# Non-Magnetic Circulator Based on a Time-Varying Phase Modulator
## 3 Circulator Implementation  
A circulator device is implemented by the above-mentioned gyrator and multiple reciprocity devices [19]. As shown in Figure 9, the conduction resistance of the switch is set to 0.1 \(\Omega\). The center frequency of the circulator is 1 GHz.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Reference** & **[**19**]1 ** & **[**15**]** & **[**16**]** & **This Work** \\ \hline Architecture & Magnetic ferrite & N-path filter & Conductivity & TVPM \\ ANT port & Single end & Single end & Differential end & Single end \\ Frequency & 1 GHz & 0.75 GHz & 25 GHz & 1 GHz \\ BW(\(-3\) dB) & 100 MHz & 40 MHz & Infinite 2 & 400 MHz \\ Insertion loss & \(-1\) dB & \(-2\) dB & \(-0.5\) dB & \(-1.7\) dB \\ C/M ratio 3 & N/A & 1 & 3 & 5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison between different methods to achieve the gyrator.  
The transmission line is an ideal one, with the characteristic impedance is 50 \(\Omega\). The delay unit is based on the transmission line, with a delay time of 1.25 ns. The RX, ANT, and TX ports are denoted by port1, port2, and port3, respectively. According to the definition of the scattering matrix, to obtain the specific value of the scattering matrix of the three-port system, it is necessary to ensure that only one port is excited and the other two ports are perfectly matched without reflecting waves.  
\[S_{ij}=\frac{V_{i}^{-}}{v_{j}^{+}}\mid_{v_{k}^{+}=0\
eq j} \tag{7}\]  
For example, it is necessary to ensure that port2 and port3 are matched to obtain the values of the three scattering matrix elements under the excitation of port1. The time-domain waveform of the voltage values of the three ports is obtained by using the sine wave excitation of port1, and the response amplitude and phase of each port at the center frequency are obtained by Fourier transform. Using Equation (7), \(S_{11}\), \(S_{21}\), \(S_{31}\) can be obtained.  
Figure 10 shows the method of calculating the S-parameters. The matching of the port impedance has a great impact on the performance of the circulator. Theoretically, the impedance transparency of the \(\pm 90^{\circ}\) rotary device [20] can be used to calculate the matching impedance of each port. During the process of frequency conversion, it cannot be guaranteed that the impedance of each port will continue to match all of the time, so there are still some errors.  
Figure 10: Method of calculating S-parameters.  
Figure 9: Circuit of the designed circulator based on the TVPM.  
Figure 11 shows the simulation results of the time domain and scattering parameters. S11, S21, and S31 parameters are shown in Figure 11a. S12, S22, and S32 parameters are shown in Figure 11b. Figure 11c shows S13, S23, and S33. Based on the simulation results of all matrix elements of the scattering matrix, it can be demonstrated that the directional transmission of the circulator and the isolation performance between ports can be realized. The insertion loss at the center frequency is \(-1.7\) dB, and the isolation is \(-20\) dB with the antenna as the excitation port. The actual insertion loss of the circulator is smaller than \(-3\) dB. The good insertion loss performance can be attributed to the fact that the gyrator is placed between the incoming transmission lines leading to the reuse of part of the original reflected loss [21].  
The scattering matrix structure calculated by the above simulation method still has errors. However, the method of time domain conversion to the frequency domain is still a relatively simple and intuitive approach.