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電路知識：Boost Converter
簡介：Boost Converter 原理
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By No machine-readable author provided. CyrilB~commonswiki assumed (based on copyright claims). - No machine-readable source provided. Own work assumed (based on copyright claims)., CC BY-SA 3.0, Link $$ \begin{cases} V_o=V_C & \text{ if } t\ in\ T_{on}=DT \\ V_o=V_C=V_i-V_L-V_D & \text{ if } t\ in\ T_{off}=(1-D)T \\ \end{cases}\\ $$ 平均輸出電壓 $$ \begin{align*} V_{o,av}&=\frac{1}{T}\int_{0}^{t}V_o(t)\mathrm{d} t \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_o(t)\mathrm{d} t + \int_{t_{on}}^{T}V_o(t)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_L-V_D)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t - \int_{t_{on}}^{T}V_L\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t - \int_{0}^{T}V_L\mathrm{d} t + \int_{0}^{t_{on}}V_L\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t + 0 + \int_{0}^{t_{on}}(V_i-V_{sw})\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t + \int_{0}^{t_{on}}(V_i-V_{sw})\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{T}V_i\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t + \int_{0}^{t_{on}}V_i\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{0}^{T}V_i\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t \right ) \\ &= V_i + \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t \right ) \\ \end{align*} $$ $$ V_L=L\frac{\mathrm{d}I_L }{\mathrm{d} t} \\ $$ $$ \begin{align*} \Delta I_{L_{on}}&=I_{max}-I_{min} \\ &= \int_{0}^{T_{on}}\frac{V_L}{L} \mathrm{d} t \\ &= \int_{0}^{T_{on}}\frac{V_i-V_{sw}}{L}\mathrm{d} t \\ &= \int_{0}^{T_{on}}\frac{V_i}{L}\mathrm{d} t-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ &= \frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ \\\\ \Delta I_{L_{off}}&=I_{min}-I_{max} \\ &= \int_{t_{on}}^{T}\frac{V_L}{L} \mathrm{d} t \\ &= \int_{t_{on}}^{T}\frac{V_i-V_o-V_D}{L}\mathrm{d} t \\ &= \frac{(1-D)TV_i}{L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{align*} $$ $$ \begin{align*} \Delta I_{L_{on}}=I_{max}-I_{min} &= \frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ \Delta I_{L_{off}}=I_{min}-I_{max} &= \frac{(1-D)TV_i}{L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{align*} $$ 從另一方面解 \(V_{o,av}\)，因電感電流穩定
$$ \begin{align*} 0 &= \Delta I_{L_{on}} + \Delta I_{L_{off}} \\ 0 &= \frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t+ \frac{(1-D)TV_i}{L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t\\ 0 &= \frac{TV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t - \int_{t_{on}}^{T}\frac{V_o}{L}\mathrm{d} t- \int_{t_{on}}^{T}\frac{V_D}{L}\mathrm{d} t\\ 0 &= \frac{TV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t - \int_{0}^{T}\frac{V_o}{L}\mathrm{d} t+ \int_{0}^{t_{on}}\frac{V_o}{L}\mathrm{d} t- \int_{t_{on}}^{T}\frac{V_D}{L}\mathrm{d} t\\ \int_{0}^{T}\frac{V_o}{L}\mathrm{d} t &= \frac{TV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t + \int_{0}^{t_{on}}\frac{V_C}{L}\mathrm{d} t- \int_{t_{on}}^{T}\frac{V_D}{L}\mathrm{d} t\\ \frac{1}{T}\int_{0}^{T}\frac{V_o}{L}\mathrm{d} t &= \frac{V_i}{L}+\frac{1}{T}\left (\int_{0}^{t_{on}}\frac{V_C}{L}\mathrm{d} t -\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t - \int_{t_{on}}^{T}\frac{V_D}{L}\mathrm{d} t \right )\\ \frac{1}{T}\int_{0}^{T}V_o\mathrm{d} t &= V_i+\frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t -\int_{0}^{T_{on}}V_{sw}\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t \right )\\ V_{o,av}&= V_i+\frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t -\int_{0}^{T_{on}}V_{sw}\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t \right )\\ \end{align*} $$ 因電感平均電流位於最大值與最小值的中間
