# ACMS 2016 – Homework #3

ACMS 2016 – Homework #3

# ECE4893A: Analog Circuits for Music Synthesis

## Due: Thursday, February 18 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 strictly prohibited.

## Problem 1

In this problem, we will explore
VCO

of the
Bergfotron. The
Advanced VCO is, uhm, a more advanced version of the Complex VCO that
we looked at in class on Tuesday, February 11.
In particular, we will investigate a very small part of the

schematic of its waveshapers
. There’s a lot of freaky stuff on
that schematic. We will concern ourselves with the op amp labeled IC2
in the middle of the page (the TL072 is a dual op amp; we will not worry
out the op amp in the upper right corner). This IC2, along with
some additional circuity, claims to produce a sawtooth wave,
labeled “SAW OUT,”
from the triangle wave produced by the Advanced VCO’s core.

Let’s denote the output signal labeled “TRIANGLE OUT,” which is produced
by half of IC9 (not to be confused with the IC9 in the upper left
corner of the page), as V_tri (where I am using the underscores to indicate
subscripts). Let’s denote “SAW OUT” as V_saw.

The square wave produced by the Advanced VCO’s core controls the 4066
CMOS switch; for this case, focus on the switch corresponding to pins
3, 4, and 5 (pin 5 is the control). The 11K resistor and 22p capacitor
in the feedback loop of IC2 seem to form a lowpass filter; however,
computing the cutoff of this filter yields a cutoff of 658 kHz. This
is far above the range of human hearing, so I presume the 22p capacitor
is there for stability and general gremlin-eating. So, let’s ignore
(i.e. open) that capacitor.

The 20K and 68K resistors attached to pin 2 of IC2, and the 4.7M, 13K,
and 24K resistors attached to to pin 3 of IC2, will all play a role in
this problem. The circle with a + in it connected to the 4.7M resistor
is the +15 volt power supply; the cirlce with the – in it connected
to the 68K resistor is the -15 volt power supply.

a) Find V_saw as a function of V_tri when the 4066 connected
to the 13K resistor is switched OFF. (Assume its OFF resistance is
infinite.)

b) Find V_saw as a function of V_tri when the 4066 connected
to the 13K resistor is switched ON. (Assume its ON resistance is zero).

## Problem 2

In class on Tuesday, Februrary 9,
we looked at exponential converters, particularly
a current “sink” that used a matched NPN pair. In this problem, we will look
at a current “source” that uses a matched PNP pair; the same sort of thinking
we used in class also applies to this variation.

J.G.N. Bergfors, the
creator of the Bergfotron,
conducted a
VCA
shootout

comparing various VCA designs. Let’s take a look at
CA3080
VCA 1
. The exponential converter is at the top of the schematic, and
the main VCA is at the bottom part of the schematic.
The power supply voltages are not marked on the schematic or on the webpage,
but based on J.G.N.’s
<A HREF="http://hem.bredband.net/bersyn/psu.htm"power supply design,
let’s assume the VCA uses a +/- 15 V supply.

The exponential converter takes a control voltage “CV” (found in the
upper left of the schematic) and
generates a control current for the OTA of the
form I_{con} = I_{ref} exp(const*CV).

(a) What is I_{ref}?

(b) Assuming that the CV offset trim pot is set all the way to the
“right”
(i.e. at ground), what change in
CV will cause the control current to double? (Assume the PNP BJTs
draw insignificant current throught their bases).

(c) Assuming the OTA is operating in the linear region, give
an expression relating the audio
output voltage to the audio input voltage in
terms of the current at the control input pin of the OTA. (You
may ignore the offset trimming circuitry of the OTA. Assume
the positive input of the 3080 is grounded.)

(d) What is the input impedance of this VCA?

(e) What is the output impedance of this VCA? (It might be “0” –
remember we’re assuming ideal op amps.)

## Problem 3

This is a direct continuation of Problem 3 from Homework 2, continuing
to explore
Ray
Wilson’s 1V/Octave Voltage Controlled Oscillator
.

(a) Let’s get some practice in reasoning with tempco resistors. In
the example exponential converter shown in class, the tempco resistor was
used in a resistive divider just before one of the bases of
the transistor pair.
Here, Ray decided instead to use the tempco resistor in the feedback loop
of an op amp.
Examining this
schematic
, suppose
that R8, R10, R18, R23, R27 aren’t there, and we’ll focus just on the CV1
input through R15.
What is the output of U1-A (pin 1) as a function of voltage
CV1 if the tempco is at a temperature of 25 degrees celcius (the base
resistance is 2K for 25 degrees celcius)?

(b) Now suppose you’re using Ray’s VCO circuit to make sound for an art
installation at the Burning Man Project, which can get up to and above
100 degrees fahrenheit during the day. Redo problem (a), except use
temperature of 40 degrees celcius instead of 25 degrees celcius.