[Circuit] Boost Converter

電路知識：Boost Converter

Tool：: Qucs

Example Circuits

簡介：Boost Converter 原理

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設計概念

Continuous mode

\begin{aligned} V_{o, a v} & = V_{i} + \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t - \int_{t_{o n}}^{T} V_{D} d t - \int_{0}^{t_{o n}} V_{s w} d t) \\ Δ I_{L} & = \frac{D T V_{i}}{L} - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t = - \frac{(1 - D) T V_{i}}{L} + \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{L} d t \\ I_{a v} & = \frac{V_{o, a v}}{R} \\ L & \geq - \frac{R (1 - D) T V_{i}}{2 V_{o, a v}} + \frac{R}{2 V_{o, a v}} \int_{t_{o n}}^{T} (V_{o} + V_{D}) d t \end{aligned}

{\begin{cases} V_{o} = V_{C} & if t i n T_{o n} = D T \\ V_{o} = V_{C} = V_{i} - V_{L} - V_{D} & if t i n T_{o f f} = (1 - D) T \end{cases}

平均輸出電壓

\begin{aligned} V_{o, a v} & = \frac{1}{T} \int_{0}^{t} V_{o} (t) d t \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{o} (t) d t + \int_{t_{o n}}^{T} V_{o} (t) d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t + \int_{t_{o n}}^{T} (V_{i} - V_{L} - V_{D}) d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t + \int_{t_{o n}}^{T} (V_{i} - V_{D}) d t - \int_{t_{o n}}^{T} V_{L} d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t + \int_{t_{o n}}^{T} (V_{i} - V_{D}) d t - \int_{0}^{T} V_{L} d t + \int_{0}^{t_{o n}} V_{L} d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t + \int_{t_{o n}}^{T} (V_{i} - V_{D}) d t + 0 + \int_{0}^{t_{o n}} (V_{i} - V_{s w}) d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t + \int_{t_{o n}}^{T} (V_{i} - V_{D}) d t + \int_{0}^{t_{o n}} (V_{i} - V_{s w}) d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t + \int_{t_{o n}}^{T} V_{i} d t - \int_{t_{o n}}^{T} V_{D} d t + \int_{0}^{t_{o n}} V_{i} d t - \int_{0}^{t_{o n}} V_{s w} d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t + \int_{0}^{T} V_{i} d t - \int_{t_{o n}}^{T} V_{D} d t - \int_{0}^{t_{o n}} V_{s w} d t) \\ = V_{i} + \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t - \int_{t_{o n}}^{T} V_{D} d t - \int_{0}^{t_{o n}} V_{s w} d t) \end{aligned}

V_{L} = L \frac{d I_{L}}{d t}

\begin{aligned} Δ I_{L_{o n}} & = I_{m a x} - I_{m i n} \\ = \int_{0}^{T_{o n}} \frac{V_{L}}{L} d t \\ = \int_{0}^{T_{o n}} \frac{V_{i} - V_{s w}}{L} d t \\ = \int_{0}^{T_{o n}} \frac{V_{i}}{L} d t - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t \\ = \frac{D T V_{i}}{L} - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t \\ Δ I_{L_{o f f}} & = I_{m i n} - I_{m a x} \\ = \int_{t_{o n}}^{T} \frac{V_{L}}{L} d t \\ = \int_{t_{o n}}^{T} \frac{V_{i} - V_{o} - V_{D}}{L} d t \\ = \frac{(1 - D) T V_{i}}{L} - \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{L} d t \end{aligned}

\begin{aligned} Δ I_{L_{o n}} = I_{m a x} - I_{m i n} & = \frac{D T V_{i}}{L} - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t \\ Δ I_{L_{o f f}} = I_{m i n} - I_{m a x} & = \frac{(1 - D) T V_{i}}{L} - \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{L} d t \end{aligned}

