# Designing a Kelvin-Varley Potentiometer Part 3 of 3 – Kelvin Varley Dividers

Over the last two blog posts, we’ve built the foundations of a Kelvin-Varley potentiometer. Recall that a potentiometer is a measuring instrument that uses a resistor with three terminals and a wiper to form an adjustable voltage divider. This voltage divider can be used in applications such as light, volume, and fan speed controls, though the Kelvin-Varley variety is more suited to precision applications in electronics labs.

In the previous blog, we added a string of 10 identical tapped resistors to enable voltage selection in increments of 10, thus creating a Kelvin potentiometer. This method can increase resolution by a factor of 10, 100, or more, but there is a problem: Adding to an element of the previous string also requires adding buffers between the new strings, and buffers are a notorious source for imprecisions and non-linearities.

We’re well on our way toward the series goal: to create a potentiometer that functions as a laboratory-grade digital-to-analog converter (DAC). In this third and final blog, we will envision a smarter potentiometer without buffers.

## Lose the Buffers, Keep the Resistance

Isolation buffers can be removed; just choose a second string with resistor values that can be connected to the first string without changing its original value (10R). But putting something in parallel inherently changes the value of R, and it will take at least two resistors to restore it (Fig. 1).

Figure 1: Second column resistor values – paralleling 2 elements R

In order to keep every R of the first column constant, even when it is loaded by the 10R’ from the second column, we need to ensure that 2R in parallel with 10R’ still equals R.

Solving the equation: R’ = R/5

But notice that adding 10R’ (or 2R) at any position on the string replaces two elements R from the first column. We’ve lost one resistor in the string (i.e. the total resistance of the first string is not 10R anymore, but 9R). We need to compensate by adding an additional resistor R in the first string (Fig. 2).

Figure 2: Final 2-digit Kelvin-Varley potentiometer.

In the final iteration of our two-digit Kelvin-Varley potentiometer (Fig. 3), the first (main) digit is determined by a double wiper. Its two hands always point together across two resistors R on the first string. The wiper on the second string determines the second digit. It can remain singular since its elements won’t be loaded by an additional string.

## Kelvin-Varley Divider as a DAC

To use the Kelvin-Varley Divider (KVD) structure as a digital-to-analog converter (DAC), we will implement strings of eight resistors instead of 10. Using isolators in the wiper connections ensures that all columns are identical and all resistors within each column share a common value. We can use the double-wiper technique described above to eliminate the buffers between strings.

Figure 3 – KVD structure to build a two-digits DAC

In Figure 4, each double-wiper position defines three binary bits. The first column contains the three most significant bits (100) while the second column contains the three least significant bits (011). Thus, the digital inputs are 100011. If Vref=1 volt, the output will be 0.43 volts, which is well represented by 100011.

Now what if, instead of 10 or eight resistors, we tried using only one tap per column? In such a structure, the values between columns decrease by two at every step, a progression that is well suited for binary numbers. We have, in fact, created an R-2R ladder, a classic structure that can be built using the Kelvin-Varley principle!

Figure 4: Basic DAC using R-2R ladder structure derived from KVD

## Precision Applications for Kelvin-Varley Dividers

Despite of the use of only pure passive resistances, KVDs function as high-precision lab equipment used to measure voltage and resistance at PPM levels. They are often paired with other precision devices such as voltage references and null detectors or galvanometers.

In addition to acting as a DAC, KVDs can be used to determine linearity, measure voltage or resistance with high precision, and calibrate voltage, current, or resistance values. For example, here we see a KVD used to determine an unknown DC voltage, Ex. A similar setup can be made to measure unknown resistance, divider ratio, and more.

Figure 5: Use of KVD to measure unknown voltage

## Finite Regression: The Limits of a Kelvin-Varley Potentiometer

The technique from this blog can be used to create and attach a third column of 10 resistors R” in parallel with two elements R’. The cascading operation can be extended to a fourth, fifth, or sixth column for even greater precision.

Theoretically, the extension can be infinite, but quickly the resistor values in the added columns become too small to handle. They decrease as follows: R, R/5, R/25, R/125, R/5n (n being the rank of the column).

So, if you started with R=2,000 Ω, then each resistor in the second column would have a resistance of 400 Ω, then 80 Ω, then 16 Ω, then 3.2 Ω, etc. This is why Kelvin-Varley potentiometer precision generally does not exceed six digits, or in rare cases, seven. Seven-digit KVDs can cost several tens of kilo dollars apiece.

Figure 6: 7-digit Kelvin-Varley potentiometer structure

Banner image attribution: Dr. Hannes Grobe via Wikimedia Commons. Image used under the Creative Commons Attribution 4.0 International license.