## 2 Operating Principle and Modes
### Non-Inverting and Inverting Buck Operation  
This section provides detail on how the rms value of the input voltage at the output is regulated in voltage step-down capabilities with PWM control. Here the output rms voltage is adjusted from the maximum value of the input rms to some lower value. This task is achieved with PWM control of the switching transistors \(T_{r1}\) and \(T_{r3}\) for positive and negative input, respectively. The low frequency (input or output) operation of transistors \(T_{r2}\), \(T_{r4}\), \(T_{r5}\) and \(T_{r6}\) is responsible for in-phase or out-phase outputs. The circuits of Figure 5 comprehensively illustrate the voltage step-down operation indicating how the input power is supplied to load. Figure 5a-d show the power transferring loops for non-inverted output for positive and negative half-cycles input voltage, respectively. Figure 4: Generation of control signals with respect to input voltage polarity detector for voltage boost to achieve (**a**) non-inverted form of the output and (**b**) inverted form of the output. Figure 5: Equivalent power transfer loops formed by PWM control to have non-inverted voltage buck output for half-cycles of (**a**,**b**) positive source voltage; (**c**,**d**) negative source voltage. For positive input, the on-state role of the semiconductor devices \(T_{r1}\), \(D_{1}\), \(T_{r2}\), \(T_{r6}\), \(D_{6}\), and \(D_{2}\) facilitate the power flow from source to load. The voltage drop across the inductor during the conducting interval (\(t_{c}\)) of the PWM signal is the difference of the input and output voltage as observed in Equation (1). \[V_{L}(t_{c})=V_{s}-V_{o} \tag{1}\]  
The turning on of the transistor \(T_{r3}\) facilitates the continuous conduction of inductor current as the reverse bias diode \(D_{3}\) turns on in the non-conducting interval (\(t_{nc}\)) of the PWM signal. During this interval, the absorbed power of the inductor is supplied to load, as may be found from the power supply loop of Figure 5b. Now the value of the inductor voltage is changed to output voltage level \(V_{o}\). \[V_{L}(t_{nc})=-V_{o} \tag{2}\]  
A similar analysis may be used to express the voltage buck operation for negative input and determine the value of inductor voltage during conducting and non-conducting PWM intervals. Figure 5c expresses the power supply path from source to load via the conduction of \(T_{r3}\), \(D_{3}\), \(T_{r4}\), \(D_{4}\), \(T_{r5}\), and \(D_{5}\). The continuous conduction of inductor current is confirmed by maintaining the transistor \(T_{r1}\) in conducting state because the diode \(D_{1}\) is only reverse biased as long as \(T_{r3}\) is on. The transfer of the inductor's stored power is recovered by the conduction of \(T_{r1}\), \(D_{1}\), \(T_{r4}\), \(D_{4}\), \(T_{r5}\), and \(D_{5}\), as expressed in Figure 5d. The realization of the inductor voltage during the conducting and non-conducting interval is accomplished as  
\[V_{L}(t_{c})=V_{s}-V_{o} \tag{3}\]  
\[V_{L}(t_{nc})=-V_{o} \tag{4}\]  
The values of these voltages may be regarded as constant as the selected switching frequency is much higher (for example, 25 kHz) than the input or output frequency (50 Hz). The average value of the inductor voltage in the steady-state condition in one switching interval is always zero, and this result may be utilized to find the relation of output voltage with respect to the input voltage. Here \(D\) and \(T\) are the duty ratio and the period of the PWM signal, respectively. \[V_{L(avg)}=\frac{L}{T}\{i_{L}(T)-i_{L}(0)\}=0 \tag{5}\]  
This implies that in a switching interval (\(T\)), the inductor volt-second product is zero.