The Kelvin-Varley voltage divider is a precision circuit. The circuit is used in laboratory equipment and can achieve accuracy and resolution of 1 in 10 million.
The conventional voltage divider (aka Kelvin divider) uses a tapped resistor string. The fundamental disadvantage of this architecture is that resolution of 1 part in 1000 requires 1000 precision resistors.
The Kelvin-Varley divider uses a clever iterated scheme. In a typical design, each stage provides a decade of resolution and requires only 11 precision resistors. Cascading 3 stages permits any division ratio from 0 to 1 in increments of 0.001.
Each stage of a Kelvin-Varley divider consists of a tapped string of equal value resistors. Let the value of each resistor in the i-th stage be Ri ohms. For a decade stage, there will be 11 resistors. Two of those resistors will be bridged by the following stage, and the following stage is designed to have an input impedance of 2Ri. That design choice makes the effective resistance of the bridged portion to be RiThe resulting input impedance of the i-th stage will be 10Ri.
In the simple Kelvin-Varley decade design, the resistance of each stage decreases by a factor of 5: Ri + 1 = Ri / 5. The first stage might use 10 kilohm reistors, the second stage 2 kilohm, the third stage 400 ohm, the fourth stage 80 ohm, and the fifth stage 16 ohm.
The final stage of a Kelvin-Varley divider is just a Kelvin divider. For a decade divider, there will be ten equal value resistors. Let the value of each resistor be R1 Ohms. The input impedance of the entire string will be 10Rn. Alternatively, the last stage can be a two resistor bridge tap.
A shunt resistor can be used to keep the resistor values the same.
In practical designs, the resistor values will not be exactly the desired value. The initial division ratios need tight control, so designs will allow the resistor values to be trimmed.
Typically, each resistor in a string will be adjusted to have the same value. That will set an equal division ratio.
The shunt resistor also allows the string's impedance to be trimmed with one control rather than adjusting all eleven string resistors. This is a construction issue. It is easy to adjust two resistors to have the same value with a Wheatstone Bridge, but it is more difficult to adjust a resistor to have one eleventh the value of another. The latter operation requires a precision divider.
Other Design Issues
Ideally, a resistor has a constant resistance. In practice, the resistance will vary with time and external conditions. Resistance will vary with temperature.
Carbon film resistors have temperature coefficients of several 100 parts per million per degree C. Some wirewound resistors have coefficients of 10ppm/degC. Some metal foil resistors can be 1ppm/degC.
The energy dissipated in a resistor is converted to heat. That heat raises the temperature of the device. The heat is conducted or radiated away. A simple linear characterization looks at the average power dissipated in the device (watts) and the device's thermal resistance (degrees C / watt). A device that dissipates 0.5 watts and has a thermal resistance of 12 degrees per watt will have its temperature rise 6 degrees above the ambient temperature.
Kelvin-Varley dividers are used to test high voltages, and that creates a problem with self-heating. The first divider stage is often made from 10 kilohm resistors, so the divider input resistance is 100 kilohms. Total power dissipation at 1000 volts is 10 watts. Most of the divider resistors will dissipate 1 watt, but the two resistors bridged by the second divider stage will only dissipate 0.5 watt each. That means the bridged resistors will have only half the self-heating and half the temperature rise.
For the divider to maintain accuracy, the temperature rise from self-heating must be limited. Getting very low temperature coefficients keeps the effect of temperature variations small. Reducing the thermal resistance of the resistors keeps the temperature rise small.
Commercial Kelvin-Varley dividers use wirewound resistors and immerse them in an oil bath.
Thermal EMF problems
More info. is needed about this, but, particularly when working with unusual precision and accuracy (for which such dividers are well suited), one must be aware that different metals in contact become thermocouples. These generate unwanted voltages that interfere. Careful design mitigates such problems.
William Thomson, 1st Baron Kelvin
^ Fluke 720A Kelvin-Varley divider