# ECE4803B: Theory and Design of Music Synthesizers

Fall 2006

Homework #4

Due: Tuesday, October 24st at the start of class

Below, I will use underscores to indicate subscripting.

### Problem 1

Carefully read over Problem 1 of Homework #4 from the Spring 2006 offering,

and then carefully

read over the solutions. You will use the results given

there in this problem.

Pick one of the three circuits given in last Spring’s problem based

on the last digit of your GT ID number (instead of the year you were

born; that didn’t give me an even distribution).

a) Using the approximation given in part (c) of last year’s problem)

(as well as other useful results from the solutions to that problem,

what control current would give a single-stage cutoff frequency of

2000 Hz?

b) Using the control current you found in part (a), what cutoff

frequency would you compute if you now used the approximation given

in part (b) of last semester’s problem?

c) Again using the control current you found in part (a), what

cutoff frequency would you compute if you now used the non-approximate

analysis from part (a) of last semester’s problem?

I’ll say it again: exploit the results from last semester as much

as possible! There’s no need to reinvent the wheel.

### Problem 2

While doing some google searching, I ran across

this page on Moog

circuits by Theo Verelst. Scroll down to the “VCF Circuit” section,

and you’ll see he’s run some SPICE simulations (when I say SPICE, I

generically mean any circuit simulation software – if you dig below

the fancy graphical front ends, they have some sort of SPICE-like engine

running underneath), and he even provides his file! (Every Moog synth had

a different variation of the Moog filter; I’m not sure what exact

circuit he’s basing his simulation on.)

Download his Microsim schematic file and see if you can get

it running in your simulation

favorite software. If you can’t, see if you can get a free version of

this “Microsim” program and run it. If neither of those options works,

well, you may just have to re-enter the schematic yourself in your

favorite program.

Make one change to the schematic – break the connection at the 10 microfarad

cap, so we have no resonance.

If the strange “E1” differencing block is giving you trouble in your

software, basically all you need to do is subtract the voltage at the

collector of Q14 from the voltage at the collector of Q19; your software

must have some way of doing that.

a) Let’s start with some hand-written simplified DC analysis. At DC,

the caps are open circuits. 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). Also, what is the voltage at the base of the initial differential

pair? (This sets the DC bias point of the input.)

b) What does SPICE tell you the DC values at the bases of the transistors

are? (Use the same numbering scheme as in part (a)).

Qualitatively speaking, how

close are these values to what you found in part (a)?

c) Read over Problem 3 of Homework 6 from the Spring 2006 offering, and

then read over the solutions. Using the formula given for the cutoff

frequency of a single stage, what control current (I1 on the schematic)

would correspond to a single-stage cutoff frequency of 4 kHz?

d) Using the control current found in part (c), use your SPICEs

frequency domain (i.e. AC)

analysis capabilities to plot the magnitude of the

frequency respone of the filter from 100 Hz to 20 kHz.

e) Read over Problem 2 of Homework 4 from the Spring 2006 offering,

and then read over the solutions. This should tell you the half-power

point of the full four-pole cascade, given that each individual stage

has (allegedly) a half-power point at 4 kHz. How close is the half-power

point found from your graph in part (d) to this predicted half-power point?

f) The AC analysis SPICE did in part (d) made all sorts of linearizing

approximations. In this last part, let’s look at the nonlinear behavior

of the filter when driven by a strong signal. Using the same control

current

found in part (c), input a 1 kHz sinusoid. Experiment with different values

of the sinusoid amplitude. What is the smallest amplitude you find where

you can just start to see non-sinusoidal distortion in the output?

(This is a subjective question.)

Provide a graph of the output for this

value of the input amplitude.

Also provide a graph of the output for an input amplitude value

that makes the output be “obviously” distorted.

Include copies of relevant SPICE-related things (the schematic, etc.)

### Problem 3

We spent a lot of class time looking at four pole filters that had

four buffered low-pass sections in cascade,

each with the same cutoff frequency,

with a negative feedback loop around the whole cascade. This included

OTA-C style designs (as examined in Problem 1 above) and Moog BJT-C

ladder designs (as examined in Problem 2) above. The core of such

filters had an open-loop

transfer function that looked like 1/(s+1)^4, where for

simplicity of expression I’m normalizing the single-stage cutoff to

be omega_c = 1. We saw how applying negative

feedback gave it a more complicated form that allowed resonant

behavior.

We also briefly looked at diode-C style filters, such as in the Roland

TB-303 (where transistors are connected to act as diodes) and the

EMS Synthi. (Interestingly, there is one Moog-manufactured synth

that uses a diode-C ladder instead of a BJT ladder, namely the

Moog Sonic Six.) This effectively forms the equivalent of four

*unbuffered* R-C stages.

One day in class I briefly held up the five pages of

algebra it took me to find the open loop transfer function of such

a diode-C filter: 1/(s^4 + 7 s^3 + 15 s^2 + 10 s + 1).

Write a program (I would recommend using MATLAB) to plot the

magnitude of the frequency response of the OTA/BJT-C and diode-C

style filters. We’ll keep the using omega_c = 1 normalization; experiment

with the range of omega you plot so you get nice interesting graphs.

Set up your program so both plots appear on the

same graph.

a) Make the plot for the case of no feedback. What do you notice as

the main qualitative differences between the two curves?

b) Via experimentation, gradually increase the value of K until you

can just barely see a resonance “bump” appear in the OTA/BJT-C

graph. What value of K did you find? Again give the plots, and

describe what you observe.

c) Via further experimentation, gradually increase the value of K

until you can just barely see a resonance “bump” appear in the

diode-C graph. What value of K did you find? Again give the plots,

and describe what you observe. (If the K you find happens to be greater

than 4, then you may have to resort to only plotting the diode-C curve!)

d) Make a nice computer-generated complex plane plot showing

the locations of the poles for the diode-C filter

associated with parts (a) through (c). (You may want to

use different kinds

of marks to clearly

differentiate the three cases.) By experimenting with some

other values of K and looking at the locations of the poles on the screen,

make some hand sketches on your plot showing the trajectory of the poles

as K increases for the diode-C filter.

(Aaron’s aside: I tried to answer these questions analytically, but in

every case I was defeated by the amount of messy algebra that started

piling up.)