# 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。这一变换理论是由亚瑟·肯内利（Arthur Kennelly）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

$R_y = \frac{R'R''}{\sum R_\Delta}$

$R_1 = \frac{R_aR_b}{R_a + R_b + R_c}$
$R_2 = \frac{R_bR_c}{R_a + R_b + R_c}$
$R_3 = \frac{R_aR_c}{R_a + R_b + R_c}$

### 把Y形电路变换成Δ形电路

$R_\Delta = \frac{R_P}{R_\mathrm{opposite}}$

$R_a = \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_2}$
$R_b = \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_3}$
$R_c = \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_1}$

## 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

\begin{align} 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{align}

$R_T = R_a + R_b + R_c$

$R_\Delta(N_1, N_2) = \frac{R_b(R_a+R_c)}{R_T}$

Y形电路中N12的对应阻抗为

$R_Y(N_1, N_2) = R_1 + R_2$

$R_1+R_2 = \frac{R_b(R_a+R_c)}{R_T}$   (1)

$R_2+R_3 = \frac{R_c(R_a+R_b)}{R_T}$   (2)

$R_1+R_3 = \frac{R_a(R_b+R_c)}{R_T}.$   (3)

$R_1+R_2+R_1+R_3-R_2-R_3 = \frac{R_b(R_a+R_c)}{R_T} + \frac{R_a(R_b+R_c)}{R_T} - \frac{R_c(R_a+R_b)}{R_T}$
$2R_1 = \frac{2R_bR_a}{R_T}$

$R_1 = \frac{R_bR_a}{R_T}.$

$R_1 = \frac{R_bR_a}{R_T}$ (4)

$R_2 = \frac{R_bR_c}{R_T}$ (5)

$R_3 = \frac{R_aR_c}{R_T}$ (6)

### Y形负载到Δ形负载的变换方程

$R_T = R_a+R_b+R_c$.

$R_1 = \frac{R_aR_b}{R_T}$   (1)

$R_2 = \frac{R_bR_c}{R_T}$   (2)

$R_3 = \frac{R_aR_c}{R_T}.$   (3)

$R_1R_2 = \frac{R_aR_b^2R_c}{R_T^2}$   (4)

$R_1R_3 = \frac{R_a^2R_bR_c}{R_T^2}$   (5)

$R_2R_3 = \frac{R_aR_bR_c^2}{R_T^2}$   (6)

$R_1R_2 + R_1R_3 + R_2R_3 = \frac{R_aR_b^2R_c + R_a^2R_bR_c + R_aR_bR_c^2}{R_T^2}$   (7)

$R_1R_2 + R_1R_3 + R_2R_3 = \frac{(R_aR_bR_c)(R_a+R_b+R_c)}{R_T^2}$
$R_1R_2 + R_1R_3 + R_2R_3 = \frac{R_aR_bR_c}{R_T}$ (8)

$\frac{R_1R_2 + R_1R_3 + R_2R_3}{R_1} = \frac{R_aR_bR_c}{R_T}\frac{R_T}{R_aR_b},$
$\frac{R_1R_2 + R_1R_3 + R_2R_3}{R_1} = R_c,$

## 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.