# Single-Switch Non-Isolated Resonant DC-DC Converter for Single-Input Dual-Output Applications
## 2 Proposed Single-Switch SIDO ZVS-QRC Configuration Description
### Steady-State Analysis  
Assuming a ZVS-QR SEPIC-Cuk combination converter without losses (ideal components), the following assumptions can be considered:  
1. The inductors \(L_{1}\), \(L_{2}\) and \(L_{3}\) are considered current sources (currents are assumed to be constant and without any ripple), with \(L_{1}\), \(L_{2}\) and \(L_{3}>>L_{R}\). 2. The link and output capacitors are considered voltage sources (voltages are assumed to be constant and without any ripple), with C\({}_{1}\), C\({}_{2}\), C\({}^{+}_{o}\) and C\({}^{-}\)\(>>\)\(C_{R}\). 3. The \(L_{R}\)-\(C_{R}\) tank is undamped (oscillatory). 4. The resonant wave amplitude is greater than \(V_{g}+V_{o}/2\) (\(V_{m}>V_{g}+V_{o}/2\)). The following parameters are also used in the analysis:  
\[ \begin{array}{l}\text{Resonant angular frequency}\ (\text{rad}/\text{sec}): \omega_{o}=2\pi f_{o}=\frac{1}{\sqrt{L_{R}\cdot C_{R}}}\\ \text{Characteristic impedance}\ (\Omega):Z_{o}=\sqrt{\frac{L_{R}}{\text{c}_{R}}}= \omega_{o}\cdot L_{R}=\frac{1}{\omega_{o}\cdot C_{R}}\\ \text{Equivalent Loads}\ (\Omega):\frac{1}{R_{o}^{\prime}}=\frac{1}{R_{Load_{1}}}+ \frac{2}{R_{Load_{3}}}\\ \frac{1}{R_{o}}=\frac{1}{R_{Load_{2}}}+\frac{2}{R_{Load_{3}}}\\ \frac{1}{R_{o}}=\frac{1}{R_{Load_{1}}}+\frac{1}{R_{Load_{2}}}+\frac{4}{R_{Load _{3}}}\end{array}  \tag{1}\]  
A switching period is divided into four intervals, as seen in Section 2.1. Considering the initial conditions of each interval, the differential equations can be solved with the help of the corresponding equivalent circuit (Figure 4) and the duration of the interval determined. * \(t_{0}\leq t\leq t_{1}\); \(Q\), \(D_{Q}\), \(D_{1}\) and \(D_{2}\) off (Figure 4a). The corresponding equations can be written as follows: Initial Conditions: \[\begin{array}{l}\text{$i_{L_{R}}(t_{0})=I_{g}+I_{o_{1}}+I_{o_{2}}$, $\,$\,$\,$\,$\,$\,$\,$\,$\,$\,$\,$\,$\,$\,$\,$\,$\,$\,$\,$\,$\,$  
The waveforms of the ZVS-QR SEPIC-Cuk combination converter during a switching period and the four intervals analyzed are shown in Figure 5.  
Figure 5: Main waveforms of the ZVS-QR SEPIC–Cuk combination converter during a switching period.  
The output voltages can be evaluated by considering the average voltage across the freewheel diodes (\(D_{1}\) and \(D_{2}\)) for a switching period (waveforms shown in Figure 5), since these voltages are filtered by the output capacitors.  
The current conversion relation can be expressed by  
\[I_{g}=\left(I_{o_{1}}+I_{o_{2}}\right)\cdot\frac{\left(1-f_{s}\cdot\left(t_{3}- \frac{t_{1}}{2}\right)\right)}{f_{s}\cdot\left(t_{3}-\frac{t_{1}}{2}\right)}= \left(I_{o}^{+}+I_{o}^{-}+2I_{o}\right)\cdot\frac{\left(1-f_{s}\cdot\left(t_{3} -\frac{t_{1}}{2}\right)\right)}{f_{s}\cdot\left(t_{3}-\frac{t_{1}}{2}\right)} \tag{7}\]  
Conversion relations can also be expressed by defining the parameter \(m\) as:  
\[V_{o}^{+}=V_{o}^{-}=V_{g}\cdot\frac{\left(1-m\right)}{m}\Rightarrow V_{o}=2V_ {g}\cdot\frac{\left(1-m\right)}{m}\]  
\[I_{g}=\left(I_{o_{1}}+I_{o_{2}}\right)\cdot\frac{\left(1-m\right)}{m}=\left( I_{o}^{+}+I_{o}^{-}+2I_{o}\right)\cdot\frac{\left(1-m\right)}{m} \tag{8}\]  
With \(m=f_{5}\cdot\left(t_{3}-\frac{t_{1}}{2}\right)\); \(0<m<1\)  
Output Voltages:  
\[V_{D_{1}}(t)\text{ and }V_{D_{2}}(t)=\left\{ \begin{aligned} \left[\left(V_{g}+\frac{V_{g}}{2}\right)-\frac{\left(I_{g}+I_{o_{1}}+I_{o_{2}} \right)}{C_{R}}(t-t_{1})\right]& t_{o}\leq t\leq t_{1}\\ 0& t_{1}\leq t\leq t_{2}\\ 0& t_{2}\leq t\leq t_{3}\\ \left(V_{g}+\frac{V_{g}}{2}\right)& t_{3}\leq t\leq t _{4}\\ V_{o}^{+}=V_{D_{1}}(t)_{AVG}\\ V_{o}^{-}=\left|V_{D_{2}}(t)_{AVG}\right|\end{aligned}