Phasor Sum and Difference Calculator
Add or subtract up to three phasors given as magnitude and angle, such as voltages or currents at different phase angles.
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Estimate only. Not engineering or safety advice. Check the result against the applicable standard and the equipment data, and have a competent engineer review it before it is relied on. Terms of use.
Open in the calculator with these values
About this calculator
This page calculates resultant magnitude, resultant angle, resultant real part and 3 more results from phasor 1 magnitude, phasor 1 angle, phasor 2 is, phasor 2 magnitude and 4 more inputs. Change any value or its unit and the result updates. The steps below show the working for the default values, so you can check each line against your own method.
Limits of this calculator
Phasors add as vectors, not as plain numbers: 10 ∠0° plus 10 ∠90° is 14.14, not 20. Use this for the sum of branch currents at a node or of voltages around a loop when each is known by magnitude and angle. To work from rectangular values or to multiply or divide, use the complex number calculator; for a voltage, current and impedance relation use AC Ohm's law.
Cross-check: 36 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review.
Verification status: Engineering reference
Engineering reference. Formula and method documented and tested against separately coded reference calculations. Not yet Verified: the evidence listed on this page is still needed.
What this calculator checks
- The arithmetic of the published formula for the values you entered, in the units shown.
What remains unverified
- That the inputs suit your application. Take ratings, resistances and constants from the manufacturer data sheet or project specification.
Verification register entry
| Calculation author | Rahul Sharma (author) |
|---|---|
| Technical review | No independent technical review has been carried out |
| Individual professional approval | Not recorded |
| Formula reference | not recorded |
| Standard and edition | Not applicable: Pure mathematics or statistics; no electrical standard governs the method |
| Standard review state | Not applicable |
| Scope-screening note | not recorded |
| Cross-check test cases | 36 reference vectors (tests/reference_vectors.json) |
| Review date | not recorded |
| Next review date | not recorded |
| Status | Engineering reference |
Needed before this calculator can be marked Verified: reviewer name, qualification and review date; exact formula reference; formula source (textbook or standard); next review date.
Formula
Worked example
| Input | Value |
|---|---|
| Phasor 1 magnitude | 10 |
| Phasor 1 angle (degrees) | 30 |
| Phasor 2 is | added (+) |
| Phasor 2 magnitude | 5 |
| Phasor 2 angle (degrees) | -45 |
| Phasor 3 is | added (+) |
| Phasor 3 magnitude | 0 |
| Phasor 3 angle (degrees) | 0 |
- Phasor 1: + 10 ∠30° = 8.660254 + j5
- Phasor 2: + 5 ∠-45° = 3.535534 − j3.535534
- Sum = 12.195788 + j1.464466
- |R| = √(12.1958² + 1.4645²) = 12.2834 θ = atan2(1.4645, 12.1958) = 6.8473°
- Resultant = 12.2834 ∠6.847°
Assumptions
- Angles are in degrees, counter-clockwise from the reference axis; the resultant angle is given from −180° to 180° (atan2)
- All phasors have the same frequency and the same kind of value (all RMS or all peak)
- Magnitudes are plain numbers: enter volts, amps or any one quantity in one consistent unit
- Phasor 1 is always added; a magnitude of 0 leaves a phasor out
Phasors add as vectors, not as plain numbers: 10 ∠0° plus 10 ∠90° is 14.14, not 20. Use this for the sum of branch currents at a node or of voltages around a loop when each is known by magnitude and angle. To work from rectangular values or to multiply or divide, use the complex number calculator; for a voltage, current and impedance relation use AC Ohm's law.
