Due to the explosive growth of various wireless communication services, today's microwave communication systems often require multi-band operations. For example, global systems for mobile communications (GSM) operate at both 900 and 1800 MHz, IEEE 802.11a and IEEE 802.11b wireless local area networks (WLAN) products operate in the unlicensed industrial, scientific and medical (ISM) 2.4 and 5 GHz bands, respectively. The compactness, low insertion loss in passbands and wide stopbands are three main standards for judging the performance of a filter, and in order to achieve these goals, various methods have been proposed.^{1-16} Dual-band BPFs have been designed by combining two separated filters with two specific single passbands or cascading a wideband BPF and a stopband filter, but they suffer from excess sizes and high insertion losses.^{1,2} Stepped-impedance resonators (SIR) have been employed in dual-band BPFs design. However, the resonant frequencies of SIRs are dependent in some cases, which makes the filter design complicated.^{4,5}

On the other hand, microstrip filters with dual-mode response are also attractive, because each dual-mode resonator can be used as a doubly tuned resonant circuit. Thus, the number of resonators required for a given degree of filter is reduced by a half, resulting in a compact filter configuration. Therefore, dual-mode, dual-band filters have received much attention.^{8-16} Dual-mode, stub-loaded resonators (SLR) have been successfully used to design dual-band BPF with good performance for their characteristic of easy control of resonant frequencies.^{8-10} The SLR has some very important properties. First, there are two main coupling paths between source and load. Second, for the dual-mode resonators, the two modes are not coupled to each other. Thus, the dual-mode filters are special, two-order, full canonical transversal filters. The two-order full canonical transversal filter has an inherent transmission zero in the stopband.^{17} By introducing source-load coupling (S-L coupling), additional transmission zeros are created near the passband.^{18} Furthermore, due to the intrinsic characteristics of a transversal filter, the bandwidths of the two bands can be adjusted over a relatively wide range.

**Figure 1** Schematics of the OSLR (a) and SSLR (b).

In this article, a dual-mode, dual-band, microstrip filter incorporating an SSLR and an OSLR is proposed. Additional transmission zeros are obtained by the widely used ways of introducing S-L coupling. Consequently, a 2.4/5 GHz dual-band bandpass filter is designed with multiple transmission zeros on either side of both passbands.

### Characteristics of Stub-loaded Resonators

**Figure 1** shows the schematic models of the OSLR and SSLR, either of which comprises a half-wavelength resonator and an open stub or a short stub shunted at the midpoint. Z_{1}, L_{1}, Z_{2}, L_{2}, Z_{3}, L_{3}, Z_{4} and L_{4} are the characteristic impedances and lengths of the half-wavelength resonator and open or short stub in each SLR. The two SLRs are symmetrical in structure. Thus, odd- and even-mode analysis can be adopted.

For the odd-mode resonance of the OSLR, the symmetry plane behaves as an electric wall, along which there is a voltage null. Thus, the plane is short circuited. The resonant condition is

where θ1 = βL1 is the electric length of the half-wavelength resonator in the OSLR, and f_{oO} is the odd-mode resonant frequency of the OSLR, given as:

where c is the speed of light in free space, and ε_{eff} denotes the effective dielectric constant of the substrate.

For the even-mode resonance of the OSLR, the symmetry plane, on the contrary, behaves as a magnetic wall, through which no current flows. The resonant condition in this case will be

where θ2 = βL1 is the electric length of the open stub, and f_{eO} is the even-mode resonant frequency of the OSLR. For the special case Z_{1} = 2Z_{2}, it can be deduced that

As for the SSLR, the first resonant frequency is an even-mode. The resonant condition is given by

where θ3 and θ4 are the electric length of the half-wavelength resonator in the SSLR and the short stub, respectively. f_{eS} is the even-mode resonant frequency of the SSLR. For the special case of Z_{3} = 2Z_{4}, the resonant frequency here can be derived as

For the odd mode in the SSLR, the resonant characteristic is almost the same as that of the OSLR, which is mainly determined by the half-wavelength resonator. Therefore, the odd mode resonant frequency of the SSLR can be given by

Therefore, by properly choosing the value of L_{3} and L_{4}, using Equations 6 and 7, the even- and odd-modes of the SSLR together, can generate the first passband of the filter. By properly choosing the value of L_{1} and L_{2} using Equations 2 and 4, the even- and odd-modes of the OSLR together, can generate the second passband of the filter.

### Design of the Dual-mode, Dual-band BPF with Multiple Transmission Zeros

The configuration of the proposed dual-mode, dual-band filter is shown in **Figure 2**. The SLRs used here are SSLR and OSLR with source-load coupling. The SSLR is operated at an even and odd resonance in the first passband and the OSLR is operated in the second band. Namely, the resonant frequencies in the first band are f_{eS} and f_{oS}, and f_{oO} and f_{eO} in the second band. The signal is coupled to the OSLR and the SSLR at the same time, while no coupling between resonators is introduced, providing two main paths to load for each passband signal.

