# A Comprehensive Loss Model and Comparison of AC and DC Boost Converters

^{1}

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## Abstract

**:**

## 1. Introduction and Motivation

#### 1.1. AC and DC Converters

_{rms}AC to 380 V DC). As converters are generally more efficient at higher voltage [3], it is unclear whether DC systems are inherently more efficient or are simply analyzed at a higher voltage.

#### 1.2. Boost Converters

## 2. Deriving Conduction Loss Models

## 3. Conduction Loss Component Currents

#### 3.1. Input and Duty Cycle

#### 3.2. Inductor Current

#### 3.3. Diode Bridge Current

#### 3.4. Switch Current

#### 3.5. Boost Diode Current

#### 3.6. Capacitor Current

## 4. Switching Loss in the Switch (Q)

#### 4.1. Hard-Switching Loss

- The gate driver charges ${C}_{GS}$. The gate voltage, ${v}_{GS}$, increases to the gate-threshold voltage, ${V}_{TH}$.
- The gate driver continues to charge ${C}_{GS}$. ${v}_{GS}$ continues to increase as ${i}_{DS}$ rises to ${I}_{DS,max}$.
- The gate driver now discharges ${C}_{GD}$. ${v}_{GS}$ remains constant at the gate-plateau voltage, ${V}_{GP}$, as ${v}_{DS}$ falls to near-zero.

- 4.
- The gate driver discharges ${C}_{GS}$. ${v}_{GS}$ decreases to ${V}_{GP}$.
- 5.
- The gate driver charges ${C}_{DS}$. ${v}_{GS}$ remains constant at ${V}_{GP}$ as ${v}_{DS}$ rises to ${V}_{DS,max}$.
- 6.
- The gate driver discharges ${C}_{GS}$. ${v}_{GS}$ decreases to ${V}_{TH}$ and ${i}_{DS}$ falls to near-zero.

#### 4.2. Output-Capacitance Loss

## 5. Switching Loss in the Diode (D)

#### 5.1. Reverse Recovery Loss

#### 5.2. Junction-Capacitance Loss

## 6. Model Validation

#### 6.1. Simulation Validation

#### 6.2. Experimental Validation

## 7. Efficiency Comparison of AC vs. DC

## 8. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Model Equations

#### Appendix A.2. P_{X,cond}: Conduction Loss

Parameter | AC PFC Model Formula | DC Model Formula |
---|---|---|

${I}_{L,rms}$ ${I}_{B,rms}$ | $\frac{\sqrt{2}{P}_{o}}{{V}_{pk}}$ | $\frac{{P}_{o}}{{V}_{pk}}$ |

${I}_{B,avg}$ | $\frac{4}{\pi}\frac{{P}_{o}}{{V}_{pk}}$ | − |

${I}_{Q,rms}$ | $\frac{{P}_{o}}{\sqrt{{V}_{o}}{V}_{pk}}\sqrt{2{V}_{o}-\frac{16}{3\pi}{V}_{pk}}$ | $\frac{{P}_{o}}{\sqrt{{V}_{o}}{V}_{pk}}\sqrt{{V}_{o}-{V}_{pk}}$ |

${I}_{D,rms}$ | $\frac{4}{\sqrt{3\pi}}\frac{{P}_{o}}{\sqrt{{V}_{o}{V}_{pk}}}$ | $\frac{{P}_{o}}{\sqrt{{V}_{o}{V}_{pk}}}$ |

${I}_{D,avg}$ | $\frac{{P}_{o}}{{V}_{o}}$ | $\frac{{P}_{o}}{{V}_{o}}$ |

${I}_{C,rms}$ | $\frac{{P}_{o}}{{V}_{o}\sqrt{{V}_{pk}}}\sqrt{\frac{16}{3\pi}{V}_{o}-{V}_{pk}}$ | $\frac{{P}_{o}}{{V}_{o}\sqrt{{V}_{pk}}}\sqrt{{V}_{o}-{V}_{pk}}$ |

