fra0105 Summary of Complex Numbers for Electronics Technology

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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:  cq

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

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 = 15.429

 Learn From More Examples

  1.  10 +j10 = 14.145
  2.  10 -j10 = 14.1-45
  3.  1 + j3 = 3.1671.6
  4. 2 + j20 = 20.184.3
  5.  20 +j2 = 20.15.7
  6. 10 = 100
  7.  j10 = 1090
  8.  3 +j5 = 5.859
  9.  5 +j2 = 5.421.8
  10.  3 -j10 = 10.4-73.3

David L. Heiserman, Editor

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Revised: June 06, 2015