Wideband quasi-reflectionless bandstop filters using open/shorted coupled lines

Abstract

Two wideband bandstop filters (BSFs) for single and dual-band are proposed and then extended to reflectionless BSFs based on the analysis from input impedance/admittance perspective. Also, topologies of higher-number-stopband input-reflectionless BSF are provided to broaden the design scope. Open/shorted coupled lines are adopted to obtain multi transmission zeros and desired stopband bandwidth by adjusting the even-/odd-mode impedance of coupled lines. Resistor-loaded coupled lines are connected with Port 1 to absorb unwanted signals and obtain input-reflectionless behavior. For validation of the proposed theory analysis, BSFs with corresponding absorptive prototypes are constructed and measured.

Introduction

With the development of modern communication systems, there is an urgent need for bandstop filters (BSFs) to provide effective suppression of undesired frequency band signals in radio frequency/microwave front end. On the one hand, BSFs are responsible for rejecting unwanted powers in a certain frequency range, and a lot of research has been devoted to exploring novel practical structures of BSFs to meet the market’s demands [1–12]. On the other hand, reflectionless/absorptive filters also play an important role in wireless systems compared to conventional filters, which may cause serious consequences to previous stages due to the inevitable reflection. It is gradually becoming a hot trend since reflectionless filter can provide superior performance without degrading the function of adjacent circuit components. Thus, scholars in recent years have paid much attention to investigating absorptive BSF (ABSF) using various methodologies. A miniaturized multiband ABSF is realized in literature [7] by replacing quarter-wavelength transmission lines with bridged T-coils and consequently reducing the circuit size. In [8], a single-port reflectionless bandpass filter composed of a bandpass filter shunted by BSF section is proposed first. Then after cascading N numbers of one-cell reflectionless bandpass filter, a high-order filtering response is realized such that higher stopband rejection (SR) levels will be realized as N increase. Narrow- and wideband reflectionless bandpass filters are reported in [9], whose absorptive auxiliary channels are composed of a resistively terminated bypath transversal filtering architecture, and both attain input-reflectionless behavior as well as high-frequency signal preselection. In [10], the authors demonstrate that the stepped-impedance short-circuited stub can be used for equalizing the passband insertion loss response, which is especially advantageous for cascaded ABSF design to realize higher SR or more stopbands. A balanced quasi-reflectionless bandpass filter is reported in [11], which makes use of complementary transfer function of Kth order BSF branches. In differential-mode operation, energy is absorbed by a loading resistor, while in common-mode operation, it maintains reflective-type nature. In addition, differential-mode selectivity and in-band common-mode suppression can be improved by increasing the order of branch. The work in [12] presents a new filtering power divider topology with all-port reflectionless response, which is realized by exploiting an admittance inverter into the sub-circuit. There are other works that are devoted to all-port reflectionless behavior with symmetric structure property as in [13, 14], by adding absorptive branches at both input and output ports to achieve bilateral quasi-reflectionless response or straightforwardly transform the BSF section into absorptive one, as in [15].

In this paper, two BSFs including wideband and dual-band types have been introduced initially. Then, the corresponding ABSFs are provided with resistor-loaded coupled lines connected to Port 1 to absorb unwanted signals and achieve input quasi-reflectionless response. Also, topologies of higher-number-stopband-ABSF are given to broaden the design scope. The four structures are all made of simple configurations and are easy for practical implementation. Simulation and measurement results are verified for their abilities to reject undesired signals by four design prototypes centered at 2.2 GHz. Specifically, the proposed wideband and dual-band ABSFs perform high SR and wideband reflectionless bandwidth.

Analysis of the proposed BSFs

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 [3] 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 [16], 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.

Analysis of proposed ABSFs

To transform into absorptive filters, a lumped resistor is loaded at the end of the coupled lines at Port 1 of each BSFs to exhaust all the energy from input. Then, two ABSF topologies acquired are presented in Fig. 3 and can be analyzed from input impedance/admittance perspective.

