Star-Delta Conversion Calculator
Enter three resistor values in either star or delta configuration to instantly get the equivalent values in the other configuration.
Reviewed by the ToolNestr Editorial Team — July 2026
How star-delta conversion works
Star-delta (Y-Δ) conversion is a mathematical technique for transforming a three-terminal resistor network between star (wye) and delta (pi) configurations. The equivalent networks are electrically identical from the perspective of the three terminals. For star-to-delta: each delta resistor equals the sum of all pairwise products of star resistors divided by the opposite star resistor.
For delta-to-star: each star resistor equals the product of the two adjacent delta resistors divided by the sum of all three delta resistors. When all three resistors are equal, the conversion simplifies to RΔ = 3 × RY and RY = RΔ / 3.
Worked example
Resistance ranges comparison
Star and delta resistor values span different ranges depending on the application. The chart below shows common resistance ranges used in electronics.
Electrical Engineer
Analyze complex resistor networks and simplify bridge circuits for precise measurement and control system design.
Power Systems Engineer
Convert between wye and delta configurations for three-phase transformer connections, motor windings and load analysis.
Electronics Student
Master the star-delta transformation technique used extensively in circuit analysis textbooks and exam problems.
Circuit Designer
Simplify impedance matching networks and filter designs by converting between equivalent network topologies.
| Conversion | Formula | Equal values | Notes |
|---|---|---|---|
| Star \u2192 Delta | R\u2081\u2082 = (R\u2090R\u2091 + R\u2091R\u2093 + R\u2093R\u2090) / R\u2093 | R_\u0394 = 3 \u00d7 R_Y | Each delta resistor = sum of products / opposite star |
| Star \u2192 Delta | R\u2082\u2083 = (R\u2090R\u2091 + R\u2091R\u2093 + R\u2093R\u2090) / R\u2090 | R_\u0394 = 3 \u00d7 R_Y | Same numerator, divided by opposite star resistor |
| Star \u2192 Delta | R\u2083\u2081 = (R\u2090R\u2091 + R\u2091R\u2093 + R\u2093R\u2090) / R\u2091 | R_\u0394 = 3 \u00d7 R_Y | Same numerator, divided by opposite star resistor |
| Delta \u2192 Star | R\u2090 = (R\u2081\u2082 \u00d7 R\u2083\u2081) / (R\u2081\u2082 + R\u2082\u2083 + R\u2083\u2081) | R_Y = R_\u0394 / 3 | Product of adjacent deltas / sum of all deltas |
| Delta \u2192 Star | R\u2091 = (R\u2081\u2082 \u00d7 R\u2082\u2083) / (R\u2081\u2082 + R\u2082\u2083 + R\u2083\u2081) | R_Y = R_\u0394 / 3 | Product of adjacent deltas / sum of all deltas |
| Delta \u2192 Star | R\u2093 = (R\u2082\u2083 \u00d7 R\u2083\u2081) / (R\u2081\u2082 + R\u2082\u2083 + R\u2083\u2081) | R_Y = R_\u0394 / 3 | Product of adjacent deltas / sum of all deltas |
How to use the Star-Delta Conversion Calculator
Choose conversion direction
Select either \u201cStar \u2192 Delta\u201d or \u201cDelta \u2192 Star\u201d depending on which configuration you have and what you need to find.
Enter three known values
Input the three resistor values in your current configuration. All values must be in ohms (positive numbers only).
Read equivalent values
The output section instantly displays the three equivalent resistor values for the other configuration.
Tips for star-delta conversion
Check symmetry first
If all three resistors are equal in star (R_Y), the delta equivalent is simply 3 \u00d7 R_Y. For equal delta (R_\u0394), star is R_\u0394 / 3. This shortcut saves time in balanced circuits like three-phase systems.
Verify your conversion
After converting, measure the resistance between any two terminals in both configurations \u2014 they should be identical. For example, R_ab(Y) = R_a + R_b should equal R_ab(\u0394) = R_12 || (R_23 + R_31).
Use star-delta for bridge circuits
Wheatstone bridge circuits cannot be simplified using series-parallel rules. Apply star-delta conversion to one side of the bridge (usually the \u0394 on one side) to transform it into a solvable network.
Watch for floating point precision
When very large and very small resistor values are combined, precision loss can occur. Use consistent units and be aware that extremely unbalanced networks may give results that are sensitive to rounding.
When to use star-delta conversion
Star-delta conversion is essential when analyzing circuits that cannot be simplified using only series and parallel resistor combinations. The classic example is the Wheatstone bridge \u2014 a five-resistor network that forms a delta configuration. By converting one delta to a star, the entire circuit becomes a simple series-parallel network that can be solved with Ohm\u2019s Law. This technique is also used in three-phase power system analysis, filter design, and impedance matching.
