fra0105 Summary of Complex Numbers for Electronics Technology

Strengthening Skills   Building Confidence

Converting Complex Numbers
Rectangular to Polar

Here are  the  formal expression for converting rectangular to polar coordinates:

 c = √ a2 + b2

For the sake of this lesson, lets assume that the two forms of complex numbers are expressed this way:
Rectangular:  a +jb
Polar:  cÐq

The objective is to express a +jb in in the form of cÐq.

Recall how the same complex number can be represented two different ways:

 Citing a complex number with rectangular coordinates. Citing the same complex number with polar coordinates.

 Overlay the two coordinate system to show that there is a right triangle made up of all the important parts of the complex number. Converting from rectangular to polar coordinates is a matter of solving the triangle for the hypotenuse (c) and angle q. a = the length of the base of the triangle b = the height of the triangle c = the hypotenuse of the triangle q = the angle

From the Pythagorean theorem:

 c = √ a2 + b2

That gives us the length component of the polar form in terms of the rectangular components, a and b.

From basic trigonometry:

That provides the angle component of the polar form, also in terms of the rectangular components.

Converting from rectangular to polar coordinates for any complex number is a matter of solving those two equations.

Example

Convert 13.5 +j7.5 to polar form

Step 1. Identify the components

a = 13.5
b = 7.5

Step 2. Solve for the polar length

Step 3. Solve for the polar angle:

Step 4. Put it all together

13.5 +j7.5 = 15.4Ð29°

Learn From More Examples

1.  10 +j10 = 14.1Ð45°
2.  10 -j10 = 14.1Ð-45°
3.  1 + j3 = 3.16Ð71.6°
4. 2 + j20 = 20.1Ð84.3
5.  20 +j2 = 20.1Ð5.7°
6. 10 = 10Ð0°
7.  j10 = 10Ð90°
8.  3 +j5 = 5.8Ð59°
9.  5 +j2 = 5.4Ð21.8°
10.  3 -j10 = 10.4Ð-73.3°