從另一方面解

V_{o, a v}

，因電感電流穩定

\begin{aligned} 0 & = Δ I_{L_{o n}} + Δ I_{L_{o f f}} \\ 0 & = \frac{D T V_{i}}{L} - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t + \frac{(1 - D) T V_{i}}{L} - \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{L} d t \\ 0 & = \frac{T V_{i}}{L} - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t - \int_{t_{o n}}^{T} \frac{V_{o}}{L} d t - \int_{t_{o n}}^{T} \frac{V_{D}}{L} d t \\ 0 & = \frac{T V_{i}}{L} - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t - \int_{0}^{T} \frac{V_{o}}{L} d t + \int_{0}^{t_{o n}} \frac{V_{o}}{L} d t - \int_{t_{o n}}^{T} \frac{V_{D}}{L} d t \\ \int_{0}^{T} \frac{V_{o}}{L} d t & = \frac{T V_{i}}{L} - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t + \int_{0}^{t_{o n}} \frac{V_{C}}{L} d t - \int_{t_{o n}}^{T} \frac{V_{D}}{L} d t \\ \frac{1}{T} \int_{0}^{T} \frac{V_{o}}{L} d t & = \frac{V_{i}}{L} + \frac{1}{T} (\int_{0}^{t_{o n}} \frac{V_{C}}{L} d t - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t - \int_{t_{o n}}^{T} \frac{V_{D}}{L} d t) \\ \frac{1}{T} \int_{0}^{T} V_{o} d t & = V_{i} + \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t - \int_{0}^{T_{o n}} V_{s w} d t - \int_{t_{o n}}^{T} V_{D} d t) \\ V_{o, a v} & = V_{i} + \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t - \int_{0}^{T_{o n}} V_{s w} d t - \int_{t_{o n}}^{T} V_{D} d t) \end{aligned}

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

Δ I_{L_{o n}}

可得下面兩式

{\begin{matrix} I_{m a x} + I_{m i n} & = & 2 * I_{a v} \\ I_{m a x} - I_{m i n} & = & \frac{D T V_{i}}{L} - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t \end{matrix} {\begin{matrix} 2 \cdot I_{m a x} & = & 2 * I_{a v} + \frac{D T V_{i}}{L} - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t \\ 2 \cdot I_{m i n} & = & 2 * I_{a v} - \frac{D T V_{i}}{L} + \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t \end{matrix} {\begin{matrix} I_{m a x} & = & I_{a v} + \frac{D T V_{i}}{2 L} - \int_{0}^{T_{o n}} \frac{V_{s w}}{2 L} d t \\ I_{m i n} & = & I_{a v} - \frac{D T V_{i}}{2 L} + \int_{0}^{T_{o n}} \frac{V_{s w}}{2 L} d t \end{matrix}

或者根據之前推導的

Δ I_{L_{o f f}}

可得下面兩式

{\begin{matrix} I_{m a x} + I_{m i n} & = & 2 * I_{a v} \\ I_{m i n} - I_{m a x} & = & \frac{(1 - D) T V_{i}}{L} - \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{L} d t \end{matrix} {\begin{matrix} I_{m a x} + I_{m i n} & = & 2 * I_{a v} \\ I_{m a x} - I_{m i n} & = & - \frac{(1 - D) T V_{i}}{L} + \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{L} d t \end{matrix} {\begin{matrix} 2 \cdot I_{m a x} & = & 2 * I_{a v} - \frac{(1 - D) T V_{i}}{L} + \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{L} d t \\ 2 \cdot I_{m i n} & = & 2 * I_{a v} + \frac{(1 - D) T V_{i}}{L} - \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{L} d t \end{matrix} {\begin{matrix} I_{m a x} & = & I_{a v} - \frac{(1 - D) T V_{i}}{2 L} + \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{2 L} d t \\ I_{m i n} & = & I_{a v} + \frac{(1 - D) T V_{i}}{2 L} - \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{2 L} d t \end{matrix}

保持 continuous mode 的條件，就是

I_{L} \geq 0

\begin{aligned} I_{m i n} & = I_{a v} + \frac{(1 - D) T V_{i}}{2 L} - \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{2 L} d t \geq 0 \\ I_{a v} & \geq - \frac{(1 - D) T V_{i}}{2 L} + \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{2 L} d t \\ \frac{V_{o, a v}}{R} & \geq - \frac{(1 - D) T V_{i}}{2 L} + \int_{t_{o n}}^{T} \frac{V_{o} + V_{D}}{2 L} d t \\ L & \geq - \frac{R (1 - D) T V_{i}}{2 V_{o, a v}} + \frac{R}{2 V_{o, a v}} \int_{t_{o n}}^{T} (V_{o} + V_{D}) d t \end{aligned}

