Converting Complex Numbers to Polar Form: A Mathematical Explanation

[Paddy]

Firstly I do apologise, because this question is got more to do with the mathematical side of Electronic Engineering, because my mathematical classification is not that good I don't know where I would put this question on the mathematics section, if any of the moderators or whoever can, wants to move it there, I do apologise.

I am having a bit of problem with my calculator, unlike many other people in my class, my calculator can't exchange complex numbers to polar form, so I have to do it by using some mathematics.

I know how to change from polar form back to complex notation, so for example imagine a voltage of 6.08 |51.8*.

I believe that 6.08cos51.8 will get you the real component and 6.08sin51.8 is J, so in J-notation ==> 6.08 |51.8* = 3.76 + J4.78

I haven't made up these difficult numbers, but I have taken them from a class example, I suppose the method is correct because that was the answer checked by the teacher.


However I do not know how to exchange a number back to polar form, I think my best option would be buying a better calculator, but I would appreciate if someone could show me by mathematical terms, Thank You.

 

[cristo]

Let's consider a general case first, and consider the complex number x+yj, where x and y are real numbers. Now, we want to express this in polar form, which is R(cost+jsint). Here, R is the modulus of the complex number (I'm not sure whether you're familiar with this) but it is simply defined as √(x²+y²).

So, to move onto your example, we want to write 3.76+4.78j in the form R(cost+jsint). First calculate the modulus of the complex number; R=√(3.76²+4.78²)=6.08. Now, we factor this out of the complex number, to give 6.08(0.6181+0.7026j). This is nearly in polar form. The final step is to take cos^-1(0.6181) and sin^-1(0.7026). You will find that these are both 51.8, and so we have the number expressed as 6.08(cos(51.8)+jsin(51.8)), which has real part 6.08cos(51.8) and imaginary part 6.08sin(51.8), as you state in your post.

Hope this helps!

 

[berkeman]

No worries, I googled polar to rectangular conversion complex numbers, and got lots of good hits. Here's the first one:

http://www.allaboutcircuits.com/vol_2/chpt_2/5.html

Basically just think of the trigonometry involved, with an x-y graph where the +x axis is the Real axis, and the +y axis is the Imaginary axis. We use the prefex j (or i in non-EE areas) to denote the Imaginary component in the rectangular form of complex numbers, or as a prefex to the angle (in radians) in complex exponential form (which is another notation for the polar form).

If you have any other questions about this topic after reading the tutorial, please feel free to repost in this thread.

 

[berkeman]

Dang, cristo beats me to the punch again!

 

[Paddy]

Thank you So, So much.

I was along the right lines, but the relationship between the imaginary and real numbers confused me a bit.

Starting to make sense and I am starting to get the idea. Thank you.