejection level is accompanied by 0.67 of maximum fractional stopband bandwidth when k of coupled lines on two sides is 0.48.

Figure 2. Simulated results of BSFs: (a) wideband BSF and (b) dual-band BSF.

Proposed dual-band BSF

Built upon the structure of the proposed wideband BSF, the central open-circuited stub is substituted for single-ended ground-coupled lines as showed in Fig. 1(b). Through allocating different k for the central and two sides coupled lines, four TZs are realized and result in two stopbands. Similar to wideband BSF, their locations can be obtained by using (2) and controlled by adjusting Z_{e 1}, Z_{o 1}, Z_{e 2}, and Z_{o 2}. As shown in Fig. 2(b), the bandwidth gets wider as k ₁ increases and differs from k ₂, whereas SR decreases in this manner. Simulation results show that the 15-dB rejection level with 0.22/0.16 of maximum fractional bandwidth and the 10-dB rejection level with 0.23/0.17 of fractional bandwidth can be achieved when k ₁ = 0.05 if k ₂ is fixed at 0.30. In addition, 0.28 GHz of maximum frequency spacing between stopbands is obtained when the normalized center frequencies of two stopbands are 0.86 and 1.14 GHz, respectively.

Proposed wideband ABSF

Except different absorptive branches from the ABSF in [Reference Xu, Lu, Guo and Chen13] and [Reference Zhu, Cai and Chen17], single-ended ground-coupled lines at Port 2 for the proposed wideband ABSF in Fig. 3(a) ensure overall three zero responses. Detailed theory analysis are as follows.

First, the input impedance Z _T1 of T-section I can be extracted from ABCD matrix as

(3)\begin{equation}{\left[ \begin{matrix} A & B \\ C & D \end{matrix} \right]_{{\textrm{T}}1}} = \left[ \begin{matrix} {\cos \,2\theta - {{\sin }^2}\theta } & {j{Z_0}\left( {\sin 2\theta - \tan \theta {{\sin }^2}\theta } \right)} \\ {j1.5{Y_0}\,\sin 2\theta } & {\cos 2\theta - {{\sin }^2}\theta } \end{matrix} \right].\end{equation}

(4)\begin{equation}{Z_{{\textrm{T}}1}} = {Z_0}{{\cos 2\theta - {{\sin }^2}\theta + j\left( {\sin 2\theta - \tan \theta {{\sin }^2}\theta } \right)} \over {\cos 2\theta - {{\sin }^2}\theta + j1.5\,\sin \,2\theta }}\end{equation}

Based on (1), the input impedance Z _in looking into the coupled lines shunted on Port 2 equals infinite at f ₀ and assumes open-circuited. Then Z _T1 of the Z ₀-loaded T-section I equals infinite is required. Meanwhile, by extracting from Z matrix, the input admittance Y _in1 can be depicted as

(5)\begin{equation}{Y_{{\textrm{in}}1}} = {{0.5R\left( {{Z_{e1}} + {Z_{o1}}} \right){{\csc }^2}\theta + j\left[ {{R^2} - {{\left( {{Z_{e1}} + {Z_{o1}}} \right)}^2}} \right]\cot \theta } \over {{{\left( {{Z_{e1}} + {Z_{o1}}} \right)}^3} + \left( {{Z_{e1}} + {Z_{o1}}} \right){R^2}{{\cot }^2}\theta }}.\end{equation}

To theoretically match Port 1 and dissipate unwanted signals, Y _in1 should be equal to Y ₀ and expression of R can be derived as

(6)\begin{equation}R = {Y_0}{\left( {{Z_{e1}} + {Z_{o1}}} \right)^2}{\sin ^2}\theta .\end{equation}

In contrast, input signals are all directly shorted to the ground at two resonant frequencies, and in-band reflection suppression is actualized so far.

For the wideband ABSF, design example with other parameters are provided in Table 1, comparison in Fig. 4(a) illustrates that return loss for different coupling coefficient k_R of resistive terminated coupled lines vary especially within stopband. Under the same circumstances, reflection in-band reaches minimum when k_R is around 0.25. As depicted in Fig. 4(b), curves of S ₁₁ and S ₂₁ indicate the value of R significantly affects in-band reflection of wideband ABSF. Note that a transmission pole is generated when R equals 288 Ω, which means perfect match for achieving reflectionless.

Figure 4. Simulated results of various values of (a) k_R and (b) R for wideband ABSF.

Table 1. Circuit and layout parameter for each prototype

Extended from the proposed wideband ABSF, odd-number-stopband ABSF topology is illustrated in Fig. 5(a). It can obtain notch-filter response at center frequency, whereas every two coupled lines of similar coupling coefficients would attain two stopbands in symmetric distribution with regard to f ₀.

Figure 5. Topologies of (a) (2N + 1)-band ABSF and (b) (2N)-band ABSF.

