# Electronics Lab Prep: Filters & Transfer Functions

• oddjobmj
In summary, Figures 1 (a), (b), and (c) show different types of filters and their transfer function H(ω) as a function of the angular frequency ω of the input voltage. The low-pass filter has a power response function of |H(ω)|2 = \frac{ω_0^4}{(ω_0^2-ω^2)^2+ω^2(R/L)^2}, where ω0 = \frac{1}{\sqrt{LC}}. Resonance occurs when ω=ω0, with a value of |H(ω)|2=\frac{ω^2L^2}{R^2}. The value of H(ω) at
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 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 = $\frac{ω_0^4}{(ω_0^2-ω^2)^2+ω^2(R/L)^2}$, where ω0 = $\frac{1}{\sqrt{LC}}$

## Homework Equations

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

Vout/Vin=$\frac{Z_2}{Z_2+Z_1}$

## 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!

oddjobmj said:

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

## Homework Equations

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

Vout/Vin=$\frac{Z_2}{Z_2+Z_1}$

## 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

Last edited:
1 person
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?

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

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!

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.

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

## What is an electronic filter?

An electronic filter is a circuit that is designed to allow certain frequencies of electrical signals to pass through while blocking or attenuating others. It is used to shape or modify the frequency response of a circuit or system.

## What are the different types of electronic filters?

There are four main types of electronic filters: low-pass, high-pass, band-pass, and band-stop. Low-pass filters allow frequencies below a certain cutoff frequency to pass through, while blocking higher frequencies. High-pass filters do the opposite, allowing frequencies above a certain cutoff frequency to pass through. Band-pass filters allow a specific range of frequencies to pass through, while blocking others. Band-stop filters, also known as notch filters, do the opposite, blocking a specific range of frequencies while allowing others to pass through.

## How do electronic filters affect signal amplitude?

Electronic filters can affect signal amplitude in different ways. Low-pass filters generally attenuate or reduce the amplitude of higher frequencies while passing lower frequencies with minimal attenuation. High-pass filters do the opposite, attenuating lower frequencies and passing higher frequencies with minimal attenuation. Band-pass and band-stop filters can also attenuate or amplify certain frequencies within their passbands or stopbands.

## What is a transfer function in relation to electronic filters?

A transfer function is a mathematical representation of the input-output relationship of an electronic filter. It describes how the input signal is modified by the filter to produce the output signal. The transfer function is typically represented as a ratio of the output signal to the input signal and can be used to analyze and design electronic filters.

## What are some applications of electronic filters?

Electronic filters have a wide range of applications in various fields, including signal processing, audio and video equipment, communication systems, and power supplies. They are used to remove noise or unwanted signals, separate different frequency components, and shape the frequency response of a circuit. Some common examples of electronic filters include radio tuners, equalizers, and active crossovers in audio systems.

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