Why is the line voltage of a wye √3 (1.732) times the phase voltage, and not 2 times?

B. Because the two phase voltages are 120° apart, so they add as arrows (vectors), not as plain numbers

A. Because of losses in the windings

C. Because the neutral carries some of the voltage

D. It is a rounding convention from the utility

Which connection gives you a neutral point you can bring out and use for line-to-neutral loads like 120 V or 277/347 V lighting?

D. Neither — a neutral always comes from the utility

Third year Lesson 1 of 13 · about 14 min

Wye and delta: line vs phase voltage and current

What you’ll learn

The two ways three windings (or three loads) are joined, and how to recognise each on a drawing and with a meter.
Which of voltage or current picks up the √3 in a wye, and which in a delta.
Why the factor is 1.732 and not 2 — the 120° between the phases, in one picture.
How to work line and phase values both ways, and why a mis-connected load can take three times the power.

Before this: Second year: AC fundamentals · Transformer basics

Three-phase power is three AC voltages, each one a third of a cycle (120°) behind the last. You can join three windings — or three loads — to those voltages in only two ways: all three meeting at a centre point (wye, also called star), or end to end in a ring (delta). Everything about line and phase values follows from which one you have.

Two words first, because every mistake on this topic is a mix-up between them. Phase means one winding or one load element — what sits across a single coil. Line means what you measure between the conductors that leave the connection — the lines, L1, L2 and L3. On a wye the two differ in voltage; on a delta they differ in current.

The same three windings joined two ways. In the wye each line reaches one winding, so line and phase current are the same and the voltage picks up the √3. In the delta each winding sits straight across two lines, so line and phase voltage are the same and the current picks up the √3.

Wye: the voltage picks up the √3

Follow line A in the wye drawing: it goes into one winding and nothing else. So the current in line A is the current in that winding — I line = I phase.

Now put a meter from A to B. You are across two windings in series, A-to-N and N-to-B. If the two voltages were in step you would read double. They are not — they are 120° apart — and two equal voltages 120° apart add up to 1.732 times one of them. So V line = √3 × V phase. That is where every familiar pair comes from:

System	Line to neutral (phase)	× 1.732 = line to line	Where you meet it
208Y/120	120 V	207.8 ≈ 208 V	Commercial receptacles and lighting, Canada and US
480Y/277	277 V	479.8 ≈ 480 V	US commercial and industrial — 277 V lighting
600Y/347	347 V	601 ≈ 600 V	Canadian commercial and industrial — 347 V lighting

The centre point N is why the wye is so common in buildings: bring it out as a neutral and one transformer feeds three-phase motors line to line and single-phase lighting or receptacles line to neutral.

Delta: the current picks up the √3

In the delta, each winding is connected straight between two lines. Put a meter from A to B and you are across exactly one winding — V line = V phase. A 480 V delta has 480 V across every winding, and there is no centre point to read 277 from.

The current is where the √3 turns up. Line A feeds the corner where two windings meet, so the line current is the difference of two winding currents that are 120° apart — which works out to I line = √3 × I phase. A delta motor drawing 52 A per line has only 30 A in each of its windings.

Why 1.732 and not 2

Phase voltages drawn as arrows (phasors). If they pointed the same way the line voltage would be 200; because they are 120° apart it is 173 — √3 times one phase.

You do not need to draw this on the job. You do need to remember that it is the 120°, and not anything about the equipment, that makes the number √3 — which is why the same factor turns up in the power formula, the current formula and the transformer arithmetic in the rest of this year.

Worked: the same three heaters, wye then delta

Three identical 10 Ω heating elements on a 480 V three-phase supply, first connected in wye.
Each element sees the phase voltage: 480 ÷ 1.732 = 277 V.
Current in each element (and each line, since it is a wye): 277 ÷ 10 = 27.7 A.
Power: 3 elements × 277 V × 27.7 A ≈ 23,000 W.
Now reconnect the same three elements in delta. Each element sees the full line voltage: 480 V.
Current in each element: 480 ÷ 10 = 48 A. Line current: 48 × 1.732 = 83.1 A.
Power: 3 × 480 × 48 = 69,120 W — three times the wye figure, because each element has √3 times the voltage and power goes with voltage squared.
Check with the three-phase formula: √3 × 480 × 83.1 × 1.0 (a heater's power factor) = 69,090 W. Same answer, to rounding.

Why this matters on a real job

Dual-voltage motors and many heaters are built to be connected either way. Connect for delta when the nameplate wanted wye on that supply and every winding gets 1.732 times its rated voltage — three times the heat. It will not last the shift.

Worked: going the other way

A clamp-on reads 40 A on each line feeding a balanced delta-connected load. Winding current = 40 ÷ 1.732 = 23.1 A.
A 600Y/347 V panel: the voltage across each wye winding of the transformer is 347 V; line to line is 600 V.
Rule of thumb: going from phase to line on the side that has the √3, multiply; going from line to phase, divide. Wye → voltage. Delta → current.

