# ACMS 2016 – Homework #3

ACMS 2016 – Homework #3

# ECE4893A: Analog Circuits for Music Synthesis

## Due: Wednesday, March 2 at 3:30 PM

This homework will be graded out of 100 points. Please turn it in
to my VL431 office.
Of course, you may turn it in ahead of time if
you wish.

Background music: The original
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.
(It has recently
been
re-released
.)
To put yourself in the right frame
of mind for this homework, listen to this
video
featuring former Nine Inch Nails synthesist Alessandro Cortini
performing on his Music Easel.

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.

Sorry, no late turn-ins accepted on this one: Since we’ll have Quiz 1 on
Thursday, March 3, I want to be able to get solutions up quickly (although
this material won’t be on Quiz 1, per se.)

## Problem 1

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
above.
You can print out
the schematic from
Magnus’s
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 topmost deadband circuit,
which consists of an op amp and
R25, R27, and R26.
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:

• 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
the
datasheet
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).

• 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?) It’s not relevant for the exact problem I asked; I just
thought this was an interesting observation.

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 2

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) through (g), 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 the
effect of resistor
coupling in the resonant feedback loop while working (a) and
(b)).

a) Find the voltage at the input terminal of the OTA in terms of the
voltage at the output of the buffer and voltage
at the input of the filter block. Don’t make any approximations concerning
the resistors (i.e., if you use superposition, note that you must
compute the value of the little resistor in parallel with the
big resistor to solve this.)

b) In class on Thursday, Feb. 18,
I attempted to use vigorous handwaving to
to convince you that part (a) could be approximated as

v_at_ota = (v_input + v_output) *
(little_resistor / (little_resistor + big_resistor))

Comment on how close this approximation is to what you found in (a).

c) In class, I used even more vigorous handwaving to attempt to convince you
that part (a) could be further approximated as

v_at_ota = (v_input + v_output) *
(little_resistor / big_resistor)

Comment on how close this approximation is to what you found in (a) and (b).

d) 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, using the approximation
in part (c)? (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.)

e) Given the result in (d), what value I_con would be needed for the
cutoff frequency of one stage to be 3000 Hz?

f) What single-stage cutoff frequency would you compute if you used the
I_con you computed in (e), but you used the no-approximation
technique of part (a)?

g) What single-stage cutoff frequency would you compute if you used the
I_con you computed in (e), but you used the approximation in part (b)?
Comment on how close the cutoffs computed in (f) and (g) are to 3000 Hz.

## Problem 3

In class on Tuesday, Feb. 23, 2016, we explored the consequences of cascading
four one-pole lowpass filter and adding a negative feedback loop
with a feedback gain of k. The individual one-pole filters each
had the transfer function H_1(s) = wc/(s + wc).
We showed that the maximum
usable k was k=4, at which point the filter would self-resonate.

In this problem, we will take a look at the same structure, except instead
of one-pole lowpass filters, we will cascade four highpass filters with
the transfer function H_1(s) = s / (s+wc).

a) Find H_4(s), the
transfer function of a cascade of four one-pole highpass
filters described above.

b) Find H_4F(s),
the closed-loop transfer function of H_4(s) with a
negative feedback loop with feedback gain k.
You need not expand out
terms of the form (s+wc)^4;
that just makes the expression more complicated.
(Unlike in the lowpass case, I haven’t been able to find a closed-form
solution for the pole locations in this highpass case. From my numeric
studies, it does seem like the maximum usable k is 4, at which point
the filter self-resonances. Also, there are four zeros at the origin in
the highpass case, which of course are not present in the lowpass case.)

c) Asymptotically, what is the value of
the frequency response H_4F(jw) as w approaches
infinity?

d) Now let’s do a numerical experiment with
the consider the full
k,
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 values at infinity
corresponds with the results
seem reasonable relative to the results of
the simple formula derived in part (c)
.