There are many ways to reduce electrical circuits with several elements into circuits with fewer elements that behave identically at their output terminals. In fact, Thevenin's theorem guarantees that a circuit with only linear elements (resistors, capacitors, and inductors) can be reduced to one voltage source and one resistor (with complex impedance if there are reactive elementscapacitors and inductors). I will assume the circuits contain only resistive elements in what follows, though it is easy to extend the results to circuits that contain reactive elements.
It is trivial to reduce pairs of resistors that are in series or in parallel into single resistors to simplify circuit analysis. Usually those formulae are sufficient, but it is possible to find networks of resistors where no resistors are in parallel or series. Take the circuit below as an example (R means resistor).
,
/ \
/ \
R R
/ \
/R\
\ /
\ /
R R
\ /
\ /
^
It would be useful to combine all of those resistors into one for circuit analysis purposes. To accomplish this one can make use of the deltawye transformation, which is easy to justify, shown below.
Delta shape
a
,
/ \
/ \
R1 R2
/ \
/R3\
c b
Wye shape
a


Ra


/ \
/ \
Rc Rb
/ \
/ \
c b
Transformation formulae
Ra = R1R2/(R1+R2+R3)
Rb = R2R3/(R1+R2+R3)
Rc = R1R3/(R1+R2+R3)
R1 = (RaRb+RbRc+RaRc)/Rb
R2 = (RaRb+RbRc+RaRc)/Rc
R3 = (RaRb+RbRc+RaRc)/Ra