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I am having trouble understanding when to use $\theta = \omega t$ and when to use the $\tan \phi = \frac{X_L-X_c}{R}$.

Do we only use the $\tan \phi = \frac{X_L-X_c}{R}$ when dealing with series RLC circuits? Do we use the $\theta = \omega t$ equation when dealing with any other circuit?

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  • $\begingroup$ Phase angle is $\omega t + \phi$ where the $\phi$ is the initial phase (or phase difference when you are comparing equations). The way you have formulated the question indicates that you haven't understood the definition of phase angle. Please read the Wikipedia article and clarify your question. $\endgroup$ – Yashas Mar 25 '17 at 17:56
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The phase angle of the current/voltage in a circuit having inductance, capacitance or both will be different from the phase angle of the current/voltage generated by the source. In such a case the phase angle of the voltage or current from the source will be found by using the formula: $$\theta = \omega t$$

The phase angle of the current or voltage in the circuit or through the passive circuit elements(R/L/C) will be found by the formula:

$$\theta = \tan^{-1}(\frac{X_L -X_c}{R})$$

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Consider an RLC circuit whose source voltage varies as $V = V_0 \sin (\omega t)$ and source current varies as $I = I_0 \sin (\omega t + \phi)$.

The phase difference between the voltage and the current phasors for an RLC circuit is given by:

$$\tan \phi = \frac{X_L-X_c}{R}$$

The above equation is always true. Not only does it hold good for RLC circuits, it also works for RL, RC, LC circuits. It also works for DC circuits ($\omega$ = 0) but using that formula in DC circuits is redundant.


$\Delta \phi = \omega \Delta t$

The above formula is used to calculate the phase change for the wave after a time $\Delta t$. This formula has nothing special to do with AC circuits. This equation can be used anywhere (sound waves, waves on a string, AC circuits, etc) but it isn't a special formula for AC circuits.

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