The Rectangular to Phasor Calculator is a tool designed to convert complex numbers from rectangular form to phasor form. This conversion involves determining the magnitude (amplitude) and angle of the complex number, providing valuable insights into the characteristics of the signal or system being analyzed.
Formula of Rectangular to Phasor Calculator
The conversion from rectangular to phasor form involves calculating the magnitude (amplitude) and angle of the complex number.
- Rectangular Form:
- A complex number in rectangular form is represented as: Z = x + jy where:
- Z is the complex number
- x is the real part
- y is the imaginary part
- j is the imaginary unit (j² = -1)
- A complex number in rectangular form is represented as: Z = x + jy where:
- Phasor Form:
- A phasor is a complex number expressed in terms of its magnitude (A) and angle (θ). There are two common ways to write a phasor:
- Polar Form: Z = A ∠θ
- A represents the magnitude (amplitude) of the phasor.
- θ (theta) represents the angle of the phasor.
- Exponential Form: Z = Aejθ
- A represents the magnitude.
- e (Euler’s number) = 2.71828…
- j is the imaginary unit.
- θ represents the angle.
- Polar Form: Z = A ∠θ
- A phasor is a complex number expressed in terms of its magnitude (A) and angle (θ). There are two common ways to write a phasor:
- Conversion Formula:
- Magnitude (A): A = √(x² + y²)
- Angle (θ): θ = tan⁻¹(y / x) Note: The inverse tangent function (tan⁻¹) has multiple outputs depending on the quadrant where the original number lies. Consider the signs of x and y to determine the correct angle.
General Terms Table
| Rectangular Form (Z) | Magnitude (A) | Angle (θ) |
|---|---|---|
| 1 + j0 | 1 | 0° |
| 0 + j1 | 1 | 90° |
| -1 + j0 | 1 | 180° |
| 0 – j1 | 1 | -90° |
| 1 + j1 | √2 | 45° |
| 1 – j1 | √2 | -45° |
Note: This table covers a limited set of examples. For more complex rectangular forms, you’ll need to use the conversion formulas or a rectangular to phasor calculator.
Example of Rectangular to Phasor Calculator
Suppose we have a complex number Z = 3 + 4j in rectangular form. Using the Rectangular to Phasor Calculator, we can determine its magnitude (A) and angle (θ).
- Magnitude (A): A = √(3² + 4²) = √(9 + 16) = √25 = 5
- Angle (θ): θ = tan⁻¹(4 / 3) ≈ 53.13°
So, the phasor representation of Z is A = 5 ∠53.13°.
Most Common FAQs
Converting to phasor form simplifies the analysis of AC circuits and signals, making it easier to understand their behavior in terms of magnitude and phase.
The quadrant determines the sign of the angle, which is crucial for accurately representing the phase relationship of the complex number.