### I. Introduction

*R*

*=*

_{S}*R*

*= 50 Ω) termination impedance design. Thus, designing a frequency-selective BPF capable of center frequency tuning and arbitrary termination impedance is an important step in the miniaturization of emerging next-generation wireless communication systems.*

_{L}### II. Design Method

*R*

*±*

_{S}*jX*

*and*

_{S}*R*

*±*

_{L}*jX*

*impedances, respectively. The proposed tunable BPF comprises a dual-mode resonator that provides the even- and odd-mode resonant frequencies. Fig. 1(b) presents the coupling diagram of the proposed BPF, where each node represents even-and odd-mode resonant frequencies. In this figure, the solid and dashed lines illustrate the direct coupling and cross-coupling paths, respectively. Using a lossless (*

_{L}*N*+ 2) × (

*N*+ 2) filter model, the coupling matrix of the proposed tunable BPF is given as (1), where the source and load ports are normalized to 1 Ω.

*f*

*, even-mode;*

_{e}*f*

*, odd-mode) of a dual-mode resonator can be calculated by as follows:*

_{o}*f*

*, Δ, and*

_{c}*x*indicate the center frequency, fractional BW, and tuning element of the filter, respectively.

*r*

_{S}*±jx*

*) and load impedance (*

_{s}*r*

*±*

_{L}*jx*

*), the (*

_{L}*N*+ 2) × (

*N*+ 2) coupling matrix of filter is evaluated as follows:

*R*

*and*

_{S}*R*

*are the real parts and*

_{L}*X*

*and*

_{S}*X*

*are the imaginary parts of the source and load port impedances, respectively, which are normalized with reference to 50 Ω. The*

_{L}*S-*parameters of an arbitrary impedance terminated BPF can be obtained as.

*R*] is the (

*N*+ 2) × (

*N*+ 2) is zero matrix, except for the non-zeros entries of

*R*

_{11}=1/

*r*

*and*

_{s}*R*

_{N}_{+2,}

_{N}_{+2}=1/

*r*

*. Similarly, [*

_{L}*W*] is the (

*N*+ 2) × (

*N*+ 2) identity matrix except for

*W*

_{11}=0 and

*W*

_{N}_{+2,}

_{N}_{+2}=0.

*f*

*and Δ are 2.50 GHz and 4.40%, respectively, for a Chebyshev filter with a ripple of 0.043 dB. The synthesized (*

_{c}*N*+ 2) × (

*N*+ 2) coupling matrix of the proposed tunable BPF with source and load ports of 1 Ω can be determined as.

*N*+ 2) × (

*N*+ 2) coupling matrix of the arbitrarily terminated BPF can be calculated using (3) and (7).

*x*from −8.2 to 8.2. The TZs are located at the lower and higher frequencies of the passband. The location of the TZs can be controlled by changing the source-load coupling (

*M*

*). Similarly, the TZs are also moved while tuning the center frequencies. These results indicated that even though the source and load termination impedances (*

_{SL}*R*

*and*

_{S}*R*

*) of BPF are chosen arbitrarily, the response remans identical.*

_{L}### 1. Proposed Dual-Mode Resonator

*Z*

_{2}and

*Z*

_{1}, electrical lengths of

*θ*

_{2},

*θ*

_{1}, and

*θ*

_{0}; and a shunt short-circuited stub with a characteristic impedance of

*Z*

*and an electrical length of*

_{k}*θ*

*. Fig. 3(b) and 3(c) depict the even- and odd-mode equivalent circuits. Using these circuits, the even- and odd-mode input admittances are derived as follows:*

_{k}*f*

*and*

_{e}*f*

*) can be calculated by equating*

_{o}*im*(

*Y*

*)=0 and*

_{ine}*im*(

*Y*

*)=0.*

_{ino}*C*

*. The frequencies*

_{v}*f*

*and*

_{e}*f*

*are tuned by changing the varactor diode capacitance from 1 pF to 20 pF. Similarly, Fig. 4(b) shows the simulated resonant frequencies for different values of*

_{o}*L*

*. Here, the value of*

_{k}*C*

*is maintained at 1 pF. As highlighted in this figure, the even-mode resonant frequency moves lower as the*

_{v}*L*

*increases, however, the odd-mode resonant frequency remains constant. These results confirm that the separation between the even- and odd-mode resonant frequencies can be controlled by*

_{k}*L*

*.*

_{k}### 2. External Quality Factors

*W*

*and*

_{s}*L*

*, as well as the coupled line physical parameter*

_{s}*g*

_{2}. The external Q-factors can be extracted using the method proposed in [20] as follows:

