## Personal web pages of## Tim Stinchcombe |

## Monotron Filter - Investigating its PolesThe impetus for this work stems from a particular thread at ModWiggler, 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 C in parallel with _{22}R, and not being able to simply understand how the additional pole and zero introduced by _{76}R/_{74}C 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..._{22}(Note there are 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 ## A usable model of the filterMy 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 ( C). Initially I tried making the two main capacitors in the filter equal (_{22}C and _{20}C), 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:_{21}I have represented the resistance of the transistors as The poles, \(\omega_c\), 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: \(f_c=12\text{kHz}\) for the LM324, and \(f_c=200\text{kHz}\) for the OP462 (where \(\omega_c=2\pi f_c\) of course). Gain \(k_2\) represents the resonance setting, from \(0\) through to \(1\), which represents full resonance. Impedance \(Z\) is either simply R and _{74}C with _{22}R:_{76}Finally, there is a factor, \(D\), formed by the potential divider effect of \(Z\) and Putting this all together leads to the transfer function \begin{multline*} H(s)=\frac{k_{1N}}{3k_{1D}s^2C^2_{21}R^2_Q\left(Z_N\frac{R_{68}}{R_Q}+D_D\right)}\cdots\\ \cdots\frac{\times(3sC_{21}Z_NR_{68}+D_D)}{+sC_{21}R_Q\left(3k_{1D}Z_N\frac{R_{68}}{R_Q}+4k_{1D}D_D-3Z_DR_{68}k_{1N}k_2\right)+k_{1D}(D_D+Z_D)} \end{multline*}(where I emphasize this is 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 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 VR1 setting of \(0.6\) above.If we take C 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:_{22}And again, here is an equivalent plot from the expression: ## Pole and zero lociFirst off, let's start by looking at the poles without the capacitor R), as this probably explains _{74}why it is included. This has one zero and three poles (one real, the others a complex conjugate pair). The resistance R is varied from 1k to 1M as before; using the LM324 model; the resonance factor \(k_2\) 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:_{Q}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 Now let's add the R 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):_{74}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 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 ## Using a higher bandwidth op ampNow 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, \(k_2\), to \(0.3\), we are still going to get lots of it: However if we remove the capacitor/resistor pairing R, and put \(k_2\) back up to \(0.4\), we get some nice looking loci:_{74}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 (Note in these last two plots the \(k_2\) values are not directly comparable, but were merely chosen to give some 'photogenic' plots!) [Page last updated: 16 Dec 2022] |