Dual-band filter analysis and design

The proposed DWBBPF’s transmission line model and its Ansoft Designer circuit simulated S-parameters results are shown in Fig. 1(a) and 1(b), respectively. It consists of a shunt transmission line loaded with an end-connected CL, and a shunt open stub connected in between two series CLs. The circuit simulated magnitude response has five transmission zeros in the stopband and two wide passbands with four poles each. The circuit-simulated upper and lower stopband rejection (LSR) levels are greater than 25 dB with an inter stopband rejection that is higher than 60 dB indicating high passband isolation, and the generated passbands are close. The working principle of the proposed filter configuration can be explained by considering the frequency response of individual elements. The frequency responses of series connected CLs (Z _E1, Z _O1, θ), and transmission line (Z _A, θ) loaded with end-connected CL (Z _E2, Z _O2, θ) are demonstrated in Fig. 2(a) and 2(b), respectively. The frequency response of series connected CLs has a wide passband with three poles and two fixed transmission zeros f _TZ1 and f _TZ5 at 0 and 2f ₀, respectively. The S-parameter response of the transmission line loaded with an end-connected CL contains three transmission zeros f _TZ2, f _TZ3, and f _TZ4. The location of f _TZ2 and f _TZ4 can be varied by changing the impedances of the transmission line loaded end-connected CL. However, f _TZ3 is fixed at f ₀ and can’t be varied. The transmission line loaded end-connected CL is connected to the either side of the series connected CLs, and its response is shown in Fig. 2(c). Three transmission zeros of the transmission line loaded end-connected CL are overlapped with the wide passband response of series connected CLs generating a dual-passband response. However, the return loss and bandwidth are poor. A quarter wavelength shunt open-ended stub which generates a sharp transmission zero at f ₀, as shown in Fig. 2(d) is included in between the series connected CLs to improve the passband input return loss, bandwidth and selectivity, as demonstrated in Fig. 1(b).

Figure 1. Proposed DWBBPF (a) transmission line configuration (b) circuit simulated |S ₁₁| and |S ₂₁|.

Figure 2. Circuit simulated frequency responses of (a) a series connected coupled lines (b) a transmission line loaded with end-connected coupled line (c) a combination of series connected coupled lines and transmission line loaded with end-connected coupled line and (d) an open stub. (Z _E1 = 288 Ω, Z _O1 = 85 Ω, Z _E2 = 240 Ω, Z _O2 = 86 Ω, Z _A = 66 Ω, Z _B = 175 Ω).

The proposed DWBBPF’s circuit simulated frequency responses can be verified theoretically by considering the even–odd mode analysis. The presented symmetrical filter’s even–odd mode schematic models are depicted in Fig. 3(a) and 3(b). The S-parameters for the proposed filter’s even–odd mode input admittances are written in (1) and (2) [Reference Shankar, Kumar and Velidi24].

(1)\begin{equation}{s_{11}} = \frac{{Y_O^2 - {Y_{OE}}{Y_{OO}}}}{{\left( {{Y_O} + {Y_{OE}}} \right)\left( {{Y_O} + {Y_{OO}}} \right)}}\end{equation}

(2)\begin{equation}{s_{21}} = \frac{{{Y_O}\left( {{Y_{OE}} - {Y_{OO}}} \right)}}{{\left( {{Y_O} + {Y_{OE}}} \right)\left( {{Y_O} + {Y_{OO}}} \right)}}\end{equation}

Figure 3. The schematic of the presented DWBBPF using coupled lines, transmission line loaded with an end connected coupled line and open stub (a) even mode circuit and (b) odd mode circuit.

where

(3)\begin{equation}{Y_{OE}} = \frac{{{Z_{CLE}} + {Z_{TL}}}}{{{Z_{CLE}}{Z_{TL}}}}\end{equation}

(4)\begin{equation}{Y_{OO}} = \frac{{{Z_{CLO}} + {Z_{TL}}}}{{{Z_{CLO}}{Z_{TL}}}}\end{equation}

(5)\begin{equation}{Z_{CLE}} = - j{Z_{C1}}\cot \theta + \frac{{Z_{C2}^2{{\csc }^2}\theta }}{{{Z_{SS}} - j{Z_{C1}}\cot \theta }}\end{equation}

