Objectives
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 (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:
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 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
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 VN expression in the above equation, we get
Equating the coefficients of VAB and VAC 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 VN (see eq.(6.9), we get the final expression as
Equating the coefficient of VBC 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.
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.
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 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).
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 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.
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:
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.
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 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
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1. What is the purpose of the Wye-Delta transformation? |
2. When should I use the Wye-Delta transformation? |
3. Can the Wye-Delta transformation be applied to any circuit? |
4. How do I perform the Wye-Delta transformation? |
5. What are the advantages of using the Wye-Delta transformation? |
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