Circuit FundamentalsYear 2

Circuit fundamentals: Wye-Delta conversions

Circuit fundamentals: Wye-Delta conversions

The connections considered in the previous article (8) are quite simple for calculation. However, usually electric circuit schemes appear complex where it’s not easy to find out parallel or serial connection for elements. Here below we will consider some techniques which will help us to simplify complex electric circuits.

Circuit fundamentals: Figure 13 and 14
Figure 13 and 14

Figures 13 and 14 show Y or T terminal network and ∆ or Pi network. These networks can also be a part of larger network and can be used in electrical filters or 3-phase networks.

When we meet Y or ∆ network in a circuit, Y network can be superimposed on the latter to find the equivalent resistances.

Resistance between the same pair of nodes should be the same for both schemes. Let’s calculate resistance between terminals 1 and 2.

R(T)12=R1+R3, R()12=Rb(Rc+Ra)Rb+Ra+Rc, R(T)12=R()12 means R1+R3=Rb(Rc+Ra)Rb+Ra+Rc R(T)23=R1+R2, R()23=Ra(Rc+Rb)Rb+Ra+Rc R(T)34=R2+R3, R()34=Rc(Ra+Rb)Rb+Ra+Rc R1R2=Ra(RbRc)Rb+Ra+RcR1+R2=Ra(Rc+Rb)Rb+Ra+Rc

Hence:

R1=RaRbRb+Ra+Rc

Similarly:

R2=RaRcRb+Ra+Rc

and:

R3=RbRcRb+Ra+Rc

Rule: Y to ∆ conversion. Each resistor of Y network is equal to the product of adjacent resistances of ∆ network, divided by the sum of ∆ resistances.

Circuit fundamentals: Figure 15
Figure 15

Delta-Wye conversion

 To find the reverse conversion of resistances from network to Y, we must find the following sum:

R1R2+R1R3+R2R3=RaRbRcRb+Ra+Rc

thus:

Ra=R1R2+R1R3+R2R3R3; RbR1R2+R1R3+R2R3R2; Rc=R1R2+R1R3+R2R3R1

Rule: ∆ to Y conversion. Each resistor of  network equal to the product of any possible pair of Y network resistors divided by the opposite Y network resistor.

The networks  and Y are balanced when:

R1=R2=R3=Ry, Ra=Rb=Rc=R and Ry=R3, R=3Ry

#10 Methods of analysis: nodal / mesh

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