Enter values and press Calculate.
In an AC circuit, voltage and current are not just magnitudes — they also have a phase relative to each other, caused by inductors and capacitors. Representing a sinusoidal signal as a phasor (a complex number) captures both magnitude and phase in one compact quantity, letting engineers use simple algebra (instead of calculus/trigonometric identities) to add, subtract, multiply and divide AC quantities like impedance, voltage, and current.
| Operation | Best form | Rule |
|---|---|---|
| Addition/subtraction | Rectangular | Add/subtract real and imaginary parts separately |
| Multiplication | Polar | Multiply magnitudes, add angles |
| Division | Polar | Divide magnitudes, subtract angles |
| Rectangular→Polar | — | r=√(a²+b²), θ=atan2(b,a) |
| Polar→Rectangular | — | a=r·cosθ, b=r·sinθ |
r=√(a²+b²) for the magnitude, and θ=atan2(b,a) for the angle (using the two-argument arctangent to get the correct quadrant). For example, 3+4i becomes 5∠53.13°.
a=r×cos(θ) for the real part, and b=r×sin(θ) for the imaginary part. For example, 10∠30° becomes 8.66+5i.
Adding complex numbers in rectangular form is just adding the real and imaginary parts separately — simple and direct. Multiplying in rectangular form requires FOIL expansion and simplifying i²=-1, which gets messy; polar form reduces multiplication to just multiplying magnitudes and adding angles, which is far simpler.
A phasor is a complex number representing a sinusoidal signal's magnitude and phase angle at a single frequency, letting you use algebra instead of calculus/trigonometry to analyze AC circuits — voltage, current, and impedance are all commonly represented as phasors.
Add them in rectangular form: Z_total = (R1+R2) + (X1+X2)i, where R is resistance and X is reactance (positive for inductive, negative for capacitive).
Convert both impedances to polar form, then use Z_parallel = (Z1×Z2)/(Z1+Z2) — the multiplication is easy in polar form, but the addition in the denominator still needs rectangular form, so parallel impedance combination typically requires converting back and forth between forms.
It represents the phase shift of that sinusoidal quantity relative to a reference (often the source voltage) — a positive angle means the quantity leads the reference in time, and a negative angle means it lags.
Since i = 1∠90°, dividing by i is equivalent to subtracting 90° from the angle (per the polar division rule), which is exactly the -90° phase shift seen in capacitive reactance calculations (Xc = 1/(jωC) = -j/(ωC)).
It isn't fundamentally different mathematically — this calculator is specifically designed around the rectangular/polar workflow and terminology used in AC circuit analysis (phasors, impedance), with a live phasor diagram to visualize the result, which general scientific calculators typically don't provide.
Yes — three-phase quantities are commonly represented as phasors 120° apart, and this calculator's add/subtract and multiply/divide operations directly support the phasor arithmetic used in unbalanced three-phase analysis and symmetrical components.
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