ECE4893A: Analog Circuits for Music Synthesis

Spring 2016

Homework #2

Due: Thursday, Feb. 11, at the start of class

This homework will be graded out of 100 points.

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 expressly forbidden. Look, here I am
expressing how forbidden it is. Forbidden! Forbidden!!!

Late penalty: If you show up really late, suggesting that you
were doing the homework during class, then I may take off up to 10
points based on my mood. If you turn it in by 4:00 on Friday, Feb. 12,
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 after that deadline on Friday
since I will send out solutions shortly thereafter.
(If you have some severe
extenuating circumstance, i.e., family emergency, major health issues,
court 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

Now, my young padawans, we will delve into the mysteries of the triangle
core of the Principle Oscillator of the
Buchla 259 Programmable Complex Waveform Generator.
The Bergfotron
circuit
we saw in class is based on this, although
the Buchla original is more complicated. I had to get advice
from many folks to figure this out, so I’ll guide you through it step by step.
Head to

Magnus’s Buchla page, and get the schematic
for the “B2590-2A” “Principle Oscillator.”
I’d recommend zooming in on it and
and printing just the part of the diagram you need. Reading Buchla schematics
takes some practice. Most schematics drawn by anyone besides Don Buchla
have most op amp outputs pointing towards
the right; Buchla draws his pointing all over the place, and he usually
doesn’t bother labeling the negative input.
In Buchlaese, 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. He often uses a “0” for ground instead
of the usual ground symbol. He also
sometimes has two kinds of grounds, denoted Q (quiet, for audio
signal paths) and N (noisy, for digital logic, LED power supplies, etc.)

IC 20 is a TL082, which has the high input impedance (and hence low
unwanted input currents) that you’d want in an op amp playing the role of an
integrator.
Note that Buchla uses a voltage divider to put 7.5 V on the positive input
terminal of IC 20, likely for biasing purposes.
Anyway, you don’t need to worry about it; this 7.5 V will not show up in
any of your answers. (Remember that
there’s nothing that guarantees that any integrator starts out with an output
at zero volts when you fire up a synth anyway).

IC 21 implements a special feature that allows you to lock this
oscillator to another oscillator; this feature will be unused in our analysis,
so assume that the output of IC 21 is zero volts.

Fasten your seat belts – this may get bumpy:

a) Let’s look IC 22, which is an LM311 comparitor.
C20A (220 pF) and R164A (10K) seem like the usual goo you find hanging off
of an LM311. Let’s ignore C20A and R164A, treat the comparitor as ideal
(i.e. effectively infinite input impedance), and image that the output of
the comparator is tied to its + terminal.
The LM311 has an
“open collector” output. If the voltage at the
– terminal is greater than the voltage at the + terminal, the output is
forced to the negative supply rail of the LM311, which here is 0 V. If the
voltage at the – less than the voltage at the + terminal, then the LM311
“lets go” of the output, and the output goes to the voltage set by the
resistive voltage
divider consisting of R155 (4.99K to ground), R156 (10K to 15 V),
and R164 (24K to 0 V).

Show that these assumptions give 4.39 V, unlike the 4.29 V shown
on the schematic. Let’s use that 4.39 V number, since it gives a
better-centered triangle in the end. 🙂

b) The triangle wave output is taken from the output of IC 20, which forms
the integrator along with C20.
Note that the output of the integrator is fed to the the – terminal of IC 22
via a resistive divider (R153, 4.99K from the output of the integrator
to the – terminal of the comparator, and R154, 28.7K from the – terminal of
the the comparator to the +15 V supply). (When I first did my analysis, I
couldn’t read the 28.7K; I thought it was 20.7K or 26.7K and came up with
strange answers. After some confusion, I contacted People In The Know and
found out that it’s 28.7K.)

Let’s call the output of IC20 “vtri” (where “tri” is a subscript). What
values of vtri correspond to values of 0 V and 4.39 V at the – input to
the comparitor? This tells you what voltages the triangle wave swings
between.

c) What is the DC value of the triangle wave? What percentage is this
relative to the full peak-to-peak swing of the triangle wave?

d) Pretending for a moment that Q7 had infinite input impedance at its
base, find is the voltage at the base of Q7 defined by the resistive divider
consisting of R131 (let’s suppose the hard to read smudge is 22K) and R132
(100K).
Note that
the difference between 0 V and this value, and 4.39 V and this value, are
both well outside of the +/- 10 mV “linear” range of most OTA. Let’s call
the current flowing into the collector of the right transistor of the Q5 NPN
matched pair (that’s an AD811, which is nearly impossible to find now;
if you were trying to build your own Buchla 259, you’d want to use
something like a MAT02, SSM2210, LM394, or LS318 instead) “Icon” (where
the “con” is a subscript); let’s also suppose that the OTA is being driven
so hard that it essentially acting as a switch (we’re so far into the
“tanh” function that it’s approximately one), so a current of Icon is either
flowing into out out of C20, depending on which way the triangle is going.
Note that Q7 will typically sink some current through it’s base, so the voltage
won’t be what we computed, but our overall conclusion, that the OTA is
operation in saturation, still applies.

e) No units are specified for C20, but after looking at many other VCO
designs, I strongly suspect that C20 is 4.7 nF; let’s use that value.
Find the Icon that would
result in a triangle wave with a pitch of 261.63 Hz, which is “middle C.”
Remember that for a full period of the triangle wave, the wave has to travel
both up and down.

