View source for 1.49 (a,b,f,i) (ECE301Summer2008asan) - Rhea

[[Category:ECE301Summer08asan]]
[[Category: ECE]]
[[Category: ECE 301]]
[[Category: Summer]]
[[Category: 2008]]
[[Category: asan]]
[[Category: Homework]]
Express each of the following complex numbers in polar form, and plot them in the complex plane, indicating the magnitude and angle of each number. 

A) <math> 1 + j\sqrt{3}</math>

<math> r = \sqrt{1^2 + \sqrt{3}^2} = \sqrt{4} = 2</math>

<math>\theta = arctan(\sqrt{3}/1) = arctan(\sqrt{3}) = \frac{\pi}{3}</math>

Therefore the polar form of this complex number is: <math>2e^{j\frac{\pi}{3}}</math>

B) <math> -5 </math>

<math> r = 5 </math>

<math> \theta = \pi </math>

Therefore the polar form of this complex number is: <math>5e^{j\pi}</math>

F) <math> (1 + j)^{5} </math>

<math> r = \sqrt{1^2 + 1^2} = \sqrt{2} </math>

<math> \theta = \frac{\pi}{4} </math>

<math> (1 + j) = \sqrt{2}e^{j\frac{\pi}{4}} </math>

<math> (1 + j)^{5} = (\sqrt{2}e^{j\frac{\pi}{4}})^{5} = 2^{\frac{5}{2}}e^{j\frac{5\pi}{4}} = 4\sqrt{2}e^{j(\pi + \frac{\pi}{4})} =4\sqrt{2}e^{j\pi}e^{j\frac{\pi}{4}} = -4(\sqrt{2}e^{j\frac{\pi}{4}}) = -4(1 + j)</math>

Therefore the polar form of this complex number is: <math> -4(\sqrt{2}e^{j\frac{\pi}{4}}) </math>

I) <math> \frac{1 + j\sqrt{3}}{\sqrt{3} + j} </math>

<math> r = 2 </math>

<math> Equation 1 = 1 + j\sqrt{3}  =>  \theta_{1} = \frac{\pi}{3} </math>

<math> Equation 2 = \sqrt{3} + j  =>  \theta_{2} = \frac{\pi}{6} </math>

<math> \frac{2e^{j\frac{\pi}{3}}}{2e^{j\frac{\pi}{6}}} = \frac{e^{j\frac{\pi}{3}}}{e^{j\frac{\pi}{6}}} = e^{j(\frac{\pi}{3} - \frac{\pi}{6})} = e^{j\frac{\pi}{6}}</math>

Therefore the polar form of this complex number is: <math> e^{j\frac{\pi}{6}}</math>