# Non-Magnetic Circulator Based on a Time-Varying Phase Modulator
## 2 Gyrator Design
### Circuit of Gyrator  
The phase-shifting unit can be operated instantaneously without any additional frequency. To realize the function of the gyrator, it is necessary to ensure that the phase change after the output is 180\({}^{\circ}\) when the carrier wave enters in different directions. It is assumed that the phase-shifting units of the two TVPM phase-shifting functions \(\varphi_{1}(t),\varphi_{2}(t)\) are the same, as shown in Figure 3. The phase-shifting of each TVPM can be \(\varphi_{A},\varphi_{B}\). The difference lies in the phase-shifting of one TVPM being delayed by the other, with the delay time of the buffering unit \(\tau\). The total phase change of the input signal is affected by the delay transmission line, which acts as the buffering unit. However, this section explores the difference between the phase change of the propagating transmission and the opposite transmission. Therefore, unless otherwise mentioned, the phase change of the signal caused by the intermediate buffering unit is ignored in the analysis of the gyrator in the following. The relationship between the functions of the two TVPMs for phase modulation is expressed by the following:  
\[\varphi_{1}(t)+\varphi_{2}(t-\tau)=\varphi_{2}(t)+\varphi_{1}(t-\tau)\pm\pi\ \tau=\Delta L/v_{g} \tag{2}\]  
The left side of Equation (2) is the phase change caused by the propagating transmission of the gyrator, while the right side is the phase change caused by the opposite transmission, with a phase difference of 180\({}^{\circ}\). Figure 3: Structure of gyrator. The nonlinear multivariate equation can be solved by the numerical analysis method. The period Tc controlling of the TVPM signal can be normalized into unit 1. Phase shift \(\varphi_{A},\varphi_{B}\) are 0, 1, separately. The periods of the control signals are the same, and both are periodic signals with a duty cycle of 50%. The two TVPMs operating the phase transformation of the carrier wave at different times are:  
\[\varphi_{1}(t)=\varphi_{2}(t-t_{1})=\{0,1\} \tag{3}\]  
\(t_{1}\) is the delay time caused by the buffering unit between the right and left TVPM. It is advisable to set the _PWM_ signal function with a duty cycle of 50%, amplitude of 0,1, and initial value of 0 as _PWM_ (\(t\)); the equation after subtraction is as follows:  
\[PWM(t)+PWM(t-\tau-t_{1})=PWM(t-t_{1})+PWM(t-\tau)\pm 1 \tag{4}\]  
Four sets of solutions are finally solved, as shown in Table 1:  
When \(\tau=t_{1}=0.25\), the delay of the buffering unit and the delay of the right to left TVPM are one-quarter of the controlling signal period _Tc_. When the carrier wave is propagating, as shown in Figure 4a, it is assumed that the carrier wave enters the port in the first half period of the control signal and it enters the right arm TVPM after _Tc_/4. Currently, the change of the carrier phase on the right TVPM is \(\varphi_{A}\) as same as that on the left ones. Therefore, the total phase change of the carrier wave from the port is \(2\varphi_{A}\). When the carrier enters the port in the second half period of the control signal, the carrier output after _Tc_/4 and the total phase change is \(2\varphi_{B}\). When the carrier is transmitted in the opposite way, as shown in Figure 4b, compared with the analysis of forward transmission, the total phase change of the carrier in the output port of the first and second half periods of the control signal is \(\varphi_{A}+\varphi_{B}\). Above all, if \(\varphi_{A}-\varphi_{B}=\pi\), the circuit can achieve the performance of the gyrator. The phase difference between the two carriers transmitted in different directions is \(\pi\). The other three sets of solutions are the same as the above analysis, and all can achieve the function of the gyrator. The 180\({}^{\circ}\) differential phase shifter mentioned above can be used to construct the TVPM, and the circuit structure of the gyrator can be further obtained. Figure 5 shows the circuit structure of the gyrator. The TVPM of the left and right arms are differential 180\({}^{\circ}\) phase shifters with the same structure.