# 降压变换器

（重定向自降壓轉換器

## 概念

### 連續模式（CCM）

• 當上述的開關導通（圖2的上圖），電感兩側的電壓為${\displaystyle V_{L}=V_{i}-V_{o}}$ ，流過電感的電流會線性增加，因為二極體被電壓源V反向導通，理想上二極體不會有電流。
• 當上述的開關斷路（圖2的下圖），二極體被正向導通，電感兩側的電壓為${\displaystyle V_{L}=-V_{o}}$ （不考慮二極體壓降），而電感電流${\displaystyle I_{L}}$ 會線性下降。

${\displaystyle E={\frac {1}{2}}L\cdot I_{L}^{2}}$

${\displaystyle I_{L}}$ 的變化率可以表示如下：

${\displaystyle V_{L}=L{\frac {dI_{L}}{dt}}}$

${\displaystyle \Delta I_{L_{\mathit {on}}}=\int _{0}^{t_{\mathit {on}}}{\frac {V_{L}}{L}}\,dt={\frac {\left(V_{i}-V_{o}\right)}{L}}t_{\mathit {on}},\;t_{\mathit {on}}=DT}$

${\displaystyle \Delta I_{L_{\mathit {off}}}=\int _{t_{\mathit {on}}}^{T=t_{\mathit {on}}+t_{\mathit {off}}}{\frac {V_{L}}{L}}\,dt=-{\frac {V_{o}}{L}}t_{\mathit {off}},\;t_{\mathit {off}}=(1-D)T}$

{\displaystyle {\begin{aligned}&\Delta I_{L_{\mathit {on}}}+\Delta I_{L_{\mathit {off}}}=0\\&{\frac {V_{i}-V_{o}}{L}}t_{\mathit {on}}-{\frac {V_{o}}{L}}t_{\mathit {off}}=0\end{aligned}}}

{\displaystyle {\begin{aligned}&(V_{i}-V_{o})DT-V_{o}(1-D)T=0\\&V_{o}-DV_{i}=0\\\Rightarrow \;&D={\frac {V_{o}}{V_{i}}}\end{aligned}}}

### 不連續模式（DCM）

${\displaystyle \left(V_{i}-V_{o}\right)DT-V_{o}\delta T=0}$

${\displaystyle \delta ={\frac {V_{i}-V_{o}}{V_{o}}}D}$

${\displaystyle {\bar {I_{L}}}=I_{o}}$

{\displaystyle {\begin{aligned}{\bar {I_{L}}}&=\left({\frac {1}{2}}I_{L_{max}}DT+{\frac {1}{2}}I_{L_{max}}\delta T\right){\frac {1}{T}}\\&={\frac {I_{L_{max}}\left(D+\delta \right)}{2}}\\&=I_{o}\end{aligned}}}

${\displaystyle I_{L_{Max}}={\frac {V_{i}-V_{o}}{L}}DT}$

${\displaystyle I_{o}={\frac {\left(V_{i}-V_{o}\right)DT\left(D+\delta \right)}{2L}}}$

${\displaystyle I_{o}={\frac {\left(V_{i}-V_{o}\right)DT\left(D+{\frac {V_{i}-V_{o}}{V_{o}}}D\right)}{2L}}}$

${\displaystyle V_{o}=V_{i}{\frac {1}{{\frac {2LI_{o}}{D^{2}V_{i}T}}+1}}}$

### 連續模式和不連續模式的邊界

{\displaystyle {\begin{aligned}&DT+\delta T=T\\\Rightarrow \;&D+\delta =1\end{aligned}}}

${\displaystyle I_{o_{lim}}={\frac {I_{L_{max}}}{2}}\left(D+\delta \right)={\frac {I_{L_{max}}}{2}}}$

${\displaystyle I_{o_{lim}}={\frac {V_{i}-V_{o}}{2L}}DT}$

${\displaystyle V_{o}=DV_{i}}$

${\displaystyle I_{o_{lim}}={\frac {V_{i}\left(1-D\right)}{2L}}DT}$

• 正規化電壓，定義為${\displaystyle \left|V_{o}\right|={\frac {V_{o}}{V_{i}}}}$ 。若輸出電壓${\displaystyle V_{o}=0}$ ，正規化電壓也會是0，若輸出電壓等於輸入電壓，正規化電壓也是1。
• 正規化電流，定義為${\displaystyle \left|I_{o}\right|={\frac {L}{TV_{i}}}I_{o}}$ 。其中的${\displaystyle {\frac {TV_{i}}{L}}}$ 等於電感器在一個週期可以增加的電流最大值，也就是在占空比D=1，電感電流的增加量。因此在變換器穩態運作下，若沒有輸出電流，${\displaystyle \left|I_{o}\right|}$ 等於0，若輸出電流為最大輸出電流，則${\displaystyle \left|I_{o}\right|}$ 會是1。

