Convert from Phasor to Rectangular Form
Use magnitude and phase angle to find the real and imaginary components of a complex value.
Formula Used
A phasor in polar form is written as M ∠ θ. Rectangular form is written as a + jb.
z = M ∠ θ = M(cos θ + j sin θ)
a = M cos θ
b = M sin θ
z = a + jb
When θ is entered in degrees, the calculator first uses θ radians = θ degrees × π / 180.
How to Use This Calculator
- Enter a label if you want a named result.
- Type the phasor magnitude in the magnitude field.
- Enter the phase angle and select degrees or radians.
- Choose the angle direction. Use the entered option for signed angles.
- Select j or i for the imaginary part.
- Set decimal precision and press the convert button.
- Read the rectangular form above the input form.
- Use the CSV or print option to save the result.
Example Data Table
These examples show common phasor conversions. They use degrees and j notation.
| Phasor | Real part | Imaginary part | Rectangular form | Quadrant |
|---|---|---|---|---|
| 10 ∠ 30° | 8.6603 | 5 | 8.6603 + 5j | Quadrant I |
| 12 ∠ -45° | 8.4853 | -8.4853 | 8.4853 - 8.4853j | Quadrant IV |
| 5 ∠ 180° | -5 | 0 | -5 + 0j | Negative real axis |
| 20 ∠ 120° | -10 | 17.3205 | -10 + 17.3205j | Quadrant II |
| 7 ∠ 270° | 0 | -7 | 0 - 7j | Negative imaginary axis |
Understanding Phasor to Rectangular Conversion
Why Phasors Matter
Phasors make alternating signals easier to handle. They store a length and an angle. Rectangular form shows the same value with real and imaginary parts. This calculator connects both views. It helps when you need a fast check during circuit work, signal analysis, or complex number practice.
What the Two Forms Mean
A phasor is written as M ∠ θ. The value M is the magnitude. The angle θ shows rotation from the positive real axis. Rectangular form is written as a + jb. The real part is a. The imaginary part is b. Both parts may be positive or negative. Their signs depend on the angle quadrant.
Core Trigonometry
The conversion uses simple trigonometry. The real component equals M times cosine of θ. The imaginary component equals M times sine of θ. Degrees are changed to radians before calculation, because most programming functions use radians. The tool also supports radian input. This makes it useful for engineering notes and textbook examples.
Quadrant Awareness
Quadrant checks are important. An angle of 30 degrees gives a positive real part and a positive imaginary part. An angle of 150 degrees gives a negative real part and a positive imaginary part. An angle of -45 degrees gives a positive real part and a negative imaginary part. The calculator reports the normalized angle, so signs are easier to verify.
Where Rectangular Form Helps
Rectangular form is useful in addition, subtraction, and network equations. Two complex values can be added by combining real parts and imaginary parts. That is why many impedance, admittance, voltage, and current calculations use rectangular form. Polar or phasor form is often easier for multiplication and division. Both forms are useful, but each serves a different step.
Advanced Options
The advanced options improve control. You can choose degrees or radians. You can mark a signal as leading or lagging. You can select the imaginary symbol. You can set decimal precision for neat answers. You can also view the conjugate and a magnitude check. These extra values help detect entry mistakes.
Accuracy Tips
For best accuracy, enter the original magnitude without rounding. Use the correct angle reference. A lagging current is often shown with a negative angle. A leading signal is often shown with a positive angle. When uncertain, keep the angle as entered and compare the quadrant.
Practical Use
This calculator is designed for clear learning and practical checking. It does not replace a full circuit solver. It focuses on one reliable task. It turns magnitude and phase into real and imaginary components. The result can then be copied into formulas, spreadsheets, reports, or lab notes.
Unit Consistency
Use the result with care when units matter. A voltage phasor keeps voltage units. A current phasor keeps current units. The rectangular parts share the same unit as the magnitude. If the magnitude is RMS, both components are RMS based. If the magnitude is peak, both components are peak based. Consistent units keep the calculation meaningful.
This is vital in mixed homework and design checks today.
FAQs
What is phasor to rectangular conversion?
It changes magnitude and angle into real and imaginary parts. The output is normally written as a + jb or a + ib.
What formula does this calculator use?
It uses a = M cos θ and b = M sin θ. The final rectangular value is a + jb.
Can I enter the angle in radians?
Yes. Select radians in the angle unit menu. The calculator then uses the entered value directly in the trigonometric functions.
What does a negative imaginary part mean?
It means the phasor lies below the real axis. This often happens with negative angles or third and fourth quadrant positions.
Should I use i or j?
Use j for electrical engineering work. Use i for general mathematics. Both represent the same imaginary unit.
What is the leading option?
The leading option forces the angle to be positive. It is useful when a signal leads a reference waveform.
What is the lagging option?
The lagging option forces the angle to be negative. It is useful when current or voltage lags a reference signal.
Why is a magnitude check included?
It recomputes the magnitude from the real and imaginary parts. This confirms that the rectangular result matches the original phasor.
Why does the calculator normalize angles?
Normalized angles make quadrant checks easier. For example, 390 degrees becomes 30 degrees in the same direction.
Can I use this for impedance?
Yes. Enter impedance magnitude and phase angle. The real result is resistance, and the imaginary result is reactance.
Can I save the result?
Yes. After conversion, use the CSV button or print option. The print option can be saved as a PDF by your browser.