## 2 Operating Principle and Modes
### Non-Inverting and Inverting Boost Operation  
The operation in this mode helps to increase the voltage of any polarity at the output with respect to the input voltage, and this voltage stabilization depends on the PWM control. The switching arrangement to govern the on and off behavior of the transistors is shown in the control schemes of Figure 4. It has already been stated that the control schemes used in the realization of voltage buck operation can be effectively employed for voltage boost operation. The power flow loops of Figure 7a-d can be used to demonstrate the non-inverted voltage boost behavior for any value of input voltage.  
Figure 6: Equivalent power transfer loops formed by PWM control to have inverted voltage buck output for half cycles of (**a**,**b**) positive source voltage; (**c**,**d**) negative source voltage.  
For positive input voltage, the PWM control of transistor \(T_{r5}\) stores power in the inductor via the power flowing loop formed by the conduction of \(T_{r1}\), \(D_{1}\), \(T_{r2}\), \(D_{2}\), \(T_{r5}\), and \(D_{5}\) that may be seen from Figure 7a. In the non-conducting interval of the PWM control, the inductor and source get connected in series with the load through the operating devices \(T_{r1}\), \(D_{1}\), \(T_{r2}\), \(D_{2}\), \(T_{r6}\), and \(D_{6}\) that may find in the loop of Figure 7b. Here switching of transistor \(T_{r6}\) during conducting and non-conducting intervals of PWM control ensures the continuous inductor current as series diode \(D_{6}\) operates in a complementary way of transistor \(T_{r5}\). A similar operation can be explored for negative input as depicted in the power flow loops of Figure 7c,d. The value of the inductor voltage during the conducting and non-conducting intervals of the PWM control is used to find out the variation in the output voltage with respect to the constant input voltage. That is to say  
\[V_{L}(t_{c})=V_{s} \tag{17}\]  
\[V_{L}(t_{nc})=V_{s}-V_{o} \tag{18}\]  
By finding the average of inductor voltage  
\[V_{s}t_{c}+(V_{s}-V_{o})t_{nc}=0 \tag{19}\]  
\[V_{s}(t_{c}+t_{nc})=V_{o}t_{nc} \tag{20}\]  
\[V_{s}T\;=\;V_{o}(1-D)T \tag{21}\]  
\[V_{o}\;=\;\frac{V_{s}}{1-D} \tag{22}\]  
Figure 7: Equivalent power transfer loops formed by PWM control to have non-inverted voltage boost output for half cycles of (**a**,**b**) positive source voltage; (**c**,**d**) negative source voltage.  
The voltage inverting operation can also be supported with the same control signals to generate non-inverting outputs. The conduction of \(T_{r1},D_{1},T_{r4},D_{2},T_{r5},D_{6}\) and PWM control of \(T_{r6}\) during positive input helps to store power in the inductor, as may be viewed from Figure 8a. Then turning off of \(T_{r6}\) and forward biasing of \(D_{5}\) along with previously conducting devices \(T_{r1},D_{1}\), \(D_{2}\), \(T_{r4}\), \(T_{r5}\) connect the source, inductor, and load in series as may be viewed from Figure 8b. The detail of inverting behavior for negative input may view from the power flow loops of Figure 8c,d plotted during the conducting and non-conducting intervals of the PWM control, respectively.  
The validation of inverting voltage gains for both input polarities may also be formulated with the help of inductor voltage during the conducting and non-conducting intervals of the PWM control.  
\[V_{L}(t_{c})=V_{s} \tag{23}\]  
\[V_{L}(t_{nc})=V_{s}+V_{o} \tag{24}\]  
Averaging of inductor voltage is  
\[V_{s}t_{c}+(V_{s}+V_{o})t_{nc}=0 \tag{25}\]  
\[V_{s}(t_{c}+t_{nc})=-V_{o}t_{nc} \tag{26}\]  
\[V_{s}T\;=\;-V_{o}(1-D)T \tag{27}\]  
\[V_{o}\;=\;\frac{-V_{s}}{1-D} \tag{28}\]  
Figure 8: Equivalent power transfer loops formed by PWM control to have inverted voltage boost output for half cycles of (a,b) positive source voltage; (c,d) negative source voltage.  
The comparison of Equations (22) and (28) proves stabilization of the bipolar voltage capabilities of the output voltage with PWM or duty cycle control.