Solving a Series Circuit with Resistor, Capacitor and Inductor

[Pacopag]

Homework Statement

Consider a circuit consisting of a voltage source with $$V = V_0 \sin{\omega t}$$, a resistor, a capacitor, and an inductor, all connected in series. The problem is to find the steady state current as a function of omega.

Homework Equations

$$V = IZ$$
$$Z = Z_R + Z_C + Z_L$$
Take
$$Z_R = 1 \Omega$$
$$Z_C = -i2$$ Fd
$$Z_L = +i4$$ H

The Attempt at a Solution

I find the total impedance to be
$$Z = 1+2i$$.
So I'm guessing that the maximum amplitude of the current will be $$V_0 \over \sqrt{5}$$.
Now, I'm not sure how to handle the phase. From what I remember, a resistor doesn't affect the phase, a capacitor shifts the current ahead 90 degrees compared to the voltage, and an inductor shifts it 90 degrees behind. So I would think that the total phase is zero and we get $$I = {V_0\over \sqrt{5}}\sin\omega t$$.
But then what does the angle associated with Z (i.e. \tan^{-1} 2) have to do with it.

Also, will the current have the same profile (and phase) everywhere in the circuit, or does it depend on "where" we measure the current in the circuit??

Any help would be great.

 

[Pacopag]

I think I may have got it. I'd still really like it if someone can tell me if I have this right.
First of all, the total impedance should be
$$Z=1+\left(4\omega -{1\over{2\omega}}\right)i$$
or in polar form
$$Z = \sqrt{1+\left(4\omega -{1\over{2\omega}}\right)^2}e^{i\tan^{-1}\left(4\omega -{1\over{2\omega}}\right)}$$.
Now, using Ohm
$$I = V/Z$$, we get
$$I = {V_0\over{\sqrt{1+\left(4\omega -{1\over{2\omega}}\right)^2}}}\sin{\left(\omega t - \tan^{-1}\left({4\omega -{1\over{2\omega}}}\right)\right)}$$.
I'm not really sure if this is right. I'm using a textbook that uses funny notation where they just forget about the omega t and the sin and work only with the phases.

 

[buffordboy23]

Now, I'm not sure how to handle the phase. From what I remember, a resistor doesn't affect the phase, a capacitor shifts the current ahead 90 degrees compared to the voltage, and an inductor shifts it 90 degrees behind. So I would think that the total phase is zero.

Your description about the phase shifts is correct, but only for the individual circuit components. The total phase would be zero only if the impedance of the capacitor and inductor are equivalent--the two phasors would point in opposite directions and have same magnitude. Does the problem give values for the resistor, capacitor, inductor? If so, I recommend computing the individual impedances of the components, drawing a phasor diagram, and computing the total impedance of the circuit from your diagram. If not, you may have to generalize your derivation to include such arbitrary values.