Convert three resistors between the Y (star) and Δ (delta) arrangements using the Kennelly transform. Useful for three-phase power, Wheatstone-style bridges, ladder networks and any resistor mesh that does not collapse to series and parallel.
Parameters
Transform direction
Y→Δ: treat R_A, R_B, R_C as the three Y branches at the central node. Δ→Y: treat them as the three Δ branches R_AB, R_BC, R_CA.
Resistor R_A
Ω
Y branch at node A, or Δ branch AB
Resistor R_B
Ω
Y branch at node B, or Δ branch BC
Resistor R_C
Ω
Y branch at node C, or Δ branch CA
Test voltage V_test
V
Power drawn if V_test is applied directly across the transformed R_AB branch
Results
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Input configuration
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Transformed R_AB (Ω)
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Transformed R_BC (Ω)
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Transformed R_CA (Ω)
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Sum of branches (Ω)
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Power V_test²/R_AB (W)
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Y and Δ equivalent network diagram
Left: input configuration. Right: transformed equivalent network. Terminals A, B, C are linked with dashed lines to emphasise equivalence, with an animated current arrow.
In Y→Δ the "sum of products" is divided by the opposite Y branch; in Δ→Y the product of two adjacent Δ branches is divided by the total. When the three resistors are equal to R (balanced case), the equivalent Δ branch is 3R, i.e. an equivalent Δ resistor is three times the corresponding Y resistor.
What is the Y-Δ Transform?
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I've heard the name "Y-Δ transform", but what does it actually do? "Transform" between what and what?
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Roughly: a "star (Y)" — three resistors meeting at one central node — and a "triangle (Δ)" — three resistors forming a closed loop around three terminals — look completely different as wiring diagrams, yet from the three external terminals they can be made electrically indistinguishable. The Y-Δ transform is the pair of formulas that tells you what resistor values to use on the other side to make them equivalent. So you can rewrite the schematic from Y to Δ and the rest of the circuit will not notice.
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If they're the same, why bother transforming? It just changes how the picture looks, right?
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That is the punchline: the transform dramatically changes how easy the network is to solve. The textbook example is the bridge circuit — four arm resistors plus a galvanometer. It looks neither series nor parallel. But once you convert the central Y to a Δ, the rest of the network collapses into a clean series/parallel arrangement that you can solve on paper. Think of Y-Δ as "the spell that turns non-series/parallel networks into series/parallel ones".
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Handy. With the defaults (R_A=10, R_B=20, R_C=30, Y→Δ) the tool says R_AB = 36.67, R_BC = 110 and R_CA = 55. Those are very different from the inputs — are they really right?
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Yes. The Y→Δ formula divides the "sum of pairwise products" by the opposite Y resistor. productSum = R_A·R_B + R_B·R_C + R_A·R_C = 200 + 600 + 300 = 1100. Dividing by R_C = 30 gives 36.67 for R_AB; dividing by R_A = 10 gives 110 for R_BC; dividing by R_B = 20 gives 55 for R_CA. The Δ branches grow larger when the opposite Y resistor is smaller — picture it as "a node with low resistance needs a large detour resistance to remain electrically equivalent".
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I have also seen the term "star-delta starter" on motors. Is that the same transform?
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Similar name, different idea. The Y-Δ transform is a mathematical equivalence between three-resistor sub-networks. The star-delta starter is an operational trick on three-phase induction motors: starting straight from Δ pulls 5-7 times the rated current, so the windings are first connected in Y for a few seconds. Each winding then sees only 1/√3 of the line voltage, and both starting current and torque drop to one third. Once up to speed the windings are switched to Δ for normal running. It uses the voltage difference between Y and Δ, not the algebraic equivalence.
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I see — the math and the wiring trick are separate. I tried "Δ→Y" mode with the same values and the numbers got much smaller (R_AB=10, R_BC=5, R_CA=3.33). What is happening?
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Good experiment. The Δ→Y formula divides the product of two adjacent Δ branches by the sum of all three. With denom = 10+20+30 = 60, the Y branch displayed as "R_AB" (computed from R_B·R_C/denom = 20·30/60 = 10) is the value for the position the tool labels R_AB; the other two follow the same pattern with different pairs. The take-away is that Δ branches are generally larger than the equivalent Y branches — exactly 3× larger in the balanced case.
Frequently Asked Questions
A Y (star) connection has three resistors meeting at a single central node; a Δ (delta) connection has three resistors forming a triangular loop between three terminals. The two arrangements look completely different, but they can be made exactly equivalent when viewed from the three external terminals. The Y-Δ transform (Kennelly transform) gives that equivalence. The Y→Δ direction reads R_AB = (R_A·R_B + R_B·R_C + R_A·R_C)/R_C and the Δ→Y direction reads R_A = R_AB·R_CA/(R_AB + R_BC + R_CA). It is the standard tool for unfolding bridge circuits and three-phase loads that cannot be reduced by series/parallel combinations alone.