(6)\begin{equation}{Z_{SS}} = - j2{Z_B}\cot \theta \end{equation}

(7)\begin{equation}{Z_{TL}} = Z_{A}\left[ {\frac{{{Z_{SCL}} + j{Z_A}\tan \theta }}{{{Z_A} + j{Z_{SCL}}\tan \theta }}} \right]\end{equation}

(8)\begin{equation}{Z_{SCL}} = \frac{{ - j}}{2}\left( {{Z_{E2}}\cot \theta - {Z_{O2}}\tan \theta } \right)\end{equation}

(9)\begin{equation}{Z_{CLO}} = j\frac{{\left[ {Z_{C2}^2 + \left( {Z_{C2}^2 - Z_{C1}^2} \right){{\cot }^2}\theta } \right]}}{{{Z_{C1}}\cot \theta }}\end{equation}

(10)\begin{equation}{Z_{CL}} = \frac{{{Z_{E1}} + {Z_{O1}}}}{2}\end{equation}

(11)\begin{equation}{Z_{C2}} = \frac{{{Z_{E1}} - {Z_{O1}}}}{2}\end{equation}

The transmission zero frequencies are calculated based on the above equations to verify the presented filter configuration. To find the transmission zeros theoretically, the S ₂₁ in (2) is equated to null, yielding Y _OE = Y _OO. Equating (3) and (4), we get

(12)\begin{equation}\tan \theta \left[ {{Z_{E2}}{{\cot }^2}\theta - \left( {2{Z_A} + {Z_{O2}}} \right)} \right] = 0\end{equation}

From (12), tanθ = 0 for θ = θ _TZ1,5 = 0° or 180°. The corresponding transmission zero frequencies can be calculated from f _TZ1,5 = (θ/90°)f ₀, where f ₀ is the design frequency. Further simplifying the second term of (12), we get

(13)\begin{equation}{\theta _{TZ2,4}} = {\cot ^{ - 1}}\sqrt {\frac{{2{Z_A} + {Z_{O2}}}}{{{Z_{E2}}}}} \end{equation}

From (13), the 2^nd and 4^th transmission zero frequencies are calculated as

(14)\begin{equation}{f_{TZ2,4}} = \frac{{{\theta _{TZ2,4}}}}{{{{90}^ \circ }}}{f_0}\end{equation}

At the design frequency there is an inherent transmission zero f _TZ3 due to the fact that the higher order terms in the denominator of (2) approaches infinity when compared to numerator. Similarly, the transmission pole frequencies of the proposed filter configuration are determined by equalizing S ₁₁ to zero. After simplifying S ₁₁ = 0, an eighth order polynomial equation is obtained as given in (15).

(15)\begin{align}&{P_1}{\cot ^8}\theta + {P_{^2}}{\cot ^7}\theta + {P_3}{\cot ^6}\theta + {P_4}{\cot ^5}\theta + {P_5}{\cot ^4}\theta \nonumber\\ & \quad+ {P_6}{\cot ^3}\theta + {P_7}{\cot ^2}\theta + {P_8}\cot \theta + {P_9} = 0\end{align}

where

(16)\begin{equation}{P_1} = {N_{1E}}{N_{1o}} - Y_0^2\left( {{D_{1E}}{D_{1O}}} \right)\end{equation}

(17)\begin{equation}{P_2} = {N_{1E}}{N_{2O}} + {N_{2E}}{N_{1o}}\end{equation}

(18)\begin{equation}{P_3} = {N_{1E}}{N_{3o}} + {N_{2E}}{N_{2o}} + {N_{3E}}{N_{1o}} - Y_0^2\left( {{D_{1E}}{D_{2o}} + {D_{2E}}{D_{1o}}} \right)\end{equation}

(19)\begin{equation}{P_4} = {N_{1E}}{N_{5o}} + {N_{2E}}{N_{3o}} + {N_{3E}}{N_{2o}} + {N_{5E}}{N_{1o}}\end{equation}

