Equation for frequency -> Schmitt Trigger - Integrator LFO

[savethewhales]

I am really short of time, and I have a relatively simple question : what should be the equations for the frequency of this LFO?

This is "my" LFO (Schmitt-Trigger -> Integrator):

NOTE:
-the integrator caps are like they are for a matter of showing what I build into the PCB, but the equivalent cap value should be around 9 and 10 uF for the calculations.
- R32 is a 500k potentiometer
- Vref sits at 4.78 Volt.

While I was searching for the integrator cap and resistor values, I did simulations, no math. In that manner I was able to get the frequencies I wanted, between aproximately 7 Hz and 0.3 Hz.
I'm writing my project down and I needed to know what are the equations for the frequency of the LFO.. Can someone help me?
I've searched online and what I found didn't give me a reasonable answer (when I would calculate, it would always give me like 14 Hz or 0.05 Hz, with these values, which is far from the measurement and the Spice simulations).

Thank you very much and cheers,

Fred

[anotherjim]

Is this paper any help?
https://classes.engineering.wustl.edu/jee2330/Exp09.pdf
Generally, it's a "function generator" but it's difficult to get search engines to find the classic 2 opamp type.

[savethewhales]

Is this paper any help?
https://classes.engineering.wustl.edu/jee2330/Exp09.pdf
Generally, it's a "function generator" but it's difficult to get search engines to find the classic 2 opamp type.

Thank you very much for your attention!

I've just looked at it and the equation 9.5 should help me, but how can I know period/frequency with that?

But in page 9-15, we get an equation for the triangular wave LFO: T = (1.3)RC. However, there's no explanation where exactly this comes from, also it says thtat the equation depends on the schmitt-trigger resistances, which is smth that should be mentioned in the equation (probably it's taken into account).
What I got of values with my components is:
- higher freq: 6.8 Hz
- lower freq: 0.3 Hz

[PRR]

It is not an oscillator, it is a function generator. Figure the voltage trip-points. Figure the slew rate.

I reject your 2-part resistor, make it 524k. I utterly reject your E-caps: insufficient data on the action of two polar caps of two very different values with non-negligible voltage swings. I make it one ideal 10uFd cap.

I am a simple mind and I like sample voltages. To keep arithmetic simple I assumed U7 would swing +/-10.00V (Vcc may be 12V); but I already knew that changing all voltages would not change the ratios or the timing. Everything may be written by inspection and figured on a TI-30 (or slide-rule but we are out of practice and decimals will spill).

Ah.. and I assumed "Vref" is zero but the circuit is trimmed so tight that this is reckless.

[savethewhales]

It is not an oscillator, it is a function generator. Figure the voltage trip-points. Figure the slew rate.

I reject your 2-part resistor, make it 524k. I utterly reject your E-caps: insufficient data on the action of two polar caps of two very different values with non-negligible voltage swings. I make it one ideal 10uFd cap.

I am a simple mind and I like sample voltages. To keep arithmetic simple I assumed U7 would swing +/-10.00V (Vcc may be 12V); but I already knew that changing all voltages would not change the ratios or the timing. Everything may be written by inspection and figured on a TI-30 (or slide-rule but we are out of practice and decimals will spill).

Ah.. and I assumed "Vref" is zero but the circuit is trimmed so tight that this is reckless.

Hey PRR,

I can understand the conditions you've put to the circuit Vcc and to the Vref. I have to say that I was kinda forced to experiment with the 2 polar caps because I didn't have anymore and had to make testings. The result is good after all, but I don't exactly know how to explain why. No problem either way.

I'm happy cause I was able to understand your explanation, at least a bit (lol).
- I understand the Vref=0 but why is - of U8 = 0? Virtual Ground?
- And where does the equation of 0.019 mA/(10uF)=1.9 V per second come from?
- 3.48 V is peak-2-peak, so that shouldn't be 1/2 cycle?

For my circuit, I did the calculations with Vcc+=9v and Vcc-=0v, Vref=4.78V, Voutpos=7.5v and Voutneg=1.5v (these 2 because of spice simulations). It's giving me 0.342 Hz on lower freq and 7.48 Hz on higher., which is spot on. However I'm not getting that thing of the peak-2-peak.. these values were gotten with that but again, shouldn't that count as 1/2 cycle?

For the measurements, the oscilloscope actually showed more or less 0.342 Hz and 7.48 Hz also, so I'm kindof bugging now.

