# Design Considerations of Multi-Phase Buck DC-DC Converter
## 2 Power Losses in the Multi-Phase Synchronous Buck Converter
Output ripple current peak-to-peak amplitude \(\Delta I_{Cout}\) in an \(N\)-phase converter is given by  
\[\Delta I_{Cout}=\frac{U_{0}}{f_{SW}L}\frac{N}{D}\left(D-\frac{m}{N}\right) \left(\frac{1+m}{N}-D\right). \tag{3}\]  
The graphical representation of (3) in normalized form \(\Delta I_{Cout}=\Delta I_{Cout}/\frac{U_{0}}{f_{SW}L}\) is plotted in Figure 2b. For the RMS current \(I_{QH\_rms}\) through an HS transistor, the following relation is valid:  
\[I_{QH\_rms}=\sqrt{\left[\left(\frac{I_{0}}{N}\right)^{2}+\left(\frac{\Delta I _{L}^{2}}{12}\right)\right]D}. \tag{4}\]  
Hence, the conduction losses of \(P_{Qon\_H}\) of the HS transistor can be written as follows:  
\[P_{Qon\_H}=I^{2}{}_{QH\_rms}R_{Dson\_H}=\left[\left(\frac{I_{0}}{N}\right)^{2} +\left(\frac{\Delta I_{L}^{2}}{12}\right)\right]DR_{Dson\_H}. \tag{5}\]Switching losses \(P_{QSW\_H}\) in the high-side MOSFET are given by  
\[P_{QSW\_H}=\frac{1}{2}U_{IN}f_{SW}\bigg{[}\bigg{(}\frac{I_{0}}{N}+\frac{\Delta I _{L}}{2}\bigg{)}t_{r\_H}+\bigg{(}\frac{I_{0}}{N}-\frac{\Delta I_{L}}{2}\bigg{)}t _{f\_H}\bigg{]}. \tag{6}\]  
where \(t_{r\_H}\) is the rise time and \(t_{f\_H}\) is the fall time of the high-side MOSFET. For the RMS current \(I_{QL\_rms}\) through an LS transistor, the following relation is valid:  
\[I_{QL\_rms}=\sqrt{\bigg{[}\bigg{(}\frac{I_{0}}{N}\bigg{)}^{2}+\bigg{(}\frac{ \Delta I_{L}^{2}}{12}\bigg{)}\bigg{]}(1-D)}. \tag{7}\]  
The conduction losses of \(P_{Qon\_L}\) of the LS transistor are given by  
\[P_{Qon\_L}=I^{2}{}_{QL\_rms}R_{DSon\_L}=\bigg{[}\bigg{(}\frac{I_{0}}{N}\bigg{)} ^{2}+\bigg{(}\frac{\Delta I_{L}^{2}}{12}\bigg{)}\bigg{]}(1-D)R_{DSon\_L}. \tag{8}\]  
At the beginning and the end of lower MOSFET's conduction intervals, losses in the freewheeling diode must be considered during the dead time intervals. Furthermore, dead time conduction losses in the body diode \(P_{QL\_td}\) are expressed as follows:  
\[P_{QL\_td}\approx U_{SD}f_{SW}\bigg{[}\bigg{(}\frac{I_{0}}{N}+\frac{\Delta I_{ Cout}}{2}\bigg{)}t_{d1}+\bigg{(}\frac{I_{0}}{N}-\frac{\Delta I_{Cout}}{2} \bigg{)}t_{d2}\bigg{]}. \tag{9}\]  
where \(U_{SD}\) is the body diode voltage, and \(t_{d1}\) and \(t_{d2}\) indicate dead time for rising and dead time for falling, respectively. These losses depend on the duration of the dead time (\(t_{d1}\) and \(t_{d2}\)), switching frequency\(f_{SW}\), diode forward voltage \(U_{SD}\), and current \(I_{0}/N\). Reverse recovery losses of the low-side MOSFET are expressed as follows:  
\[\Delta I_{L}=\frac{U_{0}(1-D)}{Lf_{SW}}, \tag{1}\]0  
where \(Q_{RR}\) is the body diode reverse recovery charge. In the discussed circuit there are also other types of losses, such as gate charge losses and output capacitance losses in the MOSFETs, which are typically much smaller than the losses considered previously. \[\Delta I_{L}=\frac{U_{0}(1-D)}{Lf_{SW}}, \tag{1}\]1 Normalized ripple currents in an N-phase converter: (**a**) Normalized input ripple current \(I_{IN\_rms}/I_{0}\); (**b**) normalized output ripple current peak-to-peak amplitude \(\Delta I_{Cout}\). Gate charge losses \(P_{GH}\) of the transistor \(Q1\) can be expressed by  
\[\Delta I_{L}=\frac{U_{0}(1-D)}{Lf_{SW}}, \tag{1}\]2  
where \(U_{GS}\) is the gate-source voltage and \(Q_{GH}\) is the gate charge of the high-side MOSFET. Output capacitance losses in the high-side MOSFET \(P_{\text{C}_{OSS\_H}}\) are given by  
\[\Delta I_{L}=\frac{U_{0}(1-D)}{Lf_{SW}}, \tag{1}\]3  
where \(\text{C}_{OSS\_H}\) is the output capacitances of the high-side MOSFET. The total losses in the HS transistor, in one stage, are the sum of the losses determined by (5), (6), (10), (11), and (12), and are calculated using the following [38]:  
\[\Delta I_{L}=\frac{U_{0}(1-D)}{Lf_{SW}}, \tag{1}\]4  
The total losses in the HS MOSFETs \(P_{QHN}\) in an \(N\)-phase synchronous buck converter is \(N\) times \(P_{QH}\).