Free Online Calculators
A phasor calculator converts sinusoidal signals into complex numbers and performs AC circuit operations like addition, subtraction, multiplication, impedance calculation, and rectangular-to-polar conversion without manual trigonometry. It simplifies AC steady-state analysis by replacing time-domain equations with basic complex number algebra, widely used in electrical engineering and power systems.
In-phase component
Quadrature component (j)
Z = 3 + j4
Every phasor exists in two mathematically equivalent forms. Rectangular form writes the phasor as a + jb, where a is the real (in-phase) component and b is the imaginary (quadrature) component. Polar form writes the same phasor as R∠θ, where R is the magnitude and θ is the phase angle measured counterclockwise from the positive real axis.
Engineers use j — not i — to denote √(−1), because the letter i already represents current in circuit equations, following the convention established in IEEE standards. The mathematics is identical to the mathematical imaginary unit; only the label differs.
Conversion Formulas:
Plain arctan(b/a) fails when a is zero or negative because it cannot distinguish between opposite quadrants. The atan2(b, a) function returns the correct angle in all four quadrants. Always use atan2.
Example:
Z = 3 + j4 → R = √(9+16) = 5, θ = atan2(4,3) = 53.13°
→ Result: 5∠53.13°
Reverse check: a = 5·cos(53.13°) = 3.0, b = 5·sin(53.13°) = 4.0 ✓
The table below shows unit-magnitude phasors (R = 1) at standard angles with their exact rectangular components. These values appear in virtually every AC circuit, three-phase power, and signal processing problem. For any other magnitude, multiply both x and y by R.
| Angle | Rectangular Form |
|---|---|
| 0° | 1 + j0 |
| 30° | 0.8660 + j0.5000 |
| 45° | 0.7071 + j0.7071 |
| 60° | 0.5000 + j0.8660 |
| 90° | 0 + j1 |
| 120° | −0.5000 + j0.8660 |
| 180° | −1 + j0 |
| 270° | 0 − j1 |
In three-phase power systems, the standard phase offsets are 0°, 120°, and 240° (equivalently −120°), which is why 120° is a key reference value.
Write the time-domain signal
v(t) = Vm·cos(ωt + φ)
Convert to phasor domain
V = Vm∠φ
Apply complex algebra
V = I × Z
Convert back to time domain if required
This flow eliminates all differential equations. A derivative in the time domain becomes multiplication by jω in the phasor domain, and an integral becomes division by jω. This is the foundation of all AC steady-state analysis taught in Nilsson & Riedel's Electric Circuits and Hayt & Kemmerly's Engineering Circuit Analysis.
Use rectangular form for addition and subtraction — add or subtract real and imaginary parts separately. Use polar form for multiplication and division — multiply or divide magnitudes and add or subtract angles. This single rule eliminates the most common source of phasor arithmetic errors.
Operations:
Quick Examples:
The standard reference in AC phasor analysis is the cosine function. Convert all signals to cosine before extracting the phasor. If your signal uses sine, subtract 90° from the phase angle first — because sin(θ) = cos(θ − 90°).
Conversion Rules:
Examples:
Impedance Z = R + jX is the AC generalization of DC resistance. R is resistance (real part) and X is reactance (imaginary part). Inductive reactance XL = ωL is positive. Capacitive reactance XC = −1/(ωC) is negative. Phasor Ohm's law V = I × Z applies to all AC circuit elements — solve for any unknown by multiplying or dividing phasors in polar form.
Component Impedances:
Memory rule — ELI the ICE man: In an inductor (L), voltage (E) leads current (I). In a capacitor (C), current (I) leads voltage (E).
Key Formulas:
Apparent power S = V·I* where I* is the complex conjugate of the current phasor. This gives real power P = |V||I|·cos(φ) in watts and reactive power Q = |V||I|·sin(φ) in VAR. A power factor of 1 means the load is purely resistive. Industrial facilities with large inductive loads typically have power factors between 0.7 and 0.85 — utilities penalize this because low power factor increases line current without delivering useful work.
Example:
V = 10∠0° V, Z = 5∠36.87° Ω
→ I = 10∠0° / 5∠36.87° = 2∠−36.87° A
Current lags voltage — load is inductive.
A phasor diagram plots phasors as arrows on the complex plane. The horizontal axis is the real (in-phase) component and the vertical axis is the imaginary (quadrature) component. Each arrow's length represents magnitude and its angle represents phase.
In a series RLC circuit: VR points along the positive real axis (in phase with current), VL points upward at +90°, VC points downward at −90°, and VS is the vector sum of all three. The angle between VS and the real axis is the circuit's power factor angle. At resonance (ω = 1/√LC), VL and VC cancel completely, leaving VS = VR and power factor = 1.