Parameter | Model Formula |
---|---|

${I}_{L,rms}$ ${I}_{B,rms}$ | $\frac{\sqrt{576\pi {L}^{2}{P}_{o}^{2}{V}_{o}^{2}{f}^{2}+12\pi {V}_{o}^{2}{V}_{pk}^{4}-64{V}_{o}{V}_{pk}^{5}+9\pi {V}_{pk}^{6}}}{12\sqrt{2}\sqrt{\pi}L{V}_{o}{V}_{pk}f}$ |

${I}_{B,avg}$ | − |

${I}_{Q,rms}$ | $\frac{\sqrt{\begin{array}{c}2880\pi {L}^{2}{P}_{o}^{2}{V}_{o}^{3}{f}^{2}-7680{L}^{2}{P}_{o}^{2}{V}_{o}^{2}{V}_{pk}{f}^{2}+60\pi {V}_{o}^{3}{V}_{pk}^{4}\dots \\ -480{V}_{o}^{2}{V}_{pk}^{5}+135\pi {V}_{o}{V}_{pk}^{6}-128{V}_{pk}^{7}\end{array}}}{12\sqrt{10}\sqrt{\pi}L{V}_{o}^{\frac{3}{2}}{V}_{pk}f}$ |

${I}_{D,rms}$ | $\frac{\sqrt{3840{L}^{2}{P}_{o}^{2}{V}_{o}^{2}{f}^{2}+80{V}_{o}^{2}{V}_{pk}^{4}-45\pi {V}_{o}{V}_{pk}^{5}+64{V}_{pk}^{6}}}{12\sqrt{5}\sqrt{\pi}L{V}_{o}^{\frac{3}{2}}\sqrt{{V}_{pk}}f}$ |

${I}_{D,avg}$ | $\frac{{P}_{o}}{{V}_{o}}$ |

${I}_{C,rms}$ | $\frac{\sqrt{\begin{array}{c}3840{L}^{2}{P}_{o}^{2}{V}_{o}^{2}{f}^{2}-720\pi {L}^{2}{P}_{o}^{2}{V}_{o}{V}_{pk}{f}^{2}\dots \\ +80{V}_{o}^{2}{V}_{pk}^{4}-45\pi {V}_{o}{V}_{pk}^{5}+64{V}_{pk}^{6}\end{array}}}{12\sqrt{5}\sqrt{\pi}L{V}_{o}^{\frac{3}{2}}\sqrt{{V}_{pk}}f}$ |

Parameter | Model Formula |
---|---|

${I}_{L,rms}$ ${I}_{B,rms}$ | $\frac{\sqrt{12{L}^{2}{P}_{o}^{2}{V}_{o}^{2}+{T}^{2}{V}_{o}^{2}{V}_{pk}^{4}-2{T}^{2}{V}_{o}{V}_{pk}^{5}+{T}^{2}{V}_{pk}^{6}}}{2L{V}_{o}{V}_{pk}\sqrt{3}}$ |

${I}_{B,avg}$ | − |

${I}_{Q,rms}$ | $\frac{\sqrt{\left({V}_{o}-{V}_{pk}\right)\left(12{L}^{2}{P}_{o}^{2}{V}_{o}^{2}+{T}^{2}{V}_{o}^{2}{V}_{pk}^{4}-2{T}^{2}{V}_{o}{V}_{pk}^{5}+{T}^{2}{V}_{pk}^{6}\right)}}{2L{V}_{pk}\sqrt{3{V}_{o}^{3}}}$ |

${I}_{D,rms}$ | $\frac{\sqrt{12{L}^{2}{P}_{o}^{2}{V}_{o}^{2}+{T}^{2}{V}_{o}^{2}{V}_{pk}^{4}-2{T}^{2}{V}_{o}{V}_{pk}^{5}+{T}^{2}{V}_{pk}^{6}}}{2L\sqrt{3{V}_{o}^{3}{V}_{pk}}}$ |

${I}_{D,avg}$ | $\frac{{P}_{o}}{{V}_{o}}$ |

${I}_{C,rms}$ | $\frac{\sqrt{\left({V}_{o}-{V}_{pk}\right)\left(12{L}^{2}{P}_{o}^{2}{V}_{o}+{T}^{2}{V}_{o}{V}_{pk}^{4}-{T}^{2}{V}_{pk}^{5}\right)}}{2L\sqrt{3{V}_{o}^{3}{V}_{pk}}}$ |

