Phase of current and voltage in LCR network

[Froskoy]

Homework Statement

A network consists of an inductor, capacitor and resistor are connected in parallel and an alternating sinusoidal voltage is placed across the network. Sketch the phase of the total current relative to the voltage as a function of the angular frequency.

The Attempt at a Solution

I've got an expression for the impedance of the network in the form $a+ib$. This is equal to $\frac{V}{I}$. So is the phase equal to $tan^{-1}\left({\frac{b}{a}}\right)$ - or do you have to do something more? Is there a better way to calculate the phase using exponentials?

To sketch it, I don't understand how you determine if the voltage lags behind the current or the current lags behind the voltage? Once you've determined the phase, $\theta$, do you just sketch the sin wave for the voltage and then sketch a sin wave for the current that lags an angle $\theta$ behind the voltage, or is there more to it? How do you determine that the current is indeed a sin shape and not some other shape?

With very many thanks,

Froskoy.

 

[gneill]

Homework Statement

A network consists of an inductor, capacitor and resistor are connected in parallel and an alternating sinusoidal voltage is placed across the network. Sketch the phase of the total current relative to the voltage as a function of the angular frequency.

The Attempt at a Solution

I've got an expression for the impedance of the network in the form $a+ib$. This is equal to $\frac{V}{I}$. So is the phase equal to $tan^{-1}\left({\frac{b}{a}}\right)$ - or do you have to do something more? Is there a better way to calculate the phase using exponentials?

No, the complex number approach is about as simple as it gets.

Note that, as you say, Z = V/I, so that I = V/Z. If the voltage source is taken as the phase reference and if angle(Z) is the angle of the impedance, then the current will have a phase angle that is -angle(Z).

To sketch it, I don't understand how you determine if the voltage lags behind the current or the current lags behind the voltage? Once you've determined the phase, $\theta$, do you just sketch the sin wave for the voltage and then sketch a sin wave for the current that lags an angle $\theta$ behind the voltage, or is there more to it? How do you determine that the current is indeed a sin shape and not some other shape?

With very many thanks,

Froskoy.

So many questions
Whether or not the current lags or leads the voltage will depend upon the relative sizes of the reactances of the inductor and capacitor. But not to worry; the sign of the phase angle that you calculate tells you whether the current leads or lags.

For the sketch, it looks like they want you to plot θ(f).

 

[Froskoy]

Thank you so much! It's all really clear now!