Electronics Lab Prep: Filters & Transfer Functions

1. The problem statement, all variables and given/known data

Figures 1 (a), (b), and (c) show low-pass, bandpass, and high-pass filters. Write the transfer function H(ω) for each of these filters, showing the ratio Vout/Vin as a function of the angular frequency ω of the input voltage.

-The low-pass filter calculations:
Show that the low-pass filter in Fig. 1 (a) above has a power response function:

|H(ω)|2 = [itex]\frac{ω_0^4}{(ω_0^2-ω^2)^2+ω^2(R/L)^2}[/itex], where ω0 = [itex]\frac{1}{\sqrt{LC}}[/itex]

2. Relevant equations

Treating the filters as voltage dividers with impedances instead of resistances:

Vout/Vin=[itex]\frac{Z_2}{Z_2+Z_1}[/itex]

3. The attempt at a solution

To be clear, these aren’t actually homework problems. I have my electronics lab on Thursday and I am trying to prepare for it beforehand as much as possible. I believe putting the transfer functions together is rather straight forward. I am having a hard time equating the low pass filter with the form they provided though.

(a) H(ω) = [itex]\frac{\frac{1}{jωC}}{R+jωL+\frac{1}{jωC}}[/itex]

(b) H(ω) = [itex]\frac{R}{R+jωL+\frac{1}{jωC}}[/itex]

(c) H(ω) = [itex]\frac{jωL}{R+jωL+\frac{1}{jωC}}[/itex]

There is more to the lab that I am having trouble with but I guess it would be better to take it one step at a time and break it into different posts for different concepts.

Any suggestions are welcome. Thank you!

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