*Q*

*and*

_{e}*Q*

*indicate for even and odd-mode external Q-factors, respectively. Similarly,*

_{o}*τ*

_{s}_{11}is the group delay at

*f*

*and*

_{e}*f*

*.*

_{o}*W*

*and*

_{s}*g*

_{2}, respectively. The external Q-factors increases with the values of

*W*

*and*

_{s}*g*

_{2}and the desired external Q-factors are therefore obtained by controlling

*W*

*and*

_{s}*g*

_{2}. Similarly, Fig. 6(c) shows the extracted Q-factor as a function of the center frequency. As indicated in this figure, the extracted Q-quality factors are nearly constant across a wide range of center frequencies. A summary of the step-by-step design method for the proposed BPF is provided in Fig. 7.

*j*10-to 50 Ω microstrip line BPFs as shown in Figs. 8 and 9, are compared with the coupling matrix synthesis results. The two TZs located at the lower and upper sides of the passband are generated by source-load coupling, which is implemented through a coupled line. The simulation results of microstrip line BPFs are consistent with the coupling matrix synthesis results.

*C*

*from 1.16 pF to 60 pF. Moreover, the two TZs are also tuned as the center frequency is tuned.*

_{v}*g*

*) between the coupled lines, as shown in Fig. 10, are analyzed. As shown in the figure, the TZs move slightly away from the passband as the value of*

_{c}*g*

*increases.*

_{c}### III. Simulation and Measurement Results

*j*10-to-50 Ω) are fabricated and measured using the Taconic substrate (dielectric constant

*ɛ*

*= 2.20 and thickness*

_{r}*h*= 0.787 mm, and loss tangent tanδ = 0.0009). Each tunable BPF was designed using a Chebyshev response with a passband return loss of 20 dB for an FTR between 2.10 GHz and 3 GHz. Variable capacitances are implemented using varactor diode SMV 1233-079LF (Skyworks Corporation), which provides diode capacitance between 1.1 pF and 60 pF at 2 GHz by varying the reverse bias-voltage between 15 and 0 V. The physical dimensions of the fabricated the BPFs are shown in Table 1.

*R*

*=*

_{S}*R*

*= 50 Ω,*

_{L}*X*

*=*

_{s}*X*

*= 0 Ω) tunable BPF. The measurement results are consistent with those of the simulation results, confirming that the center frequency is tuned from 2.1 GHz to 3.02 GHz (920 MHz or an FTR of 35.94%), while the insertion loss varies from 2.82 dB to 1.66 dB and the 3-dB BW varies from 238 to 265 MHz. Similarly, the measured return losses are better than 12.5 dB in the overall FTR.*

_{L}*R*

*= 25 Ω,*

_{S}*R*

*= 50 Ω and*

_{L}*X*

*=*

_{s}*X*

*= 0 Ω) tunable BPF. The measured center frequency is tuned from 2.2 GHz to 3.02 GHz (820 MHz) with an FTR of 31.42%. Similarly, the measured insertion loss varies from 2.4 dB to 1.67 dB whereas the 3-dB BW varies from 249 to 277 MHz. The measured return losses in the overall FTR are better than 12.5 dB.*

_{L}*j*10-to-50 Ω (

*R*

*= 20 Ω,*

_{S}*Xs*= 10 Ω and

*R*

*= 50 Ω,*

_{L}*X*

*= 0 Ω) tunable BPF. The center frequency is tuned from 2.10 GHz to 3.01 GHz (910 MHz) with an FTR of 35.62%. Similarly, the measured insertion loss varies from 2.55 dB to 1.76 dB whereas 3-dB BW varies from 242 to 282 MHz. The measured return losses in the overall FTR are better than 12 dB. Photographs of the fabricated filters are shown in Fig. 14.*

_{L}*N*+ 3 coupling based BPF design by incorporating a transistor model for small-signal as input conjugately matched input impedance. In [30], the arbitrary input and output port complex impedances BPF without any TZs is designed and fabricated using an SIR. However, since these previous studies [27–30] have experimentally demonstrated arbitrarily terminated BPF at a fixed center frequency, they might have faced difficulty in the physical realization of arbitrary input and output port impedance tunable BPFs. In contrast, the present study demonstrates arbitrarily terminated port impedance tunable BPFs (real-to-real and real-to-complex port impedances BPFs) over a wide FTR as well as two TZs. The result shows that the impedance transformer and frequency-selective tunable BPF can be integrated within a single circuit.

### IV. Conclusion

*j*10-to-50 Ω BPF) are designed and fabricated. The measurement results revealed that the center frequency is tuned across a wide frequency range.