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Monotron Filter - Investigating its Poles
The impetus for this work stems from a particular thread at MuffWiggler, in which Scott Willingham reported that when he replaced the LM324 used in the filter with an Analog Devices OP462 (in order to lessen the filter noise), he discovered there was a deleterious affect on the filter's resonance response. This is undoubtedly caused by the much larger unity-gain bandwidth of the OP462, at 15MHz compared to the 1.2MHz claimed of TI's LM324; however even within the original circuit there is obviously some issue connected with the resonance, witness R74/C22 in parallel with R76, and not being able to simply understand how the additional pole and zero introduced by R74/C22 might affect the resonance, nor on why the larger bandwidth impacts this, I decided to study it all in greater detail, and started scribbling algebraic equations and drawing pictures...
(Note there are other issues around using the OP462 as a replacement for the LM324—see Scott's post here.)
Much of what follows will of course translate across to the Korg35-type MS-10/MS-20 lowpass filter, since this is what the Monotron filter is based upon, having a very similar circuit set-up. Note also that I have given no explicit plots of the phase of the filter at all—this is because I felt that I got what I needed by way of explanation without any, and of course the phase is implicit in all the pole plots in any case!
A usable model of the filter
My first task was to come up with a model that would show the effects I was looking into, but which was hopefully not too involved and cumbersome. The standard Sallen-Key topology is easy to start with, but to that I needed to add the pole from the amplifier, and also the pole/zero from the extra cap and resistor in the feedback loop (R74/C22). Initially I tried making the two main capacitors in the filter equal (C20 and C21), but it was clear their inequality was needed to be accounted for, as the responses I got from them being equal didn't cut it. Here is the circuit I eventually ended up analyzing:
I have represented the resistance of the transistors as RQ, and made them the same (in reality, they will turn out a little different—see my in-depth study of the MS-10/MS-20 filters). The amplifier, k1, was modelled as a simple first-order lowpass, with gain A=70:
The poles, ωc, were simply determined empirically by running simulations with the above model until the 1-pole model followed the open-loop plots for the appropriate op amp SPICE model:
giving values of: fc=12kHz for the LM324, and fc=200kHz for the OP462 (where ωc=2πfc of course).
Gain k2 represents the resonance setting, from 0 through to 1, which represents full resonance.
Impedance Z is either simply R76 on its own, or the parallel combination of R74 and C22 with R76:
Finally, there is a factor, D, formed by the potential divider effect of Z and R68, which can be usefully factorized out in several places:
Putting this all together leads to the transfer function
(where I emphasize this is one fraction—I still don't know if LaTeX can more elegantly represent the line wrap!). Note also that the degree of both the numerator and the denominator is a little hard to establish, due to the 'hidden' dependence on s within k1D, Z and D—since the pole plots etc. are done numerically within Mathematica, I was letting that sort it all out.
This expression was entered into Mathematica, whereupon I could start plotting frequency responses, pole locations etc., at will, with different values for the amplifier pole, and with the feedback capacitor C22 in or out. I validated its gross characteristics against simulation of the Monotron filter (the full circuit used is here, but note the reference designators don't follow those of the Monotron), and I'm happy that it agrees well enough to show what I want it to. First up is a series of traces from the simulation: stepping the frequency pot, 'VR2', in 30 steps, with the resonance pot, 'VR1' at 60% (for the 12kHz op amp, i.e. the LM324, and with the feedback cap in):
Here is the equivalent from Mathematica for the big transfer function expression:
The RQ values were just chosen to give similar-looking curves, but the values, 1kΩ to 1MΩ, seem reasonable. The value for k2 is 0.4, so quite a bit less than the equivalent VR1 setting of 0.6 above.
