In the latest mobile communication system in which a high data rate multimedia service is required, a miniaturized bandpass filter (BPF) with high performance is essential. Therefore, many studies have focused on the design of such BPFs, one of which employs a closed loop resonator in which the dual resonant modes exist. Since the first presentation of the dual-mode ring BPF by Wolf [1], various innovative designs have been proposed, including the dual-mode microstrip loop resonators with circular or square shapes [2–6]. In [2], dual-mode resonators made of square and circular loops were introduced. In [3], a meander loop resonator was proposed to reduce its size. However, this structure becomes complicated, and interferences among the meander lines can be generated when the size reduction is excessive. The square closed loop filter was proposed in [4]. By inserting capacitive perturbation, adjusting the bandwidth of two-pole BPFs is possible. However, having a wider bandwidth with two resonators is difficult and having many resonators to cover the wider bandwidth is desirable. In general, increasing the number of the resonator to cover the wide bandwidth increases the filter size. In [5], triple-mode BPFs with transmission zeros were presented. However, the theoretical analysis was not fully discussed.

In this study, we propose a three-pole BPF based on the square closed loop resonator. By adding a series resonator to the loop as in [5], creating a third resonant mode, which results in a three-pole BPF, and accommodating more bandwidth are possible. In case the series resonator is replaced by a T-shaped transmission line coupled to the loop resonator by a small gap, the size of the BPF remains the same as that of the conventional closed square loop resonator, which has two-pole characteristics.

The conventional closed loop resonator shown in Fig. 1(a) has two resonance modes [2]. The even–odd mode analysis is usually employed to analyze its characteristics. By adjusting the perturbation θ_a, as shown in Fig. 1(a), the electrical length of the even mode is changed to 0.5θ_low + θ_a when measured from the feed point; the odd mode is not affected by the perturbation and its length remains 0.5θ_low, as shown in the Fig. 1(b) and (c), respectively. Assuming that the total length of the resonator is λ, the electrical length of the upper line from the feeding point is π-0.5θ_low for both even and odd modes. By using the even–odd mode analysis [7], the closed loop resonator’s resonance condition is given as [8]:

Based on (1), the resonant conditions for the even and odd modes are derived as follows:

Without the perturbation (θ_a = 0), the resonance frequency of two modes are identical (θ_low = λ/4). Fig. 2 shows the transmission characteristics of the resonator with varying perturbations θ_a at 2.5 GHz. Even if θ_a is changed, the resonant frequency of the odd mode remains at 2.5 GHz, as shown in Fig. 2.

The proposed BPF that can improve filter performances without affecting the size of the conventional closed loop BPF is shown in Fig. 3(a). It employs a series resonator as a perturbation. The even- and odd-mode equivalent circuits are given in Fig. 3(b) and (c), respectively.

As the odd mode is not affected by the perturbation, its resonant frequency remains the same as that of the conventional closed loop resonator, the resonant condition of which is given by (2). However, because of the LC resonator introduced as a perturbation, the resonant characteristics of the even mode are influenced in such a way that there exist two resonant modes. Their equivalent circuits are shown in Fig. 4. The resonant condition for the split two even modes is given as

When the resonant frequency of the series resonator is identical to f_odd, the two even resonant modes are located around the odd resonant frequency f₀.

As shown in Fig. 3(b), the series resonator works as a capacitor C_r at a low resonant frequency and as an inductor L_r at a high frequency (Fig. 4).

The center frequency f₀ and the ratio of inductance and capacitance of the series resonator are newly defined as the following equations:

To determine the resonant frequencies, the admittances Y_even1 and Y_even2 are derived as follows:

Two resonant frequencies, f_even1 and f_even2, are calculated for several α using (7) and (8) and plotted as shown in Fig. 5. The coupling coefficient k between the two even modes is defined as follows:

The value of the coupling coefficient k as a function of α is shown in Fig. 6. As α decreases, the k value increases. Therefore, the proposed BPF is suitable for designing a wideband BPF. Even if the value of α is small, the lumped element circuit can be replaced with a distributed one without any difficulty.

The proposed three-pole BPF is designed at 2.4 GHz with a 100 MHz bandwidth. The 50 Ω transmission line is used except for the series resonator and the input/output coupling lines. The input/output coupling is made through a capacitor, which is replaced by the gap g_c and length l_c, as shown in Fig. 7 [9]. The series resonator composed of lumped elements is converted into an equivalent T-shaped resonator, as given in Fig. 7. The capacitance of the series resonator can be replaced by the gap with w₁ and g, and the inductance is replaced by the T resonator with length l₂ and width w₂, as shown in Fig. 7. The layout of the designed filter is illustrated in Fig. 7.

The substrate used in the design is RO3003 (ε_r = 3.0, tanδ = 0.0013) with a thickness of 60 mil, and the resonator size is 0.25λ × 0.25λ, which is the same as that of the conventional closed loop resonator. The measured frequency responses of the triple-mode BPF are presented in Fig. 8(a), and they show a good agreement with the simulated ones. The simulated frequency responses of the conventional dual mode BPF are also plotted in Fig. 8 for comparison. The insertion loss at passband was mea-sured as 2.3–2.7 dB, and the return loss at passband was mea-sured as greater than 20 dB. The two transmission zeros at the stopband are created by the asymmetrical feeding of the resonator [10], and the transmission zero at the upper stopband is created by the input/output asymmetrically coupled lines [9]. The proposed BPF has third-order filter responses and better skirt response characteristics. In Fig. 8(b), the group delay is also compared. The fabricated triple-mode BPF has a group delay of 4 ns in the operating frequency band.

In this study, a method for improving the conventional two-pole BPF made of a closed loop into a triple-mode BPF was proposed. The proposed three-pole BPF could accommodate a wider bandwidth than the conventional two-pole BPF without affecting the filter size. The lumped series resonator was used in the theoretical approach. However, at high frequencies, using the lumped element inductor is difficult because of self-resonant characteristics. For this reason, the equivalent T-shaped resonator was used instead of the series resonator. The size of the proposed BPF was the same as that of the conventional closed loop BPF. The proposed design method was applied to the BPF based on the closed loop resonator. The proposed scheme is expected to be applicable to other types of resonators.

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