Our analysis above made a lot of assumptions; in particular, there’s probably
some current sunk through Q8, and probably a bit of current sunk through the
+ terminal of the comparitor (IC22), so the top voltage of the square wave
is probably more like 4.29 V than 4.39, making the DC offset worse;
but considering the tolerences on all the various components, this isn’t that
big of a deal.

You always need a few trimpots here and there to get VCOs in tune.

Problem 2

In Problem 1,
we looked at the triangle VCO core of the
Buchla 259.
That oscillator is designed to operate at audio rates. In this problem, we
will look at a voltage-controlled VC “low frequency oscillator” (LFO), which
is a particular kind of VCO.
Although some LFOs can run at
it can run at lower audio frequencies, they’re typically not designed
with the rigorous requirements needed to play “in tune.” Instead, they’re
usually intended to provide control voltages to control other parameters
(such as the pitch of an audio VCO to create a police siren.)

Let’s look at one of
Ray
Wilson’s old VC-LFO designs.

The main triangle core is on the left half, midway between the
top and the bottom. C11 and U2-A form the integrator. (I’m not sure
why the R22 is there, so let’s ignore it in our analysis). Let’s call
the output of U2-A (pin 1)
V_tri (I’m using the underscore to indicate a subscript).

Notice that U4-A is not being used in a negative feedback configuration,
so the “golden op amp rules” do not apply. U4-A is being used as a comparator,
so the output of U4-A (pin 1) will try to snap to the
positive supply (+12 V)
if the voltage at pin
3 is greater than pin 2, and try to snap to the
negative supply (-12 V) otherwise. Now, in reality, the TL082 is not
a so-called “rail to rail” op amp. Take a look at simplified schematic
of the National TL082 datasheet on my
Datasheet
Archive –
you’ll see there’s
a NPN BJT between the output and the positive supply and a PNP BJT
between the output and the negative supply. Hence, we’d expect that the
output could swing to at most within a “diode drop” of the supply lines
(in this case, assuming a 0.7 V diode drop, -11.3 V to 11.3 V). Based on
the “output voltage swing” line on the datasheet, I’m guessing it’s closer to
something like to within 2 volts of the supply. So, let’s suppose that
the comparator outputs +10 V or -10 V.

Let’s suppose that the Tri Skew trim pot is set to the middle. Assume that
the diodes are either “off” (in which case no current flows through them) or
“on” (in which case we’ll assume a “diode drop” of 0.7 V).

Assume the OTA has infinite input impedance. Ignore the C10 cap in the
feedback loop of the comparator op amp (U4-A).

a) When the output of the comparator is +10 V, what is the
voltage at the positive
input terminal of U3-A?

b) When the output of the comparator is +10 V,
using the nonlinear “tanh” model for OTA behavior, what is the output
current of the OTA as a function of the current control input (pin 1) of
the OTA? (Note that unlike the Buchla 259 VCO circuit we looked at in
class, the OTA here does not seem to be fully saturated.)

c) When the output of the comparator is +10 V, what voltage at the output
of the integrating op amp (pin 1 of U2-A) would cause 0 V to appear at the
positive terminal of the comparator op amp (pin 3 of U4-A)? (Note that
this will tell you the maximum level of the triangle wave).

d) What is the frequency of the triangle wave as a function of the
current control input (pin 1) of the OTA?

e) Take a look at the TRI output in the middle of the page (pin 2 of R15).
What is the output impedance of the TRI output?

Problem 3

In class on Tuesday, Feb. 2,
we looked at sawtooth VCO core designs. Let’s look at
Ray
Wilson’s 1V/Octave Voltage Controlled Oscillator.
This is a very complicated circuit, so we’ll rely on Ray’s thorough
description.

Check out the LM394 in the
Page 1
schematic; this forms the core of the exponential
converter (note Ray later discovered that
the SSM2210 works better). Call the current
flowing into pin 1 of the LM394 “I_{freq}.” (Hint: you may use Ray’s
“1.1 volt” figure.)

(a) Given Ray’s description of the circuit operation, find the frequency of
the oscillator in Hertz in terms of I_freq. (To make things easy, assume
the reset time is finite.)

(b) Given you result in part (a),
what value of I_{freq} would generate a 261.63 Hz tone?