ECE4893A: Analog Circuits for Music Synthesis

Spring 2015

Homework #4

Due: Friday, March 13, by 4:30 PM

This homework will be graded out of 100 points.

Turn-in procedure: You can slip it under my Van Leer 431 office door
by 4:30 on Friday, March 13.
Of course, I will accept it earlier, such as in class on March 12.

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

Check out
Ray Wilson’s old 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.

In parts (a) and (b), 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 talking about
the effect of the resistor coupling
in the resonant feedback loop while working (a) and
(b)).

a) 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?
(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. You may use the various approximations that I used in class to reduce
the resistive divider to a R_small/R_big type of multiplier. I want
to emphasize that you don’t need to rederive anything; you can feel free
to use the final results from lecture.)

b) Given the result in (a), what value I_con would be needed for the
cutoff frequency of one stage to be 1000 Hz?

c) Now let’s consider the full four-pole cascade with feedback level
denoted as K, as in lecture. 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 value at DC corresponds with the results computed
by the simple formula derived in lecture. Please include your computer
code and give me a real printout of your curve, i.e., not something
sketched by hand.

Problem 2

In this problem,
we’ll
look at the
VCF in the Minimoog (at least the
version linked here; the Minimoog went through several revisions). The
scan I am linking to was posted by someone calling themselves
Fantasy
Jack Palance; this FJP person has
posted
a
lot more info about it, including the story about how
they acquired their Minimoog.

If you don’t see a specific unit on a capacitor, there’s usually an implied
“microfarads.”

a) The Moog transistor ladder VCF contains a cascade of four
one-pole lowpass filter sections. In class, I presented an analysis of a
hypothetical
“one-sided” ladder
to get a feel for how this kind of circuit works. The rigorous analysis
of the actual “two-sided ladder” is a bit more complicated. We’ll rely on
the analysis in Tim Stinchcombe’s
Analysis
of the Moog Transistor Ladder and Derivative Filters, particularly
Equation 13 on page 12. Using this formula,
find the cutoff frequency (in Hertz) of
one of those sections in the Minimoog’s transistor ladder
as a function of the control current
being pulled from the tied emitters of the transistor pair that feeds
the ladder.

(Note that when analyzing the Moog VCF, we don’t include a
resistive divider in the gain as we’ve done in OTA-C filter
cutoff computations;
there is typically a resistive divider right at
the first input, but it’s not important for our frequency-domain analysis.)

b) Let’s do some DC analysis.
At DC, the caps are open circuits.
For the purpose of this analysis, ignore R76 (the 330 ohm resistor) and
the R73 (the 1K regeneration calibration pot); treat them as “open” too.
Supposing that the transistors draw negligible
current through the bases, what are the voltages
at the bases of the four stages of the
ladder? (Number the stages 1 through 4, from bottom to top; the first stage
is Q23/Q24, the second is Q19/A20, the third is Q10/11, and the fourth is
Q2/Q3).

Problem 3

Problems 1 and 2 dealt with a fairly popular lowpass filter configuration,
namely a cascade of four identical single-pole lowpass filters with
negative feedback.

The VCF in the
Elka
Synthex
(see this demo
by Paul Wiffen, one of the original Synthex sound programmers)
permits this configuration, but it also allows
the musician to select a configuration with a cascade of two identical
single-pole lowpass filters with negative feedback. We will mathematically
explore this configuration. Assume each single-pole filter stage
has a transfer function of w_c / (s + w_c), where w_c is
the half-power cutoff point of a single stage (in radians). (Here, interpret
the letter w as the greek letter omega).

a) Assuming a negative feedback factor of K, find the closed-loop
transfer function of the complete filter. Write your answer so that
it has w_c^2 in the numerator and a quadratic polynomial in s
in the denominator. Make sure the highest power of s has unit coefficient,
i.e. a coefficient of 1.

b) In class on March 5, we looked at a canonical lowpass 2nd-order filter
transfer function
of the form w_n^2 / [s^2 + (w_n / Q) s + w_n^2], where w_n was the “natural
frequency” and Q was the “quality factor.” If you put your answer to
(a) in this form, what are w_n and Q in terms of w_c and K?

c) What are w_n and Q for the special case of K = 0 (i.e. no feedback)?

d) For what values of K does the filter exhibit a resonance bump,
i.e. for what values of K is Q > 1/sqrt(2)?

e) Using the quadratic formula, find the locations of the
poles of this filter as a function of w_c and K.

f) Where are the poles for the special case of K = 0?

g) Describe how the poles move as K increases.

h) Can this filter be made to self-oscillate like the
four-pole-casade-with-feedback filter explored in the pervious problems?
In other words, is there a value of K > 0 for which the poles can be made to
lie on the imaginary axis?