# تحويل واي-دلتا

(تم التحويل من Y-Δ transform)
تحليل خطي للشبكات
العناصر

المكونات

دوائر التوالي والتوازي

تحويلات المعاوقة

مبرهنات المولد مبرهنات الشبكة

أساليب تحليل الشبكات

Two-port parameters

تحويل واي دلتا, Y-Δ transform وتكتب Y-delta، Wye-delta، Kennelly’s delta-star transformation, star-mesh transformation, T-Π or T-pi transform، هي تقنية رياضية لتبسيط تحليل الشبكة الإلكترونية.

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## Basic Y-Δ transformation

Δ and Y circuits with the labels which are used in this article.

The transformation is used to establish equivalence for networks with 3 terminals. Where three elements terminate at a common node and none are sources, the node is eliminated by transforming the impedances. For equivalence, the impedance between any pair of terminals must be the same for both networks. The equations given here are valid for real as well as complex impedances.

The general idea is to compute the impedance ${\displaystyle R_{y}}$  at a terminal node of the Y circuit with impedances ${\displaystyle R'}$ , ${\displaystyle R''}$  to adjacent nodes in the Δ circuit by

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

where ${\displaystyle R_{\Delta }}$  are all impedances in the Δ circuit. This yields the specific formulae

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

The general idea is to compute an impedance ${\displaystyle R_{\Delta }}$  in the Δ circuit by

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

where ${\displaystyle R_{P}=R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}$  is the sum of the products of all pairs of impedances in the Y circuit and ${\displaystyle R_{\mathrm {opposite} }}$  is the impedance of the node in the Y circuit which is opposite the edge with ${\displaystyle R_{\Delta }}$ . The formula for the individual edges are thus

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

## Graph theory

In graph theory, the Y-Δ transform means replacing a Y subgraph of a graph with the equivalent Δ subgraph. The transform preserves the number of edges in a graph, but not the number of vertices or the number of cycles. Two graphs are said to be Y-Δ equivalent if one can be obtained from the other by a series of Y-Δ transforms in either direction. For example, the Petersen graph family is a Y-Δ equivalence class.

## Demonstration

Δ and Y circuits with the labels that are used in this article.

To relate {${\displaystyle R_{a},R_{b},R_{c}}$ } from Δ to {${\displaystyle R_{1},R_{2},R_{3}}$ } from Y, the impedance between two corresponding nodes is compared. The impedance in either configuration is determined as if one of the nodes is disconnected from the circuit.

The impedance between N1 and N2 with N3 disconnected in Δ:

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

To simplify, let's call ${\displaystyle R_{T}}$  the sum of {${\displaystyle R_{a},R_{b},R_{c}}$ }.

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

Thus,

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

The corresponding impedance between N1 and N2 in Y is simple:

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

hence:

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

Repeating for ${\displaystyle R(N_{2},N_{3})}$ :

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

and for ${\displaystyle R(N_{1},N_{3})}$ :

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

From here, the values of {${\displaystyle R_{1},R_{2},R_{3}}$ } can be determined by linear combination (addition and/or subtraction).

For example, adding (1) and (3), then subtracting (2) yields

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

thus,

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

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

For completeness:

${\displaystyle R_{1}={\frac {R_{b}R_{a}}{R_{T}}}}$  (4)
${\displaystyle R_{2}={\frac {R_{b}R_{c}}{R_{T}}}}$  (5)
${\displaystyle R_{3}={\frac {R_{a}R_{c}}{R_{T}}}}$  (6)

Let

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

We can write the Δ to Y equations as

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

Multiplying the pairs of equations yields

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

and the sum of these equations is

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

Factor ${\displaystyle R_{a}R_{b}R_{c}}$  from the right side, leaving ${\displaystyle R_{T}}$  in the numerator, canceling with an ${\displaystyle R_{T}}$  in the denominator.

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

-Note the similarity between (8) and {(1),(2),(3)}

Divide (8) by (1)

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

which is the equation for ${\displaystyle R_{c}}$ . Dividing (8) by ${\displaystyle R_{2}}$  or ${\displaystyle R_{3}}$  gives the other equations.

## المصادر

• William Stevenson, “Elements of Power System Analysis 3rd ed.”, McGraw Hill, New York, 1975, ISBN 0070612854