Potentiometers - Modifying Taper
Before reading this tech article, you may want to familiarize yourself with the different types of potentiometer tapers and ideal vs actual potentiometer tapers.
If you have done enough builds, you’ve probably had a few instances where you did not have a potentiometer with the ideal taper for the circuit you were working on. Maybe a schematic calls for an audio taper, but you only have a linear taper potentiometer.
First, it’s good to know that using a different taper will not change the way the circuit operates in any way. You can always use a linear taper pot in place of an audio/log taper pot (assuming identical specs otherwise) and vice-versa. The taper only determines how the resistance is distributed around the potentiometer’s rotational travel, but every setting that can be achieved with a linear taper pot can also be achieved with an audio taper; its just a matter of how that setting is achieved with the physical manipulation of the potentiometer.
The practical effect of using different tapers is that it will change the way the potentiometer "feels" to use. Though personal preference can play a role, usually the goal is for the potentiometer to have a nice range of control throughout its entire rotation. A poorly-selected taper will result in the useful range of controls mostly being in one small range. For example, if a volume control has a poorly-selected taper, when rotating the knob clockwise you may hear most of the increase in volume in the first half of the pot’s rotation and hear little increase in the second half. Ideally, the change would be smooth from 0-10.
Fortunately, there are some simple ways to modify potentiometer taper curves when you are in a pinch by wiring in one additional resistor. However, these methods are not without caveats that you need to be aware of.
Potentiometers as Variable Resistors
First, we will look at potentiometers that are being used as variable resistors. A potentiometer used as a variable resistor will either have lug 1 or 3 left open (or the lug can be connected to the middle lug for a functionally equivalent circuit).
Figure 1: Potentiometer wired as a variable resistor (equivalent circuits)
With a potentiometer wired as in Figure 1, there is only one value of concern, and that’s the resistance between lug 1 and lug 2. The resistance between lugs 2 and 3 does not play a role. Let’s use a 500K linear potentiometer as an example. In an ideal 500K potentiometer, when the potentiometer is turned fully clockwise, the resistance between lug 1 and 2 is 500kΩ. When the potentiometer is turned fully counterclockwise, the resistance between lug 1 and 2 is 0Ω. In-between these two points, the resistance changes linearly. At 25% rotation, the resistance value is 25% of the max (125kΩ). At 50% rotation, the resistance value is 50% of the max (250kΩ), etc.
Figure 2: Linear potentiometer resistance vs. rotation
With no parallel resistor, the formula for the actual resistance of the potentiometer as graphed in Figure 2 is:
$$R_{pot} = \text{Rotation} \times R_{max}$$Where ~R_{max}~ is the potentiometer’s max value, e.g. 500kΩ. As expected, the formula for a linear taper is very simple.
In order to change the taper, we can wire a resistor in parallel with potentiometer lugs 1 and 2.
Figure 3: Resistor wired parallel to a potentiometer operating as variable resistor
The equivalent resistance of two resistors in parallel can be determined by the following formula:
$$R_{equivalent} = \frac{R1 \times R2}{R1 + R2}$$If we consider ~R1~ to be the resistance of the potentiometer and ~R2~ to be the parallel resistor, the equation becomes:
$$R_{equivalent} = \frac{R_{pot} \times R_{parallel}}{R_{pot} + R_{parallel}}$$Substituting in ~R_{pot} = \text{Rotation} \times R_{max}~ as the resistance of the potentiometer which we determined earlier, this becomes:
$$R_{equivalent} = \frac{\text{Rotation} \times R_{max} \times R_{parallel}}{(\text{Rotation} \times R_{max}) + R_{parallel}}$$This equation defines the taper of the potentiometer, which depends on the ratio of ~R_{max}~ to ~R_{parallel}~.
Figure 4: Scaled curves for linear taper potentiometer with parallel resistor
Try an interactive version of this graph to see how changing resistor values affect the taper!
Figure 4 shows the various curves that can be achieved using different parallel resistor values with a linear potentiometer. These are similar to "reverse audio" tapers. The resistor values are selected relative to the max potentiometer resistance, e.g. for a 100kΩ potentiometer, a 10% parallel resistor is 10kΩ. As Figure 4 shows, the smaller the parallel resistance is relative to the potentiometer resistance, the more extreme the curve.
Unfortunately, figure 4 does not show the full story. Figure 4 is only showing the relative curves. In reality, wiring a resistor in parallel will also lower the max resistance of the potentiometer. The smaller the parallel resistor, the lower the equivalent maximum resistance.
Figure 5: Actual resistance curves with parallel resistor
Try an interactive version of this graph to see how changing resistor values affect the taper!
