The delta wye converter above transforms a three-element impedance network — three resistors arranged as a triangle — into the electrically identical three-resistor star, and back again. This is the network-topology transformation, sometimes written delta-Y, Y-delta, pi-T or star-mesh. It is not the three-phase supply conversion that shares the same name, and the difference matters: that other job is about line and phase voltages and currents in a power system, and it is covered by the kVA calculator and the electrical power calculator rather than by this page.
Arb Digital publishes free engineering calculators that say what they do in the first sentence. This one exists because some resistor networks cannot be reduced by series and parallel combination at all, and this transformation is the standard way out of that dead end. The tool also prints a terminal-resistance check so you can see for yourself that the two networks really are interchangeable.
What This Delta Wye Converter Does
Take three terminals, A, B and C. You can connect three elements between them in exactly two topologies. In the delta arrangement each element bridges a pair of terminals directly, forming a closed triangle. In the wye arrangement each element runs from one terminal inwards to a shared central node that has no external connection.
These look completely different and behave identically from the outside. For any pair of terminals, the resistance you measure across a correctly converted pair of networks is the same. That is the whole content of the transformation, and it is exact rather than approximate. Nothing is lost and no assumption is made about what is connected to the terminals.
The hero shows the three converted values. The grid breaks them out individually and adds a check figure: the resistance measured between terminals A and C, computed from the network you supplied. Convert in one direction and then back, and that check figure should be unchanged, which is a quick way to confirm you entered what you meant to.
How to Use It
- Choose the direction. Delta to wye is the more common need, because it is what unlocks a bridge circuit for series-parallel reduction.
- Label your terminals consistently. The delta legs are named by the pair of terminals they bridge; the wye branches are named by the single terminal they attach to. Getting the correspondence wrong is the main source of error.
- Enter all three values in the same unit. The transformation is scale-free, so kilohms in gives kilohms out, but mixing units within one network gives nonsense.
- Read the check figure. It confirms that the terminal-to-terminal resistance is preserved, which is the property the whole transformation rests on.
- Substitute and reduce. Once the awkward delta is a wye, the surrounding network usually collapses into ordinary series and parallel combinations.
The Formula: How the Transformation Works
Going from delta to wye, each wye branch is the product of the two delta legs that touch its terminal, divided by the sum of all three delta legs. With delta legs RAB, RBC and RCA, and their sum S, the wye branches are RA = RABRCA/S, RB = RABRBC/S and RC = RBCRCA/S.
Going the other way, form P = RARB + RBRC + RCRA. Then each delta leg is P divided by the wye branch at the terminal it does not touch: RAB = P/RC, RBC = P/RA and RCA = P/RB. That "opposite terminal" rule is the part people misremember most often.
Both sets follow from writing down the resistance between each pair of terminals for each topology and demanding they match. For the delta, the resistance from A to B is RAB in parallel with the series combination of the other two. For the wye it is simply RA + RB. Three such equations in three unknowns give the transformation uniquely. OpenStax University Physics Volume 2, section 10.2 on resistors in series and parallel, sets out the series and parallel reduction that this transformation exists to make possible, and section 9.4 on Ohm's law covers the linear relation the whole method depends on.
Work the defaults through by hand. With RAB = 10, RBC = 20 and RCA = 30 Ω, the sum is 60. So RA = 10 × 30 / 60 = 5 Ω, RB = 10 × 20 / 60 = 3.333 Ω and RC = 20 × 30 / 60 = 10 Ω. Checking the A-to-C resistance: in the delta it is 30 in parallel with 30, which is 15 Ω; in the wye it is 5 + 10, which is also 15 Ω. Converting back, P = 5 × 3.333 + 3.333 × 10 + 10 × 5 = 100, so RAB = 100/10 = 10, RBC = 100/5 = 20 and RCA = 100/3.333 = 30, exactly what we started with.
Why This Transformation Exists at All
Most textbook resistor networks reduce by inspection. You spot two resistors in series, replace them with their sum, spot two in parallel, replace those, and repeat until one resistor remains. The method feels universal because most exercises are constructed so that it works.
A Wheatstone bridge breaks it. Five resistors, four nodes, and no two of them are in series or in parallel with each other, because the bridging element ruins every pairing. You can attack it with mesh or nodal analysis and grind through simultaneous equations, or you can convert one of the two triangles into a wye, at which point the whole thing collapses into series and parallel steps and finishes in three lines.
That is the real value here. It is not that the transformation reveals new physics; it is that it converts a problem requiring simultaneous equations into one requiring arithmetic. In filter design the same move maps between pi and T sections, and in power engineering it lets an impedance network be redrawn into whichever form makes the fault calculation easier.
Impedance Networks Versus Three-Phase Supplies
Two quite different jobs are both called delta-wye, and confusing them wastes a lot of time. This page does the network transformation: three passive elements, one topology mapped onto another, exact equivalence at three terminals.
