I. Introduction
The arbitrary termination impedance (ATI) bandpass filter (BPF) plays an important role in modern communication systems. Fig. 1(a) shows the conventional radio frequency (RF) transmitter of a wireless system [1]. The ATI BPF can directly match the output impedance of the power transistor or the input impedance of the antenna to reduce the overall circuit size, insertion loss, cost, and complexity of the network as can be seen in Fig. 1(b). An unequal termination impedance power divider with a bandpass filtering response and a very low impedance transforming ratio (r) were proposed in [2]. The analysis was based on the characteristics of couple transmission lines (TLs).
Similarly, an arbitrary real-to-real termination impedance parallel-coupled TL BPF was presented in [3]. The BPF consisted of the multi-stage parallel-coupled TLs with different even- and odd-mode impedances. Recently, ATI BPF matching networks using λ/4 stepped impedance resonators were proposed in [4]. The structure consisted of open-ended parallel-coupled TLs and T-type TLs that were cascaded alternately. However, the aforementioned BPFs can be designed with Butterworth or Chebyshev responses which do not consider transmission zeros (TZs) for high out-of-band signal suppression. Moreover, those BPFs were analyzed and realized by using couple TLs, which may be difficult to fabricate when the center frequency (f0) is increased to a higher operation band.
Substrate-integrated waveguide (SIW) filters with either real or complex termination impedance (CTI) are important circuits widely used in microwave applications, with a relatively high quality factor and low insertion loss (IL). As demand increases for a broader channel bandwidth and high out-of-band signal suppression, the demand for a quasi-elliptic filter (QEF) with finite TZs also grows. An equal termination impedance QEF with a broadband post-loaded electric coupling structure on a single-layer SIW was proposed in [5]. Further, an SIW QEF with controllable mixed electric and magnetic couplings using a three-layer printed circuit board (PCB) was proposed in [6]. Compact frequency-selective cascading SIW cavities with a quasi-elliptic response were also proposed based on the dual-mode filter theories found in [7]. Most recently, an SIW cavity QEF with slot coupling and non-adjacent cross-coupling was presented in [8]. In their designs, the slots etched on the top metal plane of the SIW cavity were used to produce electrical coupling, and cross-coupling was realized by the microstrip TL on the SIW cavity.
However, slots etched on the SIW cavity may increase the IL of the filter due to an increase in radiation loss. On the other hand, the SIW QEFs presented in [5–8] were designed with equal termination impedances of 50 Ω to 50 Ω. In [6–8], the circuits were implemented on a multi-layer PCB, which was a very difficult design process and had a high fabrication price. Recently, a synthesis approach for the design of a Chebyshev BPF with complex frequency variant loads was proposed in [9]. The proposed approach was based on the power wave renormalization theory and legitimate assumptions.
In this paper, a new SIW QEF with arbitrary real and/or complex termination impedances is introduced. The proposed SIW QEF can be designed easily by adding a parallel J-inverter between the first and last resonators, which is represented as cross-coupling. Furthermore, the location of the pair TZs is controllable for a good frequency-selective response. The proposed unequal real-to-real and complex-to-real termination impedance SIW QEFs are designed on a single-layer PCB that provides simpler design processes and a cheaper fabrication price.
II. Design Equations and Procedures
Fig. 2(a) shows an equivalent circuit with shun LC resonators and admittance inverters of the proposed ATI QEF, in which the ATIs are ZS =RS ± jXS and ZL = RL ± jXL. Fig. 2(b) presents the modified equivalent circuit of Fig. 2(a), which is based on the conventional ATI BPF presented in [10]. This QEF can be designed for ATIs with even coupled resonators. As mentioned in [10], matching the reactive parts of the termination impedances requires detuning the first and last resonators. In terms of capacitive or inductive compensations, the detuning frequency of the CTI using series reactive components is more beneficial than the addition of shunt reactive components. For fourth-order QEF, the first (fS1) and last (f4L) resonance frequencies are detuned from the f0 on the basis of termination impedances and can be defined as follows:
where RS,L and XS,L are the real and imaginary components of the source and load impedances, respectively. FBW is the fractional bandwidth of the passband, while gi (i = 0, 1, 4, 5) is the element value of the low-pass prototype. The intermediate resonators are not changed by the CTI. Ci =1/ω2Li
The capacitance of the parallel resonator can be calculated with Ci =1/ω2Li by choosing an arbitrary value for the inductor (Li) of the shunt parallel resonator. The slope parameters of the first, intermediate, and last resonators can be calculated using the following equations:
where ωS1 and ω4L are the detuned angular frequencies (which can be found from Eq. (1)) and ω0 is an angular center frequency.
