PW-A2 · Three-phase AC

Line and phase values: where √3 comes from

That the line voltage is √3 times the phase voltage is not a number to memorize but comes from the geometry of the difference of two phasors. Get a feel for why that √3 arises and whether it attaches to voltage or to current.

Subtract two phase voltages into a line voltage

The two phase voltages V_a and V_b turn together, and their difference, the gold arrow, is the line voltage. Change the separation angle and find the angle at which the difference is exactly √3 times as long.

The three tabs change the separation angle between the two phase voltages. The faint dashed line is the difference drawn from the tip of V_b to the tip of V_a, and the gold arrow is that difference moved to the origin. The two phasors turn together, so the difference turns too, but its length is set by the separation angle alone.

Magnitude of the difference, relative to phase

|V_ab| ≈ 1.73 · V_p

Exactly 120 degrees, so the difference is √3 times as long and leads V_a by 30 degrees. This difference is the line voltage V_ab.

√3 · Y line voltage

√3 · balanced ·

Y connection · √3 on the voltage

In a Y each line connects straight to one end of one phase's coil, so the line current is the phase current. The voltage between two lines, the line voltage, is the difference of two phase voltages, though, so it is √3 times as large, the difference of two phasors 120 degrees apart. The 380V line voltage of Korea's low-voltage distribution is exactly √3 times its 220V phase voltage.

Δ connection · √3 on the current

In a delta it is the other way round. Each phase's coil sits directly between two lines, so the line voltage is the phase voltage; instead, the current leaving on one line is the difference of the two phase currents meeting at that terminal, so the same √3 attaches to the current. The line current then lags the phase current by 30 degrees. Which side forms the difference decides where √3 attaches.

One geometry, two faces

In the end √3 comes from one fact. Subtract two phasors of equal size 120 degrees apart and the difference is √3 times as long; a Y shares a common point, the neutral, so the voltages form the difference, while a delta shares common terminals, so the currents form the difference. That is why √3 shows up in voltage for a Y and in current for a delta.

ObserveV_ab = V_a − V_b

The line voltage is the difference of two phase voltages.

Choose|V_ab| = 2 V_p sin(?)

Difference magnitude = 2 V_p sin(separation/2).

Fill in|V_ab| = 2 V_p · ? = √3 V_p

sin 60 degrees = √3/2, so √3 V_p.

On your ownΔ: |I_line| = ?

In a delta the same √3 attaches to the current.

Back to the first screen

The gold difference arrow grew to √3 times the phase voltage when the separation angle was exactly 120 degrees, and that difference is the line voltage of a Y. In a delta the same subtraction happens in the currents, making the line current √3 times the phase current. What to remember is not the number √3 but one picture: subtract two phasors 120 degrees apart and you get √3 times the length.

The line voltage is the difference of two phase voltages. The difference of two equal phasors 120 degrees apart is √3 times as long and leads by 30 degrees. Y connection: line voltage = √3 · phase voltage, line current = phase current. Δ connection: line current = √3 · phase current, line voltage = phase voltage. √3 always attaches to the side that forms the difference.

The next step

With line and phase quantities in hand, the next unit (PW-A3) works out balanced three-phase power. Adding up the power of one phase three times gives P = 3 V_p I_p cosφ, and rewriting it in line quantities gives the widely used P = √3 V_L I_L cosφ. The √3 learned here goes right into the middle of the power formula.