Can someone explain this complex math problem?

[Voltux]

I'm working out an impedance matching problem from a textbook (it is not part of any coursework) and I am trying to figure out how they get the 315 term in the polar coordinates below.

Z = (XC*RL)/(XC+RL)

= (-j333*(1000))/(-j333+1000)
= 315 , -71.58*
= 100 -j300 ohms

I calculated that atan(1000/-333) = 71.58* however I do not understand where they got the 315 from. I get 249 or 499 depending on positive or negative.

I understand that 315 would be the length of the ray, and -71.58 would be the angle as this is capacitive hence -j. e.g. converting from rectangular to polar if I understand correctly.

It was explained as (333*1000)/(sqrt(333^2+1000^2)) and that indeed does give us 315.943 ~316, however, I do not understand how they used the mathematical function to get to this point and I was hoping someone could explain what I'm missing.

 

[Mark44]

I'm working out an impedance matching problem from a textbook (it is not part of any coursework) and I am trying to figure out how they get the 315 term in the polar coordinates below.

Z = (XC*RL)/(XC+RL)

= (-j333*(1000))/(-j333+1000)
= 315 , -71.58*
= 100 -j300 ohms

I calculated that atan(1000/-333) = 71.58* however I do not understand where they got the 315 from. I get 249 or 499 depending on positive or negative.

I understand that 315 would be the length of the ray, and -71.58 would be the angle as this is capacitive hence -j. e.g. converting from rectangular to polar if I understand correctly.

It was explained as (333*1000)/(sqrt(333^2+1000^2)) and that indeed does give us 315.943 ~316, however, I do not understand how they used the mathematical function to get to this point and I was hoping someone could explain what I'm missing.

Getting rid of a complex number in the denominator of a fraction is done by multiplying the fraction by 1 in the form of the conjugate of the denominator over itself.

$$\frac {a + jb}{c + jd} = \frac {a + jb}{c + jd} \cdot \frac{c - jd}{c - jd} \\
= \frac{(a + jb)(c - jd)}{c^2 + d^2} = \frac{ac + bd }{c^2 + d^2} + \frac{j(bc - ad)}{c^2 + d^2}$$