# Analysis of Subharmonic Oscillation and Slope Compensation for a Differential Boost Inverter
## 4 Continuous-Time Modeling of the Differential Boost Inverter  
### Quasi-Steady-State Analysis  
From the simulation results presented in the previous section, it has been observed that SO takes place when suitable parameters are varied. One of the widespread tools to analyze and to investigate this kind of nonlinear behavior is Floquet theory [52; 53]. Considering the switched model of the system, one can identify possible periodic orbits, their stability as well as several other important aspects of the  
Figure 5: Waveforms of the duty cycle \(d(nT)\) at steady-state operation in terms of the phase angle in [\({}^{\circ}\)] for different values of \(k_{p}\) and for \(V_{M}=2\) V.  
dynamical behavior. To apply this theory, the mathematical model is first derived. By applying KVL and KCL, the switched model of the differential boost inverter can be expressed as follows  
\[\frac{\mathrm{d}i_{1}}{\mathrm{d}t} = \frac{1}{L_{1}}(V_{g}-v_{o2}(1-u))-\frac{r_{1}}{L_{1}}i_{1}, \tag{1}\] \[\frac{\mathrm{d}i_{2}}{\mathrm{d}t} = \frac{1}{L_{2}}(V_{g}-uv_{o2})-\frac{r_{2}}{L_{2}}i_{2},\] (2) \[\frac{\mathrm{d}v_{o1}}{\mathrm{d}t} = \frac{1}{C_{1}}((1-u)i_{1}+\frac{v_{o1}-v_{o2}}{R}),\] (3) \[\frac{\mathrm{d}v_{o2}}{\mathrm{d}t} = \frac{1}{C_{2}}(ui_{2}-\frac{v_{o1}-v_{o2}}{R}), \tag{4}\]  
where \(L_{1}\) and \(L_{2}\) are the inductance of the inductors of the differential boost inverter with stray resistances \(r_{1}\) and \(r_{2}\) respectively, \(C_{1}\) and \(C_{2}\) are the capacitances of their capacitors. \(V_{g}\) is the DC input voltage and \(R\) is the AC load resistance. All other parameters appearing in (1)-(4) are shown in Figure 1. The quasi-steady-state average values of the state variables are related to the quasi-steady-state duty cycle \(D\) by the following expressions:  
\[I_{1} = \frac{V_{g}(2D-1)}{RD(1-D)^{2}},\quad I_{2}=-\frac{V_{g}(2D-1)}{RD ^{2}(1-D)} \tag{5}\] \[V_{o1} = \frac{V_{g}}{1-D},\qquad V_{o2}=\frac{V_{g}}{D} \tag{6}\]  
These expressions have been obtained by using the averaged model of the inverter within a switching period. Using (6) and the fact that \(v_{o}=v_{o1}-v_{o2}\), the voltage gain of the differential boost inverter can be expressed as follows  
\[M(D):=\frac{v_{\text{ref}}}{V_{g}}=\frac{2D-1}{D(1-D)} \tag{7}\]  
The inverter gain \(M(D)\) reaches its maximum value \(M_{\text{max}}=V_{\text{ref}}/V_{g}\) when the voltage reference \(v_{\text{ref}}\) reaches its peak value \(V_{\text{ref}}\). From the expression of \(M(D)\), the steady-state value of the duty cycle can be derived and this can be expressed as follows  
\[D(t)=\begin{cases}\frac{1}{2}-\frac{V_{g}}{v_{\text{ref}}}+\frac{ \sqrt{4V_{g}^{2}+v_{\text{ref}}^{2}}}{2v_{\text{ref}}}&\text{if $v_{\text{ref}}(t)>0$},\\ \frac{1}{2}-\frac{V_{g}}{v_{\text{ref}}}-\frac{\sqrt{4V_{g}^{2}+v_{\text{ref}} ^{2}}}{2v_{\text{ref}}}&\text{if $v_{\text{ref}}(t)<0$}.\end{cases} \tag{8}\]  
In terms of the phase angle \(\varphi\), the quasi-steady-state duty cycle can be expressed as follows  
\[D(\varphi)=\begin{cases}\frac{1}{2}-\frac{1}{M_{\text{max}}\sin( \varphi)}+\sqrt{M_{\text{max}}+\frac{1}{4}}&\text{if $\varphi\in(0,\pi)$},\\ \frac{1}{2}-\frac{1}{M_{\text{max}}\sin(\varphi)}-\sqrt{M_{\text{max}}+\frac{1 }{4}}&\text{if $\varphi\in(\pi,2\pi)$}.\end{cases} \tag{9}\]