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 lame. Unpleasantness may result from such behavior.
Late policy: If you show up really late, suggesting that you were doing the homework during class, then I'll take off up to 10 points based on my mood. If you turn it in later that day, say before 6:00 PM or so (that's when my 2025 recitation ends - you can find me in Van Leer 361 from 3 to 6). After that, I'll take off 20 per day. This isn't to be mean - it's to encourage you to get it turned in and get on with 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.
Suggested references: CA3080 and LM13700 datasheets (available from Aaron's datasheet collection)
In class session 9, 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 TRI output?
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 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.)
(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 (see class session 8 if you need help). 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 is 2K 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.
In class session 11, we looked at a nonlinear circuit used in Ken Stone's Cat Girl Synth Wave Multiplier. To find the schematic, go to http://www.cgs.synth.net, click on "Modules," and click on "Wave Multiplier" (be sure it just says "Wave Multiplier" by itself; don't click on the "Saw Pitch Shifter/Wave Multiplier"), and then click on "Grinder and Folder Schematics." The folding nonlinearities are at the bottom of the page. Note that Ken uses four in series (unlike the six in series like the Serge Wave Multiplier uses). The last one has some additional diode clipping action, but we'll ignore that.
Let's consider one of the first three stages. Use your favorite implementation of SPICE to run a simulation of one of the stages (10K resistors from input to each of the op amp terminals, 10K resistor in negative feedback configuration, and two 1N4148 diodes, facing different directions, in parallel from the positive terminal to ground). Be sure to use a 1N4148 model (if one isn't built into your SPICE, let me know) and not some sort of "idealized" diode. Make a plot of the output voltage vs. the input voltage for input voltages ranging from -1.5 to 1.5 volts. Does the nonlinearity exhibit a sharp corner, as my handwaving analysis in class suggested, or does it have a more rounded corner? Be sure to provide some sort of printout "showing your work," i.e. a SPICE schematic or netlist (if you're into typing your own netlists by hand).
The 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. In class session 11, we looked at the "timbre" nonlinearity implemented in the Buchla 259 Programmable Complex Waveform Generator. A similar timbre generator circuit is used in the Music Easel. 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 second one for the top, which consists of an op amp and R28, R32, R29, and R33. 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 formula we derived in class session 11; you don't have to do it from scratch.
Important warnings:
