How Do Phase Relationships Affect Voltage Waveforms in an LCR Circuit?

[t_n_p]

Homework Statement

A sinusoidal input is applied to the circuit below. Sketch the waveforms of each of Vr (voltage of resistor), Vc (capacitor) and Vl (inductor) on the one graph with particular attension to the phases (the amplitudes are not important) Hint: Vc is proportional to the integral of the current; the inductor voltage is proportional to the time derivative of the current.

http://img249.imageshack.us/img249/8987/untitledyp0.jpg

The Attempt at a Solution

Once again, I'm pretty much clueless. Would really appreciate it if somebody could point me in the general direction!

 

[Astronuc]

This is a series RLC circuit.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/rlcser.html


http://en.wikipedia.org/wiki/RLC_circuit

R-L-C Circuits and Resonant Circuits
http://www.physics.ohio-state.edu/~gan/teaching/summer04/Lec4.pdf

 

[t_n_p]

Had a quick browse but I'm still unsure as to how to derive the graphs. Should I be sketching the graph from equations?

 

[t_n_p]

The way I think about it, wouldn't voltage be increasing at a steady rate as it flows through each of the R, L and C. But then of course I end up with a linear graph and not a wave. So totally lost...

 

[andrevdh]

Since the input voltage is sinusoidal one can say that it is of the form

$$V_{in} = V_o \sin(\omega t)$$

We also know that the voltage over the resistor will be in phase with the current in the circuit (which is determined by whatever is going on/in the circuit). Also the current in the (series) circuit (and therefore all of the components in the circuit) are the same. Just draw a graph with some arbitrary amplitude for the voltage of the resistor.

 

[t_n_p]

So all of the components are in phase with the input signal, does that mean that the graph will be the same for the L, C and R? How exactly would I model https://www.physicsforums.com/latex_images/13/1389816-0.png ? Obviously I need to find Vo and omega, but how!? *pulling hair out*

 

[andrevdh]

No, the question do not want you to quantify the relationships. You just need to show qualitatively how the various voltages are related in phase on a graph. (Sorry, I misled you. I changed my previous post. The voltage over the resistor will not be in phase with the input voltage. So do not try and relate them in your graph - see my previous post). Once you have drawn the voltage as a function of time for the resistor then you just need to fill in the other two voltages. The connecting factor is the current in the components are the same, but the voltages are shifted in phase. How to determine the relationship? When one differentiate the current (which will be in phase with the voltage over the resistor) you get the voltage for the inductor. You can thus construct the graph for the inductor from the resistor graph point by point (it will be related by the gradient of the resistor graph).