Thanks Guys! I'm surprised I got as close as I did, taking as many shortcuts to avoid the math as I typically do! 0.08Khz difference... so excruciatingly close and yet so far
. I really need to brush up on my complex analysis and transforms but as the working has been kindly laid out, it's made it a bit easier for me to get started
Megamox
Indeed, the difference is rather small. However, as you have indicated to "brush-up" on your complex analysis perhaps you are interested at which step in your calculation you have introduced an error.
Here is my explanation:
At first, without mentioning you have assumed that the phase of the product (gain*feedback)=loop gain will be real and unity (equivalent to my setting Im=0, R=1 in my calculation).
That`s what I also have done in my
first approximation (post#11). This is a simplified but reasonable approach as long as the feedback lowpass as well as the opamp work far enough above their (first) pole frequency.
That means, you have calculated with magnitudes only (like X=1/wC) . This works as long as you have to deal with multiplications or divisions only.
However, your derivation causes one single expression that is a sum, which combines a real and an imaginary part (1+R/X=1-jwRC).
At this point, you were not consistent because you have neglected the „j“, since the correct magnitude is SQRT[1+(wRC)^2].
Thus, this is the only difference between your formula and the correct one.
The error introduced, however, is pretty small because
SQRT[1+(wRC)^2]/(1+wRC)=0.9987 >>>> error of 0.13%.
That is the reason, Roff has found that the frequency according to your formula is slightly lower than using the correct formula.
With regards
W.