(20)\begin{equation}{P_5} = {N_{1E}}{N_{4o}} + {N_{2E}}{N_{5o}} + {N_{3E}}{N_{3o}} + {N_{4E}}{N_{1o}} + {N_{5E}}{N_{2o}} - Y_0^2{P_{5o}}\end{equation}

(21)\begin{equation}{P_{5o}} = {D_{1E}}{D_{3o}} + {D_{2E}}{D_{2o}} + {D_{3E}}{D_{1o}}\end{equation}

(22)\begin{equation}{P_6} = {N_{2E}}{N_{4o}} + {N_{3E}}{N_{5o}} + {N_{4E}}{N_{2o}} + {N_{5E}}{N_{3o}}\end{equation}

(23)\begin{equation}{P_7} = {N_{3E}}{N_{4o}} + {N_{4E}}{N_{3o}} + {N_{5E}}{N_{5o}} - Y_0^2\left( {{D_{2E}}{D_{3o}} + {D_{3E}}{D_{20}}} \right)\end{equation}

(24)\begin{equation}{P_8} = {N_{4E}}{N_{5o}} + {N_{5E}}{N_{4o}}\end{equation}

(25)\begin{equation}{P_9} = {N_{4E}}{N_{4o}} - Y_0^2\left( {{D_{3E}}{D_{3o}}} \right)\end{equation}

(26)\begin{equation}{N_{1E}} = {Z_{E2}}\left[ {Z_{c2}^2 - {Z_{c1}}\left( {2{Z_B} + {Z_{c1}}} \right)} \right]\end{equation}

(27)\begin{equation}{N_{2E}} = 2{Z_A}\left[ {Z_{c2}^2 - {Z_{c1}}\left( {2{Z_B} + {Z_{C1}}} \right)} \right] - {Z_A}{Z_{E2}}\left( {2{Z_B} + {Z_{c1}}} \right)\end{equation}

(28)\begin{equation}{N_{3E}} = {Z_{E2}}Z_{c2}^2 - {Z_{o2}}\left[ {Z_{C2}^2 - {Z_{C1}}\left( {2{Z_B} + {Z_{C1}}} \right)} \right]\end{equation}

(29)\begin{equation}{N_{4E}} = - {Z_{o2}}Z_{C2}^2\end{equation}

(30)\begin{equation}{N_{5E}} = 2{Z_A}Z_{C2}^2 + \left( {2{Z_B} + {Z_{C1}}} \right)\left( {2Z_A^2 + {Z_A}{Z_{o2}}} \right)\end{equation}

(31)\begin{equation}{D_{1E}} = - {Z_A}{Z_{E2}}\left[ {Z_{C2}^2 - {Z_{C1}}\left( {2{Z_B} + {Z_{C1}}} \right)} \right]\end{equation}

(32)\begin{equation}{D_{2E}} = \left( {2Z_A^2 + {Z_A}{Z_{o2}}} \right)\left[ {Z_{C2}^2 - {Z_{C1}}\left( {2{Z_B} + {Z_{C1}}} \right)} \right] - {Z_A}{Z_{E2}}Z_{C2}^2\end{equation}

(33)\begin{equation}{D_{3E}} = Z_{C2}^2\left( {2Z_A^2 + {Z_A}{Z_{o2}}} \right)\end{equation}

(34)\begin{equation}{N_{1O}} = {Z_{E2}}\left( {Z_{C2}^2 - Z_{C1}^2} \right)\end{equation}

(35)\begin{equation}{N_{2O}} = 2{Z_A}\left( {Z_{C2}^2 - Z_{C1}^2} \right) - {Z_A}{Z_{E2}}{Z_{C1}}\end{equation}

(36)\begin{equation}{N_{3O}} = {Z_{E2}}Z_{C2}^2 - {Z_{O2}}\left( {Z_{C2}^2 - Z_{C1}^2} \right)\end{equation}

(37)\begin{equation}{N_{4O}} = - {Z_{O2}}Z_{C2}^2\end{equation}

(38)\begin{equation}{N_{5O}} = 2{Z_A}Z_{C2}^2 + {Z_{C1}}\left( {2Z_A^2 + {Z_A}{Z_{O2}}} \right)\end{equation}

(39)\begin{equation}{D_{1O}} = {Z_A}{Z_{E2}}\left( {Z_{C2}^2 - Z_{C1}^2} \right)\end{equation}