Proposed dual-band ABSF

For the dual-band ABSF in Fig. 3(b), by extracting from Z matrix, the input admittance Y _in2 can be described as

(7a)\begin{equation}{Y_{{\textrm{in}}2}} = {{\left( {a + R} \right)\,\left[ {a\left( {a + R} \right) - {b^2}} \right]} \over {\left[ {a\left( {a + R} \right) - {d^2}} \right]\left[ {a\left( {a + R} \right) - {b^2}} \right] - {{\left[ {c\left( {a + R} \right) - bd} \right]}^2}}},\end{equation}

where

(7b)\begin{align}a & = - j0.5\left( {{Z_{e3}} + {Z_{o3}}} \right)\cot \theta , b = - j0.5\left( {{Z_{e3}} - {Z_{o3}}} \right)\cot \theta ,\nonumber \\ & c = - j0.5\left( {{Z_{e3}} - {Z_{o3}}} \right)\csc \theta ,\,d = - j0.5\left( {{Z_{e3}} + {Z_{o3}}} \right)\csc \theta .\end{align}

At f ₀, the input admittance Y _in2 equals zero and the input impedance Z _in of coupled lines shunted on Port 2 equals infinite by using equation (6) and (1), respectively. So, both assumed to be open-circuited. Next, the input impedance Z _T2 of Z ₀-loaded T-section II can be extracted from ABCD matrix as below:

(8)\begin{equation}{\left[ \begin{matrix} A & B \\ C & D \end{matrix} \right]_{{\textrm{T}}2}} = \left[ \begin{matrix} {\cos \,2\theta + j{{{Z_0}} \over {2{Z_{{\textrm{in}}}}}}\sin 2\theta } & { - {{Z_0^2} \over {{Z_{{\textrm{in}}}}}}\left( {{{\sin }^2}\theta + j{Z_0}\,\sin 2\theta } \right)} \\ {{{{{\cos }^2}\theta } \over {{Z_{{\textrm{in}}}}}} + j{Y_0}\,\sin 2\theta } & {\cos 2\theta + j{{{Z_0}} \over {2{Z_{{\textrm{in}}}}}}\sin 2\theta } \end{matrix} \right],\end{equation}

(9)\begin{equation}{Z_{{\textrm{T}}2}} = {Z_0}{{\cos \,2\theta - {{{Z_0}} \over {{Z_{{\textrm{in}}}}}}{{\sin }^2}\theta + j\left( {\sin \,2\theta + {{{Z_0}} \over {2{Z_{{\textrm{in}}}}}}\sin 2\theta } \right)} \over {\cos \,2\theta + {{{Z_0}} \over {{Z_{{\textrm{in}}}}}}{{\cos }^2}\theta + j\left( {\sin \,2\theta + {{{Z_0}} \over {2{Z_{{\textrm{in}}}}}}\sin 2\theta } \right)}},\end{equation}

To ensure all the powers are transmitted to output at f ₀, Z _T2 = Z ₀ should be satisfied. Otherwise, input signals are all directly shorted to the ground at four resonant frequencies and ultimately annihilate reflection. For the dual-band ABSF design example, Fig. 6(a) shows the out-band response with different coupling coefficient k_R and resistive terminated coupled lines. Considering in-band reflection suppression and maintaining symmetry of waveform, it is desired to set k_R around 0.25. Results in Fig. 6(b) reveal that reflection is suppressed drastically in two stopbands with subtle change of SR. When R nears 260, two quasi-poles appear and perform high match level.

Figure 6. Simulated results of various values of (a) k_R and (b) R for dual-band ABSF.

Even-number-stopband ABSF topology as in Fig. 5(b), which is developed from the proposed dual-band ABSF, also exploits every two coupled lines of similar coupling coefficients to attain two stopbands in symmetric distribution around f ₀. What’s more, as described in Fig. 6, k_R and R both have an influence on insertion loss within the passband range between stopbands. The phenomenon can be attributed to the absorptive branch, which is merely quasi-complementary to the BSF section over the entire passband, thus, affects power matching.

To further reduce reflection within passband, extension to higher order is a helpful way as employed in [Reference Gómez-García, Muñoz-Ferreras, Feng and Psychogiou11] that arbitrary order can be realized by increasing number of poles. For the proposed wideband ABSF, higher-order topology can be achieved through increasing open-circuited lines with different impedance as depicted in Fig. 7(a). Theoretical examples of the proposed higher-order wideband ABSF and corresponding responses are given as in Fig. 7(b). It reveals deeper suppression level and better selectivity since steeper slopes and sharper stopband skirts are obtained by extending to higher order. Besides, note that the number of poles in passbands grows and reflection near stopbands becomes lower as order increases, exhibiting different power matching level.

Figure 7. (a) Topology of the Nth-order wideband ABSF and (b) theoretical power transmission and reflection responses of first-, second-, and third-order design examples for the proposed Nth-order wideband ABSF (N = 1: Z ₁ = Z ₀; N = 2: Z ₁ = Z ₀, Z ₂ = 1.4Z ₀; N = 3: Z ₁ = Z ₀, Z ₂ = 1.4Z ₀, Z ₃ = 1.8Z ₀).

Based on the above analysis, the procedure of designing proposed input-reflectionless ABSFs can be concluded as follow:

1. 1. select center frequency f ₀ and decide demand bandwidth;

2. 2. choose proper even-/odd-mode impedance of each coupled lines and impedance of the rest lines to form BSFs;

3. 3. further extend to corresponding ABSFs based on the theory derivation in above section;

4. 4. obtain relevant dimensions and realize physically using microstrip lines; and

5. 5. conduct simulations and then carry out optimizations for final layouts.