Parameter | Model Formula |
---|---|

${P}_{L,cond}$ | ${I}_{L,rms}^{2}{R}_{L}$ |

${P}_{B,cond}$ | ${I}_{B,avg}{V}_{B}+{I}_{B,rms}^{2}{R}_{B}$ |

${P}_{Q,cond}$ | ${I}_{Q,rms}^{2}{R}_{Q}$ |

${P}_{D,cond}$ | ${I}_{D,avg}{V}_{D}+{I}_{D,rms}^{2}{R}_{D}$ |

${P}_{C,cond}$ | ${I}_{C,rms}^{2}{R}_{C}$ |

#### Appendix A.3. P_{Q,sw,hs} and P_{Q,sw,c}: Hard Switching and Output Capacitance Loss

Timing | Formula |
---|---|

${T}_{IR}$ | ${R}_{G}{C}_{ISS}\mathrm{ln}\left(\frac{{V}_{GS,max}-{V}_{TH}}{{V}_{GS,max}-{V}_{GP}}\right)$ |

${T}_{VF}$ | ${R}_{G}\frac{{Q}_{GD,0}}{{V}_{DS,0}}\frac{{V}_{DS,max}}{{V}_{GS,max}-{V}_{GP}}$ |

${T}_{VR}$ | ${R}_{G}\frac{{Q}_{GD,0}}{{V}_{DS,0}}\frac{{V}_{DS,max}}{{V}_{GP}}$ |

${T}_{IF}$ | ${R}_{G}{C}_{ISS}\mathrm{ln}\left(\frac{{V}_{GP}}{{V}_{TH}}\right)$ |

Model | Average Loss Power |
---|---|

AC PFC (simple) | $\frac{2{P}_{o}{V}_{o}f\left({T}_{off}+{T}_{on}\right)}{\pi {V}_{pk}}$ |

DC (simple) | $\frac{{P}_{o}{V}_{o}f\left({T}_{off}+{T}_{on}\right)}{2{V}_{pk}}$ |

AC PFC (ripple) | $\frac{\left(\begin{array}{c}16L{P}_{o}{T}_{off}{V}_{o}f+16L{P}_{o}{T}_{on}{V}_{o}f+4{T}_{off}{V}_{o}{V}_{pk}^{2}\dots \\ -\pi {T}_{off}{V}_{pk}^{3}-4{T}_{on}{V}_{o}{V}_{pk}^{2}+\pi {T}_{on}{V}_{pk}^{3}\end{array}\right)}{8\pi L{V}_{pk}}$ |

DC (ripple) | $\frac{\left(\begin{array}{c}2L{P}_{o}{T}_{off}{V}_{o}f+2L{P}_{o}{T}_{on}{V}_{o}f+{T}_{off}{V}_{o}{V}_{pk}^{2}\dots \\ -{T}_{off}{V}_{pk}^{3}-{T}_{on}{V}_{o}{V}_{pk}^{2}+{T}_{on}{V}_{pk}^{3}\end{array}\right)}{4L{V}_{pk}}$ |

#### Appendix A.4. P_{D,sw,hs} and P_{D,sw,c}: Diode Reverse Recovery and Junction Capacitance Loss

Model | Average Loss Power |
---|---|

AC PFC (simple) | $\frac{{K}_{Q}\sqrt{{P}_{o}}{V}_{o}f\left(48-{\pi}^{2}\right)}{24\sqrt{2}\sqrt{{V}_{pk}}}$ |

