Library / Foundations: Maths / FMA-42
Foundations: Maths
Argument (phase) of a complex number
Angle of a complex number a + jb from the positive real axis, found with the two-argument arctangent so that the quadrant is correct.
- Ref
- FMA-42
- Rev
- v1
- Field
- Foundations: Maths
- Result
- θ [°]
The argument θ of a complex number is its angle from the positive real axis, measured anticlockwise. The two-argument arctangent atan2(b, a) uses the signs of both parts to put the angle in the right quadrant, which the ordinary arctangent of b/a cannot do. In a spreadsheet the arguments are reversed: atan2(y, x) in mathematics is ATAN2(x, y) in Excel. The phase of zero is undefined.
Angle of a complex number a + jb from the positive real axis, found with the two-argument arctangent so that the quadrant is correct. For a phasor it is the phase.
Calculator
- Modulus |z|5
- Argument in radians0.9273 rad
- Angle given by the plain arctangent of b/a (can be wrong by 180°)53.13 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The number 3 + j4 has phase 53.13°. A phasor with a positive real and imaginary part lies in the first quadrant, so its phase is between 0° and 90°.
- Reversing both signs, −3 − j4, points the other way: the phase becomes −126.9°, 180° less than before, because the arrow now points the opposite way. The plain arctangent of b/a would still return 53.13° and be wrong by 180°.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Imaginary part (coefficient of j) | −20 to 20 | −81.47 | 81.47 | 162.9 | +0.8 | +4.86 ° |
| a Real part | −20 to 20 | 11.31 | 168.7 | 157.4 | +0.8 | −6.66 ° |
Sweep of Imaginary part (coefficient of j)
| Imaginary part (coefficient of j) | Argument (°) |
|---|---|
| −20 | −81.47 |
| −15 | −78.69 |
| −10 | −73.3 |
| −5 | −59.04 |
| 0 | 0 |
| 5 | 59.04 |
| 10 | 73.3 |
| 15 | 78.69 |
| 20 | 81.47 |
Sweep one variable
Variables and how each one contributes
Argument
What the equation gives you. Everything else feeds it.
Real part
The real part a of the complex number a + jb, a plain number. It is the horizontal coordinate on the complex plane. Moves the angle towards 0° as it grows and towards 90° as it shrinks, because a larger real part pulls the arrow towards the real axis. Raising it by 1 (to 4) moves the result by −15.3 %, from 53.13 to 45 °.
Imaginary part (coefficient of j)
The imaginary part b of the complex number a + jb, the number multiplying j (where j² = −1). It is the vertical coordinate on the complex plane. Raises the angle, because a larger imaginary part tilts the arrow upward. Raising it by 1 (to 5) moves the result by +11.1 %, from 53.13 to 59.04 °.
Worked example with the values above
- Use both signs to find the quadrant: a = 3, b = 4.
- θ = atan2(4, 3) = 0.9273 rad.
- Convert to degrees: θ = 53.13°.
Assumptions and limits
- The real and imaginary parts are in the same units.
- A single number: for a sinusoid in time, the phase is relative to the chosen reference.
Common mistakes
- Using ATAN(b/a) in a spreadsheet and losing the quadrant.
- Swapping the argument order in ATAN2 between mathematics and Excel.
Statistics and uncertainty
The angle is most sensitive to errors when |z| is small, because a fixed error in a or b then tilts the arrow by a larger angle: the error is about u/|z| radians for an error u in either part.
Where it is used
- Phase of an AC phasor from its real and imaginary parts.
- Phase of a frequency response, for example the phase margin in control design.
Maths you need
- Arctangent: the angle whose tangent is a given ratio
- Complex plane: real part across, imaginary part up