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Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE) PDF Download

Objectives 

  • A part of a larger circuit that is configured with three terminal network Y (or Δ) to convert into an equivalent Δ (or Y) through transformations.  
  • Application of these transformations will be studied by solving resistive circuits.

Introduction

There are certain circuit configurations that cannot be simplified by series-parallel combination alone. A simple transformation based on mathematical technique is readily  simplifies the electrical circuit configuration. A circuit configuration  shown below   

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

is a general one-port circuit. When any voltage source is connected across the terminals, the current entering through any one of the two terminals, equals the current leaving the other terminal. For example, resistance, inductance and capacitance acts as a one-port. On the other hand, a two-port is a circuit having two pairs of terminals. Each pair behaves as a one-port; current entering in one terminal must be equal to the current living the other terminal. 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Fig.6.1.(b) can be described as a four terminal network, for convenience subscript 1 to refer to the variables at the input port (at the left) and the subscript 2 to refer to the variables at the output port (at the right). The most important subclass of two-port networks is the one in which the minus reference terminals of the input and output ports are at the same.  This circuit configuration is readially possible to consider the ‘π or Δ ’ – network also as a three-terminal network in fig.6.1(c). Another frequently encountered circuit configuration that shown in fig.6.1(d) is approximately refered to as a threeterminal Y connected circuit as well as two-port circuit. 

  Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

The name derives from the shape or configuration of the circuit diagrams, which look respectively like the letter Y and the Greek capital letter Δ. 

Delta (Δ) – Wye (Y) conversion 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

These configurations may often be handled by the use of a Δ − Y or Y − Δ transformation. One of the most basic three-terminal network equivalent is that of three resistors connected in “Delta ” and in “Wye (”. These two circuits identified in fig.L6.1(e) and Fig.L.6.1(f) are sometimes part of a larger circuit and obtained their names from their configurations. These three terminal networks can be redrawn as four-terminal networks as shown in fig.L.6.1(c) and fig.L.6.1(d). We can obtain useful expression for direct transformation or conversion from Δ to Y or Y to Δ by considering that for equivalence the two networks have the same resistance when looked at the similar pairs of terminals.

Conversion from Delta (Δ) to Star or Wye (Y)

Let us consider the network shown in fig.6.1(e) (or fig. 6.1(c) →) and assumed the resistances (RAB ,RBC,and RCA) Δ network are known. Our problem is to find the values of RA, RB, and RC in Wye (Y) network (see fig.6.1(e))  that will produce the same resistance when measured between similar pairs of terminals. We can write the equivalence resistance between any two terminals in the following form. 

Between A & C terminals: 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)
Between C &B terminals: 
Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

By combining above three equations, one can write an expression as given below. 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Subtracting equations (6.2), (6.1), and (6.3) from (6.4) equations, we can write the express for unknown resistances of Wye (Y) network as  

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Conversion from Star or Wye (Y) to Delta (Δ)

To convert a Wye (Y) to a Delta (Δ), the relationships must be RAB, RBC, and R3 must be obtained in terms of the Wye (Y) resistances RA, RB, and RC (referring to fig.6.1 (f)). Considering the Y connected network, we can write the current expression through RA resistor as 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Appling KCL at ‘N’ for Y connected network (assume A, B,C terminals  having higher potential than the terminal N) we have, 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)
Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

For Δ -network (see fig.6.1.(f)), Current entering at terminal A = Current leaving the terminal ‘ A ’ 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Using the VN expression in the above equation, we get 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)
Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Equating the coefficients of  VAB and VAC in both sides of eq.(6.11), we obtained the following relationship. 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)
Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)
Similarly, IB for both the networks (see fig.61(f)) are given by
Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Equating the above two equations and using the value of VN (see eq.(6.9), we get the final expression as 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Equating the coefficient of VBC in both sides of the above equations we obtain the following relation 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

When we need to transform a Delta (Δ) network to an equivalent Wye (Y) network, the equations (6.5) to (6.7) are the useful expressions. On the other hand, the equations (6.12) – (6.14) are used for Wye (Y) to Delta (Δ) conversion.

Observations 

In order to note the symmetry of the transformation equations, the Wye (Y) and Delta (Δ) networks have been superimposed on each other as shown in fig. 6.2. 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

  • The equivalent star (Wye) resistance connected to a given terminal is equal to the product of the two Delta (Δ) resistances connected to the same terminal divided by the sum of the Delta (Δ) resistances (see fig. 6.2).
  • The equivalent Delta (Δ) resistance between two-terminals is the sum of the two star (Wye) resistances connected to those terminals plus the product of the same two star (Wye) resistances divided by the third star (Wye (Y)) resistance (see fig.6.2). 

Application of Star (Y) to Delta (Δ) or Delta (Δ) to Star (Y) Transformation 

Example:  Find the value of the voltage source (VS) that delivers 2 Amps current through the circuit as shown in fig.6.3. 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Solution:  

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Convert the three terminals Δ -network (a-c-d & e-f-g) into an equivalent Y -connected network. Consider the Δ -connected network ‘a-c-d’ and the corresponding equivalent Y -connected resistor values are given as 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)
Similarly, for the -connected network ‘e-f-g’ the equivalent the resistances of Y -connected network are calculated as  Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)
Now the original circuit is redrawn after transformation and it is further simplified by applying series-parallel combination formula. 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

The source Vs that delivers 2 A current through the circuit can be obtained as Vs = I× 3.2 = 2 × 3.1 = 6.2Volts .