Figure 3. Topologies of ABSFs: (a) wideband ABSF and (b) dual-band ABSF.

Proposed wideband ABSF

Except different absorptive branches from the ABSF in [13] and [17], 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}

where Z _in is denoted in (1).

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 [11] 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.

Simulation and experimental results

To demonstrate the analysis method mentioned in the sections “Analysis of proposed BSFs” and “Analysis of proposed ABSFs”, two pairs of BSFs operating at 2.2 GHz are fabricated on the substrate with dielectric constant ε _r = 2.65, loss tangent tanδ = 0.003, substrate thickness h = 0.508 mm, as shown in Fig. 8. The reference impedance Z _o of Port 1 and Port 2 are set to 50 Ω.

Figure 8. Layout of the proposed bandstop filters: (a) wideband BSF, (b) dual-band BSF, (c) wideband ABSF, and (d) dual-band ABSF.

For the proposed wideband BSF, within the frequency band from 1.89 to 2.58 GHz and 31% for the fractional bandwidth, the measured insertion loss is larger than 21 dB, as shown in Fig. 9(a). The maximum group delay within passband is 0.99 ns. For the proposed dual-bandbandstop filter, Fig. 9(b) shows higher SR that 35-dB insertion loss ranged from 1.82 to 1.94 GHz, and 2.57–2.69 GHz is achieved with 58.1% for the fractional bandwidth. Enhanced return loss is accompanied by other deteriorated properties that higher insertion loss over passband and lower reflection suppression on two sides may weaken its frequency selectivity. The maximum out-band group delay is 0.94 ns.

Figure 9. Photographs, simulated, and measured results of the BSFs: (a) wideband BSF and (b) dual-band BSF.

Measured results in Fig. 10(a) indicate that S ₁₁ of the proposed wideband ABSF below −10 dB is from 1.02 to 2.71 GHz; 10-dB inferred input-reflectionless bandwidth is 1.69 GHz, whereas SR is over 22 dB. Three TZs fall on 1.98, 2.22, and 2.56 GHz, respectively. The maximum group delay is 0.99 ns. In Fig. 10(b), measured S ₁₁ of the proposed dual-band ABSF below −10 dB is from 1.64 to 2.67 GHz and results in 1.03 GHz of 10-dB inferred input-reflectionless bandwidth, whereas reflection remains lower than −10 dB at upper passband. The measured SR is better than 31 dB from 1.77 to 1.94 GHz for the first stopband, and it is from 2.56 to 2.7 GHz for the second stopband. The minimum in-band insertion loss is as low as 31 dB with four TZs located at 1.8, 1.92, 2.58, and 2.68 GHz, respectively. What’s more, the maximum group delay is 0.93 ns.

Figure 10. Photographs, simulated, and measured results of the ABSFs: (a) wideband ABSF and (b) dual-band ABSF.

To further show the advantages of the two types of ABSFs, Tables 2 and 3 shows the measurement result comparisons between different works. Compared with other wideband and dual-band ABSFs, the proposed ABFSs have advantages of wider absorptive bandwidth, more TZs, and simpler circuit structure.

Table 2. Comparisons of recent studies on the wideband ABSFs

Estimated from graph.

Table 3. Comparisons of recent studies on the dual-band ABSFs

Estimated from graph.

Conclusion

In this paper, two structures of BSFs have been introduced and then extended to input-reflectionless ABSFs by rigorous analysis. To explain how to achieve no reflection, this paper intends to elaborate on theoretical derivation and results analysis. By flexibly exploiting the input impedance/admittance of each unit at a certain frequency, ideal response can be realized. Complete design procedure of ABSF is given in the section “Analysis of proposed ABSFs”. The proposed structures all possess simple configurations and are easy for practical implementation. Two examples of the proposed BSFs and corresponding absorptive prototypes operating at 2.2 GHz are constructed and measured.