تحويل واي-دلتا

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

المقاومةCapacitor button.svgInductor button.svgمفاعلةمعاوقةVoltage button.svg
مواصلةElastance button.svgBlank button.svgSusceptance button.svgمسامحةCurrent button.svg

المكونات

Resistor button.svg Capacitor button.svg Inductor button.svg Ohm's law button.svg

دوائر التوالي والتوازي

Series resistor button.svgParallel resistor button.svgSeries capacitor button.svgParallel capacitor button.svgSeries inductor button.svgParallel inductor button.svg

تحويلات المعاوقة

Y-Δ transform Δ-Y transform star-polygon transforms Dual button.svg

مبرهنات المولد مبرهنات الشبكة

Thevenin button.svgNorton button.svgMillman button.svg

KCL button.svgKVL button.svgTellegen button.svg

أساليب تحليل الشبكات

KCL button.svg KVL button.svg Superposition button.svg

Two-port parameters

z-parametersy-parametersh-parametersg-parametersAbcd-parameter button.svgS-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.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 at a terminal node of the Y circuit with impedances , to adjacent nodes in the Δ circuit by

where are all impedances in the Δ circuit. This yields the specific formulae

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

The general idea is to compute an impedance in the Δ circuit by

where is the sum of the products of all pairs of impedances in the Y circuit and is the impedance of the node in the Y circuit which is opposite the edge with . The formula for the individual edges are thus

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 {} from Δ to {} 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:

To simplify, let's call the sum of {}.

Thus,

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

hence:

  (1)

Repeating for :

  (2)

and for :

  (3)

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

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

thus,

where

For completeness:

(4)
(5)
(6)


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Y-load to Δ-load transformation equations

Let

.

We can write the Δ to Y equations as

  (1)
  (2)
  (3)

Multiplying the pairs of equations yields

  (4)
  (5)
  (6)

and the sum of these equations is

  (7)

Factor from the right side, leaving in the numerator, canceling with an in the denominator.

(8)

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

Divide (8) by (1)

which is the equation for . Dividing (8) by or gives the other equations.

انظر أيضا

الهوامش


المصادر

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

وصلات خارجية