Rectangular To Polar
1. The problem is to convert a complex number representing the final node voltage from rectangular form to polar form.
2. Rectangular form of a complex number is given by $z = x + jy$, where $x$ is the real part and $y$ is the imaginary part.
3. Polar form is expressed as $z = r \angle \theta$, where $r$ is the magnitude and $\theta$ is the phase angle.
4. To convert from rectangular to polar form, use the formulas:
$$r = \sqrt{x^2 + y^2}$$
$$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$
5. Calculate the magnitude $r$ by taking the square root of the sum of squares of the real and imaginary parts.
6. Calculate the angle $\theta$ by taking the arctangent of the ratio of the imaginary part to the real part.
7. The angle $\theta$ is usually expressed in degrees or radians depending on the context.
8. This conversion helps in analyzing the voltage magnitude and phase in AC circuit analysis.
9. Example: If the final node voltage is $3 + j4$, then
$$r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$
$$\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ$$
10. So, the polar form is $5 \angle 53.13^\circ$.