# Little Circuit Math Problem Challenge

#### MrAl

Joined Jun 17, 2014
7,872
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  Last edited:

#### t_n_k

Joined Mar 6, 2009
5,455
Haven't been here for ages but I saw your interesting challenge problem. Not easy.
I have the following values to several significant figures which should indicate whether I'm on the right track or not:
R1 = 999.798448 Ohms
R2 = 100.181417 kOhms
C1 = 7.92744991 uF
C2 = 159.506040 pF
I can post my solution if you are interested.

#### MrAl

Joined Jun 17, 2014
7,872
Haven't been here for ages but I saw your interesting challenge problem. Not easy.
I have the following values to several significant figures which should indicate whether I'm on the right track or not:
R1 = 999.798448 Ohms
R2 = 100.181417 kOhms
C1 = 7.92744991 uF
C2 = 159.506040 pF
I can post my solution if you are interested.
Hello,

Welcome back Yes that's very good. It is amazing that this is possible i think.
Could you find another solution too? There may or may not be another one but when i did the same problem with the lower frequency 20Hz again but upper frequency 120Hz i got two solutions.
Also, can you find a solution for low 20Hz and high 100Hz? There may be no solution for that one maybe because the two frequencies are too close together. So there is probably a limit to this.

If you like you might describe how you found these.
What i did was found each value one by one, which takes a little doing algebraically.
First C1, then R1, then C2, then finally R2.
But i'd like to hear your method too.

#### Analog Ground

Joined Apr 24, 2019
446
An additional challenge is to use a non-ideal op amp. The calculation must take in the full loop gain and getting the exact input impedance is a bit more complicated.

#### t_n_k

Joined Mar 6, 2009
5,455
Hopefully the attachment shows how I have derived the general relationships from which the components for a particular case can be specified. I have omitted a lot of detail lest it become even more over-complicated.

#### Attachments

• 804.8 KB Views: 15

#### t_n_k

Joined Mar 6, 2009
5,455
What the equations do indicate in particular is that where the relationship (n^2-6n+1)<0 no solution is possible. So a solution doesn't exist say where f1=20 Hz and f2=100 Hz - as Mr Al proposed. Stated exactly, if n=f2/f1<(3+2sqrt(2)) no solution exists.

#### Himakinecity

Joined May 5, 2020
1
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 View attachment 205573
You have provide tips on little circuit math which is very simple to understand and easy to calculate

#### MrAl

Joined Jun 17, 2014
7,872
You have provide tips on little circuit math which is very simple to understand and easy to calculate
It turned out to be interesting at first i was not sure if it could be solved in an exact form because the -3db points might not allow it, but it did work, for all by the tightest bandwidth. As the bandwidth narrows it looks like it becomes impossible to get all three frequency specs at the same time as well as the input impedance spec.

The really interesting part is that when the bandwidth spec narrows, the Bode approximation gets less effective with the specs being much farther off than with the wide bandwidth of 20Hz to 10kHz. That means that an exact method would work much much better with narrow bandwidth as long as it was not too narrow.

#### t_n_k

Joined Mar 6, 2009
5,455
.......That means that an exact method would work much much better with narrow bandwidth as long as it was not too narrow.
As I noted in my earlier reply #6, the spec cannot be met if the upper -3dB frequency f2 is less than (3+2*sqrt(2))*f1, where f1 is the lower -3dB frequency. So the minimum BW (f2-f1) for which spec can be met is (2+2*sqrt(2))*f1.