fra0106 Summary of Complex Numbers for Electronics Technology

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Converting Complex Numbers
Polar to Rectangular

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

b = c(sinq)
a = c(cosq)

where:

b is the imaginary component of the rectangular coordinate
a is the real component of the rectangular coordinate

 

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

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

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

Citing a complex number with polar coordinates.

Citing the same complex number with rectangular 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 polar to rectangular coordinates is a matter of solving for the two sides of the triangle in terms of its hypotenuse 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 basic definition for the sine of an angle: b = c(sinq)

That provides the imaginary component of the rectangular form in terms of the two polar components.

From the basic definition for the cosine of an angle: a = c(cosq)

That gives us the real component of the complex number in rectangular form.

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

 

 Example

Convert 15.429  to rectangular form

Step 1. Identify the components

c = 15.4
q = 29

Step 2. Solve for the imaginary component

b = c(sinq) = 15.4(sin 29) = 7.5

Step 3. Solve for the real component

a = c(cosq) = 15.4(cos 29) = 13.5

Step 4. Put it all together

15.429 = 13.5 +j7.5

 Learn From More Examples

  1. 1045 = 7.07 +j7.07 
  2. 10-45 = 7.07 -j7.07
  3. 3.1671.6  = 1 + j3
  4. 20.184.3 = 2 + j20 
  5. 20.15.7 = 20 +j2
  6. 100 = 10
  7. 1090 = j10
  8. 5.859 = 3 +j5
  9. 5.421.8 = 5 +j2
  10. 10.4-73.3 = 3 -j10

David L. Heiserman, Editor

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Revised: June 06, 2015