# Why Use an Antilog Taper Pot?

Quote of the Day

I have always imagined that paradise will be a kind of library.

— Jorge Luis Borges. I understand this feeling – I still love being in a library.

## Introduction

Figure 1: Sound Pressure Level versus Frequency
and Perceived Sound Level (Wikipedia).

I have received a number questions lately on the use of log and antilog taper potentiometers. Because of these questions, I thought it might be useful to review why these tapers are used.

These tapers are primarily used with audio systems. While I am not an audio aficionado, I do appreciate controls that vary linearly, i.e. a small rotation of the knob makes a correspondingly small change in an output. The log and antilog taper potentiometers are used to ensure that audio systems controls have linear characteristics as perceived by the human ear.

## Background

### Definitions

phon
The phon is a unit that is related to dB by the psychophysically measured frequency response of the ear. At 1 kHz, readings in phons and dB are, by definition, the same. For all other frequencies, the phon scale is determined by the results of experiments in which volunteers were asked to adjust the loudness of a signal at a given frequency until they judged its loudness to equal that of a 1 kHz signal (source).
Sound Pressure Level (SPL)
Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average, or equilibrium) atmospheric pressure, caused by a sound wave. In air, sound pressure can be measured using a microphone, and in water with a hydrophone. The Pascal (Pa) is the SI unit of sound pressure (source).

### Overview

Understanding Figure 1 is the key to understanding the role of the log and antilog tapers in potentiometers. The phon scale is intended to give you a perceived loudness scale that is linear, i.e. a 20 phon level that increases to 40 phon will be perceived as twice as loud. Looking closely at Figure 1, we can see that a  20 phon level  @ 1 kHz increase in corresponds to a 20 dB increase in SPL. Another way to view this relationship is to say that an increase of one phon increases the SPL by 1 dB or 12.2 % – SPL must increase geometrically to increase the perceived sound level linearly.

## Analysis

### Assumptions

• The speaker drive voltage is proportional to the resistance of the potentiometer.
• The speaker output power is proportional to the square of the speaker drive voltage.
• The SPL level in Pascals (not dB) is proportional to the square root of the output power, i.e. sound power is proportional to the square of the pressure.
• The percentage of the full-scale resistance will be expressed in terms of the percentage of full-scale wiper position.
• For my example here, I will use the W taper potentiometer discussed in this post. I will slightly change the resistance characteristic derived previously to eliminate the constant I added to ensure the resistance was 0% of full-scale at 0% wiper position. This will give me an ideal antilog characteristic, i.e. $R\left( x \right)={{R}_{0}}\cdot {{e}^{{{{R}_{1}}\cdot x}}}=0.003\cdot {{e}^{{3.693\cdot x}}}$

### Calculations

#### Resistance Range in dB

Figure 2 shows how the dynamic range of our ideal antilog potentiometer, which is 29 dB.

Figure 2: Overall Potentiometer Resistance Variation.

#### Phon Level Versus Wiper Position

Figure 3 shows that our perceived sound level (in phons) varies linearly with our ideal exponential potentiometer's wiper position. I have also include how the performance of a potentiometer with an ideal antilog taper differs from that of a typical, real, antilog taper potentiometer.

Figure 3: Phon Variation with Wiper Position. Observe how the dynamic range of the sound level equals the dynamic range of the potentiometer.

## Conclusion

This is just a quick note to illustrate why an antilog taper can be useful in giving an audio system a linear perceived sound level with potentiometer wiper position.

There are two worthwhile videos on this topic on Youtube. Here is a very good video that shows the difference in sound from a guitar using a linear versus a log/antilog potentiometer.

Here is a video that illustrates how to measure the resistance characteristic of a log/antilog potentiometer.

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### One Response to Why Use an Antilog Taper Pot?

1. timhughes@ieee.org says:

Quite often people fake various pot tapers by paralleling the slider with one or two resistors.
Here for example is a website that has some details on making log/antilog pots:
http://www.geofex.com/article_folders/potsecrets/potscret.htm
You could make a nice mathcad document to choose the best resistor and plot the error function from ideal.

This has the advantage of not requiring a hard to get non-linear pots and it is a continuous function,not piecewise approximation. It also may have better channel to channel matching for stereo, as the taper pots require more complex manufacture and may not match as well.

The Fletcher-Munson loudnes curves shown, also illustrate why it is difficult to do A-B comparisons, when Hi-Fi enthusiasts compare speakers or other equipment empirically. Modest changes in amplitude between equipment can substantially change the frequency response perceived. In addition if blinded,with the same equipment compared with itself ,in a fake A-B compaison, people will often choose the few dB louder setting as superior.

Because of the loudness curves,people doing mixing will often set the spl on the monitor speaker at around 88dB or so, to try to get the flatest perceived frequency response.
Tim Hughes