# ECE4893A: Analog Circuits for Music Synthesis

## Spring 2015

## Homework #4

## Due: Friday, March 13, by 4:30 PM

This homework will be graded out of 100 points.

**Turn-in procedure:** You can slip it under my Van Leer 431 office door

by 4:30 on Friday, March 13.

Of course, I will accept it earlier, such as in class on March 12.

**Ground rules:** You are free to discuss approaches to

the problems with your fellow students, and talk

over issues when looking at schematics,

but your solutions should be your own. In particular, you should never

be looking

at another student’s solutions at the moment

you are putting pen to paper on your

own solution. That’s called “copying,” and it is *bad*.

Unpleasantness,

including referral to the Dean of Students for investigation,

may result from such behavior.

**In particular, the use
of “backfiles” of solutions from homeworks and quizzes assigned in
previous offerings of this course is strictly prohibited.**

## Problem 1

Check out

Ray Wilson’s old Voltage Controlled Low Pass Filter (Four Pole 24db/Oct):

The input and feedback resistors are 100K; it looks like the divider is

made with a 1K to ground.

In parts (a) and (b), we will consider the gain of just

**one** of the filter stages,

either the second, third, or fourth (they are all the same; I’m not

including the first one so we can avoid talking about

the effect of the resistor coupling

in the resonant feedback loop while working (a) and

(b)).

a) Assume that the transductance gain

of the OTA is 19.2**I_con*, where *I_con*

is the current flowing into the

control pin of the OTA.

What is the cutoff frequency of

the filter block in terms of (*I_con*) in Hertz?

(Remember that the transconductance gain just takes the place of

1/R in the usual single-pole cutoff freuqency calculation, and for convenience

we include the scaling of the resistive divider as part of the transconducance

gain. You may use the various approximations that I used in class to reduce

the resistive divider to a R_small/R_big type of multiplier. I want

to emphasize that you don’t need to rederive anything; you can feel free

to use the final results from lecture.)

b) Given the result in (a), what value *I_con* would be needed for the

cutoff frequency of one stage to be 1000 Hz?

c) Now let’s consider the full four-pole cascade with feedback level

denoted as K, as in lecture. Let the

cutoff frequency of a single stage be 1000 Hz. Using MATLAB, Mathematica,

Maple, or some similar tool, on the same plot, show the

magnitude of the frequency response (with the horizontal axis in Hertz),

from DC to some value that you think best shows off the curves, for four

cases: K=0, K just big enough so that you can just barely see a resonance

“bump” in the curve, K close to 4 (but not so big that it swamps your other

curves), and a K somewhere between the last two cases that you think is

interesting. **Make sure the value at DC corresponds with the results computed
by the simple formula derived in lecture.** Please include your computer

code and give me a real printout of your curve, i.e., not something

sketched by hand.

## Problem 2

In this problem,

we’ll

look at the

VCF in the Minimoog (at least the

version linked here; the Minimoog went through several revisions). The

scan I am linking to was posted by someone calling themselves

Fantasy

Jack Palance; this FJP person has

posted

a

lot more info about it, including the story about how

they acquired their Minimoog.

If you don’t see a specific unit on a capacitor, there’s usually an implied

“microfarads.”

a) The Moog transistor ladder VCF contains a cascade of four

one-pole lowpass filter sections. In class, I presented an analysis of a

hypothetical

“one-sided” ladder

to get a feel for how this kind of circuit works. The rigorous analysis

of the actual “two-sided ladder” is a bit more complicated. We’ll rely on

the analysis in Tim Stinchcombe’s

Analysis

of the Moog Transistor Ladder and Derivative Filters, particularly

Equation 13 on page 12. Using this formula,

find the cutoff frequency (in Hertz) of

*one* of those sections in the Minimoog’s transistor ladder

as a function of the control current

being pulled from the tied emitters of the transistor pair that feeds

the ladder.

(Note that when analyzing the Moog VCF, we don’t include a

resistive divider in the gain as we’ve done in OTA-C filter

cutoff computations;

there is typically a resistive divider right at

the first input, but it’s not important for our frequency-domain analysis.)

b) Let’s do some DC analysis.

At DC, the caps are open circuits.

For the purpose of this analysis, ignore R76 (the 330 ohm resistor) and

the R73 (the 1K regeneration calibration pot); treat them as “open” too.

Supposing that the transistors draw negligible

current through the bases, what are the voltages

at the bases of the four stages of the

ladder? (Number the stages 1 through 4, from bottom to top; the first stage

is Q23/Q24, the second is Q19/A20, the third is Q10/11, and the fourth is

Q2/Q3).

## Problem 3

Problems 1 and 2 dealt with a fairly popular lowpass filter configuration,

namely a cascade of four identical single-pole lowpass filters with

negative feedback.

The VCF in the

Elka

Synthex

(see this demo

by Paul Wiffen, one of the original Synthex sound programmers)

permits this configuration, but it also allows

the musician to select a configuration with a cascade of two identical

single-pole lowpass filters with negative feedback. We will mathematically

explore this configuration. Assume each single-pole filter stage

has a transfer function of w_c / (s + w_c), where w_c is

the half-power cutoff point of a single stage (in radians). (Here, interpret

the letter w as the greek letter omega).

a) Assuming a negative feedback factor of K, find the closed-loop

transfer function of the complete filter. Write your answer so that

it has w_c^2 in the numerator and a quadratic polynomial in s

in the denominator. Make sure the highest power of s has unit coefficient,

i.e. a coefficient of 1.

b) In class on March 5, we looked at a canonical lowpass 2nd-order filter

transfer function

of the form w_n^2 / [s^2 + (w_n / Q) s + w_n^2], where w_n was the “natural

frequency” and Q was the “quality factor.” If you put your answer to

(a) in this form, what are w_n and Q in terms of w_c and K?

c) What are w_n and Q for the special case of K = 0 (i.e. no feedback)?

d) For what values of K does the filter exhibit a resonance bump,

i.e. for what values of K is Q > 1/sqrt(2)?

e) Using the quadratic formula, find the locations of the

poles of this filter as a function of w_c and K.

f) Where are the poles for the special case of K = 0?

g) Describe how the poles move as K increases.

h) Can this filter be made to self-oscillate like the

four-pole-casade-with-feedback filter explored in the pervious problems?

In other words, is there a value of K > 0 for which the poles can be made to

lie on the imaginary axis?