Hi. I'm trying to work out how to design a band-boost/band-attenuate filter for an audio equalizer. I've come across what seems to be a pretty good circuit to use in an application note from TI but unless I'm misunderstanding something, the design equations it provides are inconsistent.

The app note is "An audio circuit collection, Part 3" from the July 2001 edition of their Analog Applications Journal, available here.

On page 37, it gives the following formulas:

\[ X_L=2\pi\times f_0\times L \]
\[ \displaystyle Q =\frac{X_L}{R_4} \]
\[ L = (R_5 - R_4)\times R_4\times C_1 \]

Putting them together and isolating C1:

\[ Q = \displaystyle\frac{X_L}{R_4} = \frac{2\pi\times f_0 \times L}{R_4} = \frac{2\pi\times f_0 \times (R_5 - R_4)\times R_4\times C_1}{R_4}\]
\[ C_1 = \displaystyle\frac{Q}{2\pi\times f_0\times (R_5 - R_4)} \]

This is different from the formula that the app note provides for C1:

\[ C_1 = \displaystyle\frac{Q\times R_4}{2\pi\times f_0\times (R_5 - R_4)} \]

What am I missing here? If I'm not missing anything and there is an error, is anyone able to say exactly where the error lies?

 

The last equation is wrong. The R4 in the numerator shouldn't be there.

 

Hi. I'm trying to work out how to design a band-boost/band-attenuate filter for an audio equalizer. I've come across what seems to be a pretty good circuit to use in an application note from TI but unless I'm misunderstanding something, the design equations it provides are inconsistent.

The app note is "An audio circuit collection, Part 3" from the July 2001 edition of their Analog Applications Journal, available here.

On page 37, it gives the following formulas:

\[ X_L=2\pi\times f_0\times L \]
\[ \displaystyle Q =\frac{X_L}{R_4} \]
\[ L = (R_5 - R_4)\times R_4\times C_1 \]

Putting them together and isolating C1:

\[ Q = \displaystyle\frac{X_L}{R_4} = \frac{2\pi\times f_0 \times L}{R_4} = \frac{2\pi\times f_0 \times (R_5 - R_4)\times R_4\times C_1}{R_4}\]
\[ C_1 = \displaystyle\frac{Q}{2\pi\times f_0\times (R_5 - R_4)} \]

This is different from the formula that the app note provides for C1:

\[ C_1 = \displaystyle\frac{Q\times R_4}{2\pi\times f_0\times (R_5 - R_4)} \]

What am I missing here? If I'm not missing anything and there is an error, is anyone able to say exactly where the error lies?

We know that the last equation (the equation in the app note) is wrong because the units don't work out.

Q is dimensionless (it's the ratio of a reactance to a resistance). Then it has a resistance in the numerator and the difference of two resistances in the denominator, so those units cancel. All that is left is the frequency, which has units of inverse-seconds. Since this is in the denominator, the right-hand side ends up having units of seconds. But the left hand side has units of capacitance, which is charge per volt. Charge has units of current*time, so that makes the left-hand side have units of (current/volt)(time). Voltage per current is resistance, that the ends result is that the units of capacitance are time/resistance, not time.

The last line of your work has the proper units.

This is yet another example of an easily caught mistake that got through because most engineers are too lazy (and, being fair, where not required in school) to properly track and check their units.

Imagine how many people have blindly used those equations from that app note and wasted who knows how much time because they, too, were too lazy to check the units and just plugged numbers into the equation and tacked on the units that they wanted and expected the result to have.

 

We know that the last equation (the equation in the app note) is wrong because the units don't work out.

Q is dimensionless (it's the ratio of a reactance to a resistance). Then it has a resistance in the numerator and the difference of two resistances in the denominator, so those units cancel. All that is left is the frequency, which has units of inverse-seconds. Since this is in the denominator, the right-hand side ends up having units of seconds. But the left hand side has units of capacitance, which is charge per volt. Charge has units of current*time, so that makes the left-hand side have units of (current/volt)(time). Voltage per current is resistance, that the ends result is that the units of capacitance are time/resistance, not time.

The last line of your work has the proper units.

This is yet another example of an easily caught mistake that got through because most engineers are too lazy (and, being fair, where not required in school) to properly track and check their units.

Imagine how many people have blindly used those equations from that app note and wasted who knows how much time because they, too, were too lazy to check the units and just plugged numbers into the equation and tacked on the units that they wanted and expected the result to have.

I feel so silly. I've been told to check my units since high school and I should have known to do it here. Thank you for a question answered and a lesson learned.

 

I feel so silly. I've been told to check my units since high school and I should have known to do it here. Thank you for a question answered and a lesson learned.

Not much need to feel too silly. Yeah, get in the habit of looking at those units as a matter of course, but at least you did something that most people wouldn't do and that was to actually walk through the work to see if you understood where the equations came from. So good on you.