Once the detuned frequencies and slope parameters have been determined, the J-inverters of the coupled-resonator can be defined as follows:
The transfer function and design examples of the conventional QEF were presented in [11]. A similar approach can be used to determine the frequency location of the pair TZs:
where Ωa and ωTZ–H are the frequency variable normalized to ω0 and the angular frequency of TZ at the higher side, respectively.
The quasi-elliptic response can be achieved through the addition of a parallel J14 between the first and fourth resonators. The pair TZs are located at the frequency fTZ = f0 ± a. Furthermore, J14 is the function of Ωa (or fTZ–H) and can be determined through Eq. (5):
The filter will be mismatched once J14 is added. To maintain the required return loss (RL) at the operating band, the values of J12 and J34 must be slightly modified. Fig. 3 presents the correction factor values used to calculate the new
J 12 ′ J 34 ′
where the correction factors are α12 for J12 and α34 for J34. These correction factors are critical to determining the location of TZs near the f0. If the TZs are located slightly far from f0 or a > 1, J14 exerts only a small effect on the RL of the filter and results in very small cross-coupling. The external quality factors and coupling coefficients of the resonators are defined as follows [8]:
where QS1 and Q4L are the external quality factors of the first and fourth resonators, respectively. Then, Ki, i+1 is an inline coupling coefficient, while K14 is cross-coupling coefficient of the resonators. To demonstrate the analysis, the QEFs are designed with the CTI and the different locations of TZs. The lossless elements of the J-inverter and the LC resonators are used in the simulation. The circuit model is simulated with the Advanced Design System (ADS) software. Fig. 4 provides the S-parameters of the QEFs (FBW = 5% and |S11| = 20 dB), according to the location of the TZs. The QEFs were designed with source termination of ZS = 15 + j19 Ω and the load termination impedance of ZL = 50 Ω. The required RL can be obtained by using the proper correction factors.
III. Design of SIW QEF with CTI
The SIW QEFs with ATI are designed using positive and negative couplings with via-hole windows. The proposed SIW QEF was designed with f0 of 10 GHz, FBW of 5 %, and |S11| of 20 dB. The frequency locations of TZ are 8.6 GHz and 11.4 GHz (a = 1.4). For an unequal real-to-real case, the ATIs are ZS = 20 Ω and ZL = 50 Ω. Further, L1= L2 = L3 = L4 = 2 nH was selected, while C1 = C2 = C3 = C4 = 0.1266 pF is calculated from Eq. (2). Using Eq. (3)J-inverters can be determined as J01 = 0.004616, J12 = 0.000362, J23 = 0.000278, J34 = 0.000362, and J45 = 0.00292. During the final step, J14 = −0.000088 is calculated using Eq. (5).
For a complex-to-real case, the ATIs are ZS = 17 − j40 Ω and ZL = 50 Ω. Further, fS1 is calculated from Eq. (1) and detuned to 10.65 GHz. Then, L1= L2 = L3 = L4 = 2 nH was selected, while C1 = 0.1116 pF and C2 = C3 = C4 = 0.1266 pF. Similarly, J01 = 0.004852, J12 = 0.000351, J23 = 0.000278, J34 = 0.000362, J45 = 0.00292, and J14 = −0.00000761 are calculated. Because a > 1, the modifications of J12 and J34 for maintaining |S11| = 20 dB are not required.
