Hi there,

This came from a problem that came up in another section.

I started working with the simple circuit shown in the attachment. It's a bandpass filter. The -3db points are:

20Hz and 10000Hz. The center gain is supposed to be 100 (40db). The input impedance is found from the series combination of R1 and C1 as usual for two series element, and it has to be 1000 Ohms at 1000Hz.

So to repeat the specs:

-3db at 20Hz and 10kHz,

center gain (absolute maximum in the passband)=100

Zin=1k Ohms at 1kHz.

Now since the two frequencies are so far apart a Bode approximation can be used to calculate the components, along with the spec of 1k input Z at 1kHz. This technique gets VERY close to the exact specifications.

Here is the challenge...

The problem is, the Bode approximation is still just an approximation, so the -two -3db gains come out about 0.0002 percent (fractional 0.000002) in error. Likewise the center gain is also off by some small fractional amount similar to that.

So the challenge is, calculate the EXACT values of all four components to say 12 decimal places (it is acceptable to use 16 digit floating point for all the calculations) so that ALL THREE gain specifications and the input Z all come out EXACT, again to say 12 digits, although they may come out even closer than that using regular floating point, or do they?

This is an interesting exercise.

What troubled me a little was could we calculate the exact values and is it even possible to meet all three frequency specs and also the input impedance spec at the same time.

If you like math and you like exact theoretical solutions you might find this interesting.

You may want to start with the Bode approximation and verify that the three frequencies do not come out exact although the input Z is pretty easy to get exact.

Good luck

This came from a problem that came up in another section.

I started working with the simple circuit shown in the attachment. It's a bandpass filter. The -3db points are:

20Hz and 10000Hz. The center gain is supposed to be 100 (40db). The input impedance is found from the series combination of R1 and C1 as usual for two series element, and it has to be 1000 Ohms at 1000Hz.

So to repeat the specs:

-3db at 20Hz and 10kHz,

center gain (absolute maximum in the passband)=100

Zin=1k Ohms at 1kHz.

Now since the two frequencies are so far apart a Bode approximation can be used to calculate the components, along with the spec of 1k input Z at 1kHz. This technique gets VERY close to the exact specifications.

Here is the challenge...

The problem is, the Bode approximation is still just an approximation, so the -two -3db gains come out about 0.0002 percent (fractional 0.000002) in error. Likewise the center gain is also off by some small fractional amount similar to that.

So the challenge is, calculate the EXACT values of all four components to say 12 decimal places (it is acceptable to use 16 digit floating point for all the calculations) so that ALL THREE gain specifications and the input Z all come out EXACT, again to say 12 digits, although they may come out even closer than that using regular floating point, or do they?

This is an interesting exercise.

What troubled me a little was could we calculate the exact values and is it even possible to meet all three frequency specs and also the input impedance spec at the same time.

If you like math and you like exact theoretical solutions you might find this interesting.

You may want to start with the Bode approximation and verify that the three frequencies do not come out exact although the input Z is pretty easy to get exact.

Good luck

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