# ECE4893A: Analog Circuits for Music Synthesis

## Spring 2016

## Homework #4

## Due: Wednesday, March 2 at 3:30 PM

This homework will be graded out of 100 points. **Please turn it in
to my VL431 office.** Of course, you may turn it in ahead of time if

you wish.

**Background music**: The original

Buchla Music Easel,

which consists of a Buchla 208 Programmable Sound Source and a

Buchla 218 Model Keyboard together in a single case, is one of the rarest

and most coveted of the Buchla designs.

(It has recently

been

re-released.)

To put yourself in the right frame

of mind for this homework, listen to this

video

featuring former Nine Inch Nails synthesist Alessandro Cortini

performing on his Music Easel.

**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.**

**Sorry, no late turn-ins accepted on this one**: Since we’ll have Quiz 1 on

Thursday, March 3, I want to be able to get solutions up quickly (although

this material won’t be on Quiz 1, per se.)

## Problem 1

In class, we looked at the

“timbre” nonlinearity implemented in the

Buchla 259 Programmable Complex Waveform Generator.

A similar timbre generator circuit is used in the Buchla Music Easel described

above.

You can print out

the schematic from

Magnus’s

Buchla page;

search for the “B2080-9A” “Complex Oscillator 3/3” link.

You’ll see five of those “Buchla diodeless deadband” circuits.

Let’s analyze the topmost deadband circuit,

which consists of an op amp and

R25, R27, and R26.

**Calculate the positive edge of the
deadband**

(i.e., what is the largest input voltage for which the output stays

zero?), and

**calculate the slope of the output/input curve past that point**.

As in lecture, let’s define the “output” as the voltage at the negative input

of the op amp forming the deadband circuit,

and the “input” as the voltage at the output at the op amp

just above resistor R20 on the schematic. You may

adapt the formulas we derived in class; you don’t have to

do them from scratch.

Important warnings:

- Remember in Buchlaese, that 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. - The Buchla 259 used CA3160 op amps, which enjoy “rail to rail” output

swings due to their CMOS output stage, run with “voltage starved” supplies of

6 V and -6 V. The Easel appears to use RC4136’s

instead, and although the power supplies are not explicitly marked, I’m

told they run off Buchla’s

usual +15 V and -15 V. With the exception of one JFET,

the rest of the circuit for the RC4136 shown on

the

datasheet

seems to be all bipolar,

so I doubt it can do the “rail to rail” business that the CA3160 can.

Elsewhere on the sheet, I see that the “maximum peak output voltage swing”

is listed as being “minimum +/- 12 V” and “typical +/- 14 V” for a 10K

load. The resistors I see on the sheet are all higher than 10K,

suspect they’re running more towards what’s listed as “typical.” Looking at

the schematic on the

datasheet, I see that the output is sandwitched between two BJT’s between

the supply rails, so there’s at least a diode drop there from the possible

output to the rails. So… let’s use -14 V and 14 V as the output voltage

limits (as opposed to the -6 V and -6 V volts we saw in the case of the

259). - Notice a few of the “resistors” are actually a couple resistors in

parallel. (Do you get the impression that Buchla might have started with

a basic design, and then tweaked it by throwing in a few more resistors

here and there?) It’s not relevant for the exact problem I asked; I just

thought this was an interesting observation.

Interestingly, the 259 had both “timbre” (amplitude of sinewave going in)

and “symmetry” (DC offset on sinewave going in) controls; the Easel appears

to just have a timbre control.

## Problem 2

Check out

Ray Wilson’s Voltage Controlled Low Pass Filter (Four Pole 24db/Oct).

The input and feedback resistors are 100K; it looks like the divider is

made with a 1K to ground. (I find it interesting that he chooses to use

TL084 op amps as buffers instead of the buffers built in to the LM13700.

Maybe this is to avoid

having to deal with the weird 1.4 V drop you get from the LM13700 buffers?

The TL084 also are probably better quality than just the simple Darlington

pair in the LM13700.)

