Ground rules on this homework: You may verbally discuss approaches to the problems with each other while looking at the schematics, and are encouraged to do so; but you may not look at each other's written solutions or ask "what did you get on part XYZ of problem ABC." (In future homeworks, I will allow varying degrees of explicit collaboration on certain problems.)
Below, I will use underscores to indicate subscripting.
While four-pole lowpass VCFs are extremely common in synthesizer designers, four-pole highpass VCF are relatively rare. The one ones I know of are the Moog 904b, which is a complicated discrete-component design, and the Polyfusion highpass VCF. Grab the schematic of the Polyfusion lowpass VCF from here. (Interestingly, Polyfusion was started by several ex-Moog employees; they had to come up with their own 4-pole VCF design since Moog had a patent on his!)
Compute the cutoff frequency (in Hertz) of a single one-pole stage of the polyfusion highpass VCF as a function of the current at the control input of the OTA. We'll make all our usual approximations: ideal OTA, big resistor in parallel with small resistor may be approximated as the small resistor, etc. Treat the JFET with the 1K and 10K resistors as just forming a perfect voltage buffer; in a modern redesign, I'd probably replace that with a TL07x or some other JFET-input op amp. (Ignore the 741 near the input; that's just adding some buffering. I just want to focus on one of the OTA filter cells.)
The quickest way to do this problem is to remember that the OTA transconductance gain is taking place of 1/R in the usual RC highpass filter design, and to just include the attenuation of the resistive divider in the transconductance gain.
In class, we looked at the behavior of second-order lowpass and bandpass filters. In this problem, we'll look at the behavior of second-order highpass filters. Consider the transfer function:
s^2
H(s) = -------------------------
w_c
s^2 + ----- s + (w_c)^2
Q
a) What is the gain of this highpass filter, i.e. what is the magnitude as w goes to infinity? (This should not require extensive calculations.)
b) Find |H(j w)|^2. You may express your answer in some convenient form of your choosing.
c) Where is the half-power point in terms of w_c and Q, i.e., for what w (let's call it w with a 1/2 subscript) is |H(j w)|^2 = 1/2?
d) For what values of Q does |H(j w)|^2 exhibit a peak? For such Q, where is the peak located in terms of w_c and Q?
(If you get stuck on this problem, rewatch the lecture on second-order transfer functions.)