## 2 Adler's Equation and Phase Locking and Synchronization of Gyrotrons
When the oscillator locks to the external injection signal, the beat frequency vanishes, resulting in the locking condition  
\[\sin\theta=2Q\frac{E}{E_{1}}\frac{\Delta\omega_{0}}{\omega_{0}}. \tag{2}\]  
Since \(|\sin\theta|\leq 1\), the maximum locking range of the oscillator is given by  
\[|\frac{\Delta\omega_{0}}{\omega_{0}}|<\frac{1}{2Q}\frac{E_{1}}{E}. \tag{3}\]  
Analytical equations (analogous to [PERSON]'s equation) and their solutions for predicting injection locking have been proposed in [53]. This method is very suitable for analyzing the injection-locking phenomenon in LC and ring oscillators because, unlike [PERSON]'s equation, it does not require evaluation of the Q factor. The [PERSON]'s equation can be used to analyze the injection pulling that occurs when the frequency of the injected signal falls slightly out of the lock range. Following [54], this equation can be solved by assuming that \(|\Delta\omega_{0}|>[\omega_{0}/2Q](E_{1}/E)=\omega_{L}\) and noting from Equation (1) that \(d\theta/dt\) reaches a maximum of \(\Delta\omega_{0}+\omega_{L}\), which is a small value compared to \(\omega_{0}\), i.e., \(\theta\) varies slowly. Therefore, the [PERSON]'s equation can be written as follows:  
\[\frac{d\theta}{\Delta-\omega_{L}\sin\theta}=dt. \tag{4}\]Integrating this equation (see [54]) gives  
\[\tan\frac{\theta}{2}=\frac{\omega_{L}}{\Delta\omega_{0}}+\frac{\omega_{b}}{\Delta \omega 0}\tan\frac{\omega_{b}t}{2}, \tag{5}\]  
where \(\omega_{b}=\sqrt{(\Delta\omega_{0})^{2}-\omega_{L}^{2}}\). Using the notations \(C=E_{1}\omega_{0}/2 QE\), \(\alpha=\Delta\omega_{0}/A\), and \(\tau=Ct\), the [PERSON]'s equation (Equation (1)) can be written in the following dimensionless form:  
\[\theta^{\prime}=\frac{d\theta}{d\tau}=\alpha-\sin\theta. \tag{6}\]  
Depending on the value of the parameter \(\alpha\), three cases of phase dynamics are possible, as shown in Figure 1. The black and white circles represent stable and unstable fixed points, respectively. The arrows indicate the direction of the phase change towards or away from the fixed points if they exist. In this illustration, the phase-locked oscillator is represented by Figure 1b. This case is also referred to as entrainment, which is defined by a temporal locking process in which the frequency of one oscillator entrains the frequency of another oscillator. The situation when the oscillators are unlocked is illustrated by the phase portrait in Figure 1c. Many studies and experimental measurements conclude that the estimation of the locking band roughly agrees with the predictions of [PERSON]'s phase-locking equation (see e.g., [39]). However, in the search for a more accurate analysis, a number of advanced concepts and theoretical models have been developed to describe phase locking in gyrotrons [55]. Although a comprehensive overview of the advances in the implementation of phase locking in gyrotrons is beyond the scope of this article, at least given the limited space we have available here, some notable recent work should be mentioned here at the end of this section. For example, in [56] it has been shown theoretically that during phase locking of the oscillations of a gyrotron with an output power of two megawatt and a frequency of 170 GHz by a relatively weak external monochromatic signal (about tens of kilowatts), the phase and frequency modulation of the gyrotron radiation due to the low-frequency fluctuations of the accelerating voltage and current decreases severalfold compared to the autonomous regime. This promises to open the way for developing a future complex of coherently radiating megawatt class gyrotrons. The window reflects radiation at the frequency of the operating mode and is transparent for the neighboring modes.