ELT 115
Complex Number Conversion

Recall that complex numbers use a plane similar to the X and Y coordinate system.  In the complex system the X-axis is replaced by the Real axis and the Y-axis by the imaginary (j) axis.  The imaginary axis is labeled "j" in the engineering world, while mathematicians usually use "i". This yields a coordinate system as below:

Complex plane

A phasor quantity can be expressed in one of two forms, polar or rectangular.  Polar notation refers to the phasor using its' magnitude (length) and angle (direction).  Rectangular notation refers to the phasor using its' real component and imaginary component.

Consider the phasor below,

In the polar form the phasor would be expressed as having a magnitude of 5 units at an angle of 53 degrees.  In the rectangular form the same phasor would be expressed as having 3 units in the positive real direction and 4 units in the positive imaginary (+j) direction.

Converting between polar and rectangular form is a technique that will be used very frequently when dealing with reactive circuit analysis and you must be equally comfortable converting from polar to rectangular and converting from rectangular to polar.

Polar To Rectangular Conversion

 When converting from polar to rectangular for the equations used are:

P to R Formula
A couple of things to remember:
    1. When entering theta use the value as measured from the positive real axis (0 degrees).  The calculator will then give you the correct sign for the real and the imaginary components.
    2. Be sure that your calculator is set to the angle units you are planning on using.  If your problem is specified in degrees, make sure your calculator is set to degrees, not radians!
    3. Remember to account for metric prefixes such as m, k etc.

Rectangular To Polar Conversion

 When converting from rectangular to polar for the equations used are:

R to P formula

The same reminders apply.  Additionally you must be sure that you specify the angle correctly.  Do not rely on the calculator to give you the correct angle.  The calculator will try and specify an angle between 0 and 90 degrees.  
Real is: and Imaginary is: Angle must be between:
Positive Positive 0 and 90
Negative Positive 90 and 180
Negative Negative 180 and 270 (-90 and -180)
Positive Negative 270 and 360 (0 and -90)

This means you may need to add 180 degrees to determine the angle correctly.

Practice, Practice, Practice

The best way to become proficient at anything is to practice!  You can make up your own problems, pick numbers for magnitude and angle the convert them to real and imaginary components.  Do the same converting the other way.  Pick values for real and imaginary components and convert to magnitude and angle.  You can check your results with the pages below.  The pages use javascript, so you may have to enable it in your browser.

Check Rectangular to Polar Check Polar to Rectangular