# 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

$Y_ \left\{AB\right\} = \ frac \left\{Y_ \left\{AT\right\}. Y_ \left\{BT\right\}\right\} \left\{Y_ \left\{AT\right\} +Y_ \left\{BT\right\} +Y_ \left\{CT\right\}\right\}$
$Y_ \left\{BC\right\} = \ frac \left\{Y_ \left\{BT\right\}. Y_ \left\{CT\right\}\right\} \left\{Y_ \left\{AT\right\} +Y_ \left\{BT\right\} +Y_ \left\{CT\right\}\right\}$
$Y_ \left\{CA\right\} = \ frac \left\{Y_ \left\{CT\right\}. Y_ \left\{AT\right\}\right\} \left\{Y_ \left\{AT\right\} +Y_ \left\{BT\right\} +Y_ \left\{CT\right\}\right\}$

### With the impedance S

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

$Z_ \left\{AB\right\} = \ frac \left\{Z_ \left\{AT\right\}. Z_ \left\{BT\right\} + Z_ \left\{BT\right\}. Z_\left\{CT\right\}+Z_\left\{CT\right\}.Z_ \left\{AT\right\}\right\} \left\{Z_ \left\{CT\right\}\right\}$
$Z_ \left\{BC\right\} = \ frac \left\{Z_ \left\{AT\right\}. Z_ \left\{BT\right\} + Z_ \left\{BT\right\}. Z_\left\{CT\right\}+Z_\left\{CT\right\}.Z_ \left\{AT\right\}\right\} \left\{Z_ \left\{AT\right\}\right\}$
$Z_ \left\{CA\right\} = \ frac \left\{Z_ \left\{AT\right\}. Z_ \left\{BT\right\} + Z_ \left\{BT\right\}. Z_\left\{CT\right\}+Z_\left\{CT\right\}.Z_ \left\{AT\right\}\right\} \left\{Z_ \left\{BT\right\}\right\}$

## 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.

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

$Y_ \left\{AT\right\} = \ frac \left\{Y_ \left\{AB\right\}. Y_ \left\{BC\right\} + Y_ \left\{CA\right\}. Y_\left\{AB\right\}+Y_\left\{BC\right\}.Y_ \left\{CA\right\}\right\} \left\{Y_ \left\{BC\right\}\right\}$
$Y_ \left\{BT\right\} = \ frac \left\{Y_ \left\{AB\right\}. Y_ \left\{BC\right\} + Y_ \left\{CA\right\}. Y_\left\{AB\right\}+Y_\left\{BC\right\}.Y_ \left\{CA\right\}\right\} \left\{Y_ \left\{CA\right\}\right\}$
$Y_ \left\{CT\right\} = \ frac \left\{Y_ \left\{AB\right\}. Y_ \left\{BC\right\} + Y_ \left\{CA\right\}. Y_\left\{AB\right\}+Y_\left\{BC\right\}.Y_ \left\{CA\right\}\right\} \left\{Y_ \left\{AB\right\}\right\}$

### With the impedance S

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

$Z_ \left\{AT\right\} = \ frac \left\{Z_ \left\{AB\right\}. Z_ \left\{AC\right\}\right\} \left\{Z_ \left\{AB\right\} +Z_ \left\{BC\right\} +Z_ \left\{AC\right\}\right\}$
$Z_ \left\{BT\right\} = \ frac \left\{Z_ \left\{AB\right\}. Z_ \left\{BC\right\}\right\} \left\{Z_ \left\{AB\right\} +Z_ \left\{BC\right\} +Z_ \left\{AC\right\}\right\}$
$Z_ \left\{CT\right\} = \ frac \left\{Z_ \left\{AC\right\}. Z_ \left\{BC\right\}\right\} \left\{Z_ \left\{AB\right\} +Z_ \left\{BC\right\} +Z_ \left\{AC\right\}\right\}$

## See too

 Random links: Szczecinek | Marcel Bardiaux | Saint-Didier-in-Brionnais | Tourinne | Daisy Chainsaw | Vannevar_Bush