High-Selectivity Bandpass Filter Based on Two Merged Ring Resonators

A high-selectivity bandpass filter (BPF) based on two merged ring resonators is presented in this paper. The structure of this proposed BPF can be seen as the two one-wavelength ring resonators merged each other by sharing the common λg/2 microstrip line. Due to symmetric structure, it can be analyzed by evenand odd-mode method and the locations of six transmission zeros are calculated using input impedance deductions. For further demonstration, a BPF example centered at 2 GHz is fabricated with high frequency selectivity. The measured 3-dB fractional bandwidth is 11% (1.89–2.11 GHz) and insertion loss is less than 2 dB in the passband. Good agreement between simulation and measurement verifies the feasibility of the design method.


Introduction
In the modern wireless communication systems, highperformance bandpass filters (BPFs) with low insertion loss in the passband and high out-of-band suppression are extremely desirable. In recent years, numerous different methods for designing high-performance BPFs have been presented, such as using coupled line structures [1], [2], ring resonators [3], [4], spoof surface plasmon polaritons [5], [6], transversal signal-interference techniques [7]. In [1], a compact seventh-order wideband BPF with sharp roll-off skirts using coupled lines and open/shorted stubs is proposed. The open/shorted stubs are utilized to introduce more transmission zeros (TZs) and acquire better performance in the stopband. In [3], a dual-mode ring resonator fed by quarter-wavelength side-coupled lines is analyzed. The resonator synthesis is developed to calculate the center frequency, bandwidth, TZs and insertion loss. In [7], transversal signal-interference concept is utilized that employing two different transmission paths for the BPF. And the TZs were generated when the two paths are out-of-phase. But this method is at the expense of a large circuit size. Besides, several novel structures are introduced to achieve higher performance, such as two pairs of twist modified split-ring resonators [8] and stepped impedance open-stub loaded ring resonator [9].
In this paper, a BPF based on two merged ring resonators with high selectivity is proposed, which is quite different from the BPF using ring resonators in [10]. Due to the symmetric structure of this proposed BPF, the distributions of six TZs can be calculated by odd-and even-mode method. The theoretical derivation of the proposed BPF is demonstrated. For validation, a BPF example centered at 2 GHz is fabricated and measured.

