The Delta and Star circuits are fundamental in the analysis of electrical systems, especially in electrical applications
three-phase alternating current. The Delta circuit, also known as Triangle or Pi, is characterized by a
configuration in which three components are connected point to point, forming a closed loop. On the other hand, the
Star circuit, or T or Y, presents a configuration in which the components are connected to a central point
common. The transformation between these two configurations is a valuable technique, as it allows you to simplify the analysis of
circuits that cannot be solved with series or parallel associations alone. The transformation equations are
based on the equivalence of impedances between circuit terminals, ensuring that the electrical properties
are preserved during conversion. This technique is essential for professionals who want to deepen their
knowledge in electrical circuits and is widely applied in electrical engineering.
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.
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.
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.
Figure 05-05Figure 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.
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.
Figure 05-07Figure 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 R1, we must
observe that on the opposite side we have the point c and at this point it is
linked to resistance Rc. Therefore, we should use
Rc 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.
Figure 05-09
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.
Figure 05-10
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.
