# ECE4803B – Fall 2006 – Homework #2

ECE4803B – Fall 2006 – Homework #2

# ECE4803B: Theory and Design of Music Synthesizers

Fall 2006

Homework #2

Due: Friday, 5:00 PM, under my Bunger Henry 323 door (but
if you get it done sooner, please turn it in in class)

Late penalty: 20% deducted if turned in by 5 PM Monday; an additional
20% for each day thereafter, until solutions are distributed.

Importance:: You must at least make an attempt at this homework and
turn it in; i.e. you may not “ditch” the homework.

Suggested references: CA3080 and LM13700 datasheets (available from
Aaron’s
datasheet collection
)

### Problem 1

In class on 9/14, 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
Ray Wilson’s VC-VCO.

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 circuit, looking into the TRI output?

### Problem 2

Jorgen Bergors, 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 Jorgen’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

In class on 9/12, 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 description.

Check out the LM394 in the schematic; this forms the core of the exponential
converter (note Ray recently found 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 440 Hz tone?

(c) Now let’s get some practice in reasoning with tempco resistors. 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
of 2K is for 25 degrees celcius)?

(d) 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 (c), except use a
temperature of 38 degrees celcius instead of 25 degrees celcius.

<!–

[MORE TO COME]

### Problem 3

This question concerns DC blocking capacitors.

a) Let’s look at the AC input
of the Serge VCA linked to above. The input capacitor
and the two resistors form a single-pole highpass filter.
Find its cutoff frequency. (Note the resistor in series with the cap gives
this configuration a non-unity gain.)

b) Check out
this link. What kinds of capacitors
does the author recommend for use
as DC blocking (i.e. “audio coupling”) capacitors?

–>