With methods like Kirchhoff's Law, Mesh Current Analysis, and Nodal Voltage Analysis, we' ve already mastered fundamental circuit analysis and can easily address problems with simple resistive networks. But how about complex three phase circuits? This is where the Star- Delta Transformation becomes so useful.
This effective method greatly reduces the mathematical complexity of balanced 3phase circuits. To handle those difficult circuits, it is like having a hidden weapon that will save your time and effort.
Star and Delta: The Two Configurations of Three-Phase Circuits
Star (Wye) and delta connections are the two major networks in the field of three- phase circuits. The circuit's resistance hierarchy is reflected in these designations.
Star (Y): A simple representation of a star network is a star with connected points.
Delta (Δ) networks are shaped like triangles, with resistances forming the sides.
Its adaptability is what makes them wonderful. The Star-Delta Transformation (or its counterpart, Delta-Star Transformation) is a useful technique that may be used to transform a 3-phase circuit, whether it' is a 3-wire supply or a load, connected in either star or delta configuration. This give s you an alternative viewpoint on the circuit analysis, which might make calculations easier.
Networks with three resistors (or impedances) can be configured in two ways:
- T-configuration: Three resistances are joined at a central junction to form the shape of the letter "T."
- Star (Y) configuration: In this case, the resistances come together at a single point, giving the appearance of a star with connected points.
These two arrangements are depicted in the diagram below.
T-Connected Configuration and Star Network Configuration
We have previously looked at how to change a T-shaped resistor network into a star network that is similar to it. However, we have one more transforming trick in store! An electrically equivalent delta (Δ) network, with resistors arranged in a triangle, may be created by taking a pi (π) network, which consists of three resistors connected in a "π" shape. See how this transformation works by looking at the diagram below.
Pi-Connected Configuration and Equivalent Delta Network Configuration
After learning about Star (Y) and Delta (Δ) linked networks, we can now make transformations between them. This technique converts a Y network to a Δ circuit, and vice versa.
The resistors in the original and transformed networks are mathematically related by the Star-Delta and Delta-Star transformations. In both star and delta arrangements, these transformations enable us to substitute the three linked resistances (or impedances) with their corresponding values measured between terminals, such as 1-2, 1-3, or 2-3.
It's crucial to remember that the generated networks are only equivalent with respect to external currents and voltages. The currents and voltages might vary internally. Nonetheless, with relation to one another, the power consumption and power factor of both networks will be equal.
Delta to Star Network Transformation
Let us examine the process of transforming a delta network, which is a circuit with a triangle design, into a star network, which has a star shape.
In order to accomplish this successfully, we must create a formula that connects the resistances in the original delta network to those in the corresponding star network.
A standard delta network is shown in the circuit below.
Let's analyze the resistance configuration difference between the pin#1 and pin#2:
P + Q = A in parallel with (B + C)
P + Q = A(B + C) / (A + B + C) --------------- Equation#1
Analyze the resistance between the terminals 2 and 3
Q + R = C in parallel with (A +B)
Q + R = C(A + B) / (A + B + C) ------------------Equation #2
Analyze the resistance between the terminals 1 and 3:
P + R = B in parallel with (A + C)
P + R = B(A + C) / (A + B + C) -------------------Equation#3
This leaves us with three equations, and if we subtract equation 2 from equation 3, we get:
Equation#3 - Equation#2 = (P + R) - (Q + R)
P + R = B(A + C) / (A + B + C) - Q + R = C(A + B) / (A + B + C)
∴ P - Q = (BA + CB) / (A + B + C) - (CA + CB) / (A + B + C)
∴ P - Q = (BA - CA) / (A + B + C)
Next, if we rewrite Equation#1, we obtain:
P + Q = (AB + AC) / (A + B + C)
Now we add Equation#1 with the result obtained from the above Equation#3 minus Equation#2, and we get:
(P - Q) + (P + Q)
= (BA - CA) / (A + B + C) + (AB + AC) / (A + B+ C)
= 2P = 2AB / (A + B + C)
Through the above calculations, we finally get the equation for calculating the resistor P, as shown below:
P = AB / (A + B + C)
To briefly recap the arithmetic above, we can now state that the value of resistor P in a star network may be obtained using the formula Equation#1 + (Equation#3 – Equation#2).
Using a similar method, we can determine the value of resistor Q in the star network. We may do this by adding up equation 2 with the difference between equations 1 and 3. This may be expressed as Eq2 + (Eq1 - Eq3). This combined equation provides the transformation formula for resistor Q:
Q = AC / (A + B +C)
We may obtain resistor R's value in the star network in a manner similar to that of resistor Q. A particular formula consists of multiplying equation 2 by the difference between equations 1 and 3. Equation 3 + (Eq2 – Eq1) is the formula that we may use to determine R's equivalent resistance in the transformed star network.
