I am doing an exponential Fourier Series Analysis on a square wave. In the picture shown, the Fourier analysis is complete.
However, something that is confusing is how to figure out the phase of the initial coefficient (the DC offset in an electronics context).
If I use the formula given, I get a phase angle of zero.
But if I use the phasor diagram, I get a phase angle of +pi
Is there a general rule for when to use the formula and when to use analysis from the real and imaginary axes?
The DC offset of a square wave should be real since the square wave itself is a real function. Generally you find the phase using real and imaginary parts of the complex number but since this coefficient must be strictly real, it must lie on the real axis, so the phase is either $0$ or $\pi$. Note that $\tan \phi=\tan (\phi+n\pi)$ since the $\tan$ function is periodic with period $\pi$. You have you be careful taking the inverse tangent of a number since the range of $\tan^{-1}$ is $(-\frac \pi 2,\frac \pi 2)$ so using that formula will only give you the reference angle, you must determine the quadrant based on the sign of the real and imaginary parts of your phasor. It will either be the value that the inverse tangent function returns (in the 1st or 4th quadrant), or this value plus $\pi$ (in the 2nd or 3rd quadrant). In this case, the real part is negative so your phase must be $\pi$. I hope this is slightly intuitive since $-1=e^{i\pi}$.
Summary: Use the formula to determine the reference angle AND draw the phasor diagram, if they are in opposite quadrants add $\pi$ to your phase.