# Non-Magnetic Circulator Based on a Time-Varying Phase Modulator
## 2 Gyrator Design
### Conversion Loss  
Figure 8a shows the phase modulation of the single-frequency sine wave in the time domain by the TVPM with the 180\({}^{\circ}\) differential phase shifter. It is worth noting that during the process of switching in the TVPM, the phases of the carrier waves can be modulated by the capacitors and inductors, and time is needed to stabilize the carrier output. Such a loss cannot be avoided during the phase conversion process. A partial carrier signal could  
Figure 6: The phase difference between the forward and the reverse transmissions.  
Figure 7: The phase change is caused by the periods of the control signals.  
be converted into the clutter of other frequencies, as shown in Figure 8b. Therefore, the transmission process of the periodic carrier signal needs to be long enough to reduce the loss of carrier signal [19].  
In addition, according to Reference [16], the switching channels correspond to the mixer operation. With the frequency of the input signal \(f_{in}\), and the control signal \(f_{c}\). Figure 8b shows the frequency losses. When the input signal (black curve) passes through the first TVPM, the frequency is converted, and two frequency signals of \(f_{in}-f_{c},f_{in}+f_{c}\) are generated for output, namely the signal of the intermediate stage (blue curve). The  
Figure 8: (**a**) Switch and phase shifter loss. (**b**) Frequency loss.  
non-reciprocal device requires two different TVPMs. The input signal entering into the second TVPM is converted into other frequencies after passing through the first TVPM and buffer unit. Therefore, the bandwidth of a single TVPM seriously affects the amplitude of the output signal (red curve). The TVPM demonstrated in the paper is a differential \(180^{\circ}\) phase shifter, the bandwidth of \(-3\) dB is 0.8-1.2 GHz, and the signal with frequency at the boundary of this bandwidth has a large suppression loss (\(-3\) dB) compared with the central frequency of 1 GHz. When the signal passes through two TVPMs, the signal amplitude at the center frequency is weakened (\(-2\) dB). This can be mitigated by extending the period of the control signal or increasing the bandwidth of the phase shifter unit.  
In general, the switching process and phase shifting process could cause partial signals to be converted to other frequency components and cannot be resumed. In addition, during the mixing process, the intermediate signal is also suppressed when the frequency reaches the bandwidth boundary of the phase shifter. These two conversion losses cannot be avoided but can be mitigated by increasing the period of the control switch signal and increasing the bandwidth of the phase-shifting unit.  
In the ideal situation, the time loss can be neglected, \(-\)and the period of the control signal is \(T_{\mathrm{C}}\). The phase control function of TVPM in the time domain can be written as:  
\[\varphi(t)=\left\{\begin{array}{ll}e^{i\frac{\pi}{2}}&,nTc\leq t<(n+0.5)Tc\\ e^{-i\frac{\pi}{2}}&,(n+0.5)Tc\leq t<(n+1)Tc\end{array}\right. \tag{6}\]  
Table 2 lists the comparison with the other published results [15; 16; 19]. For scientific and effective exploration, the data listed in [15; 16], and this paper are all simulation results. The data show that the proposed differential phase shifter structure can be used to obtain a larger bandwidth than that in Reference [15]. Compared with Reference [16], the overhead area of the circuit is reduced by one degree of freedom.