Wye (Y) and Delta (Δ) Network Transformation

SEO title: Delta–Wye Conversion: Formulas & Worked Example

Meta description: Convert delta and wye resistor networks using clear formulas, a worked example, and terminal-resistance checks. Includes balanced-network shortcuts.

Delta–wye conversion replaces a three-terminal resistor network with an electrically equivalent network. Use it when a circuit cannot be simplified using ordinary series and parallel combinations.

For delta-to-wye conversion, each wye resistor equals the product of the two adjacent delta resistors divided by the sum of all three delta resistors. The reverse conversion uses the sum of pairwise wye products divided by the opposite wye resistor.

The terminal labels matter. Define them before substituting values.

Define the Delta and Wye Networks

Both networks have the same external terminals: A, B, and C.

In the delta network:

• RAB connects A to B.
• RBC connects B to C.
• RCA connects C to A.

In the wye network:

• RA connects A to the internal center node O.
• RB connects B to O.
• RC connects C to O.

The center node O is internal to the equivalent network. It is not an additional external connection.

Delta-to-Wye Formulas

First calculate:

SΔ = RAB + RBC + RCA

Then:

The numerator contains the two delta resistors touching the corresponding terminal.

For example, terminal A touches RAB and RCA, so their product belongs in the formula for RA.

Wye-to-Delta Formulas

First calculate:

PΥ = RA·RB + RB·RC + RC·RA

Then:

The denominator is the wye resistor connected to the terminal opposite the delta branch.

For the A–B branch, the opposite terminal is C, so divide by RC.

An equivalent form is:

RAB = RA + RB + RA·RB/RC

The other two expressions follow by rotating the terminal labels.

Why the Transformation Works

An equivalent network must preserve the voltage-current relationships seen at A, B, and C.

One way to derive the resistor formulas is to compare resistance between each terminal pair while leaving the third terminal open.

Between A and B, with C externally open:

• The delta network is RAB || (RCA + RBC).
• The wye network is RA + RB.

Therefore:

RA + RB = RAB·(RCA + RBC)/SΔ

Similarly:

RB + RC = RBC·(RAB + RCA)/SΔ

RC + RA = RCA·(RAB + RBC)/SΔ

Adding the first and third equations and subtracting the second isolates 2RA. Simplifying gives:

RA = RAB·RCA/SΔ

The other formulas follow by symmetry.

Worked Example: Convert Delta to Wye

Suppose:

RAB = 30 Ω

RBC = 60 Ω

RCA = 90 Ω

Step 1: Add the Delta Resistances

SΔ = 30 + 60 + 90 = 180 Ω

Step 2: Calculate Each Wye Resistance

RA = 30 × 90 / 180 = 15 Ω

RB = 30 × 60 / 180 = 10 Ω

RC = 60 × 90 / 180 = 30 Ω

The equivalent wye network is therefore:

Check the Answer Between All Three Terminal Pairs

For each check, leave the unused external terminal open.

All three checks agree.

A frequent mistake is to remove the delta branches connected to the open terminal. The terminal is externally open, but those two resistors still form a path between the other terminals.

Reverse Check: Convert the Wye Back to Delta

Calculate:

PΥ = 15×10 + 10×30 + 30×15

PΥ = 150 + 300 + 450 = 900 Ω²

Then:

RAB = 900/30 = 30 Ω

RBC = 900/15 = 60 Ω

RCA = 900/10 = 90 Ω

The original values are recovered.

Balanced-Network Shortcuts

If all delta resistors equal RΔ:

RY = RΔ/3

If all wye resistors equal RY:

RΔ = 3RY

For example, a delta containing three 90 Ω resistors is equivalent to a wye containing three 30 Ω resistors.

These shortcuts apply only when the three branch values are equal.

When Should You Use Delta–Wye Conversion?

Use the transformation when a three-terminal subnetwork prevents series-parallel reduction.

Typical examples include:

• An unbalanced resistor bridge.
• A triangular section inside a larger resistor network.
• A three-phase load analyzed using equivalent branch impedances.
• A circuit where eliminating an internal node makes the remaining analysis simpler.

For sinusoidal steady-state AC analysis, the same algebra can be applied to complex impedances at a specified frequency, provided the relevant denominators are nonzero.

Electrical equivalence at the external terminals does not imply identical internal branch currents or individual component power dissipation.

Common Mistakes

Mixing Up Adjacent and Opposite Resistors

For delta-to-wye, multiply the adjacent delta resistors.

For wye-to-delta, divide by the opposite wye resistor.

Connecting the Wye Center to Ground

The center node is internal. Grounding it introduces another connection and generally changes the network.

Using the Balanced Shortcut on an Unbalanced Network

If the three values differ, use the full formulas.

Applying Resistor Arithmetic to Impedance Magnitudes

For AC networks, use complex impedances, not just their magnitudes.

Frequently Asked Questions

Are star–delta and wye–delta the same transformation?

Yes. “Star” and “wye” refer to the same three-branch topology.

Does conversion preserve resistance between terminals?

Yes. For the equivalent resistor networks, the resistance measured between any pair of terminals with the third terminal open is unchanged.

Do I always need this transformation for a bridge circuit?

No. Nodal analysis, mesh analysis, or a balanced-bridge condition may provide a simpler solution. Choose the method that makes the circuit easiest to understand.

Next Steps

After converting the obstructing subnetwork, redraw the entire circuit before combining resistors. The new topology often reveals series and parallel groups that were not present before.

Review Ohm’s law and power dissipation to calculate the resulting currents, voltages, and resistor power ratings.

Reference and Further Reading

James M. Fiore — DC Electrical Circuit Analysis: Delta–Y Conversions.