تحويل واي-دلتا

تحليل خطي للشبكات
العناصر

المكونات

دوائر التوالي والتوازي

تحويلات المعاوقة

مبرهنات المولد	مبرهنات الشبكة

أساليب تحليل الشبكات

Two-port parameters

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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.

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.

Equations for the transformation from Δ-load to Y-load 3-phase circuit

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

R_{y} = \frac{R^{'} R^{″}}{\sum R_{Δ}}

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

R_{1} = \frac{R_{a} R_{b}}{R_{a} + R_{b} + R_{c}},

R_{2} = \frac{R_{b} R_{c}}{R_{a} + R_{b} + R_{c}},

R_{3} = \frac{R_{a} R_{c}}{R_{a} + R_{b} + R_{c}} .

Equations for the transformation from Y-load to Δ-load 3-phase circuit

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

R_{Δ} = \frac{R_{P}}{R_{o p p o s i t e}}

where $R_{P} = R_{1} R_{2} + R_{2} R_{3} + R_{3} R_{1}$ is the sum of the products of all pairs of impedances in the Y circuit and $R_{o p p o s i t e}$ is the impedance of the node in the Y circuit which is opposite the edge with $R_{Δ}$ . The formula for the individual edges are thus

R_{a} = \frac{R_{1} R_{2} + R_{2} R_{3} + R_{3} R_{1}}{R_{2}},

R_{b} = \frac{R_{1} R_{2} + R_{2} R_{3} + R_{3} R_{1}}{R_{3}},

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-load to Y-load transformation equations

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

To relate { $R_{a}, R_{b}, R_{c}$ } from Δ to { $R_{1}, R_{2}, R_{3}$ } 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:

\begin{array}{l} R_{Δ} (N_{1}, N_{2}) & = R_{b} ∥ (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{array}

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

R_{T} = R_{a} + R_{b} + R_{c}

Thus,

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

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

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

hence:

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

(1)

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

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

(2)

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

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

(3)

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

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

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}}

2 R_{1} = \frac{2 R_{b} R_{a}}{R_{T}}

thus,

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

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

For completeness:

R_{1} = \frac{R_{b} R_{a}}{R_{T}}

(4)

R_{2} = \frac{R_{b} R_{c}}{R_{T}}

(5)

R_{3} = \frac{R_{a} R_{c}}{R_{T}}

(6)

Y-load to Δ-load transformation equations

Let

R_{T} = R_{a} + R_{b} + R_{c}

.

We can write the Δ to Y equations as

R_{1} = \frac{R_{a} R_{b}}{R_{T}}

(1)

R_{2} = \frac{R_{b} R_{c}}{R_{T}}

(2)

R_{3} = \frac{R_{a} R_{c}}{R_{T}} .

(3)

Multiplying the pairs of equations yields

R_{1} R_{2} = \frac{R_{a} R_{b}^{2} R_{c}}{R_{T}^{2}}

(4)

R_{1} R_{3} = \frac{R_{a}^{2} R_{b} R_{c}}{R_{T}^{2}}

(5)

R_{2} R_{3} = \frac{R_{a} R_{b} R_{c}^{2}}{R_{T}^{2}}

(6)

and the sum of these equations is

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

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}}

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)

\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}},

\frac{R_{1} R_{2} + R_{1} R_{3} + R_{2} R_{3}}{R_{1}} = R_{c},

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

انظر أيضا

Analysis of resistive circuits
Electrical network, single phase electric power, alternating-current electric power, three-phase power, polyphase systems for examples of Y and Δ connections
AC motor for a discussion of the Y-Δ starting technique
Nikola Tesla
John Hopkinson
Mikhail Dolivo-Dobrovolsky
Charles Proteus Steinmetz

الهوامش

المصادر

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

وصلات خارجية

Star-Triangle Conversion: Knowledge on resistive networks and resistors