# Circuits Help (Filters)

• Engineering

## Homework Statement *in picture

## The Attempt at a Solution

Not sure where to start. This is a review but I have not covered it yet or if we did, it was very brief?

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if vi were a dc voltage source, it would have the lowest frequency possible (0 Hz). How do inductors and capacitors behave when DC voltage is applied to them as you look at t -> infinity? What would the voltage across the parallel combination of an inductor and a capacitor have to be if they behave that way?

As freq -> infinity, inductors and capacitors behave oppositely to how they behave for f = 0. Apply the same reasoning as above except with the new simplifications for when f -> infinity.

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Oh yeah. I guess your first step would be understanding what the different types of filters are:

if you find that the circuit has:
nonzero 0Hz and zero inf. Hz response, it is probably a lowpass
zero 0Hz and nonzero inf. Hz response, it is probably highpass
zero for both, probably bandpass
nonzero for both, probably bandstop

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Well, it looks like a bandpass filter. It allows frequencies between a certain range, depending on the values of your circuit elements. It becomes more apparent when you find the transfer function.

If you change those circuit elements into the s-domain, and then write a node equation at [;V_o;], you get:

[;\frac{V_o - V_i}{R}+\frac{V_o}{Ls};+V_o*Cs = 0;]

Doing some algebra, you can get the transfer function,

[;\frac{V_o(s)}{V_i(s)} = \frac{s}{RC*s^2+R*s+\frac{R}{L}};]

with a little more algebra, we can get it into a useful form:

[;\frac{s\frac{1}{RC}}{s^2+s\frac{1}{C}+\frac{1}{LC}};]

For analysis, if you change it into the frequency domain ([;s = j*\omega;]), set some values for our elements and vary the frequency, we can see what will happen. I find it rather easy in MATLAB. There are equations to see where your range will be, but I don't know them off the top of my head. Something about 3dB. This is where you can get into design.

I guess your best bet is to do a frequency response and graph the frequencies from like 0Hz to 1MHz. You'll see your bandpass

Also, we can see as we increase R, the output will decrease. So I think it's safe to say that increasing R will increase your damping factor.

Hopefully this helped.

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