**Figure 2** Layout of the proposed dual-mode, dual-band filter.

**Figure 3** Layout of the corresponding coupling scheme of each band.

The corresponding coupling scheme for each band is shown in **Figure 3**, where 1 and 2 represent the odd and even modes, respectively. The signal is coupled to each resonator at the same time, providing two main paths for the signal between the source and load, and no coupling between each mode is introduced. Therefore, the two-order full canonical transversal filter theory can explain the dual-mode resonator.^{17,18} In each band, a different resonator operates at an even and odd mode, respectively. The coupling matrix can be written as

The dual-mode resonator exhibits symmetry, so the relationship M_{S1} = −M_{1L} and M_{S2} = M_{2L} holds. An inherent transmission zero can be created near each band due to two main path signals counteraction, as explained by Cameron,^{18} which can be provided in a lowpass prototype as follows:

Thus, the inherent transmission zero can be shifted from one side of the passband to the other side by properly choosing the relative values of M_{S1} and M_{S2} as well as the signs of M_{11} and M_{22}. The additional transmission zeros can be created by introducing capacitive S-L coupling. Therefore, multiple transmission zeros can be achieved on both sides of the passband, which can improve the selectivity of the passband and reject unwanted signals above the passband. The design process is simple, because there is no coupling between each mode. Each band can be designed, respectively, by the coupling matrix without changing the other band response.

**Figure 4** The photograph of the fabricated dual-mode, dual-band BPF.

A 2.4 and 5 GHz, with 200 and 100 MHz, 3 dB absolute bandwidths, dual-band filter is designed to validate the concept. The coupling matrix is shown in Equation 8. The substrate used here is Duroid 5880 with a thickness of 0.508 mm. The final structure parameters of the bandpass filter are as follows: W_{o} = 1.52 mm, W_{1} = 0.3 mm, W_{2} = 1.95 mm, W_{3} = 1 mm, W_{4} = 0.4 mm, W_{5} = 2.8 mm, W_{6} = 1.3 mm, W_{7} = 0.6 mm, W_{p} = 1.6 mm, W_{p1} = 0.3 mm, W_{p2} = 0.3 mm, R_{1} = 0.3 mm, L_{1} = 9.4 mm, L_{2} = 6.2 mm, L_{3} = 5 mm, L_{4} = 1.4 mm, L_{5} = 1.6 mm, L_{6} = 5.6 mm, L_{7} = 1.9 mm, L_{8} = 2 mm, S_{o} = 0.3 mm, S_{1} = 0.2 mm, S_{2} = 0.2 mm. The total area of the proposed filter is 9.4 × 14.2 mm, which corresponds to a size of 0.1 λ × 0.15 λ, where λ is the guided wavelength at the center frequency 2.4 GHz. Thus, the proposed filter is very compact. **Figure 4** shows the photograph of the fabricated filter.

**Figure 5** Measured and simulated frequency responses (a) wideband, (b) narrowband at 2.4 GHz and (c) narrowband at 5 GHz.

### Simulation and Measurement Results

**Figure 5** shows the simulated and measured results, which are in good agreement. There are two transmission poles inside each passband, which correspond to the two resonance modes of the dual-mode resonators. The measured minimum insertion losses for the two passbands are 1.5 and 1.7 dB, respectively. There exists multiple transmission zeros with better than 45 dB suppression on the outsides of each passband, as expected. The measured 3 dB absolute bandwidths for the lower and upper passbands are 200 and 90 MHz, respectively. Furthermore, the spurious frequencies are suppressed from 5.1 GHz up to 8.7 GHz, with better than 20 dB suppression. There is a slight response discrepancy at 5 GHz between the simulated and measured results. This phenomenon is due to the resonant frequency shift of the resonators, which might be due to a variation of the material characteristic at higher frequencies and manufacturing effect. It can be rectified by slightly adjusting the dimensions of the open stub loaded dual-mode resonator. Thus, the proposed filter is characterized with low insertion loss, compact size and high selectivity.

### Conclusion

A new dual-mode, dual-band, microstrip bandpass filter has been presented. The filter employs stub loaded dual-mode resonators together with S-L coupling to control the transmission zeros. Multiple transmission zeros are created to improve the performance of the filter by utilizing source-load coupling and two main paths signal counteraction. One sample filter with two passbands located at 2.4 and 5 GHz has been designed and measured for demonstration. Results indicate that the proposed filter has the properties of compact size, low passband insertion loss and high selectivity. With all these good features, the proposed filter is applicable to modern wireless communication systems.

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