In praxis, shunt LC resonators can be constructed in various forms, such as a TL resonator, waveguide resonator, or SIW resonator [12, 13]. The external quality factor of the source (QS) and the load (QL) can be extracted from the electromagnetic (EM) simulation. In addition, QS and QL can be calculated using Eq. (8).
where Δf±3dB is a 3-dB bandwidth.
The coupling coefficients between resonators can also be extracted from the EM simulation using Eq. (9) for synchronously tuned coupled resonators.
where fH and fL denote the higher and lower resonant frequencies, respectively. A quality factor of around 500 can be obtained from an eigen-mode simulation.
Fig. 5 illustrates the inline coupling coefficient (Ki,i+1) of two rectangular SIW cavities. The Ki, i+1 is increased as the width of the iris window (W1) expands. Similarly, Fig. 6 shows the leakage or cross-coupling coefficient (K14) according to the viahole wall spacing (l1) between the first and fourth resonators. K14 has been shown to increase rapidly as l1 grows.
IV. Simulation and Measurement Results
The SIW QEFs with unequal real-to-real and complex-to-real ATIs were designed and implemented on RT/Duriod 5880 substrate with ɛr = 2.2 and h = 0.787 mm.
The layouts with dimensions of the proposed unequal real-to-real ATI SIW QEFs, as well as a photograph, are provided in Fig. 7. The measurement results are consistent with those obtained by the simulation. Fig. 8(a) depicts the measured matching impedance point of the proposed real-to-real ATI SIW QEF on a Smith chart. The measured real source termination impedance is well matched to that obtained by the EM simulation. Similarly, the S-parameters within frequency range of 2–18 GHz are likewise presented in Fig. 8(b); a measured 3 dB bandwidth range between 9.75 and 10.25 GHz (FBW = 5%) is obtained. The measured minimum |S11| is 17.4 dB and maximum |S21| is 0.93 dB in the passband. The spurious response is produced at approximately 14.5 GHz (1.45f0). The stopband attenuation better than 35 dB at the lower side of the passband is between 2 GHz and 9 GHz and at the higher side of the passband is between 11 GHz and 14.3 GHz. Similarly, a minimum attenuation better than 18.5 dB is obtained between 14.4 GHz and 18 GHz.
The layouts with dimensions of the proposed complex-to-real ATI SIW QEFs, as well as a photograph, are provided in Fig. 9. Further, Fig. 10(a) depicts the measured matching impedance point of the proposed complex-to-real ATI SIW QEF on a Smith chart. The measured complex source termination impedance is well matched to that obtained by the EM simulation. The S-parameters within the frequency range of 2–18 GHz are likewise presented in Fig. 10(b); a measured 3 dB bandwidth range between 9.74 and 10.25 GHz (FBW = 5.1%) is obtained. The measured minimum |S11| is 18 dB and maximum |S21| is 1.2 dB in the passband. The spurious response is produced at approximately 14.5 GHz (1.45f0). The stopband attenuation better than 40 dB occurs at the lower side of the passband between 2 GHz and 9 GHz and at the higher side of the passband, between 11 GHz and 14 GHz. Similarly, a minimum attenuation better than 16 dB is obtained between 14.1 GHz and 18 GHz.
The performances of the proposed SIW QEF with ATIs are compared with state-of-the-art BPFs and SIW QEFs in Table 1. The proposed SIW QEFs are designed on a single-layer PCB with unequal real-to-real and complex-to-real ATIs, which are inexpensive and uncomplicated to fabricate. In view of the conductor and dielectric losses, which are related to the operating frequency and FBW, the proposed SIW QEF provides a smaller IL in the passband than conventional SIW QEFs.
V. Conclusion
A new design for an SIW QEF with ATI is proposed and investigated in this paper. The proposed SIW QEFs provide high frequency selectivity and out-of-band signal suppression. The SIW QEF with ATI may be useful in, for example, the input and/or output matching networks of the power amplifier, which may reduce the insertion loss and complexity of the transmitter in the RF front end. In addition, the proposed SIW QEFs design method can be applied to a microwave circuit and system designs.
PDF Links
PubReader
ePub Link
Full text via DOI
Download Citation
Print