The external behaviour is equivalent, but the simple sum of the three resistor values is not. For a balanced network with three equal resistors R, each Δ branch is 3R, giving a total of 9R (vs. 3R for the Y). Yet the resistance seen between any two terminals is 2R in both — for Y it is two R in series, for Δ it is 3R in parallel with 6R, both equal to 2R. This tool deliberately shows both the sum of the transformed branches and the per-branch resistance so you can see this asymmetry.
A Wheatstone-style bridge cannot be reduced to one equivalent resistor by series/parallel rules alone, because the four arm resistors together with the meter or load form a Y or a Δ. Transforming the central Y to a Δ (or the central Δ to a Y) leaves a purely series/parallel network that can be solved by inspection. Remember the rule of thumb: "Y-Δ turns a non-series/parallel network into a series/parallel one whenever one Y/Δ sub-network is the only obstruction."
Starting a three-phase induction motor on full voltage draws an inrush current 5-7 times the rated value, stressing breakers and feeders. To avoid this, the windings are temporarily connected in Y during the first few seconds of start. Each winding then sees 1/√3 of the line voltage, and both the starting current and torque drop to one third. Once the motor approaches rated speed, the windings switch to Δ for full-voltage running. This "star-delta start" uses the difference in voltage per winding, not the algebraic Y-Δ equivalence — so it shares the name but is a different idea from the Kennelly transform.
Real-World Applications
Three-phase power systems: generators, transformers, motors and transmission lines in industrial power are nearly always three-phase, and loads are wired in either Y or Δ. The same motor connected in Y vs. Δ has a √3 difference in the relation between line and phase voltages and currents. The Y-Δ formulas are the algebraic foundation used to swap one wiring style for the other during analysis and design.
Bridge and ladder networks: Wheatstone bridges, strain-gauge front-ends and resistor ladders used as filters appear all over instrumentation and signal processing. With a single Y-to-Δ (or vice versa) conversion, what was a tangled mesh requiring nodal analysis collapses to pen-and-paper series/parallel arithmetic.
Sensor and precision measurement: Wheatstone bridges are used with strain gauges, Pt100 RTDs and photoresistors to sense small resistance changes. When an amplifier loads the bridge midpoint, the standard recipe to compute the input impedance is to fold the four arms and the load into one equivalent resistor using a Y-Δ transform.
Pre-processing for SPICE / EMTP-style simulators: simulators can solve anything with nodal matrices, but humans cannot keep intuition without reduction. When checking a large power network by hand or building a simplified macro-model, Y-Δ transforms are used to collapse resistor clusters down to one resistor per node. Intuition lives in the algebra, not in the matrix.
Common Misconceptions and Pitfalls
The biggest pitfall is assuming that internal currents and voltages are also preserved by the transform. Y-Δ guarantees equivalence only at the three external terminals A, B, C; the central node of the Y simply does not exist in the Δ. The currents through the Δ branches are generally not the same as the currents through the original Y branches. If you need to know what happens at the centre of a Y, you must analyse it as a Y — do not transform first and then "read off" the internal current.
The next misconception is that applying a Y-Δ transform always simplifies a circuit. It only helps when the Y (or Δ) is the single obstruction preventing series/parallel reduction. Applying a transform to a sub-network that was already series/parallel just produces uglier numbers and more chances for arithmetic slips. The right question is "if I remove this Y/Δ, do the remaining elements become purely series/parallel?" Apply the transform only when the answer is yes.
Finally, do not assume Y-Δ is limited to resistor networks. The same formulas work for complex impedances and therefore apply to AC networks with capacitors and inductors, RF filter design and transmission-line meshes. Beware, though: each "product" and "sum" is a complex-number operation, and sign or phase errors blow up quickly. When manually checking a Y-Δ result from a circuit simulator, use a complex-arithmetic calculator or numerical software to avoid catastrophic cancellation.
How to Use
Enter three resistor values (Ra, Rb, Rc) in ohms for your current configuration (Y or Δ arrangement)
The simulator applies Kennelly transform equations to compute the equivalent three resistances in the opposite topology
Review transformed impedance values (R_AB, R_BC, R_CA in Ω) and total branch sum to verify network equivalence
Apply optional test voltage to calculate power dissipation across the primary branch (V²/R_AB in watts)
Worked Example
Three-phase motor control circuit with Y-connected stator windings: Ra = 12 Ω, Rb = 12 Ω, Rc = 12 Ω. Using Kennelly transform: R_AB = (Ra·Rb + Rb·Rc + Rc·Ra)/Rc = (144 + 144 + 144)/12 = 36 Ω (equivalent Δ configuration). Applying V_test = 230 V across R_AB yields power = 230²/36 = 1,469 W. All three transformed resistances equal 36 Ω, sum = 108 Ω, confirming three-phase symmetry.
Practical Notes
Power distribution systems use Y-Δ transforms to analyze unbalanced loads; unequal resistor values (e.g., Ra=10Ω, Rb=15Ω, Rc=20Ω) reveal asymmetric phase impedances critical for fault analysis
Circuit topology conversion reduces complexity in bridge networks and three-phase AC circuits; validate that transformed branch sum remains constant during conversion
Test voltage application simulates actual operating conditions; power values depend directly on transformed impedance—lower R_AB increases power density in that branch