# 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:**

National

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

Serge

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

schematic

is available here (and

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

series 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

frequencies).

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

configuration,

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

ideal,

i.e., they have zero output impedance).

## Problem 5

I’ve set up an

ECE4893A:

Analog Circuits for Music Synthesis site on

piazza. 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

classes,

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.)