1. What is Phase Calculation?

Phase calculation determines the angle of a complex number in the complex plane using the arctangent function with two arguments (atan2). It provides the angle between the positive real axis and the line connecting the origin to the point (Re, Im).

2. How Does the Calculator Work?

The calculator uses the phase calculation formula:

Where:

• \( Im \) — Imaginary part of the complex number
• \( Re \) — Real part of the complex number

Explanation: The atan2 function computes the principal value of the argument function applied to the complex number Re + iIm, returning the result in radians between -π and π.

3. Importance of Phase Calculation

Details: Phase calculation is essential in signal processing, electrical engineering, and physics for analyzing complex numbers, representing waveforms, and understanding phase relationships in oscillatory systems.

4. Using the Calculator

Tips: Enter both real and imaginary parts of the complex number. The calculator will compute the phase angle in both radians and degrees for convenience.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between atan and atan2?
A: atan2 takes two arguments (y, x) and returns the angle in the correct quadrant, while atan(y/x) can only return values between -π/2 and π/2.

Q2: What is the range of phase values?
A: The phase angle ranges from -π to π radians (-180° to 180°) using the atan2 function.

Q3: How is phase used in practical applications?
A: Phase is crucial in analyzing AC circuits, signal processing, control systems, and representing complex numbers in polar form.

Q4: What happens when both Re and Im are zero?
A: The phase is undefined (singularity) when both real and imaginary parts are zero, as this represents the origin point.

Q5: Can phase values exceed ±180 degrees?
A: No, the atan2 function normalizes the phase to be within the principal value range of -π to π radians.