There are circuits that have a differentiated topology and which can not be
solving them with common serial and parallel circuit techniques. Those
circuits, depending on their structure, are named
"Delta Circuits (or Triangle, or Pi )"
and "Star Circuits (or T, or Y )".

However, there is the possibility of transforming a Delta
in a Star circuit and vice versa, as we will study later. We will see that
this circuit resolution tool is indispensable for those who want a bigger
deepening in the study of electrical circuits.

2. Delta Circuits

In the Figure 05-01 we see the topology of a circuit Delta or Triangle.
In the Figure 05-02 is represented the topology of a circuit Pi. Note that the
two circuits are identical, only changes the way of drawing them and the denomination.

3. Star Circuits

In the Figure 05-03, we see the topology of a Star circuit.
In the Figure 05-04 is represented the topology of a circuit T. Again, note that the
two circuits are identical, only changes the way of drawing them and the denomination.

4. Equivalence between Delta-Star Circuits

Let's study how we can transform a Delta or Triangle circuit into a circuit
Star or Y. Note that after using the transform formulas, the Delta or Triangle circuit is replaced by the Star or Y circuit, that is, we replaced the circuit shown in Figure 05-05 with the circuit shown in
Figure 05-06.

The equations for transforming the Delta or Triangle circuit into Star or Y
are shown below.

eq. 05-01

eq. 05-02

eq. 05-03

Note that the denominator of the three equations are identical, that is, it is the sum of the three
resistors that make up the circuit. The numerator is formed by the product of the two
resistors adjacent to which we want to calculate their value.

5. Equivalence Between Star and Delta Circuits

Now let's look at how we can transform a Star or Y circuit into a circuit
Delta or Triangle. Note that after using the transform formulas, the Star or Y circuit is replaced by the Delta or Triangle circuit, that is, we replaced the circuit shown in Figure 05-07 with the circuit shown in Figure 05-08.

The equations that make it possible to transform the Star (or Y ) circuit into
Triangle (or Delta ), are shown below.

eq. 05-04

eq. 05-05

eq. 05-06

Note that the numerator of the three equations are identical, that is, it is the sum of the product
of the resistors that make up the circuit, two to two. The denominator is formed
only by the value of the resistance that is on the side opposite the resistance that we want
calculate its value.
In other words, if we want to calculate the value of R_{1}, we must
observe that on the opposite side we have the point c and at this point it is
linked to resistance R_{c}. Therefore, we should use
R_{c} in the denominator. For the calculation of the other resistances
we use the same reasoning, which facilitates memorization.

6. Equivalence between Delta - Star Circuitsfor Capacitors

When we have Triangle-Star circuits with capacitors we use the same principle as
we use for resistors, only making the substitution in the equations of R
by 1 / C.
See in the Figure 05-09 the transformed circuit and its equations to calculate
the value of the capacitors.

This equation can be worked algebraically to achieve a simpler form.
See below the three equations already transformed.

eq. 05-07

eq. 05-08

eq. 05-09

Note that these equations are similar to those used with resistances for the transformation case
Star-Triangle.

7. Equivalence between Star - Delta Circuits for Capacitors

For the Star-Triangle circuit follows the same principle as explained in the previous item.
In the Figure 05-10 we see the transformed circuit and the respective equations for the calculation of the
capacitors.

This equation can be worked algebraically to achieve a simpler form.
See below the three equations already transformed. Equations similar to those used for resistances
in the Triangle-Star transformation.