Phasor Addition Formulas: Sum and Difference
Phasor addition formulas for engineers: add and subtract voltages or currents given as magnitude and angle, with worked examples including line voltage.
The formulas
| To find | Formula |
|---|---|
| Rectangular parts of a phasor | x = m cos θ, y = m sin θ |
| Sum of phasors | Re = Σ mₖ cos θₖ, Im = Σ mₖ sin θₖ |
| Difference | Use −1 on the phasor being subtracted: subtract its x and y parts |
| Resultant magnitude | |R| = √(Re² + Im²) |
| Resultant angle | θ = atan2(Im, Re) |
| Two phasors, angle φ apart | |R|² = m₁² + m₂² + 2 m₁ m₂ cos φ |
Phasors add as vectors. Two equal phasors 90° apart sum to 1.414 times one of them, not twice. The magnitudes add arithmetically only when every phasor has the same angle. The phasors must have the same frequency, and all be RMS or all be peak.
Worked example: 10 ∠30° + 5 ∠−45°
- Phasor 1: x = 10 cos 30° = 8.66 and y = 10 sin 30° = 5
- Phasor 2: x = 5 cos(−45°) = 3.536 and y = 5 sin(−45°) = -3.536
- Sum: Re = 12.2 and Im = 1.464, so the sum is 12.2 + j1.464
- |R| = √(12.2² + 1.464²) = 12.28 and θ = atan2(1.464, 12.2) = 6.847°, so R = 12.28 ∠6.847°
Worked example: line voltage from two phase voltages
In a 230 V per phase supply the phase voltages are 230 ∠0° and 230 ∠−120°. The line voltage between them is the difference.
- V12 = 230 ∠0° − 230 ∠−120°, which is 230 − (-115 − j199.2)
- The result is 345 + j199.2, so |V12| = 398.4 V at 30°
- This equals √3 × 230 = 398.4 V, the familiar line voltage of a three-phase system
Frequently asked questions
Why is 10 + 10 not 20 for phasors?
A phasor has direction as well as size. Two phasors add to the plain sum of their magnitudes only if they point the same way. At 90° apart the sum is √2 times one of them.
How do I subtract a phasor?
Reverse it (add 180° to its angle) and add it, or subtract its x and y parts. The calculator has a plus or minus choice for phasors 2 and 3.
What does a result of zero mean?
The phasors cancel. A balanced three-phase set of three equal phasors 120° apart sums to zero, which is why the neutral of a balanced load carries no current.
Can I add RMS and peak phasors together?
No. Convert them to the same basis first. The sum is only meaningful when all the phasors use the same value type and the same frequency.
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