电阻星形连接与三角形连接的等效变换公式

电阻星形连接与三角形连接的等效变换公式

在三相电路和复杂电阻网络中，星形（Y形）与三角形（$\Delta$形）连接的等效变换是简化电路分析的重要工具。

一、变换条件

等效变换的条件：变换前后，三个端子之间的电阻值保持不变。文章源自公式库网-https://www.gongshiku.com/html/201911/dianzuxingxinglianjieyusanjiaoxinglianjiededengxiaobianhuangongshi.html

二、三角形→星形（$\Delta \to$ Y）

$R_a = \dfrac{R_{ab} \cdot R_{ca}}{R_{ab} + R_{bc} + R_{ca}}$文章源自公式库网-https://www.gongshiku.com/html/201911/dianzuxingxinglianjieyusanjiaoxinglianjiededengxiaobianhuangongshi.html

$R_b = \dfrac{R_{bc} \cdot R_{ab}}{R_{ab} + R_{bc} + R_{ca}}$文章源自公式库网-https://www.gongshiku.com/html/201911/dianzuxingxinglianjieyusanjiaoxinglianjiededengxiaobianhuangongshi.html

$R_c = \dfrac{R_{ca} \cdot R_{bc}}{R_{ab} + R_{bc} + R_{ca}}$文章源自公式库网-https://www.gongshiku.com/html/201911/dianzuxingxinglianjieyusanjiaoxinglianjiededengxiaobianhuangongshi.html

口诀：星形电阻 = 相邻两三角形电阻的乘积 ÷ 三个三角形电阻之和文章源自公式库网-https://www.gongshiku.com/html/201911/dianzuxingxinglianjieyusanjiaoxinglianjiededengxiaobianhuangongshi.html

三、星形→三角形（Y $\to \Delta$）

$R_{ab} = R_a + R_b + \dfrac{R_a R_b}{R_c}$文章源自公式库网-https://www.gongshiku.com/html/201911/dianzuxingxinglianjieyusanjiaoxinglianjiededengxiaobianhuangongshi.html

$R_{bc} = R_b + R_c + \dfrac{R_b R_c}{R_a}$文章源自公式库网-https://www.gongshiku.com/html/201911/dianzuxingxinglianjieyusanjiaoxinglianjiededengxiaobianhuangongshi.html

$R_{ca} = R_c + R_a + \dfrac{R_c R_a}{R_b}$文章源自公式库网-https://www.gongshiku.com/html/201911/dianzuxingxinglianjieyusanjiaoxinglianjiededengxiaobianhuangongshi.html

四、特殊情况：对称变换

当三个电阻相等时：文章源自公式库网-https://www.gongshiku.com/html/201911/dianzuxingxinglianjieyusanjiaoxinglianjiededengxiaobianhuangongshi.html

$R_Y = \dfrac{R_{\Delta}}{3}$（三角形变星形除以3）文章源自公式库网-https://www.gongshiku.com/html/201911/dianzuxingxinglianjieyusanjiaoxinglianjiededengxiaobianhuangongshi.html

$R_{\Delta} = 3R_Y$（星形变三角形乘以3）

在三相电路中，若每相负载电阻为 $R$：星形连接线间等效电阻为 $2R$，三角形连接线间等效电阻为 $\frac{2}{3}R$。