Finding phase angle of AC

Question

After solving for a system of complex equations, I got the following currents:

$$I1= -.80669-.35675j$$ $$I2= -1.8546+1.30730j$$

Firstly, I found the current magnitude: $$|I_1|=.88205A$$ $$|I_2|=2.26887A$$ After that, I plugged into the calculator $$tan^{-1}(complex/real)$$

So I ended up with $$\theta_1=23.85678°$$ $$\theta_2=-35.17946°$$

However, the phase angle part of my answer is incorrect; my current magnitudes are considered correct, so my complex current values are correct. Is there something else that I need to do to my phase angle values to get them to be correct?

P.S. I know that values for I1 and I2 would be in the third and second quadrants, respectively, but the answer requested is between -180°< theta <180°, so it wants the values that the calculator outputs.

Answer 1

When you divide the complex part by the real part in the tangent function, you lose information: the individual sign of each component (you instead have the relative sign between them). Therefore the quadrant information for the angle is incorrect.

The correct way to do this is either to calculate it on a calculator with a quadrant-aware tangent function (sometimes called atan2) or simply examine the signs to know what quadrant it should be in. Sometimes drawing a coordinate plane and plotting the complex numbers can help visualize this.

\$ I_1\$ has both negative real and complex parts; therefore it is in quadrant III, and the total phase angle is \$-180^\circ + atan(\frac{Im[I_1]}{Re[I_1]}) = -180^\circ + 23.85678^\circ\$. \$ I_2\$ has a negative real component and positive complex component, so it is in quadrant II, and the total phase angle is \$180^\circ + atan(\frac{Im[I_2]}{Re[I_2]}) = 180^\circ + (−35.17946^\circ)\$.

Note that just because the answer needs to be in the interval of \$[-180,180]\$ doesn't mean that it wants the value that the calculator puts out; different calculators can express numbers in different ways, and oftentimes taking the mentality of "the calculator knows best" can lead to interpretation errors like this. All it means is that you should express \$270^\circ\$ as \$-90^\circ\$.

Answer 2

Your calculator is returning the principal value of the arctangent. There are two possible angles so it returns one of them defined as the principal value.