又根據之前推導的 \(\Delta I_{L_{on}}\) 可得下面兩式
$$ \left\{\begin{matrix} I_{max}+I_{min}&=&2*I_{av}\\ I_{max}-I_{min} &=& \frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ \end{matrix}\right.\\ \left\{\begin{matrix} 2 \cdot I_{max} &=&2*I_{av} + \frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t\\ 2 \cdot I_{min} &=&2*I_{av} - \frac{DTV_i}{L}+\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ \end{matrix}\right.\\ \left\{\begin{matrix} I_{max} &=&I_{av} + \frac{DTV_i}{2L}-\int_{0}^{T_{on}}\frac{V_{sw}}{2L}\mathrm{d} t\\ I_{min} &=&I_{av} - \frac{DTV_i}{2L}+\int_{0}^{T_{on}}\frac{V_{sw}}{2L}\mathrm{d} t \\ \end{matrix}\right.\\ $$ 或者根據之前推導的 \(\Delta I_{L_{off}}\) 可得下面兩式
$$ \left\{\begin{matrix} I_{max}+I_{min} &=&2*I_{av}\\ I_{min}-I_{max} &=& \frac{(1-D)TV_i}{L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{matrix}\right.\\ \left\{\begin{matrix} I_{max}+I_{min} &=&2*I_{av}\\ I_{max}-I_{min} &=& -\frac{(1-D)TV_i}{L} + \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{matrix}\right.\\ \left\{\begin{matrix} 2 \cdot I_{max} &=&2*I_{av} -\frac{(1-D)TV_i}{L} + \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t\\ 2 \cdot I_{min} &=&2*I_{av} +\frac{(1-D)TV_i}{L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{matrix}\right.\\ \left\{\begin{matrix} I_{max} &=&I_{av} -\frac{(1-D)TV_i}{2L} + \int_{t_{on}}^{T}\frac{V_o+V_D}{2L}\mathrm{d} t\\ I_{min} &=&I_{av} +\frac{(1-D)TV_i}{2L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{2L}\mathrm{d} t \\ \end{matrix}\right.\\ $$ 保持 continuous mode 的條件，就是 \(I_L \geq 0\)
$$ \begin{align*} I_{min} &=I_{av} +\frac{(1-D)TV_i}{2L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{2L}\mathrm{d} t \geq 0\\ I_{av} & \geq -\frac{(1-D)TV_i}{2L} + \int_{t_{on}}^{T}\frac{V_o+V_D}{2L}\mathrm{d} t \\ \frac{V_{o,av}}{R} & \geq -\frac{(1-D)TV_i}{2L} + \int_{t_{on}}^{T}\frac{V_o+V_D}{2L}\mathrm{d} t \\ L & \geq -\frac{R(1-D)TV_i}{2V_{o,av}} + \frac{R}{2V_{o,av}}\int_{t_{on}}^{T}(V_o+V_D)\mathrm{d} t \\ \end{align*} $$
By No machine-readable author provided. CyrilB~commonswiki assumed (based on copyright claims). - No machine-readable source provided. Own work assumed (based on copyright claims)., CC BY-SA 3.0, Link $$ \begin{cases} V_o=V_C & \text{ if } t\ in\ T_{on}=DT \\ V_o=V_i-V_L-V_D& \text{ if } t\ in\ \delta T \\ V_o=V_i-V_D & \text{ if } t\ in\ others \\ \end{cases}\\ $$ 平均輸出電壓 $$ \begin{align*} V_{o,av}&=\frac{1}{T}\int_{0}^{t}V_o(t)\mathrm{d} t \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_o(t)\mathrm{d} t + \int_{t_{on}}^{t_{on}+\delta T}V_o(t)\mathrm{d} t + \int_{t_{on}+\delta T}^{T}V_o(t)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{t_{on}+\delta T}(V_i-V_L-V_D)\mathrm{d} t + \int_{t_{on}+\delta T}^{T}(V_i-V_D)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - \int_{t_{on}}^{t_{on}+\delta T}V_L\mathrm{d} t + \int_{t_{on}}^{t_{on}+\delta T}(V_i-V_D)\mathrm{d} t + \int_{t_{on}+\delta T}^{T}(V_i-V_D)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - \int_{t_{on}}^{t_{on}+\delta