Finding the Current in a Series RLC Circuit

Question

Question :

If the input to an RLC serise circuit is \$V=V_{in}\cos(wt)\$, what is the current I in the circuit in terms of \$V_{in}, w, t,R,L,C,\theta\$?

Answer :

$$I = \frac{V_{in}\cos(wt-\theta)}{\sqrt{R^2+(wL-1/wC)^2}}$$

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My Steps : \begin{align} Z&=R+\frac{-j}{wC}+jwL\\ &=\frac{RwC+j(w^2LC-1)}{wC} \end{align}

If I sub the Z into \$I=\frac{V}{Z}\$ to gain \$Z^{-1}\$, the result seems wrong. How to obtain the answer?

Thank you for your help.

Answer 1

The input voltage is $$V= V_{in}\cos(w t) = V_{in}\angle 0^{\circ}$$ The impedance is $$Z=R+\frac{-j}{wC}+jwL=R+j(wL-1/wC) = |Z|\angle\theta$$ Where \$|Z|=\sqrt{R^2+(wL-1/wC)^2}\$ and \$\theta=\tan^{-1}\left(\frac{wL-1/wC}{R}\right)\$.

Then the current will be $$I=\frac{V}{Z}=\frac{V_{in}\angle0^{\circ}}{|Z|\angle \theta}=\frac{V_{in}\angle(-\theta)}{|Z|}$$ $$=\frac{V_{in}\cos(wt-\theta)}{\sqrt{R^2+(wL-1/wC)^2}}$$