On the job

Tips

Say it as a pair: “wye — voltage; delta — current.” That is which quantity carries the √3.
Before you trust a drawing, meter the supply: all three line-to-line readings, then each line to neutral if there is one. Line ÷ 1.732 to neutral on all three says wye.
On a four-wire delta (high-leg, mostly in the US), one line reads about 208 V to neutral on a 120/240 V system. That is the high leg; it is marked, and it never feeds a 120 V load.
The √3 is exact geometry. Use 1.732 and the answers land within a volt or two of the nominal numbers.

Common mistakes

Putting the √3 on the wrong quantity — multiplying a wye’s current or a delta’s voltage.
Dividing by 1.732 when you should multiply (and the reverse). Sanity check: line voltage in a wye is always the bigger number.
Adding the two phase voltages as plain numbers and expecting 240 on a 120 V wye.
Assuming every three-phase supply has a neutral. A plain delta does not.
Reconnecting a dual-voltage motor or heater from the wrong half of the nameplate diagram.

Check yourself

0 right of 0 answered 8 questions · pick one answer each

1.A 208Y/120 V system measures 120 V from each line to neutral. What should you read line to line?
2.In a wye connection, how does the line current compare with the current in each winding?
3.A balanced delta load carries 30 A in each winding. What will a clamp-on read on each line?
4.On a 600Y/347 V system, what voltage appears across ONE winding of the wye?
5.Three identical heating elements are wired in wye on a 480 V supply. You reconnect the same three elements in delta on the same supply. The total heat output…
6.A 480 V delta secondary: what is the voltage across each winding?
7.Why is the line voltage of a wye √3 (1.732) times the phase voltage, and not 2 times?
8.Which connection gives you a neutral point you can bring out and use for line-to-neutral loads like 120 V or 277/347 V lighting?

Answers

B. 208 V — In a wye the line voltage is √3 times the phase (line-to-neutral) voltage: 120 × 1.732 = 207.8, called 208 V. 240 would be the two phases adding straight up, which only happens if they were in step — they are 120° apart.
A. It is the same current — Each line conductor connects to exactly one winding and nothing else, so whatever flows in the winding flows in the line. In a wye it is the VOLTAGE that picks up the √3.
C. 52 A — Each line in a delta feeds the corner where two windings meet, and their currents are 120° apart, so the line current is √3 × the winding current: 30 × 1.732 = 52 A. 17.3 A is the mistake of dividing instead of multiplying.
A. 347 V — A wye winding sits between a line and the neutral point, so it sees the line-to-neutral voltage: 347 V. The 600 V is measured between two lines, across two windings in series (at 120°).
C. goes up three times — In delta each element sees 480 V instead of 277 V — √3 times the voltage. Power in a resistor goes with voltage squared, so each element makes (√3)² = 3 times the heat. That is exactly why a mis-connected heater or motor can cook.
B. 480 V — Each delta winding is connected straight between two lines, so its voltage IS the line voltage: 480 V. There is no neutral point on a plain delta, so there is no 277 to measure.
B. Because the two phase voltages are 120° apart, so they add as arrows (vectors), not as plain numbers — Line A to line B goes through two windings whose voltages are 120° out of step. Adding them as arrows gives 2 × Vphase × cos 30° = √3 × Vphase. Only voltages that were perfectly in step would add to 2×.
B. Wye — A wye has a natural common point where the three windings meet; that point becomes the neutral. A plain delta has no such point (a centre-tapped "high-leg" delta makes one on one winding only, which is a different story).

Questions people ask

Is 208 V really 120 × √3, or 120 × 1.732?

Both — √3 is 1.7320508…, and 120 × 1.732 = 207.8, which everyone calls 208. The same arithmetic gives 277 × 1.732 = 479.8 (called 480) and 347 × 1.732 = 601 (called 600). The nominal numbers are rounded; your meter will read whatever the supply is doing that day anyway.

Can I tell wye from delta with a meter?

Often, yes. If there is a neutral (or a grounded centre point) and every line reads the same voltage to it, and that voltage is the line voltage ÷ 1.732, you are on a wye. A plain delta has no point that reads line ÷ 1.732 to all three lines. A high-leg delta shows one line reading noticeably higher to the neutral than the other two. Confirm with the drawings and the transformer nameplate — a meter tells you what is there, not what was intended.

Which is better, wye or delta?

Neither — they do different jobs. A wye gives you a neutral, so one transformer can feed motors line to line and lighting line to neutral, and it is easy to ground solidly. A delta has no neutral to share, keeps running (at reduced capacity) with one transformer out as an open delta, and traps some harmonic currents circulating in its windings. Which one a building gets is a design decision made long before you arrive.

Wye and delta (glossary) — the short definition
Three-phase power calculator — kW, kVA and amps with the √3 done for you
Delta ↔ wye impedance conversion
Single & three phase formula sheet — printable
Transformer basics — one pair of coils, before three
Rusty on three-phase math? — the one-page refresher

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