## 2 Small-Signal Model  
The flyback converter is divided into two operating modes according to the continuous or intermittent condition of the primary current: if the primary current reaches zero, it is called Discontinuous Conduction Mode (DCM); if the primary current does not reach zero, it is called continuous conduction mode. In this paper, the main power circuit diagram of a CCM single-ended flyback converter is shown in Figure 1, in which Q is the main power switch, D1 is the diode, \(C\) is the output filter capacitor, \(R_{\text{L}}\) is the load, \(L_{\text{m}}\) is the transformer excitation inductor, \(u_{\text{in}}\) is the input voltage, \(u_{\text{o}}\) is the output voltage, \(i_{1}\) and \(i_{\text{in}}\) are the excitation current and the primary input current, \(n\) is the ratio of turns on the original secondary side of the transformer, and \(d\) is the on-duty cycle of the main power switch Q.  
Figure 1: Flyback converter main power circuit diagram. When the main power switch Q is on, the primary side of the transformer stores energy, the current \(i_{1}\) rises, the diode D1 on the secondary side is in the cut-off state, and the output voltage \(u_{\text{o}}\) is kept constant by the capacitor \(C\). At this point:  
\[\left\{ \begin{array}{l}u_{\text{Q}}(t)=0\\ u_{\text{L}}(t)=u_{\text{in}}(t)\\ i_{\text{C}}(t)=-\frac{u_{\text{o}}(t)}{R_{\text{L}}}\\ i_{\text{in}}(t)=i_{1}(t)\end{array} \right. \tag{1}\]  
where \(u_{\text{Q}}\) is the drain-to-source voltage of the main power switch Q, \(u_{\text{L}}\) is the primary voltage of the transformer, and \(i_{\text{C}}\) is the current flowing through the output filter capacitor \(C\). When the main power switch Q is turned off, the primary side of the transformer releases energy, the excitation current \(i_{1}\) decreases, the diode D1 on the secondary side is in the conduction state, and the primary side transmits energy to the secondary side. At this point:  
\[\left\{ \begin{array}{l}u_{\text{L}}(t)=-u_{\text{o}}(t)n\\ i_{\text{C}}(t)=i_{1}(t)n-\frac{u_{\text{o}}(t)}{R_{\text{L}}}\\ i_{\text{in}}(t)=0\end{array} \right. \tag{2}\]  
The key waveforms of CCM flyback converter are shown in Figure 2[32]. \(T_{\text{S}}\), \(T_{\text{on}}\), and \(T_{\text{off}}\) are the switching period, turn-on time, and turn-off time of the main power switch Q separately. In this paper, the state-space averaging method is used to build the average model of the transformer primary voltage \(u_{\text{L}}\); the current \(i_{\text{C}}\) of the output filter capacitor \(C\) and the input current \(u_{\text{in}}\) is:  
\[\left\{ \begin{array}{l}u_{\text{L}}(t)=d(t)u_{\text{in}}(t)-d^{\prime}(t)u_ {\text{o}}(t)n\\ i_{\text{C}}(t)=d^{\prime}(t)i_{1}(t)n-\frac{u_{\text{o}}(t)}{R_{\text{L}}}\\ i_{\text{in}}(t)=d(t)i_{1}(t)\end{array} \right. \tag{3}\]  
Figure 2: Waveforms for CCM flyback converter. Namely  
\[\left\{ \begin{array}{l}L_{\text{m}}\frac{d\hat{i}_{1}(t)}{dt}=d(t)u_{\text{in}}(t )-d^{\prime}(t)u_{\text{o}}(t)n\\ C\frac{du_{\text{o}}(t)}{dt}=d^{\prime}(t)i_{1}(t)n-\frac{u_{\text{o}}(t)}{R_{ \text{L}}}\\ i_{\text{in}}(t)=d(t)i_{1}(t)\end{array} \right. \tag{4}\]  
Here, \(d+d^{\prime}=1\); \(d^{\prime}\) is the off-duty cycle of the main power switch Q. The system adds perturbations \(\hat{i_{1}}\), \(\hat{u_{\text{o}}}\), \(\hat{d}\), and \(\hat{u_{\text{in}}}\) so that the instantaneous value is:  
\[\left\{ \begin{array}{l}i_{1}(t)=[I_{1}+\hat{i_{1}}(t)]\\ u_{\text{o}}(t)=[V_{\text{o}}+\hat{u_{\text{o}}}(t)]\\ d(t)=[D+\hat{d}(t)]\\ u_{\text{in}}(t)=[V_{\text{in}}+\hat{u}_{\text{in}}(t)]\end{array} \right. \tag{5}\]  
Combining Formulas (4) and (5):  
\[L_{\text{m}}\frac{d[I_{1}+\hat{i}_{1}(t)]}{dt}=[D+\hat{d}(t)][V_{\text{in}}+ \hat{u}_{\text{in}}(t)]-[D^{\prime}-\hat{d}(t)][V_{\text{o}}+\hat{u}_{\text{o }}(t)]n \tag{6}\]  
where \(D+D^{\prime}=1\), and \(D\) and \(D^{\prime}\) are the on-duty cycle and off-duty cycle of the main power switch Q during the steady state of the system.