# Constructing a Microstrip Phase Shifter with Low Phase Error Using a Y-Resonator
## 3 Analysis and Design  
Figure 3 shows the schematic diagram of the equivalent circuit of the proposed phase shifter, where port 1 to port 2 are the main lines and port 3 to port 4 are the reference  
Figure 1: Topology of the Y-type dual-mode microstrip phase shifter. Figure 2: Schematic diagram of the Y-type microstrip dual-mode resonator with a parity mode resonance equivalent circuit. (**a**) Y-Resonator Topology (**b**) Even-mode resonant equivalent circuit (**c**) Odd mode resonance equivalent circuit. lines. The phase shift value of the phase shifter is the difference in the inserted phase shift between the main lines and the reference lines:  
\[\Delta\Phi=\Delta\Phi_{m}-\Delta\Phi_{r}, \tag{5}\]  
where \(\Delta\Phi_{m}\) is the main line insertion phase shift, and \(\Delta\Phi_{r}\) is the reference line insertion phase shift. According to the microwave network theory, the main and reference line insertion phase shifts of the phase shifter can be represented by the ABCD matrix as follows:  
\[\Delta\Phi_{m}=\angle S_{21}=-tan^{-1}\frac{B+CZ_{0}^{2}}{j(A+D)Z_{0}^{2}}, \tag{6}\]  
\[\Delta\Phi_{r}=\angle S_{43}=-K\theta. \tag{7}\]  
In order to keep the phase shift value constant for a certain pass band of phase shift values, the phase slopes of the main and reference lines should be the same, i.e.,  
\[\frac{d\Delta\Phi_{m}}{df}\mid_{f=f_{0}}=\frac{d\Delta\Phi_{r}}{df}\mid_{f=f_{ 0}}=-K\frac{\theta_{0}}{f_{0}}. \tag{8}\]  
For an ideal phase shifter, the phase difference between the main line and the reference line is constant over a frequency range, i.e., the phase shift value is constant. In practice, the phase difference between the two is somewhat variable over a range of operating frequencies. Assuming that the calibration phase shift value is \(\Delta\Phi_{0}\), the maximum phase shift value in the working band is \(\Delta\Phi_{max}\), and the minimum phase shift value is \(\Delta\Phi_{min}\). As shown in Figure 4, the phase deviation (PD) is defined as the maximum difference between the phase shift value and the calibration phase shift value in the working band:  
\[PD=max|\ \Delta\Phi_{max}-\Delta\Phi_{0}\mid,\mid\Delta\Phi_{min}-\Delta \Phi_{0}\mid. \tag{9}\]  
The schematic simulation results show that the proposed microstrip phase shifter can achieve a stable phase shift value of \(225^{\circ}\) over the operating bandwidth within the phase deviation PD. The reflection and transmission coefficients can be expressed as follows:  
\[S_{11}=\frac{\Gamma_{e}+\Gamma_{o}}{2}, \tag{10}\]  
\[S_{21}=\frac{\Gamma_{e}-\Gamma_{o}}{2}, \tag{11}\]  
Figure 3: Equivalent schematic of the proposed wideband phase shifter. where \(\Gamma_{e}\) and \(\Gamma_{o}\) denote the odd-mode reflection coefficient and even-mode reflection coefficient, respectively, with respect to the odd/even-mode input impedance:  
Figure 2:0  
Figure 2:1  
The parity mode input impedance in (1) and (3) is brought into (12) and (13) to yield the parity mode reflection coefficients, and then into (10) and (11) to yield the S-parameters. The ABCD matrix of the main line is obtained from (6):  
\[ \begin{bmatrix}2 jz_{1}z_{2}tan\theta_{1}^{2}-2 jz_{1}z_{2}&2z_{1}^{2}z_{2}tan \theta_{1}-z_{1}^{3}tan\theta_{1}^{2}tan\theta_{2}\\ 2z_{2}tan\theta_{1}+z_{1}tan\theta_{2}&2 jz_{1}^{2}tan\theta_{1}tan\theta_{2} \end{bmatrix} . \tag{14}\]  
The phase shift value \(\Delta\Phi\) can be obtained by bringing the required S-parameters into (6) and (7), and the transfer function F of the phase shift network is expressed as follows:  
\[|S_{21}|^{2}=\frac{1}{1+|F|^{2}}\text{ and }F=\frac{B-C}{2}. \tag{15}\]  
The transfer function F can be found by bringing the ABCD matrix of the phase shift network into the above equation. For the reference line, its ABCD matrix is, according to microwave theory, as follows:  
\[ \begin{bmatrix}cosK\theta&jz_{0}sinK\theta\\ \frac{sinK\theta}{z_{0}}&cosK\theta\end{bmatrix} .