Discontinuous mode

\begin{aligned} V_{o, a v} & = V_{i} + \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t - \int_{0}^{t_{o n}} V_{s w} d t - \int_{t_{o n}}^{T} V_{D} d t) \\ Δ I_{L} & = \frac{D T V_{i}}{L} - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t = - \frac{δ T V_{i}}{L} + \int_{t_{o n}}^{t_{o n} + δ T} \frac{V_{o} + V_{D}}{L} d t \end{aligned}

{\begin{cases} V_{o} = V_{C} & if t i n T_{o n} = D T \\ V_{o} = V_{i} - V_{L} - V_{D} & if t i n δ T \\ V_{o} = V_{i} - V_{D} & if t i n o t h e r s \end{cases}

平均輸出電壓

\begin{aligned} V_{o, a v} & = \frac{1}{T} \int_{0}^{t} V_{o} (t) d t \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{o} (t) d t + \int_{t_{o n}}^{t_{o n} + δ T} V_{o} (t) d t + \int_{t_{o n} + δ T}^{T} V_{o} (t) d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t + \int_{t_{o n}}^{t_{o n} + δ T} (V_{i} - V_{L} - V_{D}) d t + \int_{t_{o n} + δ T}^{T} (V_{i} - V_{D}) d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t - \int_{t_{o n}}^{t_{o n} + δ T} V_{L} d t + \int_{t_{o n}}^{t_{o n} + δ T} (V_{i} - V_{D}) d t + \int_{t_{o n} + δ T}^{T} (V_{i} - V_{D}) d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t - \int_{t_{o n}}^{t_{o n} + δ T} V_{L} d t + \int_{t_{o n}}^{T} (V_{i} - V_{D}) d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t - \int_{0}^{t_{o n} + δ T} V_{L} d t + \int_{0}^{t_{o n}} V_{L} d t + \int_{t_{o n}}^{T} (V_{i} - V_{D}) d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t - 0 + \int_{0}^{t_{o n}} (V_{i} - V_{s w}) d t + \int_{t_{o n}}^{T} (V_{i} - V_{D}) d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t + \int_{0}^{t_{o n}} V_{i} d t - \int_{0}^{t_{o n}} V_{s w} d t + \int_{t_{o n}}^{T} V_{i} d t - \int_{t_{o n}}^{T} V_{D} d t) \\ = \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t + \int_{0}^{T} V_{i} d t - \int_{0}^{t_{o n}} V_{s w} d t - \int_{t_{o n}}^{T} V_{D} d t) \\ = V_{i} + \frac{1}{T} (\int_{0}^{t_{o n}} V_{C} d t - \int_{0}^{t_{o n}} V_{s w} d t - \int_{t_{o n}}^{T} V_{D} d t) \end{aligned}

V_{L} = L \frac{d I_{L}}{d t}

\begin{aligned} Δ I_{L_{o n}} & = I_{m a x} - I_{m i n} \\ = \int_{0}^{T_{o n}} \frac{V_{L}}{L} d t \\ = \int_{0}^{T_{o n}} \frac{V_{i} - V_{s w}}{L} d t \\ = \int_{0}^{T_{o n}} \frac{V_{i}}{L} d t - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t \\ = \frac{D T V_{i}}{L} - \int_{0}^{T_{o n}} \frac{V_{s w}}{L} d t \\ Δ I_{L_{o f f}} & = I_{m i n} - I_{m a x} \\ = \int_{t_{o n}}^{t_{o n} + δ T} \frac{V_{L}}{L} d t \\ = \int_{t_{o n}}^{t_{o n} + δ T} \frac{V_{i} - V_{o} - V_{D}}{L} d t \\ = \frac{δ T V_{i}}{L} - \int_{t_{o n}}^{t_{o n} + δ T} \frac{V_{o} + V_{D}}{L} d t \end{aligned}

Simulation GitHub

參考

Boost converter

子風的知識庫

搜尋此網誌

[Circuit] Boost Converter

設計概念

Continuous mode

Discontinuous mode

Simulation GitHub

參考

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