ACMS 2016 – Homework #1

ACMS 2016 – Homework #1

ECE4893A: Analog Circuits for Music Synthesis

Spring 2016

Homework #1

Due: Thursday, Feb. 4, at the start of class

This homework will be graded out of 100 points.

Suggested references:
Semiconductor Application Note 31

(or pretty much any textbook that has op amp circuits in it)

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. 5,
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

Let’s look at the “AUDIO AC IN” of the
Lin/Log VCA
The input capacitor (470 nF) and the two resistors (68K and 330 ohm)
form a single-pole highpass filter whose “output” is measured by the
positive input terminal of the CA3080.
Find its cutoff frequency (the half-power
point) in Hertz
(so don’t forget to divide your cutoff in radians/second
by 2*pi). Note that the
resistor in series with the cap gives this configuration a non-unity
gain DC, so “half-power”
should be interpreted in terms of half of the power
at DC.
As usual, assume that the input is being driven by an ideal voltage
source. (Note that the 22K resistor is not relevant to this problem;
it’s there to make sure the input stays centered at ground so there’s
not a “pop” when the user plugs in a cord.)

Problem 2

In this problem, we will analyze the PAiA 9710 VCA. The
is available here
mirrored here),
although it is in squintovision, so
I would recommend opening it up in some sort of image viewing program
so you can zoom in a little.
The 9700
runs off a +/- 12V power supply, i.e. V- on the schematic means
-12V, and V+ means +12V.

The 9710 actually contains two VCAs, along with a “balanced modulator” (a
full quadrant multiplier, which is good for special effects.) We will
look at the “right” VCA. You will find the core of it along the
bottom part of the diagram. J2 is the audio input, and J11 is the output.
The circuit will we look at uses one-half of a dual OTA, the LM13600; the
one of interest to us is labeled IC5:B.

R45 and R43 create a slight voltage bias to compensate for non-ideal
characteristics of the LM13600.
Real OTAs have an output bias current, which vexingly varries with the
control current. The 9710 uses a fixed resistor network to put a small
bias voltage to try to correct for a “typical” output bias current. Many other
designers use a trim pot for this. In any case, we won’t worry about such
non-ideal effects here, so you may assume the OTA is an “ideal linear
OTA” and that the + terminal of IC5:B is at ground.

The current generating circuitry for the OTA may be found in the upper
left corner of the schematic. R14, R16 (at least I think it’s R16 – it’s the
220K resistor below R14), R12, R18 (at least I think it’s R18 – it’s
a 1500 ohm resistor to round, IC2:C (an op amp), and Q2 form a current
source similar in spirit to the one from Chamberlin, p. 203 that I presented
near the beginning of lecture on Jan. 20, so if you have difficulties
analyzing it, you will want to review your notes from that day.
R44 is a small current limiting resistor
to make sure the OTA doesn’t become an ex-OTA. D2 is for protection; we
will ignore it in our analysis. Similarly, we will ignore R48, the 18M
resistor (consider it to be infinite.)

To simplify our analysis, suppose the “pan”
pot is turned all the way towards R, so the wiper is at ground.

J5 is a “normalled” jack, meaning that if nothing is plugged into it, the
“signal” part of the jack (the top pin in the schematic, which appears
to be labled with a test point “H”)
will default to the signal going to the middle pin (which appears on
the schematic to be labeled with a test point “B”). We will assume that the
user has inserted a control signal into J5, so point “H” (the left side
of R14) will be some user-created voltage we will call “V_con”. (In
text, I will often use an underscore to indicate subscripts.)

a) Find the voltage at the output of IC3:B. You can do this quickly if
you realize that this op amp is acting in a standard “inverting mixer”
configuration. (Remember we are assuming the wiper of the “pan” pot is
at ground).

b) Given all the notes above and the assumption that the current through
the base of Q2 is negligible, so we may approximate its collector and
emitter currents as being equal, find the current input to the OTA
(I_con) as
a function of V_con.

