Phase Difference Between dynamic voltage and input current

Question

I have a RLC circuit, and I know if we wish to calculate the phase difference between Voltage and Input current then the formula is

$$ \phi = \tan^{-1} \frac{X_L -X_C}{R} $$

But as we already know, phase difference between inductor voltage and input current is 90°. I know how to calculate it through waveform. $$\phi = 360 f \,\mathrm dT $$

But I have no idea how do I calculate it through first formula. In short, what will be the question mark in the following expression and why? $$\phi = \tan^{-1} \frac{X_L}{?}$$.

Also for the capacitor voltage and input current.

I know am weak in electronics and my basics too that's why am seeking help from you guys, kindly correct me if I was wrong in any of my presumptions.

Answer 1

But I have no idea how do I calculate it through first formula.

For a pure inductor the phase relationship between voltage and current is that current always lags voltage by 90°. This comes from the basic but fundamental inductor, voltage and current equation: -

$$V = L\dfrac{\mathrm di}{\mathrm dt}$$

Or, if you insist on using your original formula, ARCTAN(infinity)=90 degrees. It's infinity because \$X_L/0\$ is infinity and this is because resistance is 0 Ω.

Similarly for the pure capacitor, the phase relationship is that current leads voltage by 90°.