In parts (a) through (g), we will consider the gain of just

**one** of the filter stages,

either the second, third, or fourth (they are all the same; I’m not

including the first one so we can avoid the

effect of resistor

coupling in the resonant feedback loop while working (a) and

(b)).

a) Find the voltage at the input terminal of the OTA in terms of the

voltage at the output of the buffer and voltage

at the input of the filter block. Don’t make any approximations concerning

the resistors (i.e., if you use superposition, note that you must

compute the value of the little resistor in parallel with the

big resistor to solve this.)

b) In class on Thursday, Feb. 18,

I attempted to use vigorous handwaving to

to convince you that part (a) could be approximated as

v_at_ota = (v_input + v_output) *

(little_resistor / (little_resistor + big_resistor))

Comment on how close this approximation is to what you found in (a).

c) In class, I used even more vigorous handwaving to attempt to convince you

that part (a) could be further approximated as

v_at_ota = (v_input + v_output) *

(little_resistor / big_resistor)

Comment on how close this approximation is to what you found in (a) and (b).

d) Assume that the transductance gain

of the OTA is 19.2**I_con*, where *I_con*

is the current flowing into the

control pin of the OTA.

What is the cutoff frequency of

the filter block in terms of (*I_con*) in Hertz, using the approximation

in part (c)? (Remember that the transconductance gain just takes the place of

1/R in the usual single-pole cutoff freuqency calculation, and for convenience

we include the scaling of the resistive divider as part of the transconducance

gain.)

e) Given the result in (d), what value *I_con* would be needed for the

cutoff frequency of one stage to be 3000 Hz?

f) What single-stage cutoff frequency would you compute if you used the

*I_con* you computed in (e), but you used the no-approximation

technique of part (a)?

g) What single-stage cutoff frequency would you compute if you used the

*I_con* you computed in (e), but you used the approximation in part (b)?

Comment on how close the cutoffs computed in (f) and (g) are to 3000 Hz.

## Problem 3

In class on Tuesday, Feb. 23, 2016, we explored the consequences of cascading

four one-pole lowpass filter and adding a negative feedback loop

with a feedback gain of *k*. The individual one-pole filters each

had the transfer function *H_1*(*s*) = *wc*/(*s + wc*).

We showed that the maximum

usable *k* was *k=4*, at which point the filter would self-resonate.

In this problem, we will take a look at the same structure, except instead

of one-pole lowpass filters, we will cascade four **highpass** filters with

the transfer function *H_1*(*s*) = *s* / (*s+wc*).

a) Find *H_4*(*s*), the

transfer function of a cascade of four one-pole highpass

filters described above.

b) Find *H_4F*(*s*),

the closed-loop transfer function of *H_4*(*s*) with a

negative feedback loop with feedback gain *k*.

You need not expand out

terms of the form (*s*+*wc*)^4;

that just makes the expression more complicated.

(Unlike in the lowpass case, I haven’t been able to find a closed-form

solution for the pole locations in this highpass case. From my numeric

studies, it does seem like the maximum usable *k* is 4, at which point

the filter self-resonances. Also, there are four zeros at the origin in

the highpass case, which of course are not present in the lowpass case.)

c) Asymptotically, what is the value of

the frequency response *H_4F*(*jw*) as *w* approaches

infinity?

d) Now let’s do a numerical experiment with

the consider the full

four-pole cascade with feedback amount

*k*,

Let the cutoff frequency of a single stage be 1000 Hz.

Using MATLAB, Mathematica, Maple, or some similar tool, on the same plot,

show the magnitude of the frequency response (with the horizontal axis in

Hertz), from DC to some value that you think best shows off the curves, for

four cases: *k*=0, *k*

just big enough so that you can just barely see a

resonance “bump” in the curve, *k*

close to 4 (but not so big that it swamps

your other curves), and a k somewhere between the last two cases that you

think is interesting. **Make sure the values at infinity
corresponds with the results
seem reasonable relative to the results of
the simple formula derived in part (c)**.

Please include your

computer code and give me a real printout of your curve, i.e., not something

sketched by hand.