Y-Δ transform
The
Y-Δ transform, also written
Y-delta,
Wye-delta,
Kennelly’s delta-star transformation,
star-mesh transformation,
T-Π or
T-pi transform, is a mathematical technique to simplify the analysis of an
electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like the letter Y and the Greek capital letter
Δ. In the
United Kingdom, the wye diagram is sometimes known as a
star. This circuit transformation theory was published by
Arthur Edwin Kennelly in 1899
-Y Transformation 
The
configuration can be converted to
and vice versa. To relate
and
, the resistances
between terminals a and b of
should be equal to
of
, and the same is true for the other two resistances, i.e.,
- Convert to : Given ,
and 
of a , the three equations can be solved for 
, 
and 
of the corresponding . For example, subtracting the 3rd equation from the sum of the first two, we get expression for . The solutions are:
- Convert to : Reversely, given , and of a , the same three equations can also be solved for , and of the corresponding to get:
The following circuit can be converted into either of the two low equivalent circuits.
The
formed by
,
, and
can be converted to a Y, which can then be combined with
to get a Y (bottom left) with:
Alternatively, the Y formed by
,
, and
can be converted to a
, which can then be combined with
to get a
(bottom right) with:
The resulting
and Y circuits are equivalent as it can be shown they can also be converted to each other with the same system variables.