Wye, Delta, and the Square-Root-of-3 Factor

9 min · difficulty 5/10

You meter a 120/208 V wye panel and read 120 V from one phase to neutral. A technician next to you bets the phase-to-phase reading is 240 V, since two 120s feel like they should add. Before you put your leads across two phases, what do you expect to read?

The answer is 208 V, not 240. Two phases in a wye system are 120 degrees apart, not lined up, so they do not simply add. The voltage between any two of them is the line-to-neutral voltage times the square root of 3, which is 1.732. That is 120 times 1.732, about 208 V, and it is exactly why the system carries the name 120/208. The 240 guess belongs to a single-phase center-tapped service, a different animal entirely.

This 1.732 relationship is the one number that defines three-phase work. In a wye (Y) system the line-to-line voltage equals the line-to-neutral voltage times 1.732. That is why standard systems come in matched pairs: 120/208 and 277/480. The smaller number is what you read from a phase to neutral, the larger is what you read across two phases, and 1.732 is always the bridge between them.

Wye and delta connections and the square-root-of-three factor On the left, a wye connection: three phase legs A, B, and C meet at a common grounded neutral. The voltage from neutral to one phase is 120 V line-to-neutral. The voltage between two phases is 208 V line-to-line, because 120 times 1.732 equals 208. On the right, a delta connection of three windings: line voltage equals phase voltage, and the 1.732 factor appears in the current instead. A note states 120/208 V and 277/480 V are matched wye pairs, while 120/240 V is a single-phase center-tapped system, not a wye. Wye, delta, and the 1.732 factor Wye (Y) neutral A B C V_LN = 120 V V_LL = 208 V 208 = 120 × 1.732 Delta (Δ) V_line = V_phase the 1.732 factor shifts to the current 120/208 V and 277/480 V are wye pairs (the 1.732 ratio) 120/240 V is single-phase center-tapped, a different system, not a wye 240
In a wye system the line-to-line voltage is the line-to-neutral voltage times 1.732, so 120 V gives 208 V (not 240). In a delta the line and phase voltages are equal and the 1.732 factor lands on the current.

Delta is the other connection you will meet, and it moves the factor. In a delta the windings join end to end, so each line conductor sits directly across one full phase winding. That means line voltage equals phase voltage in delta, with no 1.732 on the voltage at all. The square-root-of-3 factor never disappears, it just moves from the voltage in a wye to the current in a delta.

The same 1.732 carries into power. To get three-phase apparent power from line quantities, S in kVA equals 1.732 times the line-to-line volts times the line amps, divided by 1,000. A 480 V feeder pulling 100 A works out to 1.732 times 480 times 100, about 83 kVA. Get comfortable spotting that 1.732 in a formula, because its presence is what tells you the math is three-phase and not single-phase.

Power factor penalties, reactive charges, and %Z mismatches show up as real line items on a utility bill or a bid rejection. DistroForge Insider applies these fundamentals to pricing and equipment-selection intelligence.

Question 1 of 4

You meter a 120/208 V wye panel and read 120 V from one phase to neutral. Without metering across two phases, what is the phase-to-phase (line-to-line) voltage?

Educational material only. This is not engineering, safety, or procurement advice. Confirm any value against manufacturer documentation and a licensed professional before specifying equipment.