How to calculate magnitude and phase difference of impedance

Question

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}\$.

Answer 1

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) \$

Answer 2

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.

Answer 3

Let me add a picture for the benefit of future visitors to this page.