Analysis of the PS

The structure of the proposed wideband 180° PS consisting of main and reference branches is shown in Figs 1(a)–1(c), which consists of two-layered substrate and three metal layers. Four λ/4 (λ is the wavelength) microstrip lines (MLs) and two ML stepped-impedance resonators (SIRs) are placed at the top and bottom layers, respectively, whereas five λ/4 slotlines (SLs) are etched on the common ground layer. To realize the high-impedance SLs at the two sides of the structure and decrease the radiation losses, four shunt-connected low-impedance SLs with the width of w_{s 1} are used to replace two short-circuited high-impedance SLs [Reference Yang, Zhu, Zhang, Wang, Choi, Tam and Gómez-García20]. The MLs at the top and bottom layers share the same physical dimensions. The difference between reference and main branches is only the direction of ML feedlines at the bottom layer.

Fig. 1. Structure of the proposed PS: (a) 3D layout of main branch, (b) top view of main branch, and (c) top view of reference branch.

For better understanding of phase shifting mechanism in the proposed PS, the evolution of E-field in the reference and main branches is depicted in Fig. 2. In Fig. 2, the signals of reference and main branches are fed from port 1 in the same orientation and guided to port 2 in the opposite orientations. According to the cross-view of electrical field distribution shown in Figs 2(a) and 2(b), the input E-field is converted from the −y direction to the +x direction in the ML to SL transition and propagates as SL waves until SL to ML transition. In the SL to ML transition, the E-field of main (reference) branch is converted from the +x direction to the –y (+y) direction. Thus, the E-fields at the outputs of reference and main branches propagate in the opposite direction, which amounts to 180° inserted phase. Hence, the feature of frequency independent 180° PS can be obtained by changing the coupled orientation of ML feedlines in the SL to ML transition.

Fig. 2. Evolution of E-field: (a) main branch and (b) reference branch (cyan dotted line: signal propagation path; black solid line: E-field distribution).

The transmission-line model of the proposed wideband PS is shown in Fig. 3(a) and the four shunt-connected SLs are replaced with two SLs with the impedance of Z_{s 1}. The electrical length of the transmission-line section is assumed to be θ (= 90° at the center frequency f ₀) and two pairs of impedance transformers with turns ratios of N ₁ and N ₂ are utilized in this design to equal to discontinuity of MLs and SLs [Reference Yang, Zhu, Choi, Tam, Zhang and Wang18].

Fig. 3. (a) Transmission-line model of the proposed PS, (b) odd-mode circuit model, and (c) even-mode circuit model.

By absorbing the transformers and normalizing characteristic impedances in Fig. 3(a), even-mode and odd-mode equivalent circuits can be obtained as shown in Figs 3(b) and 3(c), respectively. The relationships between the characteristic impedances of Figs 3(a), 3(b), and 3(c) are shown in Figs 3(b) and 3(c). The odd-mode input impedance can be expressed as

(1)$$z_{ino} = \displaystyle{{\,jz_1z_2\lpar {z_{o1}\tan \theta + jz_2{\tan }^2\theta } \rpar } \over {z_2z_{o1} + jz_2\lpar {z_1 + z_2} \rpar \tan \theta -z_1z_{o1}{\tan }^2\theta }}$$

where

(2)$$z_{o1} = z_{o2} + z_{SIR}$$

(3)$$z_{SIR} = {{\,jz_3\lpar {z_3 + z_4} \rpar \tan \theta } / {\lpar {z_3-z_4{\tan }^2\theta } \rpar }}$$

(4)$$z_{o2} = {{\,jz_{s1}z_{s2}\tan \theta \tan \lpar {{\theta / 2}} \rpar } / {\lpar {z_{s1}\tan \theta + z_{s2}\tan \lpar {{\theta / 2}} \rpar } \rpar }}$$

Similarly, the odd-mode input impedance can be obtained as

(5)$$z_{ine} = \displaystyle{{\,jz_1z_2\lpar {z_{e1}\tan \theta + jz_2{\tan }^2\theta } \rpar } \over {z_2z_{e1} + jz_2\lpar {z_1 + z_2} \rpar \tan \theta -z_1z_{e1}{\tan }^2\theta }}$$

where

(6)$$z_{e1} = z_{e2} + z_{SIR}$$

(7)$$z_{e2} = {{\,jz_{s1}z_{s2}\tan \theta } / {\lpar {z_{s2}-z_{s1}\tan \theta \tan \lpar {{\theta / 2}} \rpar } \rpar }}.$$

According to (1)–(7), the S-parameters of the proposed filter can be obtained as [Reference Zhu, Sun and Li21]

(8a)$$S_{21} = {{\lpar {z_{ine}-z_{ino}} \rpar } / {\lsqb {\lpar {z_{ine} + 1} \rpar \lpar {z_{ino} + 1} \rpar } \rsqb }}$$

(8b)$$S_{11} = {{\lpar {z_{ine}z_{ino}-1} \rpar } / {\lsqb {\lpar {z_{ine} + 1} \rpar \lpar {z_{ino} + 1} \rpar } \rsqb }}.$$

By calculating S ₂₁ = 0, the four transmission zeros (TZs) in the stopband at the frequency range from 0 to 2f ₀ can be obtained as follows:

(9)$$\left\{\matrix{\,f_{tz1} = 0\semicolon \;f_{tz2} = \lpar {{{2f_0} / \pi }} \rpar \tan^{{-}1}\sqrt {{{z_3} / {z_4}}} \hfill \cr f_{tz3} = 2f_0{\rm \semicolon \;\ }f_{tz4} = 2f_0-\lpar {{{2f_0} / \pi }} \rpar \tan^{{-}1}\sqrt {{{z_3} / {z_4}}} \hfill} \right.$$

Based on (9), the positions of the two TZs f_{tz 2} and f_{tz 4} can be controlled by tuning the ratio z ₃/z ₄, which is also the impedance ratio of SIR.

According to (8), the characteristic function F can be formulated as [Reference Zhu, Sun and Li21]

(10)$$F = {{S_{11}} / {S_{21}}} = {{\lpar {z_{ine}z_{ino}-1} \rpar } / {\lpar {z_{ine}-z_{ino}} \rpar }}.$$

Figure 4(a) shows the ideal magnitude of S-parameters for the proposed PS with a quasi-elliptic equal-ripple response. In Fig. 4(a), the two TZs generated by the SIRs are f_{tz 2} and f_{tz 4}, the lower cutoff frequency is f_c, and RL is the return loss. The equal-ripple fractional bandwidth (FBW) can be defined as FBW = 2(f ₀ – f_c)/f ₀.

Fig. 4. (a) Ideal S-parameters of the proposed PS, (b) S-parameters with the variable z_{s 1} when z_{s 2} = 1.50 and z ₄ = 2.90, (c) S-parameters with the variable z_{s 2} when z_{s 1} = 2.87 and z ₄ = 2.90, and (d) S-parameters with the variable z ₄ when z_{s 1} = 2.87 and z_{s 2} = 1.50.

In order to design this quasi-elliptic equal-ripple response with desired ripple FBW and RL, the following condition should be satisfied:

(11)$$\vert F \vert _{\theta = \theta _0} = \vert F \vert _{\theta = \theta _c} = \varepsilon $$

where

(12)$$\theta _0 = {\pi / 2}{\rm \semicolon \;\ }\theta _c = \lpar {{{\,f_c} / {\,f_0}}} \rpar {\pi / 2}{\rm \semicolon \;\ }\varepsilon = {1 / {\sqrt {{10}^{{{RL} / {10}}}-1} }}.$$

With (11), the RL at f ₀ and f_c can be fixed. According to (11), it can be found that:

(13)$$\vert F \vert _{\theta = \theta _0} = \vert {{{\lpar {z_{s2}^2 -z_2^4 } \rpar } / {2z_2^2 z_{s2}}}} \vert = \varepsilon $$

(14)$$\vert F \vert _{\theta = \theta _c} = \vert {{{\lpar {n_0 + n_1z_1 + n_2z_1^2 } \rpar } / {mz_1^2 }}} \vert = \varepsilon $$

where

(15a)$$n_0 = az_2^2 -z_2^4 \tan ^2\theta _c-bz_2^3 \tan \theta _c$$

(15b)$$n_1 = {-}2\lpar {az_2 + z_2^3 } \rpar \tan ^2\theta _c-bz_2^2 \tan \theta _c\lpar {1-{\tan }^2\theta_c} \rpar $$

(15c)$$n_2 = \lpar {a-z_2^4 } \rpar \tan ^4\theta _c + z_2^2 \lpar {a-1} \rpar \tan ^2\theta _c-bz_2\lpar {z_2^2 -1} \rpar \tan ^3\theta _c$$

(15d)$$m = cz_2^2 \tan ^2\theta _c\lpar {1 + {\tan }^2\theta_c} \rpar $$

(15e)$$a = {z_{o1}z_{e1}} \vert _{\theta = \theta _c} = {-\lsqb {{\mathop{\rm Im}\nolimits} \lpar {z_{o1}} \rpar \times {\mathop{\rm Im}\nolimits} \lpar {z_{e1}} \rpar } \rsqb } \vert _{\theta = \theta _c}$$

(15f)$$b = {{{ {\lpar {z_{o1} + z_{e1}} \rpar } \vert }_{\theta = \theta _c}} / j} = {\lsqb {{\mathop{\rm Im}\nolimits} \lpar {z_{o1}} \rpar + {\mathop{\rm Im}\nolimits} \lpar {z_{e1}} \rpar } \rsqb } \vert _{\theta = \theta _c}$$

(15g)$$c = {{{ {\lpar {z_{o1}-z_{e1}} \rpar } \vert }_{\theta = \theta _c}} / j} = {\lsqb {{\mathop{\rm Im}\nolimits} \lpar {z_{o1}} \rpar -{\mathop{\rm Im}\nolimits} \lpar {z_{e1}} \rpar } \rsqb } \vert _{\theta = \theta _c}.$$