EDIT: I'm mistaken, my calculations resulted in 1.37 Hz on lower freq and 30.3 Hz on higher freq regarding the way you mentioned the 1 cycle. But the oscilloscope was really showing 0.342 Hz and 7.48 when I did the testing..

[Rob Strand]

Two opamp Schmitt-trigger - Integrator circuit:

Period,

T = 4 * RC * (Ri/Rf) 

where
         Integrator parts:    R, C
         Schmitt trigger parts: Ri = input resistor, Rf = output resistor

For example in Fig 9.12 of PDF, Rf = 12k, Ri = 3.9k.  T = (4  * 3.9 /12) * RC  = 1.3 RC  ; as given in the PDF.

[savethewhales]

T = 4 * RC * (Ri/Rf) 
For example in Fig 9.12 of PDF, Rf = 12k, Ri = 3.9k.  T = (4  * 3.9 /12) * RC  = 1.3 RC  ; as given in the PDF.

Thanks!!!
Is there any paper/document which cites this equation besides this pdf ?
I got this "The Triangular and Square Wave Oscillator, from Electronics Circuits II Laboratory" but it seems like it's a laboratory of electronics, idk..

With 10*10^-6 F I get 0.274 Hz and 5.98 Hz

With 9.09*10^-6 F I get 0.3 Hz and 6.58 Hz

Testing of the circuit: 7.4 Hz and 0.345 Hz (I'm guessing it is the polarized capacitors problem, for sure)

[Rob Strand]

Is there any paper/document which cites this equation besides this pdf ?
I got this "The Triangular and Square Wave Oscillator, from Electronics Circuits II Laboratory" but it seems like it's a laboratory of electronics, idk..

You should be able to derive it:
- Schmitt trigger output swing +Vout/-Vout
- The constant +Vout/-Vout  into the RC integrator ramps the output up and down  I = Vout/R  = C dV / dt
- Schmitt trigger input threshold are +Vth / -Vth.
  So that means the cap is initially at -Vth and charges towards +Vth for one half cycle
  Then the cap is initally at +Vth and charges towards -Vth for one half cycle
- By using the fact Vth is related to Vout in the schmitt trigger equations the Vout's or Vth's will cancel out.

With 10*10^-6 F I get 0.274 Hz and 5.98 Hz

With 9.09*10^-6 F I get 0.3 Hz and 6.58 Hz

Testing of the circuit: 7.4 Hz and 0.345 Hz (I'm guessing it is the polarized capacitors problem, for sure)

Electrolytic caps have large tolerances.  The mismatch between theory and calculations could be explained by tolerances.
(The only way to come at a different conclusion is to measure the caps.)

[savethewhales]

You should be able to derive it:
- Schmitt trigger output swing +Vout/-Vout
- The constant +Vout/-Vout  into the RC integrator ramps the output up and down  I = Vout/R  = C dV / dt
- Schmitt trigger input threshold are +Vth / -Vth.
  So that means the cap is initially at -Vth and charges towards +Vth for one half cycle
  Then the cap is initally at +Vth and charges towards -Vth for one half cycle
- By using the fact Vth is related to Vout in the schmitt trigger equations the Vout's or Vth's will cancel out.

"- Schmitt trigger output swing +Vout/-Vout" Shouldn't it be +vout minus -vout?
But overall pretty interesting theory, if I had time I would have derived it in my work for sure.. Your help is unmeasurable, really.
It ends up I already sent the work to my teacher so this part couldn't be there, no problem either way.

Electrolytic caps have large tolerances.  The mismatch between theory and calculations could be explained by tolerances.
(The only way to come at a different conclusion is to measure the caps.)

Hmm it could be.. It's unfortunate I had to blame this on the reverse series different caps. I said I didn't have the time to be redoing the LFO and buying no-polar caps.. MAY not be a problem!

[Rob Strand]

"- Schmitt trigger output swing +Vout/-Vout" Shouldn't it be +vout minus -vout?
But overall pretty interesting theory, if I had time I would have derived it in my work for sure..

Actually the meaning of "/" here is just listing the values; +Vout and -Vout.
Depending on how you derive it you could end-up with a subtraction since for example "dV" goes from -Vth to +Vth.

[savethewhales]

Actually the meaning of "/" here is just listing the values; +Vout and -Vout.
Depending on how you derive it you could end-up with a subtraction since for example "dV" goes from -Vth to +Vth.

Ok, sure!
Sent you a PM.

[Rob Strand]

No problem, thanks.