ECE4803B: Theory and Design of Music Synthesizers
Due: Wednesday, March 15th at the start of class
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
ideal OTA, big resistor in parallel with small resistor may be approximated
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
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
In class, we looked at the behavior of second-order lowpass and bandpass
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,
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
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