Buck转换器的电流纹波系数

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<div class=WordSection1> <h1>Buck转换器的电流纹波系数</h1> Current Ripple Factor of a Buck Converter <h2 id="__Bookmark__0">摘要</h2> 电感和电容在 Buck 转换器中构成低通滤波器，此 LC 滤波器的转折频率总是设计得较低以便滤除开关纹波。常用的设计规则是，电感电流纹波被设计成电感平均电流的 30% 左右。本文引入表达电感纹波电流和其平均电流的比值的纹波系数的概念并由它导出电感尺寸的表达式。通过利用面积积的概念和方法来设计电感，可以获得一个优化的纹波系数范围，这对进行转换器设计时选择合适的电感具有很大的帮助。 <h2 id="__Bookmark__1">1. 介绍 </h2> Buck 转换器被广泛应用于大量的降压应用中，板上负载点转换器是这种应用的典型例子。从原理上看，它就是用一个功率开关和一个自动运行的续流二极管构成一个斩波电路将输入电压变成一个矩形波，再通过一个低通 LC 滤波器滤除高频开关纹波和噪声以后生成一个基本上是纯粹直流电压供给负载使用的电路。图1 显示了典型的Buck 电路和他的开关波形。 <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image001_zh-cn.jpg" ><img width=680 height=174 src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image001_zh-cn.jpg" alt="Technical Document Image Preview"></a> 图1. (a) 典型的 Buck 电路 (b) 开关波形 当功率开关 Q1 导通时，续流二极管反向偏置，输入电流直接通过 LC 滤波器流向负载；当 Q1 关闭时，电感电流 iL 强制 D1 正向偏置导通。开关电压波形如图1所示为矩形脉冲。经过 LC 滤波以后，假设 LC 滤波器的转折频率远低于开关工作频率，输出电压基本上是纯粹的直流状态。 从滤波器理论很容易理解到，增加电感、减小电容可以导致同样的输出纹波效果，但是大电感会导致大体积和高成本的结果，反之，使用小电感则要求使用大电容，这并不是一个纯粹的设计取舍问题。 为了探讨此问题，让我们从稳态下的电感电流开始思考。 当功率开关导通 (ON) 时，加在电感上的电压是输入电压和输出电压之间的电压差： <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image002_zh-cn.jpg" ><img width=452 height=27 id="圖片 1" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image002_zh-cn.jpg" alt="Technical Document Image Preview"></a> 电感电流将从 iL(0) 开始线性增加： <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image003_zh-cn.jpg" ><img width=441 height=37 id="圖片 10" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image003_zh-cn.jpg" alt="Technical Document Image Preview"></a> 当功率开关截至 (OFF) 时，加在电感上的电压与输出电压相同，但极性相反： <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image004_zh-cn.jpg" ><img width=463 height=24 id="圖片 11" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image004_zh-cn.jpg" alt="Technical Document Image Preview"></a> 在此期间，电感电流将以斜率从 iL(Ton) 开始线性减少： <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image005_zh-cn.jpg" ><img width=463 height=36 id="圖片 12" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image005_zh-cn.jpg" alt="Technical Document Image Preview"></a> 根据电感伏秒平衡的特性，很容易从式 (1) 和 (3) 得到电压传输比： <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image006_zh-cn.jpg" ><img width=463 height=37 id="圖片 13" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image006_zh-cn.jpg" alt="Technical Document Image Preview"></a> 这也就是占空比。与图1(b) 进行比较，LC 滤波器工作起来就像一个平均函数，而占空比 D 就定义为开关管的导通时间和一个开关周期的比。 <h2 id="__Bookmark__2">2. 纹波系数 </h2> 图2 所示为电感电流波形。因为电感所受电压是矩形脉冲波，所以电感电流是包含一个支流分量的三角波。 <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image007_zh-cn.jpg" ><img width=355 height=189 id="圖片 3" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image007_zh-cn.jpg" alt="Technical Document Image Preview"></a> 图2. 电感电流波形电感纹波电流值被定义为 <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image008_zh-cn.jpg" ><img width=451 height=29 id="圖片 15" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image008_zh-cn.jpg" alt="Technical Document Image Preview"></a> 很明显，负载电流可表达为 <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image009_zh-cn.jpg" ><img width=448 height=38 id="圖片 16" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image009_zh-cn.jpg" alt="Technical Document Image Preview"></a> 纹波系数可被定义为 <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image010_zh-cn.jpg" ><img width=446 height=36 id="圖片 17" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image010_zh-cn.jpg" alt="Technical Document Image Preview"></a> 当纹波系数小于2时，转换器工作在连续导通模式 (Continous Conduction Mode, CCM)，否则就是非连续导通模式 (Discontinuous Conduction Mode, DCM)。由于连续导通模式下功率元件所受电流应力较低，工作在满载状态下的 Buck 转换器一般都被设计成工作在这种模式下。因此，本文也只对连续导通模式进行讨论。 等式(8)可以被表现为电压相关的形式： <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image011_zh-cn.jpg" ><img width=483 height=54 id="圖片 18" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image011_zh-cn.jpg" alt="Technical Document Image Preview"></a> 对于一个固定的电感量而言，输入电压越高，纹波系数就越高。当输入电压固定时，电感量越小，纹波系数就越高。纹波系数越高，意味着流过电容的纹波电流越大，对于相同的纹波电压需求来说，就需要更大的电容量。 <h2 id="__Bookmark__3">3. 电感的面积积 </h2> 从前面的描述中我们已经知道，开关导通期间电感储存能量，开关截至期间电感释放能量。从原理上讲，负载电流会流过电感，所以足够的线圈空间是必须的。假如设计了较低的纹波系数（或是较高的电感量），较多的线圈匝数就是必须的，这将导致更大的电感尺寸。