Lecture Notes:- http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-002-circuits-and-electronics-spring-2007/lecture-notes/
A simpler way to get V_p is explored in this lecture. First divide the numerator and denominator by sC, and what is obtained is something that looks like a voltage divider relationship. 1/sC can be replaced with Z_c which will make it look more like one.
Next looking at the impedance models of the resistor, capacitor and inductor, we can see how the impedance is part of Ohm's law where Z_c and Z_L are the impedances for a capacitor and an inductor. For a drive of the form Vi e raised to st, the complex amplitude, Vc of the response, is related to the complex amplitude Ic algebraically by a generalization of Ohm's Law.
Looking at the RC circuit, and replacing the capacitor with the impedance model Z_c, we will see something interesting when finding V_c. We will find that V_c is the famed complex amplitude V_p that we have been trying to derive except the method now is much simpler. All circuit methods can be used to further analyze the circuit.
Signal notation is discussed-there are four of them including the complex amplitude notation.
Here is a summary of the impedance method: -
(1) First step we replace the sinusoidal sources by their complex or real amplitudes.
(2) As a second step, we replace circuit elements by boxes. These boxes are the impedance boxes, or impedances.
(3) Determine the complex amplitudes of the voltages and currents at the various node points and the branches.
(4) The fourth step is not really necessary - obtain the time variables from the complex amplitudes.
For the series RLC circuit, we have to find out what is V_r.
Next we try to get the magnitude of the transfer function of V_r/V_i. To get the frequency response, sketch the graph of magnitude against omega. The way to do this is to figure out the asymptotes. We will need to find out the values of the magnitude for small and high values of omega. We also need to look at the value of the magnitude where a certain part of the equation goes to zero.
So when omega is very small, what happens? So when omega is very small, , and similarly, an omega squared values can be ignored. So what we are left with is approximately omega RC.
Now what does omega RC look like? Of course, at omega equals 0, it will be 0 at first. But for very low values of omega, as we increase it, it begins to go up in a linear manner.
Next, what happens for large values of omega? So when omega is very large, then 1 can be ignored in relation to omega squared LC. So we get approximately R divided by omega L for very large values of omega.
So what happens at that omega equals 1 by square root of LC? We will get a value of 1.
When we sketch the graph, we will get some kind of bandpass filter which allow signals to pass within a certain band of frequencies.