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:  cÐq
Rectangular:  a +jb

So the objective is to express cÐq 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 15.4Ð29°  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

15.4Ð29° = 13.5 +j7.5

Learn From More Examples

1. 10Ð45° = 7.07 +j7.07
2. 10Ð-45° = 7.07 -j7.07
3. 3.16Ð71.6°  = 1 + j3
4. 20.1Ð84.3 = 2 + j20
5. 20.1Ð5.7° = 20 +j2
6. 10Ð0° = 10
7. 10Ð90° = j10
8. 5.8Ð59° = 3 +j5
9. 5.4Ð21.8° = 5 +j2
10. 10.4Ð-73.3° = 3 -j10