PW-A4 · Three-phase AC

Rotating phasors: why the instantaneous power is constant

In a balanced three-phase set the instantaneous power does not swing with time but stays constant, because each phase pulsation (2ω) is 120 degrees apart and the three sum to zero (this is A1). Against single-phase, which pulses to zero every cycle, build the intuition for why three-phase makes smooth, constant torque.

Add phases and the pulsation vanishes

The phasors rotate together. The curve on the right is the instantaneous power summed over the included phases. Raise the count 1 → 2 → 3. Single-phase pulses down to zero, but with all three the curve goes flat.

Tap to switch how many phases are summed.

Ripple of the summed instantaneous power

0%

The three pulsations 120 degrees apart sum to zero. The curve is perfectly flat. Ripple 0%.

Constant · steady ·

Single-phase pulses to zero

One phase of instantaneous power is voltage times current, so it pulses at 2ω: p = V_p I_p cosφ + V_p I_p cos(2ωt − φ). The first term is the steady average, the second a pulsation at twice the frequency. A single-phase motor takes this pulsation straight into its torque, which is why it judders.

The three pulsations erase each other

The three pulsation terms cos(2ωt − φ), cos(2ωt − φ − 240°) and cos(2ωt − φ − 480°) are 120 degrees apart in 2ωt. Three equal quantities 120 degrees apart sum to zero — the very fact from A1. The pulsation cancels and only the three average terms remain.

Observep_a = V_p I_p cosφ + V_p I_p cos(2ωt − φ)

Phase power = average + 2ω pulsation.

Choose2ωt · ?

The three pulsations are 120 degrees apart.

Fill incos + cos + cos = ?

Three cosines 120 degrees apart sum to 0 (A1).

On your ownp(t) = ? = √3 V_L I_L cosφ

What remains is 3 V_p I_p cosφ = √3 V_L I_L cosφ.

Constant power = smooth torque

What remains is the sum of the three averages, 3 V_p I_p cosφ, the same value as the √3 V_L I_L cosφ from A3 — except now it is the value at every instant, not the average. Constant instantaneous power means constant torque, and this is the root reason a three-phase motor runs without the vibration of a single-phase one.

Back to the first screen

The curve went flat at three phases because the three pulsations, 120 degrees apart, erased each other. That cancellation is the same fact as the three phasors summing to zero in A1, and the flat height is the √3 V_L I_L cosφ of A3. Three rotating phasors produce the constant power that single-phase never has.

The three-phase instantaneous power is constant in time. Each phase pulsation (2ω) is 120 degrees apart so the sum is zero (A1), and only the average remains, holding p(t) = 3 V_p I_p cosφ = √3 V_L I_L cosφ at every instant. Single-phase pulses to zero every cycle, but three-phase gives constant power → constant torque.

The next step

So far everything was perfectly balanced, so the sum was always zero. Real systems drift off balance. The next unit (PW-A5) looks at what happens when balance breaks: the three phasors no longer sum to zero, so current flows in the Y neutral, and the once-constant instantaneous power begins to pulse again.