# ECE4803B – Homework #5

ECE4803B – Homework #5

# ECE4803B: Theory and Design of Music Synthesizers

## Homework #5

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.

### Problem 1

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.

### Problem 2

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

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.)