gaussmarkov: diy fx

Having talked about pots and variable resistors, let’s put them in parallel and in series with a fixed resistor and see how these can be useful mods. First we’ll look at variable resistors and then at pots. There’s more to say than you might expect.

NOTE: I relied heavily upon R. G. Keen’s article, The Secret Life of Pots, to learn much of the material presented here.

Variable Resistors

Let’s say the day arrives when you need a 5K variable resistor in your AMZ Mosfet Booster for the gain control. But all you have is a 20K linear pot. “Hah!” you say to yourself. “No problem. I have a 7.5K resistor and if I put it parallel to my pot then I will have a 5.5K variable resistor.” And you will be correct. Your pot will vary the resistance from 0Ω to 5.5KΩ.

But something else quite wonderful will also happen. The feel of your pot will be different. Instead of the gain bunching up so much at the end of the rotation, the gain increase will be spread out a little. This is caused by the nonlinear way in which parallel resistors combine their resistance.

On resistors: in parallel there is an explanation of the resistance for two parallel resistors, valued R₁ and R₂:

In this case, one of the resistors is variable. Let’s say its the first one and that its resistance is x R_p, where x is the fraction of rotation. Then the resistance of the variable resistor in parallel with a fixed resistor is

which depends nonlinearly on x. To summarize the effect for various resistor and pot values, it is helpful to express the fraction of resistance as a function of the amount of rotation:

In this expression, the scale of the resistors does not matter. Only the ratio of the two resistor values deterimines the shape of the pattern. As R₂ gets smaller relative to R_p, the resistance increases faster at the beginning of the rotation and slower at the end.

So a fixed resistor parallel to a variable resistor evens out a gain control that has most of its increase at the end of its rotation. This is an example of nonlinear taper. Unfortunately, there is no way to flip the taper around so that the resistance increases slowly at first and then accelerates. You can reverse the terminals so that the resistance decreases slowly at first, but you cannot turn this concave path into a convex path.

Another popular approach to the gain bunching problem is to put a variable resistor in series with a fixed resistor. On resistors: in series, we explain that the resistance of two resistors in series is the sum of the individual restistances. Using a 2K variable resistor in series with a 3.3K resistor combines to make a variable resistor that goes from 3.3K to 5.3K. If that is a good approximation to where there is action in the gain control then focussing the variable resistance in that range will help make the gain pot seem more responsive.

You can also see a resistor in series with a pot in the way some builders wire their Fuzz Face gain control. Much of the gain for this pot, when it is linear, also appears late in the pot rotation.

Pots

We just described how the effective taper of linear variable resistors bows up when they are placed in parallel with a fixed resistor. The taper of pots can also be changed using fixed resistors, but it works differently because pots are voltage dividers. It becomes possible to obtain voltage (not resistance) patterns that approximate both audio/log and reverse audio/log pot behavior.

The first thought might be to put a resistor across lugs 1 and 3 of the pot. This merely creates a current divider because the resistor and the pot are in parallel. Voltage differences are unaffected (according to Kirchoff’s voltage law).

Instead, one places a fixed resistor across lugs 1 and 2 or lugs 2 and 3. First, we’ll look at putting a resistor across lugs 1 and 2 of a linear taper pot. In the Taper section of resistors: pots, we explain how the action of a voltage divider causes a linear pot to give

where V_OUT is the voltage at lug 2, V_IN is the supply voltage, and x is the fraction of pot rotation. The voltage changes proportionately with the throw of the pot. If we put a resistor R₂ across lugs 1 and 2, then the resistance across lugs 1 and 2, originally x R_p, is replaced by the resistance of x R_p and R₂ in parallel:

Putting this into the voltage divider formula with the (1 - x) R_p resistance between lugs 2 and 3 gives the voltage at lug 2 as

To summarize this relationship, we rewrite the fraction of voltage attenuation as a function of pot rotation and the ratio of the resistances and create the plot on the right

This is the sort of voltage attenuation that an audio pot delivers. At the start of the rotation, voltage increases quite slowly. But after the half way point, voltage accelerates until it is increasing quite quickly. When R₂/R_p = 1/16, the taper is closest to an audio taper, where the attenuation at the half way point is around 10%.

If we had placed R2 across lugs 2 and 3 instead, similar logic leads to the voltage attenuation expression

This taper is shown on the right.

You can take all of this one step further and put taper resistors across both pairs of lugs. Joe Davisson provides an on-line calculator for customizing your linear pot this way. Follow the “Linear Pot” and “Tapered Pot” links on his Electronics Math Helper page.

I am going to leave things there for now. I plan to add an explanation about using these mods under the conditions described by R.G. on his page, The Secret Life of Pots:

Unfortunately there’s a gotcha in there. It’s true that the voltage division ratio of this rig is arbitrarily close to that of a log taper pot. However, neither the load seen by whatever drives Vin or the source resistance as seen by the input of whatever is connected to Vout is close to what would exist for a real log pot of value R. In fact, the load on Vin varies from 1/(1+1/b)*R up to R. That means that if we’re trying to do a log taper with b = 1/4, the load on Vin will be as much as 0.2* R. This may be OK, but you have to keep it in mind.

Read his page to understand what this means.