Complex Numbers to Exponential Phasor Form Calculator

Calculator

Real part a

Example: 3

Imaginary part b

Use negative values for -j terms.

Decimal precision

Allowed range: 0 to 12.

Angle mode

Power n for zⁿ

Uses De Moivre form.

Optional unit label

Useful for circuit phasors.

Example Data Table

Real a	Imaginary b	Magnitude r	Angle θ	Exponential Form	Phasor Form
3	4	5	53.130102°	5e^j0.927295	5 ∠ 53.130102°
5	-12	13	-67.380135°	13e^j-1.176005	13 ∠ -67.380135°
-7	24	25	106.260205°	25e^j1.854590	25 ∠ 106.260205°
-8	-15	17	-118.072487°	17e^j-2.060754	17 ∠ -118.072487°

Formula Used

For a complex number:

z = a + jb

The magnitude is:

r = √(a² + b²)

The angle is:

θ = atan2(b, a)

The exponential phasor form is:

z = re^jθ = r ∠ θ

The conjugate is:

z* = a - jb = re^-jθ

The reciprocal is:

1 / z = (a - jb) / (a² + b²)

How to Use This Calculator

Enter the real part of the complex number.
Enter the imaginary part without the j symbol.
Choose the decimal precision for rounded output.
Select the angle style you prefer.
Enter a power value if you want z raised to n.
Add an optional unit label for engineering work.
Press Calculate to see the result above the form.
Use CSV or PDF buttons to export the current result.

Complex Numbers and Phasor Conversion Guide

What This Tool Does

This calculator changes a complex number from rectangular form into exponential phasor form. Rectangular form uses a real part and an imaginary part. Exponential form uses a magnitude and an angle. Both forms describe the same point on the complex plane. The difference is only the way the value is written.

Why Phasor Form Matters

Phasor form is very useful in engineering. It is common in alternating current circuits. It is also used in signal analysis. A phasor shows size and direction at the same time. The magnitude gives the size. The angle gives the phase. This makes multiplication, division, and powers easier.

Understanding the Input

A complex number is written as a plus jb. The letter a is the real part. The letter b is the imaginary part. The symbol j is often used in electrical work. Mathematicians may use i instead. The meaning is the same. Enter only the number for b. Do not type the j symbol in the input box.

Magnitude and Angle

The magnitude is the distance from the origin. It is found with the Pythagorean rule. The angle is found with atan2. This function is better than a simple tangent inverse. It detects the correct quadrant. That prevents many sign errors. The calculator can show a principal angle. It can also show a positive angle.

Exponential Form

Exponential form is based on Euler’s identity. It connects sine, cosine, and exponential notation. The expression re raised to j theta is compact. It is easy to read in advanced calculations. It also supports clear phase comparison. A larger magnitude means a stronger value. A larger angle means more phase rotation.

Extra Outputs

This tool also calculates the conjugate. The conjugate changes the sign of the imaginary part. It also calculates the reciprocal when possible. Zero has no reciprocal. The power result uses De Moivre’s rule. That rule raises the magnitude to the power. It also multiplies the angle by the power.

Export Options

The CSV button downloads a spreadsheet friendly result. The PDF button downloads a simple report. These options are useful for assignments. They are also useful for lab notes. You can keep records without copying each value by hand. Recalculate after changing inputs. Then export the new result.

FAQs

1. What is exponential phasor form?

It is a way to write a complex number using magnitude and angle. It usually looks like re^jθ or r ∠ θ.

2. What does j mean?

The symbol j means the imaginary unit. It is used instead of i in electrical engineering to avoid confusion with current.

3. What is the magnitude?

The magnitude is the distance of the complex number from the origin. It is calculated as √(a² + b²).

4. What is the phase angle?

The phase angle is the direction of the complex number from the positive real axis. It is measured in degrees or radians.

5. Why use atan2?

atan2 uses both real and imaginary parts. It finds the correct quadrant and avoids common angle sign mistakes.

6. Can the zero complex number have an angle?

No. Zero has no direction on the complex plane. Its magnitude is zero, but its angle is undefined.

7. What is principal angle mode?

Principal angle mode keeps the angle between -180° and 180°. It is common in many math and engineering problems.

8. What is positive angle mode?

Positive angle mode keeps the angle between 0° and 360°. It is helpful when negative phase angles are not desired.

9. What is a complex conjugate?

The conjugate changes a + jb into a - jb. It reflects the value across the real axis.

10. What is the reciprocal of a complex number?

The reciprocal is one divided by the complex number. It is undefined when the complex number is zero.

11. How is z raised to a power?

The calculator uses De Moivre’s rule. It raises the magnitude to the power and multiplies the angle by the power.

12. Can I use this for AC circuits?

Yes. It is useful for voltage, current, impedance, and phase calculations in sinusoidal steady state circuit work.

13. What does the CSV download include?

The CSV file includes the input, magnitude, angle, exponential form, phasor form, conjugate, reciprocal, and power result.

14. What does the PDF download include?

The PDF file gives a simple printable report of the same calculated result and the main formulas used.