Inputs and results
| Input | Unit | Allowed or shown | Note |
|---|---|---|---|
| Phasor 1 magnitude | (number) | 0 or more | RMS or peak value, as long as all three phasors use the same one. Volts, amps or any consistent quantity. |
| Phasor 1 angle (degrees) | (number) | any number | Measured from the reference axis, counter-clockwise positive. A lagging quantity has a negative angle relative to the reference. |
| Phasor 2 is | (choice) | added (+); subtracted (−) | |
| Phasor 2 magnitude | (number) | 0 or more | Enter 0 to leave phasor 2 out. |
| Phasor 2 angle (degrees) | (number) | any number | |
| Phasor 3 is | (choice) | added (+); subtracted (−) | |
| Phasor 3 magnitude | (number) | 0 or more | Enter 0 to leave phasor 3 out. |
| Phasor 3 angle (degrees) | (number) | any number |
| Result | Unit | Allowed or shown | Note |
|---|---|---|---|
| Resultant magnitude | (number) | Shown to 6 decimal places | |
| Resultant angle (degrees) | (number) | Shown to 4 decimal places | |
| Resultant real part | (number) | Shown to 6 decimal places | |
| Resultant imaginary part (coefficient of j) | (number) | Shown to 6 decimal places | |
| Resultant angle (radians) | (number) | Shown to 6 decimal places | |
| Sum of the magnitudes (all in phase) | (number) | Shown to 6 decimal places |
How it is calculated
Each input is converted to the base unit of its quantity, the formula above is applied in the order shown in the worked example, and each result is shown in the unit you choose. Results are held at full precision and rounded only for display, except where a step selects a standard size, as the worked example shows. A value outside its allowed range is refused with a message instead of giving a result.
How the result changes with phasor 1 magnitude
| Phasor 1 magnitude | Resultant magnitude | Resultant angle | Resultant real part |
|---|---|---|---|
| 1 | 5.346793 | -34.5921 | 4.401559 |
| 2.5 | 6.1417 | -21.8473 | 5.700597 |
| 5 | 7.933533 | -7.5 | 7.865661 |
| 10 | 12.2834 | 6.8473 | 12.195788 |
| 20 | 21.834922 | 17.2212 | 20.856042 |
| 40 | 41.575565 | 23.3292 | 38.17655 |
| 100 | 101.409166 | 27.2702 | 90.138074 |
Other inputs are the same as in the worked example above.
Standards and further reading
Complex numbers, rectangular and polar form and angle conversion are standard mathematics, not the subject of a standard. Standards apply when the result feeds an electrical design, and they set the method there.
Frequently asked questions
What do I need to enter?
Enter phasor 1 magnitude, phasor 1 angle (degrees), phasor 2 is, phasor 2 magnitude, phasor 2 angle (degrees), phasor 3 is, phasor 3 magnitude and phasor 3 angle (degrees). Change the unit of a value with its selector and the result stays the same.
What does this calculator assume?
Angles are in degrees, counter-clockwise from the reference axis; the resultant angle is given from −180° to 180° (atan2). All phasors have the same frequency and the same kind of value (all RMS or all peak). Magnitudes are plain numbers: enter volts, amps or any one quantity in one consistent unit. Phasor 1 is always added; a magnitude of 0 leaves a phasor out.
Which calculators should I use next?
Common next steps are Complex Numbers (Rectangular and Polar), AC Ohm's Law (Phasors), Series RLC Impedance and Three-Phase Power. The related links at the bottom of this page open them.
How accurate are the results?
Every figure comes from the same tested engine as the Elecmetric calculators and is meant for estimating. 36 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review. Check design values against the standard and the equipment data you work to.
Formula guides and reference
Longer explanations with worked examples and tables, written for the same topic.
- Phasor Addition Formulas: Sum and Difference: Phasor addition formulas for engineers: add and subtract voltages or currents given as magnitude and angle
- Complex Number Formulas: Polar and Rectangular: Complex number formulas for engineers: rectangular and polar form, conversion with atan2, addition
- Three-Phase Power Formulas: Three-phase power formulas for kW, kVA, kVAR and line current, star and delta relations
More Engineering Mathematics calculatorsAll engineering calculatorsPhasor Sum and Difference (Polar Inputs) in Elecmetric