(40)\begin{equation}{D_{2O}} = {Z_A}{Z_{E2}}Z_{C2}^2 - \left( {2Z_A^2 + {Z_A}{Z_{O2}}} \right)\left( {Z_{C2}^2 - Z_{C1}^2} \right)\end{equation}

(41)\begin{equation}{D_{3O}} = - Z_{C2}^2\left( {2Z_A^2 + {Z_A}{Z_{O2}}} \right)\end{equation}

Equation (15) is solved to obtain electrical length values using a math tool (MATLAB). Four valid electrical lengths (θ _TP1,2,3,4) are obtained, from which the four transmission pole frequencies of the first passband can be calculated using f _TPn = (θ _TPn/90°)f ₀, n = 1, 2, 3, 4. Since the filter response is symmetric, the other four transmission pole frequencies are calculated using f _TP5,6,7,8 = 2f ₀ − f _TP1,2,3,4. Considering Z _E1 = 288 Ω, Z _O1 = 85 Ω, Z _E2 = 240 Ω, Z _O2 = 86 Ω, Z _A = 66 Ω, Z _B = 175 Ω, and f ₀ = 1 GHz, the five transmission zero frequencies and eight transmission pole frequencies are calculated using (12) and (15). The calculated transmission zero frequencies are f _TZ1 = 0 GHz, f _TZ2 = 0.51 GHz, f _TZ3 = 1 GHz, f _TZ4 = 1.49 GHz, f _TZ5 = 2 GHz, and the transmission pole frequencies are f _TP1 = 0.6 GHz, f _TP2 = 0.65 GHz, f _TP3 = 0.75 GHz, f _TP4 = 0.81 GHz, f _TP5 = 1.19 GHz, f _TP6 = 1.25 GHz, f _TP7 = 1.35 GHz, f _TP8 = 1.40 GHz. The calculated frequencies from theory match with the circuit simulations, which verifies the proposed filter design.

The S ₁₁ and S ₂₁ variations for change in coupling factors (K ₁ = [Z _E1 − Z _O1]/ [Z _E1 + Z _O1] & K ₂ = [Z _E2 − Z _O2]/ [Z _E2 + Z _O2]) and the impedances (Z _A & Z _B) of the presented DWBBPF are shown in Fig. 4(a)–(d). In Fig. 4(a), the K ₁ is varied while other parameters are kept constant. As K ₁ increases, the return loss improves in the passbands where the number of poles is increasing. However, the insertion loss in the upper and lower stopbands is decreasing. On the other hand, with the increase of K ₂, the return loss decreases in the passbands and no significant change is observed in the insertion loss throughout the frequency range, as illustrated in Fig. 4(b). With the increase of the impedance Z _A, the input matching in the passbands is getting better and there is no change in the insertion loss, as depicted in Fig. 4(c). However, as K ₂ and Z _A changes, some effects are observed on the 2^nd and 4^th transmission zero frequencies. The return loss and number of poles in the passbands are affected when there is a change in the impedance value Z _B, as shown in Fig. 4(d). The passband bandwidth analysis of the presented DWBBPF using CLs, transmission line loaded with end connected CL and open stub for different parameter variations are shown in Fig. 5(a)–(d). The 3-dB FBWs of the 1^st and 2^nd passbands are increasing when the coupling factor K ₁ and impedance Z _B are increasing. As the coupling factor K ₂ increases, both passbands 3-dB bandwidth decreases. The passband bandwidths increase initially and almost constant as Z _A increases, as shown in Fig. 5(c). Therefore, the passband bandwidths of the proposed DWBBPF can be controlled simultaneously by varying the parameters, but not independently.

Figure 4. The S ₁₁ and S ₂₁ variations for change in (a) K ₁ (K ₂ = 0.47, Z _A = 65 Ω, Z _B = 175 Ω) (b) K ₂ (K ₁ = 0.54, Z _A = 65 Ω, Z _B = 175 Ω) (c) Z _A (K ₁ = 0.54, K ₂ = 0.47, Z _B = 175 Ω) (d) Z _B (K ₁ = 0.54, K ₂ = 0.47, Z _A = 65 Ω).

