Delta and Wye Circuits

Circuits

Delta (also called pi)
Wye (also called tee)
- The center node is called n
- If all the resistances are the same, the current at n is 0
Shared features
- $V_{a} = V_{b} = V_{c}$
- a, b, c are labeled in a clockwise direction
- When going between the two, the voltages between a, b, and c stay the same

A technique used to simplify resistive networks that cannot be handled by parallel or series equations
Delta-to-Wye
- $R_{1} = \frac{R _{b} R _{c}}{R _{a} + R _{b} + R _{c}}$
- $R_{2} = \frac{R _{c} R _{a}}{R _{a} + R _{b} + R _{c}}$
- $R_{3} = \frac{R _{a} R _{b}}{R _{a} + R _{b} + R _{c}}$
Wye-to-Delta
- $R_{a} = \frac{R _{1} R _{2} + R _{2} R _{3} + R _{3} R _{1}}{R _{1}}$
- $R_{b} = \frac{R _{1} R _{2} + R _{2} R _{3} + R _{3} R _{1}}{R _{2}}$
- $R_{c} = \frac{R _{1} R _{2} + R _{2} R _{3} + R _{3} R _{1}}{R _{3}}$
Special case: Balanced system
- When $R_{a} = R_{b} = R_{c}$
- $R_{Y} = \frac{R _{Δ}}{3}$