# Y-Δ变换

Δ形电路和Y形电路

Y-Δ变换或稱為星角變換，是一种把Y形电路转换成等效的Δ形电路，或把Δ形电路转换成等效的Y形电路的方法。它可以用来简化电路的分析。这一变换理论是由於1899年发表。[1]

## 基本的Y-Δ变换

### 把Δ形电路变换成Y形电路

${\displaystyle R_{y}={\frac {R'R''}{\sum R_{\Delta }}}}$

${\displaystyle R_{1}={\frac {R_{a}R_{b}}{R_{a}+R_{b}+R_{c}}}}$
${\displaystyle R_{2}={\frac {R_{b}R_{c}}{R_{a}+R_{b}+R_{c}}}}$
${\displaystyle R_{3}={\frac {R_{a}R_{c}}{R_{a}+R_{b}+R_{c}}}}$

### 把Y形电路变换成Δ形电路

${\displaystyle R_{\Delta }={\frac {R_{P}}{R_{\mathrm {opposite} }}}}$

${\displaystyle R_{a}={\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{2}}}}$
${\displaystyle R_{b}={\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{3}}}}$
${\displaystyle R_{c}={\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{1}}}}$

## 推导

### Δ形负载到Y形负载的变换方程

Δ形电路中N3断开後，N1N2间的阻抗为

{\displaystyle {\begin{aligned}R_{\Delta }(N_{1},N_{2})&=R_{b}\parallel (R_{a}+R_{c})\\[8pt]&={\frac {1}{{\frac {1}{R_{b}}}+{\frac {1}{R_{a}+R_{c}}}}}\\[8pt]&={\frac {R_{b}(R_{a}+R_{c})}{R_{a}+R_{b}+R_{c}}}.\end{aligned}}}

${\displaystyle R_{T}=R_{a}+R_{b}+R_{c}}$

${\displaystyle R_{\Delta }(N_{1},N_{2})={\frac {R_{b}(R_{a}+R_{c})}{R_{T}}}}$

Y形电路中N12的对应阻抗为

${\displaystyle R_{Y}(N_{1},N_{2})=R_{1}+R_{2}}$

${\displaystyle R_{1}+R_{2}={\frac {R_{b}(R_{a}+R_{c})}{R_{T}}}}$    (1)

${\displaystyle R_{2}+R_{3}={\frac {R_{c}(R_{a}+R_{b})}{R_{T}}}}$    (2)

${\displaystyle R_{1}+R_{3}={\frac {R_{a}(R_{b}+R_{c})}{R_{T}}}.}$    (3)

${\displaystyle R_{1}+R_{2}+R_{1}+R_{3}-R_{2}-R_{3}={\frac {R_{b}(R_{a}+R_{c})}{R_{T}}}+{\frac {R_{a}(R_{b}+R_{c})}{R_{T}}}-{\frac {R_{c}(R_{a}+R_{b})}{R_{T}}}}$
${\displaystyle 2R_{1}={\frac {2R_{b}R_{a}}{R_{T}}}}$

${\displaystyle R_{1}={\frac {R_{b}R_{a}}{R_{T}}}.}$

${\displaystyle R_{1}={\frac {R_{b}R_{a}}{R_{T}}}}$  (4)

${\displaystyle R_{2}={\frac {R_{b}R_{c}}{R_{T}}}}$  (5)

${\displaystyle R_{3}={\frac {R_{a}R_{c}}{R_{T}}}}$  (6)

### Y形负载到Δ形负载的变换方程

${\displaystyle R_{T}=R_{a}+R_{b}+R_{c}}$ .

${\displaystyle R_{1}={\frac {R_{a}R_{b}}{R_{T}}}}$    (1)

${\displaystyle R_{2}={\frac {R_{b}R_{c}}{R_{T}}}}$    (2)

${\displaystyle R_{3}={\frac {R_{a}R_{c}}{R_{T}}}.}$    (3)

${\displaystyle R_{1}R_{2}={\frac {R_{a}R_{b}^{2}R_{c}}{R_{T}^{2}}}}$    (4)

${\displaystyle R_{1}R_{3}={\frac {R_{a}^{2}R_{b}R_{c}}{R_{T}^{2}}}}$    (5)

${\displaystyle R_{2}R_{3}={\frac {R_{a}R_{b}R_{c}^{2}}{R_{T}^{2}}}}$    (6)

${\displaystyle R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}={\frac {R_{a}R_{b}^{2}R_{c}+R_{a}^{2}R_{b}R_{c}+R_{a}R_{b}R_{c}^{2}}{R_{T}^{2}}}}$    (7)

${\displaystyle R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}={\frac {(R_{a}R_{b}R_{c})(R_{a}+R_{b}+R_{c})}{R_{T}^{2}}}}$
${\displaystyle R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}={\frac {R_{a}R_{b}R_{c}}{R_{T}}}}$  (8)

${\displaystyle {\frac {R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}}{R_{1}}}={\frac {R_{a}R_{b}R_{c}}{R_{T}}}{\frac {R_{T}}{R_{a}R_{b}}},}$
${\displaystyle {\frac {R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}}{R_{1}}}=R_{c},}$

## 参考文献

• William Stevenson，“Elements of Power System Analysis 3rd ed.”，McGraw Hill, New York, 1975, ISBN 0-07-061285-4
1. ^ A.E. Kennelly, Equivalence of triangles and stars in conducting networks, Electrical World and Engineer, vol. 34, pp. 413-414, 1899.