Electronics Lab Prep: Filters & Transfer Functions

[oddjobmj]

Homework Statement

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 V_out/V_in 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 = \frac{ω_0^4}{(ω_0^2-ω^2)^2+ω^2(R/L)^2}, where ω₀ = \frac{1}{\sqrt{LC}}

Homework Equations

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

V_out/V_in=\frac{Z_2}{Z_2+Z_1}

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) = \frac{\frac{1}{jωC}}{R+jωL+\frac{1}{jωC}}

(b) H(ω) = \frac{R}{R+jωL+\frac{1}{jωC}}

(c) H(ω) = \frac{jωL}{R+jωL+\frac{1}{jωC}}

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!

 

[ehild]

Homework Statement

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 V_out/V_in 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 = \frac{ω_0^4}{(ω_0^2-ω^2)^2+ω^2(R/L)^2}, where ω₀ = \frac{1}{\sqrt{LC}}

Homework Equations

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

V_out/V_in=\frac{Z_2}{Z_2+Z_1}

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) = \frac{\frac{1}{jωC}}{R+jωL+\frac{1}{jωC}}

(b) H(ω) = \frac{R}{R+jωL+\frac{1}{jωC}}

(c) H(ω) = \frac{jωL}{R+jωL+\frac{1}{jωC}}

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!

Your H(ω)-s are correct, but you can simplify them by multiplying both the numerator and denominator by jωC. Use the notation ω₀²=1/(LC).
The problem asks |H(ω)|². What have you got? The given answer might be wrong. ehild

 

[oddjobmj]

Ah, great! The simplification was enough to get me there, actually. After finding |H(ω)|² I was able to get to the suggested solution. Thank you for the tip on using ω₀² also, that helped quite a bit!

Since that wasn't as difficult as I expected I'll post a couple more related pieces here if that's okay:

"Does resonance occur near ω=ω₀? Explain why or why not."

When ω=ω₀, |H(ω)|²=\frac{ω^2L^2}{R^2}

I'm not sure what to make of this though. How do I relate this to capacitive and inductive reactance?

 

[flatmaster]

If there is a resonance, what should happen to the value of H(w) when w-->w0 from either direction on the w axis?

 

[oddjobmj]

Hmm, well, I know what -does- happen in this case. I'm not sure what -should- happen in the case of resonance. I do recall that you can have higher output voltages than input so I guess H(w), being the ratio of out/in, would be greater than 1.

Anywho, thanks for the hint!

 

[flatmaster]

You're getting there. You need to think about what the graph function would look like around the resonance in addition to the value of H(w) at the resonance. At the resonance, you would expect the value of H(w) to be at a local maximum. How can you determine if your function is at a maximum? Have you had calculus yet?
What would be the value of the derivative dH(w) / dt at…
The resonance…
before the resonance
after the resonance.

 

[ehild]

Check your notes/book what is called resonance in an electric circuit. Usually resonance is defined at the frequency when the input impedance is real. It is not sure that |H| > 1. At what frequency is it the highest? What is the condition that the maximum |H| is greater than 1?

ehild