# Non-Magnetic Circulator Based on a Time-Varying Phase Modulator
## 2 Gyrator Design
### Time-Varying Phase Modulator (TVPM)  
If a single unit of the phase shifting is placed on a single transmission path, it is a linear time-invariant system. In the system, the transmission of the signal satisfies the reciprocity and the time-reversal symmetry. By connecting two different phase shifters in parallel, when the transmitted carrier wave passes through different phase shifters at different times, the time-reversal symmetry of the phase shifting system can be broken.  
The principle of the time-varying phase modulator is shown in Figure 2a. The cylindrical green and purple parts are the units of the actual phase modulator, which can, respectively, change the phase of the transmission wave with \(\varphi_{A},\varphi_{B}\) and the change of phase varies with time. The signal controlling the rotating device can be used to change the modulation time of the phase modulator. The function of the TVPM acting on the phase of the carrier wave can be written as:  
\[\varphi(t)=\left\{\mathbf{e}^{i\varphi_{A}},\mathbf{e}^{i\varphi_{B}}\right\} \tag{1}\]  
The duration of the phase shift of the carrier wave with a single TVPM can be determined by the controlling signal and the magnitude of the phase, which makes the phase modulator more flexible.  
Figure 2b shows a circuit realized by a time-varying phase modulator. The phase-shifting unit is composed of switches and lumped elements. The TVPM can be constructed  
Figure 2: Structure of the proposed TVPM. (**a**) The diagram of the proposed TVPM. (**b**) Circuit realization method and key parameters of the proposed TVPM.  
based on the elements. The circuit is a three-platform T-type 180\({}^{\circ}\) high/low pass phase shifter in which C1 = C2 = 3.18 pF, L1 = L2 = 7.96 nH. Notably, it is still a structurally symmetric reciprocity device but breaks the time-reversal symmetry. Using NMOS as the switch, the control signal can be loaded on the gate of NMOS to control the port. During a single period, the control signal has only two states (high and low). When the switch is on at a high level, the signal can be passed. When the switch is off at a low level, the signal is completely reflected.  
As shown in Figure 2b, the two control signals are Ctr and \(\overline{\text{Ctr}}\), which are inverse to each other. The two signals are loaded on the NMOS gates of the upper and lower arms, respectively. When a signal is transmitted in the first half period (green part), Ctr is at a high level and \(\overline{\text{Ctr}}\) is low. So, the switches SW1 and SW2 are off, and SW3 and SW4 are on. The signal can pass through the upper arm of the TVPM. Similarly, in the second half period (purple part), the signal can pass through the lower arm. The upper and lower channels can be opened alternatively at different times. The upper and lower channels can generate 90\({}^{\circ}\) and \(-\)90\({}^{\circ}\) phase shifts for the 1 GHz sine wave, respectively. During one period of the control signal, the upper channel in the first half period is switched on, while the lower channel is off. At this time, the phase change of the carrier wave is determined by the low-pass network. Similarly, the phase change of the carrier wave in the second half cycle is determined by the high-pass network in the lower channel. The phase change of the carrier wave in the first half period is \(-\)90\({}^{\circ}\), and the phase change in the second half period is 90\({}^{\circ}\). In addition, the phase-shifting element of the differential construction guarantees a considerable bandwidth for the composed gyrator, which will be described further in the following.