• 在連續模式下：
${\displaystyle \left|V_{o}\right|=D}$
• 在不連續模式下：
{\displaystyle {\begin{aligned}\left|V_{o}\right|&={\frac {1}{{\frac {2LI_{o}}{D^{2}V_{i}T}}+1}}\\&={\frac {1}{{\frac {2\left|I_{o}\right|}{D^{2}}}+1}}\\&={\frac {D^{2}}{2\left|I_{o}\right|+D^{2}}}\end{aligned}}}

{\displaystyle {\begin{aligned}I_{o_{lim}}&={\frac {V_{i}}{2L}}D\left(1-D\right)T\\&={\frac {I_{o}}{2\left|I_{o}\right|}}D\left(1-D\right)\end{aligned}}}

${\displaystyle {\frac {\left(1-D\right)D}{2\left|I_{o}\right|}}=1}$

### 非理想電路下的情形

• 輸出電容器夠大，因此在驅動純電阻負載時其電壓沒有顯著的變化。
• 二極體的順向壓降為0。
• 開關及二極體都沒有切換損失。

#### 輸出電壓漣波

${\displaystyle dV_{o}={\frac {idT}{C}}}$

${\displaystyle dT_{on}=DT={\frac {D}{f}}}$

${\displaystyle dT_{off}=(1-D)T={\frac {1-D}{f}}}$

## 特殊結構

### 同步整流

${\displaystyle P_{D}=V_{D}(1-D)I_{o}}$

• VD是負載電流為Io時，二極體的電壓降
• D是占空比
• Io是負載電流

${\displaystyle P_{S2}=I_{o}^{2}R_{DSON}(1-D)}$

## 影響效率的因素

• 電晶體或是MOSFET在導通時的電阻。
• 二極體順向電降（若是肖特基二极管，約為0.4 V0.7 V）。
• 電感繞組的電阻
• 電容器的等效串聯電阻

• 電壓-電流重疊的損失
• 切換頻率*CV2的損失
• 反向延遲損失
• 因為驅動MOSFET閘及控制器本身耗能產生的損失。
• 電晶體漏電流損失，以及控制器待機功耗[5]

## 阻抗匹配

${\displaystyle \displaystyle V_{o}I_{o}=\eta V_{i}I_{i}TbH_{L}ads}$

• Vo為輸出電壓
• Io為輸出電流
• η為效率（數值在0到1之間）
• Vi為輸入電壓
• Ii為輸入電流

${\displaystyle \displaystyle I_{o}=V_{o}/Z_{o}}$
${\displaystyle \displaystyle I_{i}=V_{i}/Z_{i}}$

• Zo為輸出阻抗
• Zi為輸入阻抗

${\displaystyle V_{o}^{2}/Z_{o}=\eta V_{i}^{2}/Z_{i}}$

${\displaystyle \displaystyle V_{o}=DV_{i}}$

• D為占空比

${\displaystyle (DV_{i})^{2}/Z_{o}=\eta V_{i}^{2}/Z_{i}}$

${\displaystyle \displaystyle D^{2}/Z_{o}=\eta /Z_{i}}$

${\displaystyle D={\sqrt {\eta Z_{o}/Z_{i}}}}$

## 參考資料

1. ^ Mammano, Robert. "Switching power supply topology voltage mode vs. current mode." Elektron Journal-South African Institute of Electrical Engineers 18.6 (2001): 25-27.
2. ^ Keeping, Steven. Understanding the Advantages and Disadvantages of Linear Regulators. DigiKey. 2012-05-08 [2016-09-11]. （原始内容存档于2016-09-23）.
3. ^ Guy Séguier, Électronique de puissance, 7th edition, Dunod, Paris 1999 (in French)
4. ^ Tom's Hardware: "Idle/Peak Power Consumption Analysis". [2016-09-12]. （原始内容存档于2019-08-14）.
5. ^ iitb.ac.in - Buck converter (PDF). （原始内容 (PDF)存档于2011-07-16）. 090424 ee.iitb.ac.in
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• H. Chiacchiarini, P. Mandolesi, A. Oliva, and P. Julián, "Nonlinear analog controller for a buck converter: Theory and experimental results", Proceedings of the IEEE International Symposium on Industrial Electronics (ISIE’99), Bled, Slovenia, 12–16 July 1999, pp. 601–606.
• M. B. D’Amico, A. Oliva, E. E. Paolini y N. Guerin, "Bifurcation control of a buck converter in discontinuous conduction mode", Proceedings of the 1st IFAC Conference on Analysis and Control of Chaotic Systems (CHAOS’06), pp. 399–404, Reims (Francia), 28 al 30 de junio de 2006.
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• D’Amico, M. B., Guerin, N., Oliva, A.R., Paolini, E.E. Dinámica de un convertidor buck con controlador PI digital. Revista Iberoamericana de automática e informática industrial (RIAI), Vol 4, No 3, julio 2007, pp. 126–131. ISSN 1697-7912.
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