# Analysis of Subharmonic Oscillation and Slope Compensation for a Differential Boost Inverter
## 6 Stability Boundaries in the Parameter Space  
If SO boundary is of concern, the expression of the characteristic equation \(\det(\mathbf{M}-\lambda\mathbf{I})=0\) can be used by imposing that an eigenvalue \(\lambda=-1\) and solving the resulting equation in a suitable projection of the parametric space. Therefore, to determine the boundary of SO, the following equation is solved for a certain system parameter after fixing the other ones  
\[\det(\mathbf{M}+\mathbf{I})=0 \tag{25}\]  
The great advantage of using (25) is that only this equation has to be solved without the need of computing all eigenvalues of \(\mathbf{M}\) explicitly. Therefore, instead of solving for all eigenvalues of \(\mathbf{M}\), only (25) is solved, hence, the saving of computational load is significant when the stability boundary is to be determined. Figure 7 shows the stability boundary resulted from solving (25) with respect to the proportional gain \(k_{p}\) for values of the duty cycle within the operating range \((0.5,0.68)\) and for a value of the ramp compensator amplitude \(V_{M}=2\) V. Within one sinewave signal one has 2000 switching cycles. Therefore, the plot was generated using 1000 values of the duty cycles and the critical values of \(k_{p}\) were registered in terms of \(D\). In particular, for \(V_{M}=2\) V, the critical value of the proportional gain guaranteeing that all the eigenvalues lie inside the unit circle for all values of the operating duty cycle is about 0.2. This is in perfect agreement with the numerical simulations presented in Section 3. If \(V_{M}\) is increased, the critical value of the proportional gain also increases and the stability region gets wider as depicted in Figure 8. In particular, for \(V_{M}=3\) V, the critical value of the proportional gain is about 0.73, for \(V_{M}=4\) V, is about 1.28 and for \(V_{M}=5\) V, it is about 1.82. Notice that for a fixed switching period \(T\), changing the ramp amplitude is equivalent to changing its slope. Figure 6: Floquet multipliers loci by varying the quasi-steady-state duty cycle \(D\) for two different values of the proportional gain \(k_{p}\). As stated previously, in DC-AC inverters, the reference voltage is a time varying sinusoidal signal and accordingly the steady-state quasi-static duty cycle \(D\) is given by (8). In such a situation, the phase \(\varphi\) is a quasi-static parameter like \(D\). Solving (9) in terms of the phase angle \(\varphi\), one gets two critical values of the phase angle that can be expressed as follows  
\[\varphi_{1} = \sin^{-1}(\frac{V_{g}(2D-1)}{V_{\text{ref}}D(1-D)}) \tag{26}\] \[\varphi_{2} = \pi-\sin^{-1}(\frac{V_{g}(2D-1)}{V_{\text{ref}}D(1-D)}) \tag{27}\]  
Figure 8: Stability boundaries in terms of the proportional gain \(k_{p}\) and the quasi-steady-state duty cycle \(D\) for different values of the ramp amplitude \(V_{M}\). Figure 7: Stability boundaries in terms of the proportional gain \(k_{p}\) and the quasi-steady-state duty cycle \(D\) and for \(V_{M}=2\) V.  
These closed expressions for the critical phase angles at which SO develops explain the observation made in Section 3. In terms of the inverter gain \(M(D)\) and its maximum value \(M_{\text{max}}\), the expressions of the phase angles are given by  
\[\varphi_{1} = \sin^{-1}(\frac{M(D)}{M_{\text{max}}}) \tag{28}\] \[\varphi_{2} = \pi-\sin^{-1}(\frac{M(D)}{M_{\text{max}}}) \tag{29}\]  
The stability boundary of the system is plotted in Figure 9, in terms of the proportional gain \(k_{p}\) of the voltage controller and the phase angle \(\varphi\in(0,\pi)\). Vertical dashed lines in Figure 9 indicate this theoretical critical value for the set of parameter values shown in Table 1. For each specific union of \(\varphi_{1}\) and \(\varphi_{2}\) curves, it can be noted that there is a turning point at the left side of the union. The system will be stable at the left of the turning point and will exhibit an SO phenomenon at its right side.