T}V_L\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - \int_{0}^{t_{on}+\delta T}V_L\mathrm{d} t + \int_{0}^{t_{on}}V_L\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - 0 + \int_{0}^{t_{on}}(V_i-V_{sw})\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{0}^{t_{on}}V_i\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t + \int_{t_{on}}^{T}V_i\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{0}^{T}V_i\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t \right ) \\ &= V_i + \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t \right ) \\ \end{align*} $$ $$ V_L=L\frac{\mathrm{d}I_L }{\mathrm{d} t} \\ $$ $$ \begin{align*} \Delta I_{L_{on}}&=I_{max}-I_{min} \\ &= \int_{0}^{T_{on}}\frac{V_L}{L} \mathrm{d} t \\ &= \int_{0}^{T_{on}}\frac{V_i-V_{sw}}{L}\mathrm{d} t \\ &= \int_{0}^{T_{on}}\frac{V_i}{L}\mathrm{d} t-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ &= \frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ \\\\ \Delta I_{L_{off}}&=I_{min}-I_{max} \\ &= \int_{t_{on}}^{t_{on}+\delta T}\frac{V_L}{L} \mathrm{d} t \\ &= \int_{t_{on}}^{t_{on}+\delta T}\frac{V_i-V_o-V_D}{L}\mathrm{d} t \\ &= \frac{\delta TV_i}{L} - \int_{t_{on}}^{t_{on}+\delta T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{align*} $$
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簡介：Boost Converter 原理
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設計概念
Continuous mode
$$ \begin{align*} V_{o,av}&=V_i + \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t \right ) \\ \Delta I_L&=\frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t =-\frac{(1-D)TV_i}{L} + \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t\\ I_{av} &= \frac{V_{o,av}}{R}\\ L &\geq -\frac{R(1-D)TV_i}{2V_{o,av}} + \frac{R}{2V_{o,av}}\int_{t_{on}}^{T}(V_o+V_D)\mathrm{d} t \\ \end{align*} $$By No machine-readable author provided. CyrilB~commonswiki assumed (based on copyright claims). - No machine-readable source provided. Own work assumed (based on copyright claims)., CC BY-SA 3.0, Link $$ \begin{cases} V_o=V_C & \text{ if } t\ in\ T_{on}=DT \\ V_o=V_C=V_i-V_L-V_D & \text{ if } t\ in\ T_{off}=(1-D)T \\ \end{cases}\\ $$ 平均輸出電壓 $$ \begin{align*} V_{o,av}&=\frac{1}{T}\int_{0}^{t}V_o(t)\mathrm{d} t \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_o(t)\mathrm{d} t + \int_{t_{on}}^{T}V_o(t)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_L-V_D)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t - \int_{t_{on}}^{T}V_L\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t - \int_{0}^{T}V_L\mathrm{d} t + \int_{0}^{t_{on}}V_L\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t + 0 + \int_{0}^{t_{on}}(V_i-V_{sw})\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t + \int_{0}^{t_{on}}(V_i-V_{sw})\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{T}V_i\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t + \int_{0}^{t_{on}}V_i\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{0}^{T}V_i\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t \right ) \\ &= V_i + \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t \right ) \\ \end{align*} $$ $$ V_L=L\frac{\mathrm{d}I_L }{\mathrm{d} t} \\ $$ $$ \begin{align*} \Delta I_{L_{on}}&=I_{max}-I_{min} \\ &= \int_{0}^{T_{on}}\frac{V_L}{L} \mathrm{d} t \\ &= \int_{0}^{T_{on}}\frac{V_i-V_{sw}}{L}\mathrm{d} t \\ &= \int_{0}^{T_{on}}\frac{V_i}{L}\mathrm{d} t-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ &= \frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ \\\\ \Delta I_{L_{off}}&=I_{min}-I_{max} \\ &= \int_{t_{on}}^{T}\frac{V_L}{L} \mathrm{d} t \\ &= \int_{t_{on}}^{T}\frac{V_i-V_o-V_D}{L}\mathrm{d} t \\ &= \frac{(1-D)TV_i}{L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{align*} $$ $$ \begin{align*} \Delta I_{L_{on}}=I_{max}-I_{min} &= \frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ \Delta I_{L_{off}}=I_{min}-I_{max} &= \frac{(1-D)TV_i}{L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{align*} $$ 從另一方面解 \(V_{o,av}\)，因電感電流穩定
$$ \begin{align*} 0 &= \Delta I_{L_{on}} + \Delta I_{L_{off}} \\ 0 &= \frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t+ \frac{(1-D)TV_i}{L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t\\ 0 &= \frac{TV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t - \int_{t_{on}}^{T}\frac{V_o}{L}\mathrm{d} t- \int_{t_{on}}^{T}\frac{V_D}{L}\mathrm{d} t\\ 0 &= \frac{TV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t - \int_{0}^{T}\frac{V_o}{L}\mathrm{d} t+ \int_{0}^{t_{on}}\frac{V_o}{L}\mathrm{d} t- \int_{t_{on}}^{T}\frac{V_D}{L}\mathrm{d} t\\ \int_{0}^{T}\frac{V_o}{L}\mathrm{d} t &= \frac{TV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t + \int_{0}^{t_{on}}\frac{V_C}{L}\mathrm{d} t- \int_{t_{on}}^{T}\frac{V_D}{L}\mathrm{d} t\\ \frac{1}{T}\int_{0}^{T}\frac{V_o}{L}\mathrm{d} t &= \frac{V_i}{L}+\frac{1}{T}\left (\int_{0}^{t_{on}}\frac{V_C}{L}\mathrm{d} t -\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t - \int_{t_{on}}^{T}\frac{V_D}{L}\mathrm{d} t \right )\\ \frac{1}{T}\int_{0}^{T}V_o\mathrm{d} t &= V_i+\frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t -\int_{0}^{T_{on}}V_{sw}\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t \right )\\ V_{o,av}&= V_i+\frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t -\int_{0}^{T_{on}}V_{sw}\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t \right )\\ \end{align*} $$ 因電感平均電流位於最大值與最小值的中間
又根據之前推導的 \(\Delta I_{L_{on}}\) 可得下面兩式
$$ \left\{\begin{matrix} I_{max}+I_{min}&=&2*I_{av}\\ I_{max}-I_{min} &=& \frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ \end{matrix}\right.\\ \left\{\begin{matrix} 2 \cdot I_{max} &=&2*I_{av} + \frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t\\ 2 \cdot I_{min} &=&2*I_{av} - \frac{DTV_i}{L}+\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ \end{matrix}\right.\\ \left\{\begin{matrix} I_{max} &=&I_{av} + \frac{DTV_i}{2L}-\int_{0}^{T_{on}}\frac{V_{sw}}{2L}\mathrm{d} t\\ I_{min} &=&I_{av} - \frac{DTV_i}{2L}+\int_{0}^{T_{on}}\frac{V_{sw}}{2L}\mathrm{d} t \\ \end{matrix}\right.\\ $$ 或者根據之前推導的 \(\Delta I_{L_{off}}\) 可得下面兩式
$$ \left\{\begin{matrix} I_{max}+I_{min} &=&2*I_{av}\\ I_{min}-I_{max} &=& \frac{(1-D)TV_i}{L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{matrix}\right.\\ \left\{\begin{matrix} I_{max}+I_{min} &=&2*I_{av}\\ I_{max}-I_{min} &=& -\frac{(1-D)TV_i}{L} + \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{matrix}\right.\\ \left\{\begin{matrix} 2 \cdot I_{max} &=&2*I_{av} -\frac{(1-D)TV_i}{L} + \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t\\ 2 \cdot I_{min} &=&2*I_{av} +\frac{(1-D)TV_i}{L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{matrix}\right.\\ \left\{\begin{matrix} I_{max} &=&I_{av} -\frac{(1-D)TV_i}{2L} + \int_{t_{on}}^{T}\frac{V_o+V_D}{2L}\mathrm{d} t\\ I_{min} &=&I_{av} +\frac{(1-D)TV_i}{2L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{2L}\mathrm{d} t \\ \end{matrix}\right.\\ $$ 保持 continuous mode 的條件，就是 \(I_L \geq 0\)