DC (simple) | $\frac{{K}_{Q}\sqrt{{P}_{o}}{V}_{o}f}{\sqrt{{V}_{pk}}}$ |

AC PFC (ripple) | $\frac{{K}_{Q}\sqrt{{V}_{o}}\sqrt{f}\left(\begin{array}{c}-4{\pi}^{2}L{P}_{o}{V}_{o}f+192L{P}_{o}{V}_{o}f-48{V}_{o}{V}_{pk}^{2}\dots \\ +{\pi}^{2}{V}_{o}{V}_{pk}^{2}-2{\pi}^{2}{V}_{pk}^{3}+48{V}_{pk}^{3}\end{array}\right)}{48\sqrt{2}\sqrt{L}\sqrt{{V}_{pk}}\sqrt{4L{P}_{o}{V}_{o}f-{V}_{o}{V}_{pk}^{2}+{V}_{pk}^{3}}}$ |

DC (ripple) | $\frac{{K}_{Q}\sqrt{{V}_{o}}\sqrt{f}\sqrt{2L{P}_{o}{V}_{o}f-{V}_{o}{V}_{pk}^{2}+{V}_{pk}^{3}}}{\sqrt{2}\sqrt{L}\sqrt{{V}_{pk}}}$ |

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**Figure 2.**The inductor current ${i}_{L}(\theta ,t)$ tracks a reference input current ${I}_{ref}\left(\theta \right)={I}_{pk}sin\left(\theta \right)$.

**Figure 3.**On the switching timescale, $\theta $ is approximately constant. The inductor current, ${i}_{L}(\theta ,t)$, passes through the switch (green) during ${\delta}_{Q}\left(\theta \right)$, and the boost diode (orange) during ${\delta}_{D}\left(\theta \right)$.

**Figure 4.**On the switching timescale, a bilateral triangle, ${\mathsf{\Delta}}^{B}$, (left) can model the current through the inductor and bridge diode. This triangle is not necessarily isosceles. An elevated right triangle, ${\mathsf{\Delta}}^{R}$, (right) can model the current through the switch and boost diode.

**Figure 9.**Diode and switch current and voltage waveforms for the boost converter. Note that for the boost converter, ${V}_{R}={V}_{DS}={V}_{O}$ and ${I}_{F}={I}_{DS}={i}_{ref}-\mathsf{\Delta}{i}_{L,pp}$ at the time of transition.

**Figure 12.**These efficiency curves are the result of a parametric analysis of AC and DC boost converters. The AC converters have slightly higher switching loss and much higher conduction loss than their DC counterparts.

**Figure 13.**A loss analysis and itemized loss breakdown. The bar segments represent the percent loss (loss power divided by input power) that occurs in each component. Each pair of bars compares the loss in the AC (left) and DC (right) boost converters. The converters were modeled with an input voltage ${V}_{pk}$ = 170 V and an output voltage ${V}_{o}$ = 250 V.

Component | Identification Number |
---|---|

Inductor | Premo PFCA500-8H |

Diode Bridge | Diodes Inc. GBU804 |

Switch | STMicroelectronics STP9NK60Z |

Boost Diode | Power Integrations LQA08TC600 |

Capacitor (2×) | TDK Electronics B43544A6477M000 |

DC Power Supply | Chroma 62024P-600-8 |

Electronic Load | Chroma 63802 |

Revenue-Grade DC Meter | AccuEnergy AcuDC 243-600V-A1-P2-C-D |

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**MDPI and ACS Style**

Gerber, D.L.; Musavi, F.; Ghatpande, O.A.; Frank, S.M.; Poon, J.; Brown, R.E.; Feng, W. A Comprehensive Loss Model and Comparison of AC and DC Boost Converters. *Energies* **2021**, *14*, 3131.
https://doi.org/10.3390/en14113131

**AMA Style**

Gerber DL, Musavi F, Ghatpande OA, Frank SM, Poon J, Brown RE, Feng W. A Comprehensive Loss Model and Comparison of AC and DC Boost Converters. *Energies*. 2021; 14(11):3131.
https://doi.org/10.3390/en14113131

**Chicago/Turabian Style**

Gerber, Daniel L., Fariborz Musavi, Omkar A. Ghatpande, Stephen M. Frank, Jason Poon, Richard E. Brown, and Wei Feng. 2021. "A Comprehensive Loss Model and Comparison of AC and DC Boost Converters" *Energies* 14, no. 11: 3131.
https://doi.org/10.3390/en14113131