# Modeling a Dual-Mode Controller Design
## 2 QRF Converter with Dual-Mode Control
### Single-Switch Flyback Converter  
QRF converters operated in DCM are designed such that they lower the switching loss in the circuit. Because of the resonance, \(L_{lk}\), \(C_{\text{oss}}\), and a voltage spike across the drain and source of \(Q_{1}\) were observed. \(L_{p}\) is the primary inductance that comprises \(L_{m}\) and \(L_{lk}\). The primary reason for the generation of leakage inductance is that energy stored in the primary side of the converter cannot couple to the secondary-side. This energy must be transferred. Hence, when the switch turns off, the energy of \(L_{lk}\) causes a voltage spike in the drain-source voltage. \(C_{\text{oss}}\) releases the energy of \(L_{lk}\) through the discharging path if the switch is turned on, and a voltage spike is induced. Figure 2a,b display the voltage of the switches for the DCM fly-back converter and QRF converter respectively. Figure 1: Fly-back converter circuit. The analysis method of each operation mode of a QRF converter is discussed herein. The relationship between the lowest frequency and the primary inductance is also presented in the following section. When \(Q_{1}\) conducts, \(V_{g}\) begins to charge to \(L_{p}\). The current in the inductor can be expressed as follows:  
\[I_{Lp}(t)\ =\ \frac{V_{g}}{L_{p}}t \tag{1}\]  
At \(T_{on}\), the peak current in the inductor \(I_{Lp(pk)}\) can be represented as follows:  
\[I_{Lp(pk)}\ =\ \frac{V_{g}}{L_{p}}T_{on} \tag{2}\]  
When \(Q_{1}\) is turned off, the energy stored in \(L_{lk}\) cannot be transferred to the secondary side. Hence, \(C_{oss}\) begins to resonate with \(L_{lk}\). The voltage spike on the power switch (\(V_{ds(\max)}\)) is generated as follows:  
\[V_{ds(max)}\ =\ V_{g}+nV_{o}+I_{Lp(pk)}\sqrt{\frac{L_{lk}}{C_{oss}}} \tag{3}\]  
When the resonant energy stored in \(L_{lk}\) and \(C_{oss}\) reaches zero during the degaussing, the drain source voltage \((V_{ds})\) drops to the constant voltage until the magnetizing inductor releases energy. \(V_{ds}\) can be expressed as follows:  
\[V_{ds}\ =\ V_{g}+nV_{o} \tag{4}\]  
The primary current during \(T_{off}\) can be derived as follows:  
\[I_{Lp(pk)}\ =\ \frac{nV_{o}}{L_{p}}T_{off} \tag{5}\]  
In the steady state, when the secondary side does not transmit current through the transformer, the primary side behaves as an RLC resonant circuit:  
\[V_{ds}(t)\ =\ V_{g}+nV_{o}e^{-(\frac{R_{p}}{2p})t}\cos{(2\pi\frac{1}{2\pi \sqrt{L_{p}C_{oss}}}t)} \tag{6}\]  
Figure 2: Drain-source voltage of the power switch used in a discontinuous conduction mode (DCM) fly-back converter and quasi-resonant fly-back (QRF) converter with a discontinuous conduction mode: (**a**) drain-source voltage of the power switch in the DCM fly-back mode and (**b**) drain-source voltage of the power switch in the QRF operation mode. where \(R_{p}\) is the total resistance of a primary side path of transformer. The fly-back converter enters the resonant state when the power switch of the QRF converter is turned off. The first valley switching time at a value of \(T_{valley}\) can be obtained using the following representation of \(T_{valley}\):  
\[\cos\bigl{(}2\pi f_{w}T_{valley}\bigr{)}\ =\ -1 \tag{7}\]  
and  
\[T_{valley}\ =\ \frac{1}{2f_{w}}\ =\ \pi\sqrt{L_{p}C_{\text{oss}}}. \tag{8}\]  
The underdamped response case of the fly-back converter affects the drain-source voltage until the power switch conducts power again during \(T_{resonant}\). The drain source voltage of the QRF converter produces an underdamped oscillation until the power switch begins to turn on at a \(T_{resonant}\) value of \(T_{valley}\).