Delta/Π转Star/T变换正确性咨询：哪一种转换正确及原因？

Hey there! Let's clear up the confusion around Δ/Π to Y/T transformations—this is a super common circuit analysis gotcha, but once you know the core checks, spotting the correct one is straightforward.

First, Anchor on Topology Basics #

Before diving into formulas, make sure you're clear on the two topologies we're swapping between:

• Δ (Delta) / Π (Pi): Three components connected in a closed loop, with each pair of external nodes linked by one component. Π is just the admittance-based version of the Δ topology (same shape, using conductances instead of resistances/impedances).
• Y (Star) / T (Tee): Three components with one end all connected to a shared central node, and the other ends linked to the three external nodes. Y and T are identical topologies—just different names based on how you draw them.

The Correct Transformation Rules (and Why They Work) #

The only valid transformations are those that preserve port equivalence: the impedance (or admittance) between any pair of external nodes must be identical before and after the swap. Here are the standard formulas for the most common impedance-based transformations:

Δ (Impedance) → Y (Impedance) #

If your Δ network has impedances Z_AB, Z_BC, Z_CA between each pair of nodes, the corresponding Y network impedances Z_A, Z_B, Z_C (each linked to a single external node and the center) are:

Z_A = (Z_AB * Z_CA) / (Z_AB + Z_BC + Z_CA)
Z_B = (Z_AB * Z_BC) / (Z_AB + Z_BC + Z_CA)
Z_C = (Z_BC * Z_CA) / (Z_AB + Z_BC + Z_CA)

Π (Admittance) → T (Admittance) #

For a Π network using admittances Y_AB, Y_BC, Y_CA, the T network admittances Y_A, Y_B, Y_C are:

Y_A = Y_AB + Y_CA
Y_B = Y_AB + Y_BC
Y_C = Y_BC + Y_CA

(Note: This is just the admittance equivalent of the Δ→Y rule—since admittance is the reciprocal of impedance, you can also convert Π to impedance Δ first, apply the Δ→Y formula, then convert back to admittance if needed.)

How to Spot an Incorrect Transformation #

Any transformation that fails the port equivalence check is wrong. Here are two common mistakes to watch for:

• Applying a "special case" as a general rule: If someone tells you to divide each Δ impedance by 3 to get Y impedances, that only works if all three Δ impedances are identical. This is not a universal transformation.
• Ignoring the denominator in Δ→Y: Skipping the sum of all three Δ impedances in the formula will break port equivalence. For example, if you have a Δ with Z_AB=6Ω, Z_BC=6Ω, Z_CA=6Ω, the correct Y impedances are 2Ω each. If you forgot the denominator and used Z_A = Z_AB * Z_CA, you'd get 36Ω—way off, and the A-B impedance would be 72Ω instead of matching the original Δ's 4Ω.

The quick test: Pick any two external nodes, calculate the equivalent impedance of the original Δ/Π network, then calculate the equivalent impedance of the transformed Y/T network. If they don't match, the transformation is incorrect.

内容的提问来源于stack exchange，提问作者Sajjal Rana