我们要在这里引入一个面积积 (Area Product, AP) 的概念来表征电感尺寸，它是磁芯的有效横截面积和线圈窗口面积的乘积，其单位是 m4 而不是表达体积的 m3，但面积积是和磁芯的体积成正比的。 根据法拉第定律电感量 (L)、峰值电流 (ipk) 个磁芯之间的关系可以表达为 <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image012_zh-cn.jpg" ><img width=494 height=25 id="圖片 19" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image012_zh-cn.jpg" alt="Technical Document Image Preview"></a> 其中，ipk = io + 1/2 ΔiL，N 是线圈匝数，Bm 是磁芯的最大磁通密度，AC 是磁芯的有效横截面积对于线圈，有下列等式成立 <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image013_zh-cn.jpg" ><img width=497 height=36 id="圖片 20" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image013_zh-cn.jpg" alt="Technical Document Image Preview"></a> 其中，AWT 线圈导体的横截面积，J 导体中的电流密度，kW 是磁芯的填充系数，Wa 是线圈窗口面积。结合 (10) 和 (11) 式，我们可以得到 <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image014_zh-cn.jpg" ><img width=487 height=43 id="圖片 21" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image014_zh-cn.jpg" alt="Technical Document Image Preview"></a> 电感的电流有效值可用它的直流分量和交流分量来表达： <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image015_zh-cn.jpg" ><img width=495 height=47 id="圖片 22" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image015_zh-cn.jpg" alt="Technical Document Image Preview"></a> 等式 (12) 可被重写为 <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image016_zh-cn.jpg" ><img width=492 height=82 id="圖片 23" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image016_zh-cn.jpg" alt="Technical Document Image Preview"></a> 图3 显示了磁芯在不同占空比下的归一化尺寸和纹波系数之间的关系。 <a class="fancybox" href="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image017_zh-cn.jpg" ><img width=512 height=350 id="圖片 5" src="/content/dam/richtek-eds/resources/application-notes/an009/html-images-zh_cn/image017_zh-cn.jpg" alt="Technical Document Image Preview"></a> 图3. 不同占空比下纹波系数和电感尺寸之间的关系 当纹波系数很低时，磁芯尺寸显著地增加；当纹波系数较高时，磁芯尺寸几乎没有什么变化。这意味着在曲线拐弯的区域存在一个最优化的地带。从原则上讲，高纹波系数意味着滤波电容也要大，反之亦然。举例言之，当D= 0.3 时，纹波系数可以设定在 0.2~0.4 之间，这样可以得到比较合适的磁芯尺寸和电容尺寸。 <h2 id="__Bookmark__4">4. 一个设计示例 </h2> 一个开关工作频率为 300kHz 的 Buck 转换器要在下列条件下工作： Vin = 4 ~ 12V, Vout = 1.8V, Io = 6A, ΔVo = 10mV （输出电容上的纹波电压）。设计中假设功率开关和续流二极管都是理想的，表1 显示了用传统方法按照 30% 纹波系数计算出的电感量，而按照本文提案的面积积方法得出的经过优化的计算结果在表2 中列出。 表1. 传统方法 30% 纹波系数计算结果 <div> <table class=MsoNormalTable border=1 cellspacing=0 cellpadding=0 width=550 style='width:412.5pt;border-collapse:collapse;border:none'> <tr> <td width=110 style='width:82.5pt;border:solid #009EB8 1.0pt;background:#009EB8; padding:.75pt .75pt .75pt .75pt'> Input Voltage (V) </td> <td width=110 style='width:82.5pt;border:solid #009EB8 1.0pt;border-left: none;background:#009EB8;padding:.75pt .75pt .75pt .75pt'> Duty Cycle </td> <td width=110 style='width:82.5pt;border:solid #009EB8 1.0pt;border-left: none;background:#009EB8;padding:.75pt .75pt .75pt .75pt'> Ripple Factor </td> <td width=110 style='width:82.5pt;border:solid #009EB8 1.0pt;border-left: none;background:#009EB8;padding:.75pt .75pt .75pt .75pt'> Ripple Current (A) </td> <td width=110 style='width:82.5pt;border:solid #009EB8 1.0pt;border-left: none;background:#009EB8;padding:.75pt .75pt .75pt .75pt'> Ripple Current RMS (A) </td> <td width=110 style='width:82.5pt;border:solid #009EB8 1.0pt;border-left: none;background:#009EB8;padding:.75pt .75pt .75pt .75pt'> Inductance (μH) </td> </tr> <tr> <td style='border:solid #009EB8 1.0pt;border-top:none;padding:.75pt .75pt .75pt .75pt'> 4 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.30 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.3 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 1.80 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.52 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 1.56 </td> </tr> <tr> <td style='border:solid #009EB8 1.0pt;border-top:none;padding:.75pt .75pt .75pt .75pt'> 8 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.15 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.3 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 1.80 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.52 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 1.89 </td> </tr> <tr> <td style='border:solid #009EB8 1.0pt;border-top:none;padding:.75pt .75pt .75pt .75pt'> 12 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.1 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.3 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 1.80 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.52 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 2.00 </td> </tr> </table> </div> 表2. 