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PW-A4 · Three-phase AC

Rotating phasors: why the instantaneous power is constant

In balanced three-phase the instantaneous power does not wobble with time but stays constant, because the pulsation of each phase (2ω) is offset by 120° and adds to 0 (=A1). Set against single phase, which pulsates down to 0 every cycle, get a feel for why three-phase produces smooth, constant torque.

Add phases and the pulsation vanishes

The phasors on the left turn together, and the curve on the right is the sum of the instantaneous power of the phases added so far. Increase the number of phases with the tabs, 1 → 2 → 3, and the curve that dipped to 0 for a single phase goes flat once three are gathered.

The tabs change how many phases are added. The figure is drawn for power factor 1, with voltage and current in phase, and the white dot is the sum of instantaneous power at this moment.
Ripple of the summed instantaneous power
0%
The three pulsations are offset by 120° and add to 0, and the curve is perfectly flat. Pulsation 0%.
Constant · steady ·

Single phase pulses down to 0

The instantaneous power of one phase is the product of voltage and current, so it pulsates at 2ω. In p = V_p I_p cosφ + V_p I_p cos(2ωt − φ), the first term is the constant average and the second is a pulsation at double frequency. With power factor 1, as in the figure, the instantaneous power drops to 0 and comes back up; with a power factor below 1 it dips below 0 twice in each cycle, and power briefly flows back toward the source. This is why the torque of a single-phase motor takes on the pulsation as it is, vibrating and humming at double frequency.

The three pulsations erase each other

The pulsating parts of the three phases, cos(2ωt − φ), cos(2ωt − φ − 240°) and cos(2ωt − φ − 480°), are offset by 120° from one another in terms of 2ωt. Three quantities of equal size offset by 120° add to 0, the very fact seen in A1, so the pulsation disappears and only the three average parts remain.

Observepa = Vp Ip cosφ + Vp Ip cos(2ωt − φ)
Phase power = average + 2ω pulsation.
Choose2ωt · ?
The phase spacing between the three pulsations is 120°.
Fill incos + cos + cos = ?
Three cosines 120 degrees apart sum to 0 (A1).
On your ownp(t) = ? = √3 VL IL cosφ
What remains is 3 Vp Ip cosφ = √3 VL IL cosφ.

Constant power = smooth torque

What remains is the sum of the three averages, 3 V_p I_p cosφ, which is the same value as the √3 V_L I_L cosφ of A3. Now, though, it is not an average but the value at every instant. Constant instantaneous power means constant torque, and that is the root reason three-phase motors run without the vibration of single-phase ones.

Back to the first screen

When three phases were added the curve went flat because the three pulsations, offset by 120°, cancelled one another. That cancellation is the same fact as the sum of three phasors being 0 in A1, and the height of the flat line is the √3 V_L I_L cosφ of A3. The three phasors turning together deliver a constant power that a single phase does not have.

Three-phase instantaneous power is constant regardless of time. The pulsating part of each phase (2ω) is offset by 120° and sums to 0 (A1), leaving only the average parts, so p(t) = 3 Vp Ip cosφ = √3 VL IL cosφ holds at every instant. A single phase pulsates to 0 every cycle, but three-phase means constant power → constant torque.
The next step

Up to now the balance was perfect, so the sum was always 0. Real systems drift a little out of balance, so the next unit (PW-A5) works out what happens when the balance breaks. The sum of the three phasors is no longer 0, current flows in the neutral of a Y, and the instantaneous power that was constant starts pulsating again.