EMS 2011 – Homework #3

EMS 2011 – Homework #3

ECE4893A: Electronics for Music Synthesis

Spring 2011

Homework #3

Due: Friday, March 18 at the start of class

This homework will be graded out of 100 points.

Background music: The

Buchla Music Easel
which consists of a Buchla 208 Programmable Sound Source and a
Buchla 218 Model Keyboard together in a single case, is one of the rarest
and most coveted of the Buchla designs. To put yourself in the right frame
of mind for this homework, watch this
featuring Charles Cohen,
who performs live exclusively using a Music

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 lame.
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.

Late penalty: If you show up really late, suggesting that you
were doing the homework during class, then I’ll take off up to 10
points based on my mood. If you turn it in on Monday, March 21, by noon
(since this will be during Spring break, we won’t have class, but you
can slip it under the door to my Van Leer 276 office; please e-mail me
to let me know if/when you do this),
I will take off 30
points. This isn’t to be mean – it’s to encourage you to get it turned in and
move on to whatever other work you have to do in other classes, even if
it’s not perfect – but also to encourage you to go ahead and do the work
and turn it in and learn some stuff and get some points,
even if you’re past the deadline. I won’t accept it after class on
since I will send out solutions shortly after. (If you have some severe
extenuating circumstance, i.e., family emergency, major health issues,
date, etc.,
e-mail me and we will work something out. On the other hand, if
you know ahead of time that you won’t be able to come to class
because of something like a job interview,
please plan to work in advance and get your homework to me ahead of time.)

Problem 1

In class, we looked at a nonlinear “folding” circuit used in
Ken Stone’s
Cat Girl Synth Wave Multiplier. To find the schematic, go to
click on “Modules,” and click on “Wave Multiplier” (be sure it just
says “Wave Multiplier” by itself; don’t click on the “Saw Pitch Shifter/Wave
Multiplier”), and then click on “Click Here for the Schematic.”
folding nonlinearities are at the bottom of the page. Note that Ken uses
four in series (unlike the six in series like the Serge Wave Multiplier
uses). The last one has some additional diode clipping action, but we’ll
ignore that.

Let’s consider one of the first three stages. Use your favorite implementation
of SPICE (OrCAD, Multisim, LTspice, whatever…) to run a
simulation of
one of the stages (10K resistors from input to each of the op amp terminals,
10K resistor in negative feedback configuration, and two 1N4148 diodes, facing
different directions, in parallel from the positive terminal to ground). Be
sure to use a 1N4148 model (if one isn’t built into your SPICE, let me know)
and not some sort of “idealized” diode.
Make a plot of the output voltage vs. the input voltage for input voltages
ranging from -1.5 to 1.5 volts. Does the nonlinearity exhibit a sharp
corner, as my handwaving analysis in class suggested, or does it have a
more rounded corner?

Be sure to provide some sort of printout “showing your work,” i.e.
a SPICE schematic or netlist (if you’re into typing your own netlists
by hand).

Problem 2

In class, we looked at the
“timbre” nonlinearity implemented in the
Buchla 259 Programmable Complex Waveform Generator
A similar timbre generator circuit is used in the Buchla Music Easel described
You can print out
the schematic from
Buchla page
search for the “B2080-9A” “Complex Oscillator 3/3” link.
You’ll see five of those “Buchla diodeless deadband” circuits.

Let’s analyze the third one from the top,
which consists of an op amp and
R31, R34, and R35
Calculate the positive edge of the

(i.e., what is the largest input voltage for which the output stays
zero?), and
calculate the slope of the output/input curve past that point.
As in lecture, let’s define the “output” as the voltage at the negative input
of the op amp forming the deadband circuit,
and the “input” as the voltage at the output at the op amp
just above resistor R20 on the schematic. You may
adapt the formulas we derived in class; you don’t have to
do them from scratch.

Important warnings:

  • Buchla sometimes has two kinds of grounds, denoted Q (quiet, for audio
    signal paths) and N (noisy, for digital logic, etc.)

  • Remember in Buchlaese, that when two lines cross without a dot, they
    don’t electrically connect; when two lines meet at a T-intersection without
    a dot, they do electrically connect.

  • The Buchla 259 used CA3160 op amps, which enjoy “rail to rail” output
    swings due to their CMOS output stage, run with “voltage starved” supplies of
    6 V and -6 V. The Easel appears to use RC4136’s
    instead, and although the power supplies are not explicitly marked, I’m
    told they run off Buchla’s
    usual +15 V and -15 V. With the exception of one JFET,
    the rest of the circuit for the RC4136 shown on
    seems to be all bipolar,
    so I doubt it can do the “rail to rail” business that the CA3160 can.
    Elsewhere on the sheet, I see that the “maximum peak output voltage swing”
    is listed as being “minimum +/- 12 V” and “typical +/- 14 V” for a 10K
    load. The resistors I see on the sheet are all higher than 10K,
    suspect they’re running more towards what’s listed as “typical”. Looking at
    the schematic on the
    datasheet, I see that the output is sandwitched between two BJT’s between
    the supply rails, so there’s at least a diode drop there from the possible
    output to the rails. So… let’s use -14 V and 14 V as the output voltage
    limits (as opposed to the -6 V and -6 V volts we saw in the case of the
    259). If you’re an ECE3050 guru and have reason to pick different output
    voltage swing range, please go ahead and use
    it and tell me your reasonsing!

  • Notice a few of the “resistors” are actually a couple resistors in
    parallel. (Do you get the impression that Buchla might have started with
    a basic design, and then tweaked it by throwing in a few more resistors
    here and there?)

Interestingly, the 259 had both “timbre” (amplitude of sinewave going in)
and “symmetry” (DC offset on sinewave going in) controls; the Easel appears
to just have a timbre control.

Problem 3

Now let’s look at the triangle-to-sine waveshaper in the Buchla Music
Easel. Once again, check out
Buchla page
, and search for the
search for the “B2080-8A” “Complex Oscillator 2/3” link. The
waveshper consists of R37, R36, R38, R39, D1, D2, and Q1. Don’t worry about R40
and all the other associated “sine adjust” circuitry. The input to the
waveshaper is at the “top” of the shaper, at the junction of R37, D1, and R39.
The output of the waveshaper is the the junction of Q1 and R38.

I don’t have a good understanding of how this circuit works; in such cases I
often turn to SPICE.
Use your favorite SPICE implementation to plot the output of the waveshaper,
sweeping the input linearly over a symmetric voltage range
(i.e from some -V to some +V).
Experiment with the input range until you find something that looks like half
of a sine wave. To small of a range will make the output look like a line;
to large of an range will make the the peaks of the output sine wave look
too wide.
(You can “eyeball” this; there’s no single perfect right answer.
It’s a judgement call.) Include an input/output voltage
plot of this specific case.

The diodes are 1N457s, and the transistor is a 2N4339. If you don’t have
1N457s in your SPICE library, try a 1N4148 or 1N914, and if you don’t have
those, try some other silicon diode model. If you don’t have a 2N4339 in your
SPICE library, try any other N-channel JFET.

Be sure to provide some sort of printout “showing your work,” i.e.
a SPICE schematic or netlist (if you’re into typing your own netlists
by hand).

Problem 4

Check out
Ray Wilson’s 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. (I find it interesting that he chooses to use
TL084 op amps as buffers instead of the buffers built in to the LM13700.
Maybe this is to avoid
having to deal with the weird 1.4 V drop you get from the LM13700 buffers?
The TL084 also are probably better quality than just the simple Darlington
pair in the LM13700.)

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

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 computer
by the simple formula derived in lecture.