优化后方法计算结果 <div> <table class=MsoNormalTable border=1 cellspacing=0 cellpadding=0 width=550 style='width:412.5pt;border-collapse:collapse;border:none'> <tr> <td width=110 style='width:82.5pt;border:solid #009EB8 1.0pt;background:#009EB8; padding:.75pt .75pt .75pt .75pt'> Input Voltage (V) </td> <td width=110 style='width:82.5pt;border:solid #009EB8 1.0pt;border-left: none;background:#009EB8;padding:.75pt .75pt .75pt .75pt'> Duty Cycle </td> <td width=110 style='width:82.5pt;border:solid #009EB8 1.0pt;border-left: none;background:#009EB8;padding:.75pt .75pt .75pt .75pt'> Optimal Ripple Factor </td> <td width=110 style='width:82.5pt;border:solid #009EB8 1.0pt;border-left: none;background:#009EB8;padding:.75pt .75pt .75pt .75pt'> Ripple Current (A) </td> <td width=110 style='width:82.5pt;border:solid #009EB8 1.0pt;border-left: none;background:#009EB8;padding:.75pt .75pt .75pt .75pt'> Ripple Current RMS (A) </td> <td width=110 style='width:82.5pt;border:solid #009EB8 1.0pt;border-left: none;background:#009EB8;padding:.75pt .75pt .75pt .75pt'> Inductance (μH) </td> </tr> <tr> <td style='border:solid #009EB8 1.0pt;border-top:none;padding:.75pt .75pt .75pt .75pt'> 4 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.30 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.42 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 2.52 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.73 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 1.11 </td> </tr> <tr> <td style='border:solid #009EB8 1.0pt;border-top:none;padding:.75pt .75pt .75pt .75pt'> 8 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.15 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.45 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 2.7 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.78 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 1.26 </td> </tr> <tr> <td style='border:solid #009EB8 1.0pt;border-top:none;padding:.75pt .75pt .75pt .75pt'> 12 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.1 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.48 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 2.88 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 0.83 </td> <td style='border-top:none;border-left:none;border-bottom:solid #009EB8 1.0pt; border-right:solid #009EB8 1.0pt;padding:.75pt .75pt .75pt .75pt'> 1.25 </td> </tr> </table> </div> 在表1中，传统的计算方法设定了相同的纹波电流值，因而输入电压较高时电感值就较大；而在表2中应用的是面积积的计算方法，不同输入电压下的电感量基本上是相同的，但输入电压较高时纹波电流也较大。在实际的高频设计实践中，常常采用 POCAP 或 MLCC 作为输出电容，它们都具有极低的串联等效电阻，因此要得到纹波电压指标是很容易的。 <h2 id="__Bookmark__5">5. 总结 </h2> 本文提出了一种 Buck 转换器设计中电感尺寸和纹波电流之间关系的理论分析方法，利用面积积方法可在不同输入电压下得到优化的纹波系数范围，可作为 Buck 转换器设计中优化电感设计的指南予以利用。 <table border="1" cellspacing="0" cellpadding="0" style="width: 637.5pt; border-collapse: collapse; border: medium none;" class="3-51"> <tbody> <tr style="height: 17pt;"> <td style="border-width: 1pt 1px 1px 1pt; border-style: solid; border-color: #009eb8; background-color: #009eb8; background-image: url(e); background-repeat: repeat; background-attachment: scroll; background-clip: border-box; background-origin: padding-box; background-size: auto auto; padding-right: 5pt; padding-left: 5pt; height: 17pt;" colspan="2"> 相关资源 </td> </tr> <tr> <td style="border-width: 1pt 1px 1px 1pt; border-style: solid; border-color: #009eb8; background-color: #ffffff; background-image: url(e); background-repeat: repeat; background-attachment: scroll; background-clip: border-box; background-origin: padding-box; background-size: auto auto; padding-right: 5pt; padding-left: 5pt; height: 17pt;"> 立锜科技电子报 </td> <td style="border-width: 1pt 1px 1px 1pt; border-style: solid; border-color: #009eb8; background-color: #ffffff; background-image: url(e); background-repeat: repeat; background-attachment: scroll; background-clip: border-box; background-origin: 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