स्टार-डेल्टा परिवर्तन

स्टार-डेल्टा परिवर्तन (Y-Δ transform) एक गणितीय तकनीक है जो किसी विद्युत परिपथ के विश्लेषण को सरल बना देता है। इसे Y-delta, वाई-डेल्टा, डेल्टा-स्टार परिवर्तन, स्टार-मेश परिवर्तन, T-Π or T-पाई परिवर्तन आदि नामों से भी जाना जाता है। इसका यह नाम विद्युत परिपथ की आकृति के आधार पर पड़ा है जो कि रोमन अक्षर Y और ग्रीक अक्षरΔ जैसे दिखती हैं। यह परिपथ परिवर्तन सन् १८९९ में आर्थर एड्विन केनेडी ने प्रकाशित किया था।

आधारभूत Y-Δ परिवर्तन

Δ और Y परिपथ तथा इससे सम्बन्धित नामकरण जो कि इस लेख में प्रयुक्त हुए हैं।

वस्तुत: यह परिवर्तन तीन-सिरों वाले दो नेटवर्कों में तुल्यता स्थापित करता है। तुल्यता के लिये आवश्यक है कि दोनो ही नेटवर्कों में किन्ही दो सिरों के बीच तुल्य प्रतिबाधा (impedance) समान होनी चाहिये।

तीन-फेजी परिपथ में Δ-लोड को Y-लोड में बदलने का समीकरण

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

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

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

$R_1 = \frac{R_aR_b}{R_a + R_b + R_c},$

$R_2 = \frac{R_bR_c}{R_a + R_b + R_c},$

$R_3 = \frac{R_aR_c}{R_a + R_b + R_c}.$

तीन-फेजी परिपथ में Y-लोड को Δ-लोड में बदलने का समीकरण

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

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

where $R_P = R_1R_2+R_2R_3+R_3R_1$ is the sum of the products of all pairs of impedances in the Y circuit and $R_\mathrm{opposite}$ is the impedance of the node in the Y circuit which is opposite the edge with $R_\Delta$. The formula for the individual edges are thus

$R_a = \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_2},$

$R_b = \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_3},$

$R_c = \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_1}.$

ग्राफ सिद्धान्त (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 graphs are a Y-Δ equivalence class.

प्रदर्शन

Δ-लोड से Y-लोड में परिवर्तन के समीकरण

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

Given the values of $R_b$, $R_c$ and $R_a$ from the Δ configuration, we want to obtain the values of $R_1$, $R_2$ and $R_3$ in the equivalent Y configuration. In order to do that, we will calculate the equivalent impedances of both configurations in N₁N₂, N₁N₃ and N₂N₃, supposing in each case that the omitted node is unconnected, and we will equal both expressions, since the resistance must be the same.

The resistance between N₁ and N₂ when N₃ is not connected in the Δ configuration is

$R(N_1, N_2) = R_b \parallel (R_a+R_c) = \frac{R_b(R_a+R_c)}{R_b+R_c+R_a} = \frac{R_bR_a+R_bR_c}{R_b+R_c+R_a}.$

In the Y configuration, we have

$R(N_1, N_2) = R_1+R_2;</cmath> hence we have :<math>R_1+R_2 = \frac{R_bR_a+R_bR_c}{R_b+R_c+R_a}$   (1)

By similar calculations we obtain

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

and

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

The impedances for the Y configuration can be derived from these equations by adding two equations and subtracting the third. For example, adding (1) and (3), then subtracting (2) yields

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

and hence

$2R_1 = \frac{2R_bR_a}{R_b+R_c+R_a}$

and

$R_1 = \frac{R_bR_a}{R_b+R_c+R_a}.$

Y-लोड से Δ-लोड में परिवर्तन के समीकरण

Let $R_T = R_a+R_b+R_c$. We can write the Δ to Y equations as

$R_1 = \frac{R_aR_b}{R_T}$   (1)

$R_2 = \frac{R_bR_c}{R_T}$   (2)

$R_3 = \frac{R_aR_c}{R_T}.$   (3)

Multiplying the pairs of equations yields

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

and the sum of these equations is

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

Now we divide each side of (7) by $R_1$, leaving

$\frac{R_1R_2 + R_1R_3 + R_2R_3}{R_1} = \frac{1}{R_1}\frac{R_aR_b^2R_c + R_a^2R_bR_c + R_aR_bR_c^2}{R_T^2}.$   (8)

Using (1) in (8), we have

$\frac{R_1R_2 + R_1R_3 + R_2R_3}{R_1} = \frac{R_c(R_b + R_a + R_c)}{R_T},$

and by definition of $R_T$

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

which is the equation for $R_c$. Dividing (7) by $R_2$ and $R_3$ gives the other equations.

इन्हें भी देखें

• निकोला टेस्ला

सन्दर्भ

• William Stevenson, “Elements of Power System Analysis 3rd ed.”, McGraw Hill, New York, 1975, ISBN 0-07-061285-4

बाहरी कड़ियाँ

• Star-Triangle Conversion: Knowledge on resistive networks and resistors

en;Y-delta transformation