The other job is the three-phase supply relationship, where a delta or wye connection of source or load determines how line voltage relates to phase voltage and how line current relates to phase current, with a factor of the square root of three appearing in one place or the other depending on the configuration. That is a power-system calculation involving real supplies, apparent power and power factor rather than a topology transformation, and it belongs with the kVA calculator, the electrical power calculator and the power factor calculator.
A rule of thumb: if your inputs are three resistances or impedances, you want this page. If your inputs are a voltage, a current and a phase count, you want the power tools instead.
Extending It to Complex Impedances
The transformation holds unchanged for complex impedances, so it works for networks of resistors, capacitors and inductors at a single frequency. The algebra is identical; the arithmetic becomes complex multiplication and division. This tool handles the real case only, because that covers resistive networks and DC analysis, which is where the transformation is used most.
Two warnings apply when you extend it yourself. First, the transformation is frequency dependent once reactance is involved: a wye that is exactly equivalent at one frequency is not equivalent at another, so a converted network is only valid at the frequency you converted at. Second, a delta of pure capacitors converts into a wye that is not a set of pure capacitors in general, which is why the transformation is less useful in filter synthesis than its DC simplicity suggests.
Power dissipation is another place to be careful. The two networks are equivalent at their terminals, but the power dissipated in individual elements is completely different. If you convert a delta to a wye to simplify a calculation, do not then read off the wye branch currents as though they were the real currents in the physical resistors. They are not; only the terminal behaviour is preserved.
How This Sits Next to the Other Circuit Tools
Once the delta is a wye, the resistor combination calculator finishes the series and parallel reduction. The Ohm's law calculator turns the resulting resistance into a current or voltage, and the voltage divider calculator and current divider calculator handle the splits inside the network.
For real components, the wire resistance calculator gives the resistance of the conductors themselves and the resistance converter handles unit changes. When you extend to alternating current, the reactance calculator and the RLC impedance calculator supply the complex impedances the transformation would act on, and the power dissipation calculator tells you what each physical element is actually dissipating.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Dividing by the wrong wye branch — going wye to delta, each leg uses the branch at the terminal it does not touch, which is the opposite of what intuition suggests.
- Mislabelling terminals — the correspondence between delta legs and wye branches is fixed by geometry, and swapping two labels produces a network that is wrong but plausible-looking.
- Reading wye branch currents as real currents — only terminal behaviour is preserved; the currents inside the equivalent network are not the currents in the physical resistors.
- Converting an AC network once and reusing it — with reactance present the equivalence holds only at the frequency you converted at.
- Confusing this with the three-phase supply conversion — that calculation is about line and phase quantities and belongs with the power tools, not here.
Related Free Tools From Arb Digital
Finish the reduction with the resistor combination calculator, then apply the Ohm's law calculator, the voltage divider calculator and the current divider calculator. For component values and units, see the wire resistance calculator and the resistance converter. For alternating current, use the reactance calculator and the RLC impedance calculator, and check heat with the power dissipation calculator. Three-phase supply questions belong with the kVA calculator, the electrical power calculator and the power factor calculator. Everything is on the free online tools hub.
Frequently Asked Questions
No. This page converts a three-element impedance network from one topology to an equivalent one, which is a circuit-analysis technique. Three-phase delta-wye conversion relates line and phase voltages and currents in a power supply, involves a square root of three factor and belongs with power calculations. Both are called delta-wye, and they are different jobs.
Because some networks cannot be reduced by series and parallel combination. A Wheatstone bridge is the standard example: no two of its five resistors are in series or parallel with each other. Converting one triangle to a star breaks the deadlock and lets the rest collapse into ordinary series and parallel steps.
Exact. It is derived by demanding that the resistance between each pair of terminals is identical in both topologies, which gives three equations in three unknowns with a unique solution. No assumption is made about what is connected outside the three terminals, so the substitution is valid in any surrounding circuit.
Alternative names for the same two topologies, used mainly in filter and transmission-line work. A pi network is a delta drawn with the third element along the bottom, and a T network is a wye drawn with the central node pointing down. The transformation between them is exactly the one on this page.
Yes, with complex impedances substituted for resistances, but only at one frequency at a time. A wye that is exactly equivalent at one frequency is not equivalent at another, so the converted network is valid only where you converted it. This tool handles the real-valued case, which covers resistive and DC work.
No. Only the terminal behaviour is preserved. The central node of a wye may not physically exist, and the power dissipated in each converted element bears no relation to what the original physical resistors dissipate. Use the transformation to find terminal quantities, then work back into the original topology for internal currents.
A zero leg short-circuits two terminals together, and the resulting wye is degenerate: two branches become zero and the topology has collapsed. The tool reports this rather than dividing by something meaningless. In practice a zero-resistance leg means those two nodes are the same node and the network should be redrawn.
This tool is provided for educational and preliminary design use. It handles real-valued resistances only, preserves terminal behaviour rather than internal currents or dissipation, and does not cover three-phase supply relationships.
