-1
\$\begingroup\$

enter image description here

The task is to make a 2:degree filter with a slope of 40 dB/decade.

I have this circuit. R1=R2=10kΩ, f=1kHz I also have this formula enter image description here

What I think (probably wrong). Is that that because 40 dB/decade, then enter image description here

I'm not quite sure that I have gotten the decibel conversion (to H) right. But if that is right, how do I calculate the value of C when there is that imaginary j in the expression?

The magnitude of H is:

enter image description here

\$\endgroup\$

2 Answers 2

1
\$\begingroup\$

Your equation is the transfer function correctly of two cascaded RC low pass filters with a buffer in between them so there is very little loading effect on the first filter by the second filter.

You need to find the magnitude of the transfer function so, first get it in the form a+jb, then find the magnitude from sqrt(a^2 + b^2). Equate the expression for the magnitude of the transfer function to 0.707 which is the magnitude at the -3dB frequency then transpose to make C the subject of the equation.

\$\endgroup\$
2
  • \$\begingroup\$ I added the magnitude. But I dont see how I can take into account the 40 dB/decade in the calculation? \$\endgroup\$ Commented Apr 10, 2022 at 12:45
  • \$\begingroup\$ @Gripen I discuss some of what you may need in trying to understand how, here. \$\endgroup\$ Commented Apr 10, 2022 at 16:59
0
\$\begingroup\$

Well, in this answer I showed that the formula to find the slope (in dB/decade) is given by:

$$\mathcal{I}\left(\omega\right)=20\log_{10}\left(\lim_{\omega\to\infty}\frac{\left|\underline{\mathscr{H}}\left(10\omega\text{j}\right)\right|}{\left|\underline{\mathscr{H}}\left(\omega\text{j}\right)\right|}\right)\tag1$$

Now, in your case we have:

$$\left|\underline{\mathscr{H}}\left(\omega\text{j}\right)\right|=\frac{1}{1+\left(\text{CR}\omega\right)^2}\tag2$$

So, we get:

$$\frac{\left|\underline{\mathscr{H}}\left(10\omega\text{j}\right)\right|}{\left|\underline{\mathscr{H}}\left(\omega\text{j}\right)\right|}=\frac{\frac{1}{1+\left(10\text{CR}\omega\right)^2}}{\frac{1}{1+\left(\text{CR}\omega\right)^2}}=\frac{1+\left(\text{CR}\omega\right)^2}{1+\left(10\text{CR}\omega\right)^2}=\frac{\frac{1}{\omega^2}+\left(\text{CR}\right)^2}{\frac{1}{\omega^2}+\left(10\text{CR}\right)^2}\tag3$$

Taking the limit gives:

$$\lim_{\omega\to\infty}\frac{\frac{1}{\omega^2}+\left(\text{CR}\right)^2}{\frac{1}{\omega^2}+\left(10\text{CR}\right)^2}=\frac{0+\left(\text{CR}\right)^2}{0+\left(10\text{CR}\right)^2}=\frac{1}{10^2}=\frac{1}{100}\tag4$$

And, so you get:

$$\mathcal{I}\left(\omega\right)=20\log_{10}\left(\frac{1}{100}\right)=-40\space\text{dB/decade}\tag5$$

EDIT:

In order to find the cut-off frequency we can use:

$$\left|\underline{\mathscr{H}}\left(\omega_0\text{j}\right)\right|=\frac{1}{\sqrt{2}}\tag6$$

When solving this you will find:

$$\text{C}=\frac{\sqrt{\sqrt{2}-1}}{\text{R}\omega_0}=\frac{\sqrt{\sqrt{2}-1}}{2\pi\text{Rf}_0}\tag7$$

Using your values, you'll find:

$$\text{C}=\frac{\sqrt{\sqrt{2}-1}}{2\pi\cdot10\cdot1000\cdot1\cdot1000}=\frac{\sqrt{\sqrt{2}-1}}{20000000\pi}\approx1.024312\cdot10^{-8}\space\text{F}\tag8$$

\$\endgroup\$
5
  • \$\begingroup\$ The question is "how do I calculate the value of C?" \$\endgroup\$ Commented Apr 12, 2022 at 16:20
  • \$\begingroup\$ @ElliotAlderson well the conclusion is fairly obvious, namely the values of R and C do not matter in the case the OP is talking about, that's what I've shown. \$\endgroup\$ Commented Apr 12, 2022 at 18:03
  • \$\begingroup\$ So the OP gets a corner frequency of 1 kHz regardless of the capacitor value? \$\endgroup\$ Commented Apr 12, 2022 at 20:41
  • \$\begingroup\$ @ElliotAlderson I think Jan refers to the slope of -40 dB/dec, not fc, but I could be wrong. \$\endgroup\$ Commented Apr 13, 2022 at 9:43
  • \$\begingroup\$ @aconcernedcitizen My point was that if you and I can't figure out how to calculate the value of C from reading a wall of equations then neither can the OP. The original answer was not an answer. \$\endgroup\$ Commented Apr 13, 2022 at 9:45

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.