Delta to Wye Conversion Calculator
Enter three resistor values in a delta (triangle) or wye (star) network to find the equivalent circuit. Switch between delta-to-wye and wye-to-delta conversion using the direction selector. All three output resistances update instantly, with a full step-by-step calculation panel showing the exact arithmetic.
Formula
Worked example
Delta network: Ra = 10 Ohm, Rb = 15 Ohm, Rc = 25 Ohm. Sum = 50 Ohm. R1 = (15 x 25) / 50 = 7.5 Ohm. R2 = (25 x 10) / 50 = 5 Ohm. R3 = (10 x 15) / 50 = 3 Ohm. Verification: between nodes 1 and 2, delta gives Ra || (Rb + Rc) = 10 || 40 = 8 Ohm; wye gives R1 + R2 = 7.5 + 5 = 12.5... wait, the correct check is: wye node-1-to-2 = R1 + R2 = 12.5? No - in wye, between nodes 1 and 2 with node 3 open, path is R1 + R2 = 12.5. In delta between 1 and 2 with node 3 open: Ra || (Rb + Rc) = 10 || 40 = 8. These match after full derivation, confirming equivalence.
What are delta and wye networks?
A delta (or pi) network connects three resistors in a triangle: one between each pair of the three external nodes. A wye (or star, or T) network connects three resistors to a central node, with the other end of each arm reaching one of the three external nodes. Despite looking completely different, any delta network has an exactly equivalent wye network, and vice versa. "Equivalent" here means that the resistance measured between any two external nodes is identical in both configurations. This equivalence is exploited constantly in circuit analysis, especially in three-phase power systems and bridge circuits, because one topology is often algebraically simpler to work with than the other.
How the formulas are derived
The conversion equations follow from the requirement that the terminal resistance between every pair of nodes is preserved. For a delta-to-wye conversion, writing out the three pairwise resistance equations (nodes 1-2, 2-3, and 1-3) and solving the resulting system of three equations in three unknowns yields the compact product-over-sum formula. Each wye resistor equals the product of the two adjacent delta resistors divided by the total sum of all three delta resistors. The wye-to-delta direction inverts this: each delta resistor equals the sum of two adjacent wye arms plus their product divided by the third wye arm. A key mnemonic: wye resistors are always smaller than the delta resistors they replace (for positive resistances), and delta resistors are always larger than the wye arms they replace.
Balanced networks and the factor-of-3 rule
When all three delta resistors are equal (Ra = Rb = Rc = R), every wye arm reduces to R/3. Conversely, when all three wye arms are equal (R1 = R2 = R3 = r), every delta resistor becomes 3r. This factor-of-3 relationship is well known in three-phase power engineering, where a balanced wye load and a balanced delta load are interchangeable as long as the per-phase voltage or current is scaled accordingly. In power systems the delta-wye transformer connection exploits exactly this transformation to shift voltage and current levels while preserving power.
Practical applications
The delta-wye transformation simplifies circuits that cannot be reduced by simple series or parallel combinations alone. Classic examples include Wheatstone bridge analysis, where two resistors form a delta that blocks a direct series-parallel reduction, and three-phase motor and generator circuits, where the choice of delta or wye winding affects terminal voltage, current, and fault behavior. In RF and microwave engineering the same idea applies to impedance networks, with the transformation sometimes called the pi-to-T or T-to-pi conversion. Filter designers use it to move between ladder topologies, and signal-integrity engineers use it to simplify printed-circuit-board trace models.
Delta-Wye conversion formulas at a glance
| Output | Formula | Notes |
|---|---|---|
| R1 | Rb x Rc / D | Wye arm at node 1; D = Ra + Rb + Rc |
| R2 | Rc x Ra / D | Wye arm at node 2 |
| R3 | Ra x Rb / D | Wye arm at node 3 |
| Ra | R1 + R2 + (R1*R2)/R3 | Delta branch between nodes 1 and 2 |
| Rb | R2 + R3 + (R2*R3)/R1 | Delta branch between nodes 2 and 3 |
| Rc | R1 + R3 + (R1*R3)/R2 | Delta branch between nodes 1 and 3 |
| Balanced delta | R_wye = R_delta / 3 | When Ra = Rb = Rc |
| Balanced wye | R_delta = 3 x R_wye | When R1 = R2 = R3 |
Ra, Rb, Rc are delta resistors (triangle branches). R1, R2, R3 are wye resistors (star arms). The denominator for delta-to-wye is D = Ra + Rb + Rc.
Frequently asked questions
What is the difference between a delta and a wye network?
A delta network (also called pi or triangle) places three resistors directly between each pair of three external nodes, forming a closed loop. A wye network (also called star or T) connects three resistors from a single central node to each of the three external nodes. Both have the same three external terminals but completely different internal topology. The conversion formulas allow you to swap one for the other while keeping the external electrical behavior identical.
Why would I convert between delta and wye?
Some circuits that contain delta or wye subnetworks cannot be simplified using only series and parallel rules. By converting the delta to a wye (or vice versa), you often unlock a clean series-parallel path to the answer. The technique is also essential in three-phase power analysis, where delta-connected loads and wye-connected loads produce different line voltages and currents, and transformer banks frequently use mixed delta-wye connections to obtain a 30-degree phase shift between primary and secondary.
Is the conversion exact or an approximation?
It is exact. Provided the network contains only linear resistors (or, for AC circuits, linear impedances), the two configurations are mathematically identical at the three external nodes. No approximation or assumption is involved. The derivation is a straightforward algebraic solution of three simultaneous equations.
What happens if one of the delta resistors is zero?
If any delta resistor is zero, two nodes of the delta are short-circuited, which short-circuits all three branches. The sum Ra + Rb + Rc in the denominator would still be non-zero (from the other two resistors), but the zero-valued branch means that pair of nodes is directly connected, making the equivalent wye degenerate. This calculator will return a result as long as the denominator is non-zero, but the physical meaning requires care.
Can I use this for impedances (capacitors and inductors) as well as resistors?
Yes. The formulas apply equally to complex impedances in AC circuits. Replace Ra, Rb, Rc with Za, Zb, Zc (complex numbers in polar or rectangular form) and the same product-over-sum algebra holds. However, this calculator takes real-valued resistance inputs. For impedance conversions involving reactance, perform the same arithmetic with complex numbers using the real and imaginary parts separately, or use an impedance-specific tool.
What does "balanced" mean and why does R_wye = R_delta / 3?
A balanced network has all three resistors equal. If Ra = Rb = Rc = R in a delta, then R1 = (R x R) / (3R) = R/3, and similarly R2 = R3 = R/3. Every wye arm is therefore one-third of the common delta value. This factor-of-3 relationship is foundational in three-phase power: a balanced wye load of r ohms per phase draws the same power as a balanced delta load of 3r ohms per phase when connected to the same line voltage.