Star-delta formulas explained
The star-to-delta conversion formula for R\u2081\u2082 is: R\u2081\u2082 = (R\u2090R\u2091 + R\u2091R\u2093 + R\u2093R\u2090) / R\u2093. The numerator is the sum of all pairwise products of the three star resistors. Each delta resistor uses this same numerator but divides by the opposite star resistor. For delta-to-star: R\u2090 = (R\u2081\u2082 \u00d7 R\u2083\u2081) / (R\u2081\u2082 + R\u2082\u2083 + R\u2083\u2081). Each star resistor is the product of the two adjacent delta resistors divided by the sum of all three delta resistors.
Applications in three-phase systems
Three-phase power systems use star (wye) and delta configurations for generator and transformer windings. Star connections provide a neutral point and two voltage levels (phase and line). Delta connections provide higher current capacity without a neutral. Converting between configurations is necessary for analyzing unbalanced loads, calculating fault currents, and designing motor starters such as star-delta starters that reduce inrush current during motor startup.
Impedance matching with star-delta
In RF and audio circuits, star-delta (or pi-wye) transformations are used for impedance matching networks. A pi-network (\u0394 configuration) can be converted to a tee-network (Y configuration) and vice versa, allowing designers to choose the topology that best suits their component arrangement and parasitic management. This is particularly useful in antenna matching, filter design, and amplifier interstage coupling.
Frequently asked questions
What is star-delta conversion?
Star-delta (Y-Δ) conversion is a mathematical technique used to transform a three-terminal resistor network from a star (wye) configuration into an equivalent delta (pi) configuration, and vice versa. The equivalent networks behave identically from the perspective of the three terminals.
Why do I need star-delta conversion?
Star-delta conversion simplifies circuit analysis by transforming networks that cannot be reduced using simple series-parallel rules. It is essential for analyzing bridge circuits, three-phase power systems, and impedance matching networks.
How do I convert star to delta?
Star-to-delta formulas: R₁₂ = (RₐRₑ + RₑRₓ + RₓRₐ) / Rₓ, R₂₃ = (RₐRₑ + RₑRₓ + RₓRₐ) / Rₐ, R₃₁ = (RₐRₑ + RₑRₓ + RₓRₐ) / Rₑ. Each delta resistor equals the sum of all star product pairs divided by the opposite star resistor.
How do I convert delta to star?
Delta-to-star formulas: Rₐ = (R₁₂ × R₃₁) / (R₁₂ + R₂₃ + R₃₁), Rₑ = (R₁₂ × R₂₃) / (R₁₂ + R₂₃ + R₃₁), Rₓ = (R₂₃ × R₃₁) / (R₁₂ + R₂₃ + R₃₁). Each star resistor is the product of the two adjacent delta resistors divided by the sum of all three delta resistors.
When is star-delta conversion used in power systems?
Three-phase power systems use star (wye) or delta configurations for generator windings, transformer connections and motor windings. Star-delta conversion allows analyzing unbalanced loads, calculating equivalent impedances, and designing motor starters.
Are the star and delta networks exactly equivalent?
Yes, from the perspective of the three terminals, the star and delta networks are electrically identical. The resistance measured between any two terminals is the same in both configurations. This equivalence holds for resistors, impedances, or any linear components.
What happens if all three resistors are equal?
For equal star resistors R_Y, the equivalent delta resistors are R_Δ = 3 × R_Y. Conversely, for equal delta resistors R_Δ, the equivalent star resistors are R_Y = R_Δ / 3. This simple 3:1 ratio makes symmetrical networks easy to convert.
Can I use star-delta conversion for capacitors and inductors?
Yes. Star-delta conversion works for any linear impedances including capacitors and inductors, using the same formulas. For capacitors, impedances are 1/(jωC), and for inductors, impedances are jωL. The conversion applies to the complex impedance values.
What is the delta-star transformation also called?
The delta-star transformation is also known as the Δ-Y transformation, delta-wye conversion, or the Kennelly theorem (named after Arthur Edwin Kennelly who published the transformation in 1899). In European literature it is sometimes called the Δ-Π transformation.
Is there a mnemonic to remember the formulas?
For delta-to-star: each star resistor = (product of adjacent delta resistors) / (sum of all three delta resistors). For star-to-delta: each delta resistor = (sum of all products of star resistor pairs) / (opposite star resistor). Think: "star = product over sum; delta = sum over opposite."
Sources & references
This tool uses standard formulas and reference values from:
- • NFPA 70 — National Electrical Code (NEC), conduit fill, box fill and conductor ampacity. nfpa.org
- • IPC-2221, Generic Standard on Printed Board Design (trace width / current). ipc.org
- • NIST reference constants and unit definitions; IEEE standards where applicable. nist.gov
For educational and preliminary use. Verify against full engineering calculations and the governing standard before any real design.