Y-Δ transform
 

Y-Δ transform

The Y-Δ transform, also written Y-delta, Wye-delta, Kennelly’s delta-star transformation, star-mesh transformation, T-Π or T-pi transform, is a mathematical technique to simplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like the letter Y and the Greek capital letter Δ. In the United Kingdom, the wye diagram is sometimes known as a star. This circuit transformation theory was published by Arthur Edwin Kennelly in 1899



http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png-Y Transformation http://fourier.eng.hmc.edu/e84/lectu...res/DeltaY.gif
The http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png configuration can be converted to http://fourier.eng.hmc.edu/e84/lectures/ch2/img156.png and vice versa. To relate http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img156.png, the resistances http://fourier.eng.hmc.edu/e84/lectures/ch2/img157.png between terminals a and b of http://fourier.eng.hmc.edu/e84/lectures/ch2/img156.png should be equal to http://fourier.eng.hmc.edu/e84/lectures/ch2/img158.png of http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png, and the same is true for the other two resistances, i.e.,

http://fourier.eng.hmc.edu/e84/lectures/ch2/img159.png

• Convert http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png to http://fourier.eng.hmc.edu/e84/lectures/ch2/img156.png: Given http://fourier.eng.hmc.edu/e84/lectures/ch2/img158.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img160.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img161.png of a http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png, the three equations can be solved for http://fourier.eng.hmc.edu/e84/lectures/ch2/img162.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img163.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img164.png of the corresponding http://fourier.eng.hmc.edu/e84/lectures/ch2/img156.png. For example, subtracting the 3rd equation from the sum of the first two, we get expression for http://fourier.eng.hmc.edu/e84/lectures/ch2/img162.png. The solutions are:
  
  http://fourier.eng.hmc.edu/e84/lectures/ch2/img165.png
  
  
• Convert http://fourier.eng.hmc.edu/e84/lectures/ch2/img156.png to http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png: Reversely, given http://fourier.eng.hmc.edu/e84/lectures/ch2/img162.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img163.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img164.png of a http://fourier.eng.hmc.edu/e84/lectures/ch2/img156.png, the same three equations can also be solved for http://fourier.eng.hmc.edu/e84/lectures/ch2/img158.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img160.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img161.png of the corresponding http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png to get:
  
  http://fourier.eng.hmc.edu/e84/lectures/ch2/img166.png
  
  The following circuit can be converted into either of the two low equivalent circuits.

http://fourier.eng.hmc.edu/e84/lectu...es/DeltaY1.gif
The http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png formed by http://fourier.eng.hmc.edu/e84/lectures/ch2/img167.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img167.png, and http://fourier.eng.hmc.edu/e84/lectures/ch2/img168.png can be converted to a Y, which can then be combined with http://fourier.eng.hmc.edu/e84/lectures/ch2/img169.png to get a Y (bottom left) with:

http://fourier.eng.hmc.edu/e84/lectures/ch2/img170.png

Alternatively, the Y formed by http://fourier.eng.hmc.edu/e84/lectures/ch2/img167.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img171.png, and http://fourier.eng.hmc.edu/e84/lectures/ch2/img169.png can be converted to a http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png, which can then be combined with http://fourier.eng.hmc.edu/e84/lectures/ch2/img168.png to get a http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png (bottom right) with:
http://fourier.eng.hmc.edu/e84/lectures/ch2/img172.png

The resulting http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png and Y circuits are equivalent as it can be shown they can also be converted to each other with the same system variables.