The square root of three: every weird voltage is the same number

Three phase is three voltages, 120 degrees apart. Put that on paper and you get one of two pictures: a triangle (the delta) or a Y (the wye). Every voltage reading you'll ever take on that system is just a distance between two points on one of those pictures.

Corner to corner across one coil of a 240 delta: 240. And as the wild-leg video showed, the wild leg is the delta's height measured from the grounded X2 up to the far corner: 208.

On a wye, every coil is a transformer winding that starts on its own phase (H1) and lands on the neutral in the center (H2). Your phase-to-phase voltage runs from tip to tip, and that distance always comes out to the square root of three times the coil.

Two coils, 120 degrees apart, make a fat-bottomed triangle. Split it straight down the middle and you get two matching right triangles. Take one:

• The coil is the long side: 120 volts.
• The centerline is half a coil: 60 volts.
• Pythagorean theorem the third side: √(120² − 60²) = 104.

That 104 is half the phase-to-phase reading. Double it: 208. Same answer as 120 × 1.73. And 104 out of 120 is 86.6%. Sound familiar? Hold that thought.

Every 1.73 you will ever use is just a triangle getting cut in half somewhere. So yes: the Pythagoras crowd in the wild-leg comments was right all along.

120/208 looks like the phase-to-phase caught the square root of three. 277/480 looks like the phase-to-ground caught it. Nothing moved. The coil is built to a specific phase-to-ground voltage, and it's the same 1.73 multiplier both times:

• 120-volt coils → 120 × 1.73 = 208 phase to phase.
• 277-volt coils → 277 × 1.73 = 480 phase to phase.

The only thing that changes is which number the system gets named by. Your pot does the same thing up top: a 7200 coil on a 12470Y system is that same 1.73, right on the nameplate. On most two-bushing secondary pots, the nameplate just reads the phase-to-ground.

Thursday's Knowledge Check: a closed wye-delta (a 50 and two 25s) loses a 25 kVA power pot. And yes, for open wye the H2s have to be grounded to the system neutral or none of this works. The bank's three-phase capacity drops from 75 kVA to 43.3. Not 50. Here's the why, from the rule down:

• The open bank rule: your three-phase ceiling is 1.73 × the smallest pot. There it is again. 1.73 × 25 = 43.3 kVA.
• Per pot: split 43.3 across the two pots still working: 21.7 apiece. And 21.7 out of a 25 kVA pot is 86.6%. The same 0.866 from the triangle.
• For the bank: compare what you kept to what you had. 43.3 out of 75 is 57.7%. Square root of three, wearing its third hat.

The Knowledge Check panel showed 25 kVA of single-phase still available, before and after losing the pot. That's not a coincidence, it's simple math: the lighting pot only owes the bank what the smallest power pot can match. A 50 kVA lighting pot committing 25 to three-phase work keeps 50 − 25 = 25 kVA for the lights. Lose a power pot and that number doesn't move, because the pot you lost wasn't the lighting pot.

So where does 1.73 live? Phase-to-ground to phase-to-phase on every wye system: 7200 to 12470, 14.4 to 25 kV. The 208 wild leg on a 240 delta and the 416 wild leg on a 480. Wye secondary phase-to-phase. Open and closed bank capacity. Even the amps column on the capacity panel: turning kVA into amps on three-phase runs through 1.73 too (43.3 kVA at 240 V is 104 A). Anywhere three phases share iron, evenly spaced from each other, root three is in the math.