# ECE4803B – Homework #4

ECE4803B – Homework #4

# ECE4803B: Theory and Design of Music Synthesizers

Spring 2006

Homework #4

Due: Wednesday, March 1st at the start of class

Ground rules on this homework: You may verbally
discuss approaches to the
problems with each other while looking at the schematics, and
are encouraged to do so; but you
may not look at each other’s written
solutions or ask “what did you get on
part XYZ of problem ABC.” (In future homeworks, I will allow
varying degrees of
explicit collaboration on certain problems.)

Below, I will use underscores to indicate subscripting.

### Problem 1

Pick one of the following lowpass filter designs based on the last digit
of the year you were born. We will focus on the behavior of
just one of the four
one-pole lowpass filter blocks. Assume that the OTA is ideal (infinite
input impedance, output a perfect current source, perfectly linear
voltage-input-current-output relation).

0,3,6,9) Figure 9 of the

SSM2040 datasheet
: Look at one of the low-pass sections in
the middle, where the “big” input and feedback resistors are both 10K,
and the small resistor to ground is 200 ohms; this corresponds to Figure 1
with R1=R2 = 10K and R2 = 200 ohms, and C = 1000 pF.
(The entire circuit cuts
the gain down by 10 at the initial input and then boosts it again by 10
at the output – I want to ignore that detail and just focus on one of
the middle sections.)

1,4,7)
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.)

2,5,8) Polyfusion
Lowpass VCF
: This one might look trickier than the others, but it’s
really not. Treat the JFEsS with the 1K and 10K resistors tied to the
power rails as if they were perfect voltage buffers; treat the
impedance looking into the gate as infinite, and pretend that the
voltage at the gate magically appears at the other terminal of the FET.
There’s 20K input and feedback resistors, and the voltage is cut down
with a 100 ohm resistor to ground.

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, I attempted to use vigorous handwaving to attempt 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) Find the Laplace-domain transfer function relating the
voltage at the output of the buffer to the voltage at the
input of the filter block.
Use your approximation in part (c). 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.

e) What is the gain of the low-pass filter block (i.e. gain at DC)?

f) What is the cutoff frequency of
the filter block in terms of I_con in Hertz?

## Problem 2

Now let’s consider a four-stage cascade of single-pole lowpass filters,
each with cutoff frequency f_1 (i.e., |H(j 2 pi f_1)| = 1 / sqrt(2)).
What is the cutoff frequency of the entire four-stage cascade in terms
of f_1? Give a simple formula, with real numbers rounded to a reasonable
number of decimal places.

Remember from ECE2025 that to find the magnitude response of a cascade
of systems, you multiply the magnitude responses of the individual systems.
Here, that means we need to find the f_4 such that
|H(j 2 pi f_4)|^4 = 1/sqrt(2).

## Problem 3

The
Korg PS-3100 and

PS-3300

voltage-controlled resonator circuits that were basically a parallel
bank of three second-order bandpass filters that could be individually
tuned, and their outputs summed.
(Juergen
Haible
has built a

Korg PS series clone
from scratch!)

So far,
we’ve been focusing entirely on using OTAs to replace resistors. Some
circuits, such as that used in the Korg resonators, instead used
a combination of LDR (light dependent resistor) coupled together with
an LED; changing the current through the LED changes the amount of
light on the LDR, and hence the resistance. (Modern clones of
the Korg resonator,
such as the
MOTM-410
and the
Cyndustries
Triple Resonant Filter

typically use Vactrols, which are commercial
prepackagings of LEDs with LDRs. You can hear examples of what
the resonator sounds like on those pages. Try the “3 Zombie Tenors”
sample on the MOTM-410 page!)

Grab the schematic from
here; you want “Part 1,” which contains the main filter circuitry.
On the right part of the diagram, you’ll see the three filters, which
are each formed from a 1458 op amp, two capacitors, and two LDRs.
The LDRs are driven identically, and hence each have the same
resistance.

Let’s analyze this circuit. Call the capacitor feeding back from
the output of the op amp to the negative terminal of the op amp C2;
let’s call the other capacitor C1. Let’s suppose both LDRs have
the same resistance R.
Find the Laplace-domain transfer function describing the voltage
at the output of the op amp in relation to the input voltage
at the left terminal of C1.

(This is basically an ECE2040 problem).

An aside: I’ve looked up and down trying to find this exact filter
configuration in the literature, and haven’t been able to find it!
One web author called it a bandpass Sallen-Key, but that’s clearly
incorrect. It looks kind of like a multiple feedback topology,
but it has four passive elements instead of five, and the caps and
resistors are switched from the way they’re usually presented in
a MFB bandpass circuit. So I’m not sure what to call it! Has anyone
seen this before?