## 2 Design Procedure  
### Proposed Coupling System  
The conventional approach to creating a bandpass response is the use of capacitive gaps in the microstrip filters. However, the quality of the passband response can be significantly improved using multiple coupled transmission lines with different electrical lengths and capacitance gaps, leading to some traditional bandpass filters such as end-coupled, comb-line and hairpin-line BP filters [31, 60]. Despite their straightforward design approaches, their dimensions are undesirably large due to the large number of coupled lines required for a passband with relatively small insertion loss. In this paper, a new coupling system inspired by the end-coupled lines is proposed, and a systematic miniaturization procedure is presented, which can be used for dual-band bandpass filters with independent passband adjustability. The advantage of the proposed method is that, unlike the abovementioned methods, only one end-coupled structure is required, contributing to a compact configuration. Figure 1a shows a layout of one end-coupled transmission line to create a transmission pole at 7.66 GHz as shown in Figure 1b. The length of the coupled lines in this structure is close to half a guided wavelength at the first resonant frequency, which is inspired by the conventional end-coupled, half-wavelength resonator filters where a weak capacitive coupling is created through the gap between the two adjacent open ends. However, the conventional end-coupled filters are relatively large, due to the end to end orientation of the coupled lines [60]. To reduce the circuit size, the end-coupled lines are bent as shown in Figure 2a, resulting in a dual-band performance as depicted in Figure 2b. It needs to be mentioned that the effective length of coupled lines before and after bending remains almost same, and hence, the locations of the transmission poles remain almost unaffected by bending the coupled lines. Figure 1: End-coupled transmission line: (**a**) Layout, (**b**) electromagnetic (EM) simulation. In order to further investigate the frequency behavior of the proposed structure and predict and control the resonant frequencies, an equivalent lumped circuit model can be extracted for the structure. The use of lumped-element circuit (LC) models for analyzing microwave filters' behavior is a well-known approach and explained in [6; 56; 61; 62]. The effect of the bends on the frequency response of the coupling system can be modeled by six capacitors (C\({}_{\text{bend}}\)) as illustrated in a simplified lumped-element circuit (LC) model shown in Figure 3a. In the LC circuit, C\({}_{1}\), C\({}_{3}\), C\({}_{4}\) and C\({}_{6}\) denote the gap capacitances. C\({}_{2}\), C\({}_{5}\), C\({}_{7}\) and C\({}_{11}\) describe the capacitances of the open-end stubs. The inductances of transmission lines are also modeled by L\({}_{1}\), L\({}_{2}\), L\({}_{3}\), L\({}_{4}\) and L\({}_{5}\), and the bending capacitances are denoted by C\({}_{8}\), C\({}_{9}\) and C\({}_{10}\). The values of the lumped elements are obtained using the procedure outlined in [53; 54], and calculated as follows: C\({}_{1}\) = 0.107 pF, C\({}_{2}\) = 0.73 pF, C\({}_{3}\) = 0.3 pF, C\({}_{4}\) = 0.05 pF, C\({}_{5}\) = 1.289 pF, C\({}_{6}\) = 0.045 pF, C\({}_{7}\) = 7 pF, C\({}_{8}\) = 0.535 pF, C\({}_{9}\) = 7 pF, C\({}_{10}\) = 5 pF, C\({}_{11}\) = 0.65 pF, L\({}_{1}\) = 0.4 nH, L\({}_{2}\) = 0.34 nH, L\({}_{3}\) = 0.5 nH, L\({}_{4}\) = 0.435 nH and L\({}_{5}\) = 0.75 nH. It can be seen from Figure 3b that introducing the bending capacitors improves the passband response, reducing the passband insertion loss from 7.05 to 0.66 dB. The first and second resonant frequencies of the filter shown in Figure 3b can be calculated by equating the input impedance of the equivalent circuit model to zero as shown in (1). As a result, the equations of resonances are obtained as (2) and (3).