$$ \begin{align*} I_{min} &=I_{av} +\frac{(1-D)TV_i}{2L} - \int_{t_{on}}^{T}\frac{V_o+V_D}{2L}\mathrm{d} t \geq 0\\ I_{av} & \geq -\frac{(1-D)TV_i}{2L} + \int_{t_{on}}^{T}\frac{V_o+V_D}{2L}\mathrm{d} t \\ \frac{V_{o,av}}{R} & \geq -\frac{(1-D)TV_i}{2L} + \int_{t_{on}}^{T}\frac{V_o+V_D}{2L}\mathrm{d} t \\ L & \geq -\frac{R(1-D)TV_i}{2V_{o,av}} + \frac{R}{2V_{o,av}}\int_{t_{on}}^{T}(V_o+V_D)\mathrm{d} t \\ \end{align*} $$
Discontinuous mode
$$ \begin{align*} V_{o,av} &=V_i + \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t \right )\\ \Delta I_L&=\frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t=-\frac{\delta TV_i}{L} + \int_{t_{on}}^{t_{on}+\delta T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{align*} $$
By No machine-readable author provided. CyrilB~commonswiki assumed (based on copyright claims). - No machine-readable source provided. Own work assumed (based on copyright claims)., CC BY-SA 3.0, Link $$ \begin{cases} V_o=V_C & \text{ if } t\ in\ T_{on}=DT \\ V_o=V_i-V_L-V_D& \text{ if } t\ in\ \delta T \\ V_o=V_i-V_D & \text{ if } t\ in\ others \\ \end{cases}\\ $$ 平均輸出電壓 $$ \begin{align*} V_{o,av}&=\frac{1}{T}\int_{0}^{t}V_o(t)\mathrm{d} t \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_o(t)\mathrm{d} t + \int_{t_{on}}^{t_{on}+\delta T}V_o(t)\mathrm{d} t + \int_{t_{on}+\delta T}^{T}V_o(t)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{t_{on}}^{t_{on}+\delta T}(V_i-V_L-V_D)\mathrm{d} t + \int_{t_{on}+\delta T}^{T}(V_i-V_D)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - \int_{t_{on}}^{t_{on}+\delta T}V_L\mathrm{d} t + \int_{t_{on}}^{t_{on}+\delta T}(V_i-V_D)\mathrm{d} t + \int_{t_{on}+\delta T}^{T}(V_i-V_D)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - \int_{t_{on}}^{t_{on}+\delta T}V_L\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - \int_{0}^{t_{on}+\delta T}V_L\mathrm{d} t + \int_{0}^{t_{on}}V_L\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - 0 + \int_{0}^{t_{on}}(V_i-V_{sw})\mathrm{d} t + \int_{t_{on}}^{T}(V_i-V_D)\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{0}^{t_{on}}V_i\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t + \int_{t_{on}}^{T}V_i\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t \right ) \\ &= \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t + \int_{0}^{T}V_i\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t \right ) \\ &= V_i + \frac{1}{T}\left (\int_{0}^{t_{on}}V_C\mathrm{d} t - \int_{0}^{t_{on}}V_{sw}\mathrm{d} t - \int_{t_{on}}^{T}V_D\mathrm{d} t \right ) \\ \end{align*} $$ $$ V_L=L\frac{\mathrm{d}I_L }{\mathrm{d} t} \\ $$ $$ \begin{align*} \Delta I_{L_{on}}&=I_{max}-I_{min} \\ &= \int_{0}^{T_{on}}\frac{V_L}{L} \mathrm{d} t \\ &= \int_{0}^{T_{on}}\frac{V_i-V_{sw}}{L}\mathrm{d} t \\ &= \int_{0}^{T_{on}}\frac{V_i}{L}\mathrm{d} t-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ &= \frac{DTV_i}{L}-\int_{0}^{T_{on}}\frac{V_{sw}}{L}\mathrm{d} t \\ \\\\ \Delta I_{L_{off}}&=I_{min}-I_{max} \\ &= \int_{t_{on}}^{t_{on}+\delta T}\frac{V_L}{L} \mathrm{d} t \\ &= \int_{t_{on}}^{t_{on}+\delta T}\frac{V_i-V_o-V_D}{L}\mathrm{d} t \\ &= \frac{\delta TV_i}{L} - \int_{t_{on}}^{t_{on}+\delta T}\frac{V_o+V_D}{L}\mathrm{d} t \\ \end{align*} $$
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