c) Now let’s look at the main part of the VCA, with input at J2 and
output at J11. Find the gain of the VCA as a function of I_con. For now, you
may ignore C12 (i.e. open the cap, which is a reasonable assumption for low

d) Suppose V_con = 10V (from what I understand, the envelope generators
in the PAiA 9700 series can generate up to 10V, so that’s a reasonable
voltage to try). What is I_con in this case?

e) Take a look at the “Absolute Maximum Ratings” section of the
LM 13600 datasheet. What is the maximum rating for the
control current (which the data sheet will call I_ABC)? Is the value you
found in part (d) above or below this?

f) What is the gain of the VCA for V_con = 10 V?

g) Without going through extensive calculations – i.e., by just reasoning
your way through the circuit – what happens to the VCA gain as the wiper of
the “pan” pot is turned away from ground and toward V+.

h) Previously, we ignored C12. If we now consider it, we see that C12, R52,
and IC6:B act as a current-to-voltage single-pole lowpass filter with a
cutoff frequency (half-power point) of 1/(2*pi*RC)), in units of Hertz.
What is the cutoff
frequency of this filter? Is this cutoff frequency above or below the
limit of typical human hearing?

i) What is the input impedance of the VCA?

j) What is the output impedance of the VCA? (Assume the op
amps are all ideal,
i.e., they have zero output impedance).

Problem 3

In this problem, we’ll keep looking at the same PAiA 9710 schematic as
in Problem 2. Here, we will focus on the pan pot, and the notion of
“shaping” the curve swept by the pot.

Suppose the leg of the pot between the wiper and ground has resistance R,
where R may range from 0 to 10K;
then, the other leg (between the wiper and V+) has resistance (10K – R).
The voltage at point “C” may be found via a standard voltage divider, where
R and the 120K resistor may be simplified as a parallel combination.

On a single graph, plot (a) the voltage at “C” as a function of R,
for 0 < R < 10K, and (b) the same thing as (a), except suppose that the
120K resistor is actually a 10K resistor. I recommend using MATLAB to
make the graph, but you may use whatever computer program you prefer.
(Please actually print out your graph, don't just sketch it in pen or
pencil like some students have tried in the past.)

Which graph, (a) or (b), looks more linear?

Problem 4

In this problem,
we will
explore Dirk Lindhof’s Exponential VCA.
Notice that instead of converting
the output current of the OTA by using an op amp in an inverting
this design drops the current to ground through a resistor, and then buffers
the resulting voltage with an op amp.

The OTA, a CA3080, is drawn differently than we are used to seeing it; the
control current (let’s call it I_con) is shown as being input to pin 5.

Ignore the caps at the inputs; they just block DC.
This module has
two inputs; it looks like IN2 gives
you the option of bypassing the
DC blocking cap on that channel.
Anyway, just do the analysis for one input.
Ignore the trimming circuitry, i.e. just ground the + input on the 3080 and
assume the OTA is ideal.

a) Find the gain of the VCA as a function of I_con.

b) What is the input impedance of the VCA?

c) What is the output impedance of the VCA? (Assume the op amps are all
i.e., they have zero output impedance).

Problem 5

I’ve set up an
Analog Circuits for Music Synthesis site on
. I posted a “Question” called “Geting to know you.” Click on
“start a new followup discussion” and leave a brief post about yourself.

I’d like to know a bit about you, as it might relate to this class. For
instance: How
did you find out about the class?
Do you play any musical instruments?
Do you play in a band? If you make music, have you made any recordings?
Do you have any particular interests
in using synthesizers and/or audio production more generally,
either with or without the use of computers?
What related classes are you taking or have taken
(such as the senior level audio engineering or operational amplifier design
Music Technology classes, etc.)

You can read other people’s posts, and maybe find some common interests.

(To be clear, you don’t need
to play an instrument or anything like that to do well in
this class; no previous musical knowledge is needed.)