What is bandwidth in the context of RLC circuits?

[theBEAST]

Homework Statement

I noticed that bandwidth is the difference between the frequencies that gives half power:

But sometimes it is the difference between when the frequency is 0 and the frequency that gives half power:

So the definition in my lecture slides states that bandwidth is the difference between the upper and lower bounds of the passband where passband is the range of frequencies with nearly constant gain. So does that mean frequencies past the bandwidth will be filtered out by the network? But if you look at the graphs there is still a gain when you are at frequencies out of the bandwidth range. What does bandwidth really mean?

 

[aralbrec]

So does that mean frequencies past the bandwidth will be filtered out by the network? But if you look at the graphs there is still a gain when you are at frequencies out of the bandwidth range. What does bandwidth really mean?

The bandwidth of a filter is that portion of real frequencies that are approximately unattenuated by the network, ie that portion of signals in the frequency domain which are allowed to pass unscathed.

The cutoff is arbitrary and is most commonly (and not always!) chosen as the 3db point aka the half power level aka max passband voltage divided by root 2.

An ideal filter will act like a brick wall, have constant gain in the passband, and zero out signals beyond the intended bandwidth. No real circuit can do that and in real life there is a rounded tailing off of the filter characteristics as you've spotted. Lower order circuits will have more gradual characteristics and higher order circuits (higher sⁿ which is equivalent to more Cs and Ls) will have steeper characteristics.

The bandwidth of a signal (as opposed to a filter) gives an idea where in the frequency domain most of the signal's energy resides and is usually defined the same way. But it may be defined differently if the fidelity of the signal is important -- ie you may decide the bandwidth of a signal should contain 90% of the signal's total energy. This way you'd design your filters to allow 90% of the signal's energy to pass without much modification. Using a 3db definition may cause significant portions of the signal's energy to be distorted, which may be undesirable.

 

[CWatters]

This plot shows how the use of a higher order filter improves the steepness of the cut off..

Unfortunately life is never simple and the use of higher order filters can introduce other problems such as the need to use non standard value or very tight tollerance components.

There are several different types of filter, usually named after the person who invented it. These typically trade off one characteristic for another. For example they may improve the steepness of the cuttoff at the expense of a few ripples or peaks inside or outside the pass band. Another example..

Which you use depends on the application and if it can tollerate the side effects.