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.)