# ECE4803B: Theory and Design of Music Synthesizers

## Spring 2006

## 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

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.)