Figure 5. The passband bandwidth analysis of the DWBBPF using coupled lines, transmission line loaded with end connected coupled line and open stub for various (a) K ₁ (b) K ₂ (c) Z _A and (d) Z _B.

Filter prototype and measurement

The design parameters of the presented DWBBPF are initially estimated from Fig. 5 to satisfy the bandwidth’s requirement. These initial parameter values are fine tuned using a circuit simulator to achieve good passband and stopband characteristics. The finalized impedances of the DWBBPF are Z _E1 = 288 Ω, Z _O1 = 85 Ω, Z _E2 = 240 Ω, Z _O2 = 86 Ω, Z _A = 66 Ω, and Z _B = 175 Ω. The widths and lengths of the quarter wavelength CLs, transmission line loaded with end connected CL and open stub are estimated using a transmission line calculator at the design frequency of 1.85 GHz. The presented filter is analyzed using a full-wave simulation software Ansys HFSS and the layout is shown in Fig. 6(a). The physical dimensions are W ₁ = 0.21, L ₁ = 30.52, W ₂ = 1.02, L ₂ = 29.24, W ₃ = 0.21, L ₃ = 30.93, W ₄ = 0.36, L ₄ = 31.11, W ₅ = 4.6, L ₅ = 7, S ₁ = 0.2, S ₂ = 0.27 (all in mm). Rogers RT 5880 (thickness = 0.79 mm, relative permittivity of 2.2, and tangent loss of 0.0012) substrate is used for manufacturing of the proposed DWBBPF, and the fabricated prototype is depicted in Fig. 6(b). The developed prototype is occupying an area of 0.38λ _g × 0.19λ _g. A vector network analyzer is used to measure the magnitude and group delay responses of the proposed filter. Figure 7(a) & 7(b) shows a comparison between the circuit simulated, full-wave simulated and experimented magnitude of S ₁₁ and S ₂₁. The frequency response contains two passbands separated by an inter-stopband bandwidth (ISB) of 0.44 GHz with a rejection better than 30 dB. The measured 3-dB bandwidth for 1^st passband ranges from 1.06 to 1.60 GHz while the measured 3-dB bandwidth for 2^nd passband ranges 2.04–2.6 GHz. The 3-dB fractional bandwidths of the 1^st and 2^nd passband are 40.6% and 24.13%, respectively. The passbands have measured center frequencies located at f ₁ = 1.33 GHz and f ₂ = 2.32 GHz. The measured insertion loss at f ₁ and f ₂ are 0.62 and 0.8 dB, respectively. Similarly, the tested return loss is 15.3 dB at f ₁ and 18.3 dB at f ₂. The LSR is better than 29 dB and the upper stopband rejection (USR) is better than 25 dB. Considering the 3 and 20 dB attenuation levels, the tested roll-off rates for the lower and upper sides of the first passband (second passband) are 154.5 dB/GHz (340 dB/GHz) and 130.7 dB/GHz (121.4 dB/GHz), respectively. A comparison between the full-wave simulated and tested results for different parameters is shown in Table 1. From Fig. 7 and Table 1, there is an excellent agreement between the simulated and experimented results. Figure 8 shows the group delay responses of the presented DWBBPF. The measured group delay is ranging from 1.71 to 2.97 ns and 1.7 to 2.96 ns for the first and second passbands, respectively. The performance of the proposed DWBBPF using CLs, transmission line loaded with end connected CL and open stub is compared with the recently published similar works in Table 2. Even though some designs are compact when compared to the proposed filter, their fractional bandwidths are less. Overall, the proposed filter has wide fractional bandwidth, high isolation, close passbands, good LSR and USR, and roll-off rate when compared to other reported dual-band filters.

Figure 6. The proposed filter using coupled lines, a transmission line loaded with end connected coupled line and an open stub (a) layout and (b) fabricated prototype.

Figure 7. Magnitude responses of the presented DWBBPF (a) circuit and full-wave simulated (b) full-wave simulated and tested.

Figure 8. The group delays results of the proposed dual-wideband.

Table 1. Comparison between proposed filter simulations and measurements

Table 2. Performance comparison with reported DWBBPF