# Phasor algebra in AC circuit analysis

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1. Jan 5, 2016

### Titan97

1. The problem statement, all variables and given/known data
Find the peak value of current through the AC source of the following L-C-R circuit, if peak voltage is $V_0$ and angular frequency is $\omega_0$.

2. Relevant equations
I have learnt Vector algebra and calculus (single variable). I was taught how to use phasor diagrams for AC circuits. (only the basics).
From my book:

3. The attempt at a solution
$$i_R=\frac{V_0}{R}\sin\omega t$$
$$i_C=V_0\omega C\cos\omega t$$
$$i_L=-\frac{V_0}{\omega L}\cos \omega t$$
$$i=\frac{V_0}{R}\sin\omega t+V_0\big(\omega C-\frac{1}{\omega L}\big)\omega C\cos\omega t$$

If current is taken along positive $X$-axis, then $$i=\frac{V_0}{Z}=\frac{V_0}{R}+V_0\big(\omega C-\frac{1}{\omega L}\big)j$$

$$\vert Z\vert=\frac{1}{\sqrt{\frac{1}{R^2}+\big(\omega C-\frac{1}{\omega L}\big)^2}}$$

$$i_0=\frac{V_0}{Z}=V_0\sqrt{\frac{1}{R^2}+\big(\omega C-\frac{1}{\omega L}\big)^2}$$

These are the problems I face:
1. The answer given is: $$V_0\big({\frac{1}{R^2}+\big(\omega C-\frac{1}{\omega L}\big)^2}\big)^2$$
(probably a printing error)

2. I applied phasor algebra blindly. Hence, I am not sure if I solved it properly.

3. So should I learn phasor algebra? Can you suggest a book for learning it (or a webpage)?

2. Jan 5, 2016

### cnh1995

I believe you have solved it correctly! There's definitely a printing mistake in the answer given in the book. If you've done this problem by yourself, then I think you have a proper understanding of phasor algebra.