In a balanced three-phase system, three equal-magnitude phasors are separated by 120° each. If phase A is the reference at 0°, phase B sits at −120° and phase C at +120°. Their vector sum equals zero, which is why the neutral wire in a balanced system carries no current.
Three series voltages: V1 = 10∠0°, V2 = 8∠90°, V3 = 6∠−90°. Find VS.
Convert to rectangular: V1 = 10+j0, V2 = 0+j8, V3 = 0−j6
VS = 10 + j(8−6) = 10+j2
Magnitude: |VS| = √(100+4) = 10.20
Angle: θ = atan2(2,10) = 11.31°
Result: VS = 10.20∠11.31° V
R = 4 Ω, XL = 3 Ω, V = 10∠0° V. Find Z and I.
Z = 4+j3 → |Z| = √(16+9) = 5 Ω, θ = atan2(3,4) = 36.87°
→ Z = 5∠36.87° Ω
I = 10∠0° / 5∠36.87° = 2∠−36.87° A
Current lags voltage by 36.87° — load is inductive.
I = 2∠−30° A, Z = 5∠45° Ω
V = I × Z = (2×5)∠(−30°+45°) = 10∠15° V
V = 120∠0° V, I = 10∠−36.87° A
Power Factor = cos(0° − (−36.87°)) = cos(36.87°) = 0.8 (lagging)
Real Power P = 120 × 10 × 0.8 = 960 W
Reactive Power Q = 120 × 10 × sin(36.87°) = 720 VAR
Conversion Formulas:
R = √(a²+b²) | θ = atan2(b,a) | a = R·cos(θ) | b = R·sin(θ) | Z = R·e^(jθ)
Arithmetic Rules:
Add/Subtract → Rectangular | Multiply → Polar (× magnitudes, + angles) | Divide → Polar (÷ magnitudes, − angles)
Impedance Reference:
Resistor = R∠0° | Inductor = ωL∠90° | Capacitor = (1/ωC)∠−90°
V = I × Z | Power Factor = cos(θV − θI) | S = V·I*
Sinusoid Rules:
cos(ωt+φ) → Vm∠φ | sin(ωt+φ) → Vm∠(φ−90°) | ω = 2πf
Key Rules:
ELI the ICE man | Resonance: VL cancels VC | Always atan2, never arctan | Peak vs RMS: Vrms = Vm/√2
Using arctan instead of atan2
gives wrong angles in quadrants II, III, and IV.
Fix: always use atan2(b,a) — Casio: Arg(), TI-84: angle() under MATH CPX.
Adding phasors directly in polar form
is incorrect.
Fix: convert to rectangular first, add, then convert back to polar.
Forgetting to convert sine to cosine
before extraction.
Fix: always subtract 90° from the phase angle of any sine signal before writing its phasor — make it a fixed first step.
Mixing peak and RMS values
in the same problem.
Fix: state at the start whether you are using peak or RMS and never mix the two. Power calculations use RMS (Vrms = Vm/√2).
Dropping frequency after phasor conversion
Fix: keep ω noted separately and use it to compute XL = ωL and XC = 1/ωC before building impedance.
Mixing degrees and radians
mid-calculation.
Fix: set your calculator angle mode before starting every problem. If a result looks wrong, check mode first.
Phasors appear in any domain involving sinusoidal steady-state signals. In RF and telecommunications, signal constellations (QPSK, QAM) are plotted on the complex plane as discrete phasor points where angle encodes phase modulation and magnitude encodes amplitude modulation. In control systems, Bode plots and Nyquist plots track how a system's gain and phase shift vary with frequency — which is fundamentally phasor analysis across a frequency sweep.
In mechanical vibration, displacement, velocity, and acceleration phasors are related by successive 90° phase shifts — velocity leads displacement by 90° and acceleration leads velocity by another 90°. The same complex-number framework appears in acoustics, optics, and quantum mechanics.
In modern power grids, phasor measurement units (PMUs) installed at substations sample voltage and current phasors at 30 to 60 times per second, all GPS-timestamped. Comparing phase angles between distant nodes in real time reveals grid stress and instability far faster than traditional SCADA systems, which report data every 2 to 4 seconds.
Casio fx-991EX / fx-570:
Press Mode → 2 for complex mode. Use Pol(a,b) for rectangular to polar, Rec(R,θ) for polar to rectangular. Shift+Abs for magnitude, Shift+Arg for phase angle. Press Shift → Rect or Polar to toggle display format.
TI-84 Plus / TI-84 Plus CE:
Set angle mode under Mode first. Enter imaginary unit with 2nd → i. Use MATH → CPX for abs() (magnitude), angle() (phase), real(), imag(), and conj(). Type full complex expressions directly (3+4i)/(1−2i) returns results in complex form automatically.