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   

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. 

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. 

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 

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 (R_AB ,R_BC,and R_CA) Δ network are known. Our problem is to find the values of R_A, R_B, and R_C 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: 

Between C &B terminals: 

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

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  

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

To convert a Wye (Y) to a Delta (Δ), the relationships must be R_AB, R_BC, and R₃ must be obtained in terms of the Wye (Y) resistances R_A, R_B, and R_C (referring to fig.6.1 (f)). Considering the Y connected network, we can write the current expression through R_A resistor as 

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

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

Using the V_N expression in the above equation, we get 

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

Similarly, IB for both the networks (see fig.61(f)) are given by

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

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

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. 

• 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 (V_S) that delivers 2 Amps current through the circuit as shown in fig.6.3. 

Solution:  

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 

Similarly, for the -connected network ‘e-f-g’ the equivalent the resistances of Y -connected network are calculated as 
Now the original circuit is redrawn after transformation and it is further simplified by applying series-parallel combination formula. 

The source Vs that delivers 2 A current through the circuit can be obtained as V_s = 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). 

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. 

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). 

The circuit shown in fig.6.5(c) can further be reduced by considering two pairs of parallel branches  and the corresponding simplified circuit is shown in fig.6.4(d). 

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

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

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

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

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’.   

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

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 R_eq of the network (see fig.6.5(a)) at the terminals ‘a’ & ‘b’ Y - Δ & Δ - Y using transformations. 

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.  

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

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: 

Simplified form of the circuit is drawn and shown in fig.6.5(d) and one can easily find out the equivalent resistance R_eq 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 R_eq at the terminals ‘a’ and ‘b’ as