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.
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?
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?