Design and Analysis of the Proposed BPF
The ideal circuit of the proposed BPF, which consists of two pairs of coupled lines and five microstrip lines, is shown in Fig. 1(a). The filter structure can be seen as the two one-wavelength (λ g ) ring resonators merged each other with λ g /2 merged length in the middle and λ g /4 coupled to input and output feedlines. Due to symmetry of this BPF structure, it can be analyzed by even-and odd-mode equivalent circuits as illustrated in Fig. 1(b) and (c), respectively.
Observed from Fig. 1(b), the following equation can be established [11], e  1  1  e  e  2  2  a  e  e  3  3  e  e  b  4  4  e  e  5  5  e  6  6   e   V  I  V  I   V  I  V  I  V  I where [Z] a and [Z] b denote 4×4 and 2×2 impedance matrices of the circuit network in dash box, V n e and I n e denote the voltage and current of the corresponding nth port, respectively. 1 On the other hand, the impedance matrix [Z] b can be expressed by: Seen from Fig. 1(b), the network with impedance matrix [Z] b is constituted by two connected microstrip lines. The voltage and current of the two connected microstrip lines can be determined [11]: e e 2 2 5 5 2 2 node node j cot j csc j csc j cot where V node denotes the voltage of the connection node, I node denotes the current of the connection node, the opposite sign of I node in (3c) is due to the opposite current direction with the definition of impedance matrix.
Therefore, the impedance matrix [Z] b can be obtained by combining (3b) and (3c) to eliminate V node and I node . The elements of [Z] b are shown as follows: According to the definition of impedance matrix [9], it can be obtained that V 2 e = -Z A1 I 2 e , V 3 e = V 5 e , I 3 e = -I 5 e , V 4 e = V 6 e , I 4 e = -I 6 e in Fig. 1 Consequently, the input impedance of the even-mode equivalent circuit can be derived using inverse matrix [10], as follows: Likewise, the odd-mode input impedance can be also obtained: a  a  11  12  13  14  a  a  a  a  21  22  B1  23  24  a  a  a  a  31  32  33  B2  34  a  a  a  a  41 42 43 44 Therefore, the reflection coefficient S 11 and transmission coefficient S 21 of the proposed BPF can be calculated as [11]: 2 , 2 0e 0o 0e 0o 2 8 16 2  4  1  3  0 e  0 o   2  2  1  2  3  3  0 e 0 o   2  3  1  2  1 2  2  3   2  1  3  0e  0o  0e 0o  0e  0o   4   16 8 2   16 4  2  2 2 ,  .  Obviously, the two TZs f tz1 and f tz6 are constant, located at 0 and 2f 0 , respectively. When Z 0e = 185 Ω, Z 0o = 98 Ω and Z 1 = 50 Ω are fixed, the other TZs (f tz2 , f tz3 , f tz4 , f tz5 ) are relevant to the characteristic impedances Z 2 and Z 3 . Figure 2(b) indicates the ratio of f tz to f 0 and 3-dB fractional bandwidth (FBW) versus Z 2 . As the characteristic impedance Z 2 shifts, f tz1 and f tz6 keep fixed, f tz2 and f tz4 are almost unchanged, whereas f tz3 and f tz4 will be adjusted. To illustrate more clearly, the S 21 simulation results with different values of Z 2 are shown in Fig. 2(c). It can be seen that the 3-dB FBW will be broadened with the decrease of Z 2 . The minimum 3-dB FBW will be approached when Z 2 increases to 105 Ω under the rejection condition of over 10 dB at the stopband. In contrast, as depicted in Fig. 2(d), the locations of f tz2 and f tz5 will be moved rather than f tz3 and f tz4 , as the characteristic impedance Z 3 is changed.

Implementation Results
For demonstration, an example of the proposed BPF centered at 2 GHz is designed and fabricated. The physical dimensions of the coupled lines and those of the microstrip lines can be extracted from the corresponding electrical lengths and characteristic impedances. These dimensions of the BPF are further fine-tuned in full-wave electromagnetic simulation software Ansys HFSS to consider the unintended coupling effect. The layout of the proposed filter and its final dimensions are shown in Fig. 3(a). Figure 3(b) illustrates the photograph fabricated on a F4B substrate with relative dielectric constant of ε r = 2.65 and thickness of h = 1 mm. The occupied size of this filter is approximately 49.5 × 30.5 mm 2 , i.e., 0.49λ g × 0.30λ g , where λ g is the guided wavelength of 50 Ω microstrip line at 2 GHz.   The simulated and measured S 11 and S 21 are shown in Fig. 4, which agree reasonably well with each other. The measured insertion loss is less than 2 dB, and the return loss is better than 15 dB within the passband from 1.89 to 2.11 GHz (3-dB FBW of 11%). Moreover, the measured rejection levels are over 14 dB at lower stopband from 0 to 1.87 GHz and better than 15 dB at upper stopband from 2.19 to 5 GHz. The performance comparisons with several reported BPFs are shown in Tab. 1.

Conclusion
A high-selectivity BPF based on two merged ring resonators has been presented in this paper. Due to the characteristics of the ring structure, six transmission zeros are generated. Through analysis and calculation procedure of S-parameters and TZs, the location of TZs is determined by the formula derivation. It is of most importance that the location of TZs is adjustable with change of microstrip line width, which provides an additional measure to narrow the bandwidth of filter and improve the performance at the stopband. However, the presented structure provides the improved performance in the stopband with the relatively more occupation. However, the size of this filter is hardly decreased due to the width of the microstrip line with λ g /2 length in the middle. It is difficult to obtain more compact structure by folding the microstrip lines without effecting the frequency response. The proposed BPF can offer an alternative design idea for the application in the modern RF and wireless communication systems.