# How to calculate magnitude and phase difference of impedance

How can I use a calculator to find the phase difference and magnitude of a circuit if the circuit has an impedance of, for example, $Z = 5 + 3j\mbox{ }\Omega$.

I have the answers $|Z| = 5.83$ and $\theta = 30.96^{\circ}$.

Euler is your friend. Euler's Formula:

$e^{j \theta} = cos(\theta) + j \cdot sin(\theta)$

Euler helps you to calculate in an easy way with the complex impedance, by using the $e$ power. At the end of the calculation you can separate the complex power of $e$ into its real and imaginary parts. These agree with your resistance and reactive impedance resp.

$|Z| = \sqrt{real^2 + imaginary^2}$

and

$\theta = arctan\left( \dfrac{imaginary}{real} \right)$

Think of the real and imaginary components shown graphically on the complex plane. X is the real axis and Y the imaginary one. Now plot the vector from the origin to (5, 3). This vector nicely shows the complex impedance. Its magnitude is the impedance magnitude, and its angle from 0 is the impedance phase angle.

Now you should be able to see how the answer above was arrived at. Draw it on a piece of paper if that helps. Then please come back and show us how the answer was derived. By the way, I just did the calculations and agree that the answer you give is correct other than being expressed with excessive precision.