Figure 5 shows the actual resistance curves that are achieved with various resistors wired in parallel with a 100kΩ potentiometer. As you can see, the equivalent max resistance is substantially lower than it is when using just the potentiometer by itself. Fortunately, this can be accounted for. Let’s say we have a circuit that calls for a 100kΩ reverse audio potentiometer, which we do not have. We’d like to achieve the curve of a 25% parallel resistor (as seen in Figure 4) while still having a potentiometer with a range of 0Ω - 100kΩ.
Remember that the equation for the equivalent total resistance of a potentiometer and resistor wired in parallel is:
$$R_{equivalent} = \frac{R_{pot} \times R_{parallel}}{R_{pot} + R_{parallel}}$$We can re-arrange this formula to make it easy to solve for the Rpot value given a desired taper:
$$R_{pot} = \frac{R_{equivalent}}{R_{ratio}} + R_{equivalent}$$Where ~R_{ratio}~ is the ratio of the parallel resistance to the potentiometer resistance. For a 25% taper, ~R_{ratio} = 0.25~. We want the equivalent resistance to be 100kΩ, as the circuit called for a 100kΩ reverse audio pot. Plugging these numbers in, we get:
$$R_{pot} = \frac{100kΩ}{0.25} + 100kΩ$$$$R_{pot} = 500kΩ$$The desired parallel resistance is 0.25 * 500kΩ, or 125kΩ. Using a potentiometer value of 500kΩ and a parallel resistor of 125kΩ, we will have a potentiometer with an equivalent range of 0Ω-100kΩ while also following the 25% parallel resistor taper seen in Figure 4.
Figure 6: Original 100kΩ linear taper and taper of 500kΩ pot with 125kΩ parallel resistor
Figure 6 shows the actual resistance curves of the original 100kΩ potentiometer and the adjusted taper using a 500kΩ potentiometer and a 125kΩ parallel resistor.
You may be wondering if there is a way to adjust this taper the opposite way, turning a linear taper into an audio taper (rather than a reverse audio taper). The answer, unfortunately, is no - at least when using the potentiometer as a variable resistor. When using a potentiometer as a variable resistor, there is only one place to wire the parallel resistor and it will always make the taper more "anti-log". You can use a similar process to the above in order to make an audio taper potentiometer closer to a linear taper, but it’s more difficult to predict what the final taper will look like because it depends on variances in the audio taper (which is not exactly logarithmic) that the original potentiometer uses.
Potentiometers as Voltage Dividers
When using a potentiometer as a voltage divider (using all 3 lugs), we have more options for modifying the taper.
Figure 7: Potentiometer as voltage divider
Because there are two effective resistances in use in a voltage divider, we can wire parallel resistors in two places - between lug 1 and 2, or between lug 2 and 3 - to affect the taper in different ways. A voltage divider doesn’t necessarily have to be wired with lug 1 at 0V, but we will be looking at potentiometers wired that way because it gives us easier numbers to work with.
Linear to Audio Taper
In order to make a linear potentiometer function more like an audio potentiometer, a parallel resistor can be wired between lugs 1 and 2.
Figure 8: Linear potentiometer with parallel resistor between lugs 1 and 2
Again, the resulting taper depends on the ratio of the parallel resistor to the potentiometer value.
Figure 9: Audio-like tapers using various parallel resistor values
Try an interactive version of this graph to see how changing resistor values affect the taper!
Note that the output voltage ranges are the same in Figure 9 for all parallel resistors. Because the voltage divider depends only on the ratio of the resistance seen between lug 1 and 2 and the resistance seen between lug 2 and 3, the parallel resistor value will not affect the output voltage range.
There is a catch here too, which is that the effective total resistance of the potentiometer (the resistance between lugs 1 and 3) will change as the potentiometer is rotated, whereas it remains a static value with a potentiometer that is wired normally. When the parallel resistor is between lugs 1 and 2, the total resistance decreases as the potentiometer is rotated clockwise.
Figure 10: 100kΩ potentiometer total resistance with various parallel resistors between lugs 1 and 2
Try an interactive version of this graph to see how changing resistor values affect the taper!
As seen in Figure 10, while the total resistance for a standard potentiometer remains the same throughout its rotation, the total resistance decreases as the potentiometer is rotated clockwise when a parallel resistor is wired between lugs 1 and 2.
In some potentiometer applications, this changing "total resistance" of the potentiometer can be a problem. One example would be the Dallas Rangemaster. A standard schematic can be seen in Figure 11.
Figure 11: Schematic for the Dallas Rangemaster Treble Booster
One of the critical factors for making a Rangemaster that sounds right is making sure the transistor is biased correctly - in other words, ensuring that the idle DC voltage at the collector of the transistor is at the right level (usually around -7V in a Rangemaster). This voltage level determines how easily the transistor reaches cut-off or saturation, as well as the level of asymmetry in the clipping. The Rangemaster is not typically considered a dirt pedal, but it can contribute a small amount of pleasant soft clipping with the right input signal. If the Rangemaster is misbiased too far in either direction, it is likely to produce an uglier-sounding distortion at lower input levels.
