Vannevar Bush

The theorem of Kennelly , or transformation triangle-star , or transformation Y-Δ , or transformation T-Π , are a mathematical technique which makes it possible to simplify the study of certain electrical communications.

This theorem, named thus in homage to Arthur Edwin Kennelly, makes it possible to pass from a configuration “triangle” (or Δ, or Π, according to the way which one draws the diagram) to a configuration “star” (or, in the same way, Y or T). The diagram opposite is drawn in the form “triangle-star”; diagrams below in form T-Π.

This theorem is sometimes used in electrotechnical or electronic of power in order to simplifer of the three-phase Systèmes.

Transformation star towards triangle

With the Admittance S

The product of the adjacent admittances divided by the total sum of the admittances.

$Y\_\; \{AB\}\; =\; \backslash \; frac\; \{Y\_\; \{AT\}.\; Y\_\; \{BT\}\}\; \{Y\_\; \{AT\}\; +Y\_\; \{BT\}\; +Y\_\; \{CT\}\}$
$Y\_\; \{BC\}\; =\; \backslash \; frac\; \{Y\_\; \{BT\}.\; Y\_\; \{CT\}\}\; \{Y\_\; \{AT\}\; +Y\_\; \{BT\}\; +Y\_\; \{CT\}\}$
$Y\_\; \{CA\}\; =\; \backslash \; frac\; \{Y\_\; \{CT\}.\; Y\_\; \{AT\}\}\; \{Y\_\; \{AT\}\; +Y\_\; \{BT\}\; +Y\_\; \{CT\}\}$

With the impedance S

The sum of the products of impedances divided by the opposite impedance.

$Z\_\; \{AB\}\; =\; \backslash \; frac\; \{Z\_\; \{AT\}.\; Z\_\; \{BT\}\; +\; Z\_\; \{BT\}.\; Z\_\{CT\}+Z\_\{CT\}.Z\_\; \{AT\}\}\; \{Z\_\; \{CT\}\}$
$Z\_\; \{BC\}\; =\; \backslash \; frac\; \{Z\_\; \{AT\}.\; Z\_\; \{BT\}\; +\; Z\_\; \{BT\}.\; Z\_\{CT\}+Z\_\{CT\}.Z\_\; \{AT\}\}\; \{Z\_\; \{AT\}\}$
$Z\_\; \{CA\}\; =\; \backslash \; frac\; \{Z\_\; \{AT\}.\; Z\_\; \{BT\}\; +\; Z\_\; \{BT\}.\; Z\_\{CT\}+Z\_\{CT\}.Z\_\; \{AT\}\}\; \{Z\_\; \{BT\}\}$
”

Transformation triangle towards star

One speaks here about an equivalence of a circuit in T with a circuit in π. In practice, one more uses the transformation which consists in passing from a circuit in π to a circuit in T.

With the Admittance S

The sum of the products of the admittances divided by the opposite admittance.

$Y\_\; \{AT\}\; =\; \backslash \; frac\; \{Y\_\; \{AB\}.\; Y\_\; \{BC\}\; +\; Y\_\; \{CA\}.\; Y\_\{AB\}+Y\_\{BC\}.Y\_\; \{CA\}\}\; \{Y\_\; \{BC\}\}$
$Y\_\; \{BT\}\; =\; \backslash \; frac\; \{Y\_\; \{AB\}.\; Y\_\; \{BC\}\; +\; Y\_\; \{CA\}.\; Y\_\{AB\}+Y\_\{BC\}.Y\_\; \{CA\}\}\; \{Y\_\; \{CA\}\}$
$Y\_\; \{CT\}\; =\; \backslash \; frac\; \{Y\_\; \{AB\}.\; Y\_\; \{BC\}\; +\; Y\_\; \{CA\}.\; Y\_\{AB\}+Y\_\{BC\}.Y\_\; \{CA\}\}\; \{Y\_\; \{AB\}\}$

With the impedance S

The product of the adjacent impedances divided by the total sum of the impedances.

$Z\_\; \{AT\}\; =\; \backslash \; frac\; \{Z\_\; \{AB\}.\; Z\_\; \{AC\}\}\; \{Z\_\; \{AB\}\; +Z\_\; \{BC\}\; +Z\_\; \{AC\}\}$
$Z\_\; \{BT\}\; =\; \backslash \; frac\; \{Z\_\; \{AB\}.\; Z\_\; \{BC\}\}\; \{Z\_\; \{AB\}\; +Z\_\; \{BC\}\; +Z\_\; \{AC\}\}$
$Z\_\; \{CT\}\; =\; \backslash \; frac\; \{Z\_\; \{AC\}.\; Z\_\; \{BC\}\}\; \{Z\_\; \{AB\}\; +Z\_\; \{BC\}\; +Z\_\; \{AC\}\}$

See too

• Electrokinetic Electricity
• Quadripole

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