Example: Determine the equivalent resistance between the terminals A and B of network shown in fig.6.4 (a). 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

A ‘Δ’ is substituted for the ‘Y’ between points c, d, and e as shown in fig.6.4(b); then unknown resistances value for Yto Δ transformation are computed below. 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Next we transform ‘Δ’connected 3-terminal resistor to an equivalent ‘Y’ connected network between points ‘A’; ‘c’ and ‘e’ (see fig.6.4(b)) and the corresponding Y connected resistances value are obtained using the following expression. Simplified circuit after conversion is shown in fig. 6.4(c). 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

The circuit shown in fig.6.5(c) can further be reduced by considering two pairs of parallel branches Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE) and the corresponding simplified circuit is shown in fig.6.4(d). 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Now one can find the equivalent resistance between the terminals ‘A’ and ‘B’ as  RAB = (2.23+ 2.08) ║(1.04 + 0.93) + 0.64 = 2.21Ω. 

Example:  Find the value of the input resistance Rin of the circuit.

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

                         Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)  

Y connected network formed with the terminals a-b-o is transformed into Δ connected one and its resistance values are given below.

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)
Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Similarly, Y connected networks formed with the terminals ‘b-c-o’ and ‘c-a-o’ are transformed to connected networks.

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)
Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)
Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Note that the two resistances are connected in parallel (140║108 ) between the points ‘a’ and ‘o’. Similarly, between the points ‘b’ and ‘o’ two resistances are connected in parallel (46.66║34.6) and resistances 54.0 Ω and 29.2 Ω are connected in parallel between the points ‘c’ and ‘o’.   

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

 Now Y connected network formed with the terminal ‘a-b-c’ is converted to equivalent Δ connected network.

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Remarks: 

  • If the connected network consists of inductances (assumed no mutual coupling forms between the inductors) then the same formula can be used for Y to Δ or Δ to Y conversion (see in detail 3-phase ac circuit analysis in Lesson19).  
  • On the other hand, the Δ or Y connected network consists of capacitances can be converted to an equivalent Yor Δ network provided the capacitance value is replaced by its reciprocal in the conversion formula (see in detail 3-phase ac circuit analysis in Lesson-19).  

Example Find the equivalent inductance Req of the network (see fig.6.5(a)) at the terminals ‘a’ & ‘b’ Y - Δ & Δ - Y using transformations. 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Solution: Convert the three terminals (c-d-e) Δ network (see fig.6.5(a)) comprising with the resistors to an equivalent Y -connected  network using the following Δ − Y conversion formula.  

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)  

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Similarly, the -connected network (f-e-b) is converted to an equivalent Y- connected Network. 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

After the conversions, the circuit is redrawn and shown in fig.6.5(b). Next the series-parallel combinations of resistances reduces the network configuration in more simplified form and it is shown in fig.6.5(c). This circuit (see fig.6.5(c)) can further be simplified by transforming Y connected network comprising with the three resistors (2Ω , 4Ω , and 3.666Ω) to a Δ -connected network and the corresponding network parameters are given below: 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

Simplified form of the circuit is drawn and shown in fig.6.5(d) and one can easily find out the equivalent resistance Req between the terminals ‘a’ and ‘b’ using the series- parallel formula. From fig.6.5(d), one can write the expression for the total equivalent resistance Req at the terminals ‘a’ and ‘b’ as 

Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE)

The document Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations | Basic Electrical Technology - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Basic Electrical Technology.
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FAQs on Wye (Y) - Delta (∆) OR Delta (∆)-Wye (Y) Transformations - Basic Electrical Technology - Electrical Engineering (EE)

1. What is the purpose of the Wye-Delta transformation?
Ans. The Wye-Delta transformation, also known as the Y-Δ transformation, is used to simplify complex electrical circuits by converting a circuit from a Wye (Y) configuration to a Delta (∆) configuration or vice versa. This transformation allows for easier analysis and calculation of circuit parameters.
2. When should I use the Wye-Delta transformation?
Ans. The Wye-Delta transformation is commonly used when dealing with three-phase electrical systems. It is particularly useful when the circuit contains both Wye and Delta configurations and you need to simplify the circuit for analysis or calculation purposes. By transforming the circuit, you can easily find equivalent resistances, currents, and voltages.
3. Can the Wye-Delta transformation be applied to any circuit?
Ans. The Wye-Delta transformation can only be applied to circuits that have a balanced load, which means that the impedances or resistances in each branch of the circuit are equal. If the circuit is unbalanced, the transformation cannot be applied accurately, and other methods should be used for circuit analysis.
4. How do I perform the Wye-Delta transformation?
Ans. To convert a circuit from a Wye to a Delta configuration, you can follow these steps: 1. Identify the three branches of the Wye configuration. 2. Calculate the equivalent resistance for the Wye configuration by using the formula: R_eq = R1 + R2 + R3. 3. Calculate the equivalent impedance for the Delta configuration by using the formula: Z_eq = (Z1 * Z2 + Z2 * Z3 + Z3 * Z1) / (Z1 + Z2 + Z3). 4. Replace the Wye configuration with the Delta configuration, using the equivalent resistances or impedances calculated in the previous steps.
5. What are the advantages of using the Wye-Delta transformation?
Ans. The Wye-Delta transformation provides several advantages in circuit analysis: - It simplifies the circuit, making it easier to calculate and analyze. - It allows for the conversion of a complex circuit into a simpler and more manageable form. - It helps in determining the equivalent resistance, current, and voltage across the circuit branches. - It enables the application of various circuit analysis techniques, such as Kirchhoff's laws and Ohm's law, more effectively. - It facilitates the identification and understanding of the circuit's behavior in terms of power flow and balancing.
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