If we take R74/C22 out, the resonance is much flatter, tailing off considerably at the higher frequencies, and thus is very likely the reason why the cap is there in the first place:
And again, here is an equivalent plot from the expression:
Pole and zero loci
First off, let's start by looking at the poles without the capacitor C22 (nor R74), as this probably explains why it is included. This has one zero and three poles (one real, the others a complex conjugate pair). The resistance RQ is varied from 1k to 1M as before; using the LM324 model; the resonance factor k2 is set to 0.4. At this scale, the real pole starts central and moves left; the conjugate pair arc out from near the origin; the zero is too far off-stage to the left to be seen:
This shows why the resonance drops off at the higher frequencies—if this were a standard second-order section, at a fixed resonance setting the poles would move in a straight line away from the origin. Here they do not—there is a pronounced bending away from the imaginary axis, showing that the resonance drops away, as the frequency plots above indicate. Thus the inclusion of the capacitor to help 'restore' the resonance at the higher frequencies. Plots for this in a moment, but first let's zoom out so that we can see that lone zero (in blue):
Now let's add the C22/R74 pairing (so using the more complicated expression for Z). This introduces a pole/zero pair between the origin and the existing real pole—they are pretty close to each other, but note what happens to the shape of the conjugate pair that move out from the origin (the other zero is way out of shot again):
As the frequency is increased the zero is static of course, but the new pole moves first one way, then the other, as the following animation shows (click to animate it):
Now I was expecting that it would be the introduction of the pole/zero pair that was affecting the resonance in some way, thus bolstering it at the higher frequencies. These plots seem to illustrate that this expectation is incorrect—the actual locus of the poles itself has changed, with the poles now swinging back toward the imaginary axis, with the effect of increasing the resonance (at least for the higher audio frequencies—as it goes higher still, the poles do come back away from the imaginary axis again). Thus it is the effect the introduction of the capacitor/resistor have on the transfer function as a whole that is important, and not merely the new pole/zero pair.
The following 3D animation may help to illustrate what is going on (again, click the image for the animation). The slice down the imaginary axis is the frequency response, but note that the scale for the axis is linear, and not logarithmic as is normally used, so it doesn't quite bear a one-to-one correspondence with the Bode plots above. The sequence is for the equivalent transistor resistance, RQ, increasing and decreasing (and hence the cut-off frequency increasing and decreasing); the effect of the zero pulling the surface down can be seen when the cut-off frequency is highest; the dark blue 'sea' is when the surface drops below the 'bounding box' of the animation; the peaks look like they are 'shrinking', but this is due to loss of resolution of the plot—in reality when this happens, they are tall, thin and 'needle like':
Using a higher bandwidth op amp
Now let's see what happens when we use an op amp with a much higher bandwidth, such as the Analog Devices OP462—the single pole amplifier model I'm using now as a lone pole at 200kHz. First we plot the poles with cap and resistor in place, resonance set to 0.4:
The poles cross the imaginary axis at a much lower cut-off frequency, causing the resonance to go beserk! Even reducing the resonance setting, k2, to 0.3, we are still going to get lots of it:
However if we remove the capacitor/resistor pairing C22/R74, and put k2 back up to 0.4, we get some nice looking loci:
And a correspondingly nice looking Bode plot, with the resonant peaks looking nicely even across the useful cut-off frequency range:
If we zoom right out, drastically increase the frequency plotting range (way beyond audio), restore the cap, and overlay the plots for both the LM324 (red) and OP462 (blue), we immediately see what is happening:
The increased bandwidth has just blown the whole pole loci right out—the OP462 shows the same gross characteristics as the smaller-poled LM324, but at a much larger scale. Down in the audio frequencies which we are interested in, we don't really see the effect of the little 'wobble' that the LM324 has, and which gives the added boost to the resonance at the higher cut-off frequencies. And if we take the cap back out, we see we lose this wobble, and for the OP462 close in to the origin, the poles are nearer to being a straight line as they come away from the origin, hence the nicely behaved resonant peaks above!
(Note in these last two plots the k2 values are not directly comparable, but were merely chosen to give some 'photogenic' plots!)
[Page last updated: 24 Mar 2011]