October 23, 1997

Linear potentiometer implements logarithmic gain control

W Stephen Woodward, University of North Carolina - Chapel Hill

Trimmer potentiometers are ubiquitous components and are available in a variety of packages, resolutions, and temperature stabilities. However, none of these potentiometers implements anything but the usual linear function of resistance vs shaft position. This fact makes trouble for applications that need a wide dynamic-adjustment range.

Imagine, for example, an amplifier with gain that you set over a range of 0 to 10,000 with one 10-turn potentiometer. With a linear relationship between potentiometer position and gain, every turn represents a gain change of 1000, and every degree of shaft rotation represents an increment of about 3. If you assume that you can change settings by about 3 or 4º (0.1%), the result is minimum-gain increments of ~10 and good resolution of ~1% for gains greater than 1000.

However, the situation badly deteriorates for gains less than 100 (10%). And, if you need to set a gain of around 10, you can forget about hitting the necessary value with any meaningful accuracy. If accurate and stable trimmers with logarithmic tapers were available, it would be easy to arrange a gain-control circuit with constant resolution over the full adjustment range. Unfortunately, such trimmers are unavailable.

Fortunately, you can use a circuit that approximates a logarithmic gain control using a standard linear-taper potentiometer (Figure 1a). If P represents the wiper position (0<P<1), then the overall gain, G, is:

The interesting feature of this equation is the semilogarithmic behavior of P/(1­P). In Figure 1a, with R₃/R₁=100, a P=0.5 (midscale) gives G=100. P=0.01 gives G=1.01; a P=0.1 gives G=11; P=0.9 gives G=900; and P=0.99 produces G=9900. That gain-setting ability remains no worse than 10% over the ~10,000-to-1 range is especially significant.

This trick--using the potentiometer wiper as an input terminal--effectively moves the wiper contact inside the feedback loop of IC_1A, thus removing it as an error term and improving time and temperature stability of the gain setting. Selecting appropriate resistance values for an arbitrary gain range is easy. Choose R₂ first, on the basis of op-amp input bias and bandwidth limitations. Then set R₃=R₂ and R₁=R₂/(square root G_MAX). If dc response is unnecessary, the optional coupling capacitor, C₁, prevents gain-multiplication of amplifier offset. For any chosen cutoff frequency, F_O, C₁ must be greater than or equal to (square root G_MAX)/2 pi R₂F_O. Because only a few millivolts typically appear across C₁, use of polarized electrolytics should be OK.

For some applications, the input impedance of the amplifier in Figure 1a may be inconveniently low. If a minimum gain setting of 1.0 (rather than zero) and an overall inverting gain are acceptable, you may prefer the circuit in Figure 1b due to its high input impedance. (DI #2091)

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