## 2 Design Procedures  
This section presents the design steps of the proposed rat-race coupler. EM simulation in ADS (Advanced Design System) software, which is a powerful tool for designing and analyzing electromagnetic structures and devices in high frequency, is used for high-frequency simulation of the proposed design. The EM simulation method is highly accurate and can handle complex geometries, material properties, and boundary conditions. At the first step of the design procedure, the conventional rat-race coupler at the desired frequency is designed and explained. Since the desired main frequency is 950 MHz, a conventional RRC is designed and simulated using a Rogers 5880 substrate, where the obtained results are illustrated in Figure 1. As is seen, the conventional RRC suffers from extra-large size and also a lack of suppression band. Therefore, the aim of this research is to mitigate these disadvantages of the conventional RRC. The extra-large size of the conventional RRC corresponds to the large \(\lambda/4\) and \(3\lambda/4\) main transmission line branches between the ports. In the next steps, two compact equivalent resonators are proposed for use instead of the conventional large transmission line branches. The layout and frequency response of the \(\lambda/4\) transmission line branches are presented in Figure 2. As can be seen, in addition to the large size, there is no suppression band in the frequency response. Figure 1: First design of the conventional RRC: (**a**) layout and (**b**) frequency response at 950 MHz. Figure 2: The (**a**) layout and (**b**) frequency response of a quarter wavelength (\(\lambda/4\)) transmission line. The physical length of this transmission line is calculated based on the operating frequency of 950 MHz and the specifications of the Rogers 5880 substrate. The layout and frequency response of the \(3\lambda/4\) transmission line branches are presented in Figure 3. Since the size of the \(3\lambda/4\) transmission line is so large and as there is no suppression band in the frequency response, an equivalent filter is proposed that can be used instead of the \(3\lambda/4\) transmission line branches. By using this proposed filter, the size of the RRC can be decreased and the suppression band can be obtained. The phase characteristic of the conventional quarter wavelength (\(\lambda/4\)) and three-quarter wavelength (\(3\lambda/4\)) branches are shown in Figure 4. As can be seen, there is a 180 degree phase difference between the \(\lambda/4\) and \(3\lambda/4\) branches. Figure 4: The phase characteristic of the conventional quarter wavelength (\(\lambda/4\)) and three-quarter wavelength (\(3\lambda/4\)) branches. Figure 3: The (**a**) layout and (**b**) frequency response of a three-quarter wavelength (\(3\lambda/4\)) transmission line. The physical length of this transmission line is calculated based on the operating frequency of 950 MHz and specifications of Rogers 5880 substrate. The use of filters in microwave coupler and power divider design is an effective method for achieving compact size and harmonic suppression, in which compact filters are used instead of long conventional transmission lines. In this method, the filter and conventional line should have the same response at operating frequency. Filters are passive components that can be designed to selectively pass or reject certain frequencies depending on their configuration. One advantage of using filters in microwave coupler and power divider design is that they allow for a more compact design instead of using long conventional transmission lines, which can reduce overall size of device. This is particularly useful in applications where space is limited, such as in mobile devices or satellite communication systems. Another advantage of using filters is that they can suppress harmonics. Harmonics are unwanted frequencies that can be generated by nonlinear devices, such as amplifiers or mixers. These harmonics reduce the overall performance of the system.