By simplifying (13) and (14) with (15), the following formula can be obtained:

(16)$$z_2^4 = \left({1 \pm \sqrt {\varepsilon^4 + 4\varepsilon^2} + 2\varepsilon^2} \right)z_{s2}^2 $$

(17)$$\lpar {n_2^2 -\varepsilon^2m^2} \rpar z_1^4 + 2n_1n_2z_1^3 + \lpar {n_1^2 + 2n_0n_2} \rpar z_1^2 + 2n_0n_1z_1 = {-}n_0^2 .$$

With given RL, FBW, f_{tz 2} (f_{tz 4}), z_{s 1}, z_{s 2}, and z ₄, the values of z ₁, z ₂, and z ₃ can be calculated with (9), (16), and (17). For example, when f ₀ = 3 GHz, FBW = 127%, and f_{tz 2} = 0.86 GHz, the responses of S-parameters with the variable z_{s 1}, z_{s 2}, and z ₄ are shown in Figs 4(b), 4(c), and 4(d). The list of calculated values of z ₁ and z ₂ with the variable z_{s 1}, z_{s 2}, and z ₄ in Figs 4(b), 4(c), and 4(d) are shown in Table 1. By tuning the values of z_{s 1}, z_{s 2}, and z ₄, the desirable responses can be obtained.

Table 1. List of variables used in Fig. 4

Based on the above analysis, the values of impedance can be obtained from predetermined f ₀, FBW, RL, and TZ f_{tz 2} (f_{tz 4}). The design procedure for determinant of designed parameters can be given as follows:

1. (1) Calculate the values of θ_c, ε, and z ₃/z ₄ from predetermined f ₀, FBW, RL, TZ f_{tz 2} (f_{tz 4}) with (9) and (12).

2. (2) Based on the selected z_{s 1}, z_{s 2}, and z ₄, the values of z ₁, z ₂, and z ₃ can be calculated with θ_c, ε, and z ₃/z ₄ obtained in step (1) and (16)–(17).

3. (3) Obtain the S-parameters from the calculated values of z ₁, z ₂, z ₃, and selected z_{s 1}, z_{s 2}, and z ₄ in step (2) with (1)–(8).

4. (4) Tune the values of z_{s 1}, z_{s 2}, and z ₄ and repeat steps (2) and (3) until desirable responses are obtained.

Experimental verification

A practical high selectivity wideband 180° PS operating at 3 GHz with FBW of 127% is designed on a two-layer Arlon AD300C substrate with a permittivity of 2.97 and a thickness of 0.762 mm in each layer. The electric parameters are listed as: N ₁ = 1.06, N ₂ = 1.08, Z₁ = 112.00 Ω, Z ₂ = 58.25 Ω, Z ₃ = 30.17 Ω, Z ₄ = 129.05 Ω, Z_{s 1} = 123.03 Ω, Z_{s 2} = 64.30 Ω, and Z ₀ = 50.00 Ω. The photograph of the proposed wideband 180° PS is shown in Fig. 5. The dimension of the whole circuit is 0.51λ_g × 0.50λ_g × 2, where λ_g is the guided wavelength at f ₀.

Fig. 5. The photograph of the fabricated circuit: (a) top view and (b) bottom view.

The fabricated PS is simulated by the high-frequency structure simulator and measured by the two-port vector network analyzer of Keysight N5244A. The simulated and measured results are plotted in Fig. 6. The measured insertion loss (IL) is 0.99 dB at 3 GHz. The measured bandwidth is 120% from 1.20 to 4.80 GHz for RL higher than 19 dB and phase difference of ±2.5°. Performance comparison between this paper and the-state-of-the-art PS is shown in Table 2. As it can be seen, the designed wideband 180° PS performs well in wider bandwidth, higher selectivity, and an additional functionality of MS-to-MS VT.

Fig. 6. Simulated and measured results: (a) S-parameters and (b) phase difference (Ide: ideal response; Sim: simulated result; Ref: measured result of reference branch; Main: measured result of main branch; optimized dimensions (mm) are summarized as follows: w ₀ = 2.30, w ₁ = 0.16, w ₂ = 1.51, w ₃ = 3.95, w ₄ = 0.11, w_{s 1} = 0.56, w_{s 2} = 0.22, l ₁ = 16.38, l ₂ = 15.02, l ₃ = 16.19, l ₄ = 14.36, l_{s 1} = 19.02, l_{s 2} = 20.63, l_via = 1.50 and r_via = 0.50).

Table 2. Performance comparison with previous wideband PSs

Layers: number of substrate layers; RC: rectangle coefficient (RC = BW_{20 dB}/BW_{3 dB}, where BW_{20 dB} is the 20-dB attenuation bandwidth of S ₂₁ and BW_{3 dB} is the 3-dB attenuation bandwidth of S ₂₁).