R = BC / (A + B + C)
Interestingly, the denominator—the portion at the bottom of the fraction—remains constant throughout all transformation formulas. It is simply the total of the resistances in the original delta network (A + B + C).
Thus, we can sum up the procedures as follows to transform any delta network into a corresponding star network:
To get the equivalent resistances for each resistor in the star network (Q, R, and P), use the given formulas (see the previous section for the precise formulas).
Recall that the denominator of each of these formulas is equal to the total of the resistances (A + B + C) in the original delta network.
Formulas to Calculate Delta to Star Transformations
P = AB / (A + B + C)
Q = AC / (A + B + C)
R = BC / (A + B + C)
Things become a little easier if the three resistors in the initial delta network have the exact same value! The value of each resistor in the final star network in this particular scenario will be one-third that of a single resistor in the delta network.
This implies that the resistance (RSTAR) of each branch of the star network could be calculated as follows as follows:
RSTAR = 1/3 * RDELTA (where RDELTA is the value of any one resistor in the delta network)
Another way to write this is: (RDELTA) / 3
Solving a Delta-Star Network Transformation Problem
Our resistive network is delta. It is your job to transform it into a corresponding star network. Finding the values of the resistors in the star network that are equivalent to those in the original delta network is necessary in order to do this.
Recall that this conversion may be accomplished by using the Delta-Star Transformation formulas.
- Q = AC / (A + B + C)
- (10 * 40) / 65
- = 6.15 Ω
- P = AB / (A + B +C)
- = (10 * 15) / 65
- = 2.30 Ω
- R = BC / (A + B +C)
- = (15 * 40) / 65
- = 9.23 Ω
Actually, the process we just studied (the Delta-Star Transformation) is reversed in the Star-Delta Transformation. Recall how each resistor in the star network was the result of multiplying two particular resistors from the delta network that were connected to the same terminal when converting a delta network to a star network? In the star network, resistor P, for instance, was the result of multiplying resistors A and B (both associated to terminal 1) in the delta network.
We may construct new formulas for transforming a star network into an analogous delta network by reconfiguring the formulas we got before. This provides us with an effective tool for pursuing circuit analysis difficulties from another angle. Let's look at the following transformation formulas.
Transformation of Star to Delta Network
A formula may be used to find the value of any resistor in a delta network (Δ). This formula takes into account the corresponding star network (Y). Here's the main concept:
Consider multiplying the total of all the resistor pairs in the star network. After that, combine all of these products. Lastly, divide this total by the star network resistor that is in direct opposition to the delta resistor you are trying to solve for.
You may find the value of that particular resistor in the delta network by doing this calculation. As an illustration, consider resistor A in the delta network:
A = (PQ + QR + RP) / R
The following equation is with respect to resistor B and terminal 3:
B = (PQ + QR + RP) / Q
The following equation is in relation to resistor C and terminal 2:
C = (PQ + QR + RP) / P (with respect to termina 1).
Three separate formulas result from simplifying each of these formulas by dividing the top and bottom by the common denominator, which is the total of all the resistor products in the star network. We can transform any star network into a corresponding delta network using these equations. The simplified formulas are as follows:
Formulas for Star Delta Transformation
- A = (PQ / R) + Q + P
- B = (RP / Q) + P + R
- C = (QR / P) + Q + R
When all of the resistors in the original star network have the same value, it becomes interestingly simplified. In this particular instance, the value of each resistor in the resultant delta network will be three times greater than that of a single resistor in the star network.
This corresponds to:
RDELTA = 3 * RSTAR (where RDELTA is the value of any one resistor in the delta network and RSTAR is the value of any one resistor in the star network).
Solving a Problem to convert Star Resistive Network into an Equivalent Delta Network
Let's say, we have a star resistive network. We want to transform it into an equivalent delta network. Determine the resistor values in the delta network that correspond to those in the original star network in order to do this.
- A = (QP / R) + Q + P
- = [(90 * 75) / 30] + 90 + 75
- = 390 Ω
- B = (RP / Q) + R + P
- [(30 * 75) / 90] + 30 + 75
- = 130 Ω
- C = (QR / P + Q + R
- = [(90 * 30) / 75] + 90 + 30
- = 156 Ω
Conclusions
Have you ever been frustrated when trying to understand complicated circuits? Your hidden weapon is the Star-Delta Transformations! Using these methods, you may quickly flip between the star and delta configurations, which are two popular ways to connect impedances or resistors in circuits.
You can obtain a new understanding of the behavior of the circuit and simplify the calculations by deliberately switching between various configurations. These changes enable you to approach the analysis with more ease, regardless of whether you started with a star or delta network.
References: Y-Δ transform
A Delta-Star Transformation Approach for Reliability Evaluation