## 3 Theoretical Analysis and Simulation of the Frequency Synthesis Chain
### Analysis of the Model for the PLLs
Figure 3 illustrates the utilization of the 200 MHz clock to drive a comb generator, generating harmonics of 200 MHz. We denote the order of these harmonics as \(M_{3}\), and the \(M_{3}\) order of the 200 MHz harmonic as \(u_{m3}(t)\). Simultaneously, another 200 MHz clock serves as a sampling clock for the DDS to produce a 7.368 230 MHz clock. We denote the division factor of the DDS module as \(M_{2}\), and the signal derived from the DDS module as \(u_{m2}(t)\). The calculations for these two signals are as follows:  
\[\Theta_{m1}(s)=M_{1}\Theta_{i}(s), \tag{1}\]0  
\[\Theta_{m1}(s)=M_{1}\Theta_{i}(s), \tag{1}\]1  
where \(a_{m2}\) and \(a_{m3}\) represent the amplitude arguments of the respective signals. In PLL2, the center frequency of the DRO is denoted as \(\omega_{m2}\), and \(u_{0}(t)\) represents the output signal of the DRO. For our design, the center frequency \(\omega_{m2}\) is approximately 9.192 GHz. In the time domain, \(u_{0}(t)\) is given by the equation  
\[\Theta_{m1}(s)=M_{1}\Theta_{i}(s), \tag{1}\]2  
where \(a_{0}\) represents the amplitude and \(\theta_{0}\) is the initial phase of the DRO. The signal \(\omega_{2}\) is mixed with the 9.2 GHz signal from the comb generator to generate a clock at 7.368 MHz. This 7.368 MHz clock is then compared with the 7.368 MHz signal generated from the DDS. We define the output signal generated by the mixer as \(u_{p}(t)\):  
\[\Theta_{m1}(s)=M_{1}\Theta_{i}(s), \tag{1}\]3  
Equation (11) describes \(u_{p}(t)\) as having two parts. The first part is a higher-frequency signal, which is eliminated by a low-pass filter (LPF). The latter part represents a 7.368 MHz signal, which is compared with the 7.368 MHz signal from the DDS in PD2. The phase argument of the error signal generated by PD2 is denoted as \(\theta_{e2}(s)\). In the complex frequency domain, based on Equations (7)-(11), the phase error signal \(\theta_{e2}(s)\) of PD2 can be expressed as  
\[\Theta_{m1}(s)=M_{1}\Theta_{i}(s), \tag{1}\]4  
where \(\theta_{0}(s)\) represents the phase argument of the DRO output. The transfer function \(G_{2}(s)\) is defined as the loop gain of PLL2 shown in \[\Theta_{m1}(s)=M_{1}\Theta_{i}(s), \tag{1}\]5 Thus, the phase argument of the DRO output \(\Theta_{0}(s)\) can be expressed as \[\Theta_{m1}(s)=M_{1}\Theta_{i}(s), \tag{1}\]6 \[\Theta_{m1}(s)=M_{1}\Theta_{i}(s), \tag{1}\]7 \[\Theta_{m1}(s)=M_{1}\Theta_{i}(s), \tag{1}\]8  
In Equation (13), \(k_{d2}\) represents the detector gain of the DRO, \(k_{d2}\) is the DRO gain, and \(F_{2}(s)\) is the transfer function of the second loop filter in PLL2. Consequently, the phase-closed loop transfer function \(F_{2}(s)\) of PLL2 can be derived as  
\[\Theta_{m1}(s)=M_{1}\Theta_{i}(s), \tag{1}\]9  
Based on the aforementioned analysis, the phase-closed loop transfer function of the microwave frequency synthesis chain \(H(s)\) can be expressed as  
\[U_{e1}(s)=k_{d1}\Theta_{e1}(s)=k_{d1}(\Theta_{o1}(s)-\Theta_{m1}(s)), \tag{2}\]0 \[U_{e1}(s)=k_{d1}\Theta_{e1}(s)=k_{d1}(\Theta_{o1}(s)-\Theta_{m1}(s)), \tag{2}\]1  
In Figure 4, an active loop filter is utilized, with a transfer function gain of \(k_{p}(1/T_{i}+1)\), where \(k_{p}\) and \(T_{i}\) represent the gain and integration time of the loop filter, respectively. In this paper, a first-order passive LPF is employed, and its transfer function can be expressed as \(L(s)=1/(\tau s+1)\), where \(\tau\) denotes the time delay of the LPF. For our design, only the proportional gain is utilized in the proportional-integral (PI) regulator as shown in \[U_{e1}(s)=k_{d1}\Theta_{e1}(s)=k_{d1}(\Theta_{o1}(s)-\Theta_{m1}(s)), \tag{2}\]2 Consequently, \(k_{p}(1/T_{i}+1)\) can be simplified to \(k_{p}\). Additionally, the transfer function mode of an active loop filter can be expressed as \(F(s)=k_{p}/(\tau s+1)\). Therefore, we have  
\[U_{e1}(s)=k_{d1}\Theta_{e1}(s)=k_{d1}(\Theta_{o1}(s)-\Theta_{m1}(s)), \tag{2}\]3  
\[U_{e1}(s)=k_{d1}\Theta_{e1}(s)=k_{d1}(\Theta_{o1}(s)-\Theta_{m1}(s)), \tag{2}\]4