There are various factors that determine the DC collector voltage, one of which is the value of the collector resistor. In the Rangemaster, the potentiometer operates both as a collector resistor and a voltage divider for controlling the output level. The total effective resistance (between lugs 1 and 3) of the potentiometer contributes to the collector voltage. In a stock Rangemaster circuit, the resistance between lugs 1 and 3 never changes, and thus the bias does not change when changing the output level. But if you wanted to achieve an audio taper using a linear pot in this circuit, adding a parallel resistor to the pot is going to have the unwanted effect of changing the bias when the output level is changed.
To see why this happens, we will do DC analysis of the Rangemaster circuit (ignoring the AC aspect for now), as we are only interested in the DC voltage for biasing considerations.
Figure 12: Schematic for the Dallas Rangemaster Treble Booster, DC Analysis
When doing DC analysis, capacitors are considered to be open circuits, leaving just the schematic seen in Figure 12. With the output capacitor replaced with an open circuit, the potentiometer operates exactly like a static resistor.
Now let’s look at the equivalent DC circuit with a parallel resistor added to the potentiometer. The DC circuit now looks like Figure 13:
Figure 13: Schematic for the Dallas Rangemaster Treble Booster with parallel resistor, DC Analysis
In the schematic seen in Figure 13, the added parallel resistor is going to make the effective collector resistance variable as the volume potentiometer is rotated. In turn, the DC voltage on the transistor’s collector will change, potentially biasing the transistor outside of its desirable range. In a stock Rangemaster, the volume level should have no effect on the transistor’s bias.
Another way to visualize how the total resistance changes is by looking at the resistance per section as the potentiometer is rotated. This can be seen in Figure 14. Because the parallel resistor is only between lugs 1 and 2, the curve of the resistance between lugs 1 and 2 is non-linear, while it remains linear between lugs 2 and 3. These two resistances are in series, so they are added together to get the equivalent total resistance. As the potentiometer is rotated clockwise, the non-linear aspect of the resistance between lugs 1 and 2 continually decreases the total resistance of the pot. The result is that the total resistance is substantially lower at 100% rotation than it is at 0%.
Figure 14: 100kΩ equivalent resistance per section with parallel resistors between lugs 1 and 2
Try an interactive version of this graph to see how changing resistor values affect the taper!
Linear to Reverse Audio Taper
In order to make a linear potentiometer function more like a reverse audio potentiometer, a parallel resistor should be wired between lugs 2 and 3.
Figure 15: Linear potentiometer with parallel resistor between lugs 2 and 3
The resulting taper again depends on the ratio of the parallel resistor to the potentiometer value as seen in Figure 16.
Figure 16: Reverse audio tapers using various parallel resistors between lugs 2 and 3
Try an interactive version of this graph to see how changing resistor values affect the taper!
Like in Figure 9, the output voltage ranges in Figure 16 remain the same for any parallel resistance value.
Again, the total resistance value of the potentiometer changes depending on the parallel resistance and the rotation of the potentiometer. With this wiring, the total resistance decreases as the potentiometer is rotated counterclockwise.
Figure 17: 100kΩ potentiometer total resistance with various parallel resistors between lugs 2 and 3
Try an interactive version of this graph to see how changing resistor values affect the taper!
Figure 18 shows how the resistance of both sections of the potentiometer change as the potentiometer is rotated. Because the parallel resistor is only between lugs 2 and 3, the curve of the resistance between lugs 2 and 3 is non-linear, while it remains linear between lugs 1 and 2. As the potentiometer is rotated counter-clockwise, the non-linear aspect of the resistance between lugs 2 and 3 continually decreases the total resistance of the pot. The result is that the total resistance is substantially lower at 0% rotation than it is at 100%.
Figure 18: 100kΩ equivalent resistance per section with parallel resistors between lugs 2 and 3
Try an interactive version of this graph to see how changing resistor values affect the taper!
Final Thoughts
For a potentiometer wired as a variable resistor, as long as you calculate the required pot and parallel resistance value, the potentiometer can have a modified taper with no negative consequences to the circuit. On the other hand, when adjusting a potentiometer that is wired as a voltage divider, it’s important to consider how the changing equivalent resistance between lugs 1 and 3 will affect the rest of the circuit. If you are unsure of the effect it will have, you can always try it and see. Please be aware that in most cases this adjustment should be fine to experiment with, but experimenting with your circuit design always leaves some risk for damage to your circuit components or equipment. Make sure you have an understanding of the effects this will have on your circuit and experiment at your own risk!