Part one, the principle of current source method

**The principle: I/V method**This is the fundamental way of measure a resistor by Ohms-law:

R = V / I

In order to get a good result, the V and I should be equally good or better because any imperfection in the reading of V or I will directly affect the calculation of R.

That's is to say, if the numerator or the denominator changes by 1ppm, the result of a monomial equation will change 1ppm as well.

**The principle: I/V substitution technique**However, we can use the substitute technique, a discipline widely used in the metrology fields.

While in the old days substitution was done manually, they only substitute once or twice(probably 10 times in extreme). Modern switches and scanners allow multiple substitutions more easily.

In the diagram I only draw one-wire substitution for simplicity, but in reality a 4-wire(4-pole) switch will be used.

The good things about substitution technique, it not only tends to cancel errors like leakage or contact, but it abolishes the requirements for accurate current source or even

Of course the requirement for current source is the short term stability(and low noise), that is a much easier task to achieve without the accuracy requirement.

We do have requirement for the voltmeter not for accuracy but for low noise and good linearity instead.

Let's assume that the current I remains the same during the measurement(if not, any variation will contribute to the resulting uncertainty), and the voltage reading for Rs and Rx are Vs and Vx.

Rx=Vx/I

Rs=Vs/I

Rx = (Rx-Rs) + Rs = (Vx-Vs)/I + Rs

Because Rx is another standard resistor, the value should be very close to nominal thus close to Rs, the first term((Vx-Vs)/I) is therefore much smaller than Rs, this will require much less for how accurate the current source and how accurate the voltage reading.

Though (Vx-Vs) is very small, it is the difference of two very large quality, any variation of Vx and Vs affects the result significantly. We need a DMM of both low noise and high linearity like 3458A to achieve the expect result. Put it in another word, we rely on precision of the DMM rather than accuracy.

**The principle: I/V substitution plus voltage comparison technique**In this configuration, we use a 10V voltage standard Vr to step up the reference point of the voltmeter. Since the Vx and Vs in the previous configuration are all very close to 10V, the voltage difference between Vs/Vs and 10V is very small, allowing a reasonably good voltmeter or 5.5 digits DMM to measure in 100mV range.

Again, this 10V voltage need not to be accurate but need preciseness(stable and low noise) thus easy for us to make and there is no need to be calibrated.

Let this voltage be Vr, and the measurement of two voltage differences are dVx and dVs

Rx = (dVx-dVs)/I + Rs

(assume no change of Vr during the measurement, if it does, the sensitivity is 100%)

This looks not much change from the previous equation, but this time, dVx or dVs is much smaller than Vx or Vs allowing the lowest range of the DMM to be used. Even some hand-held DMM is capable of resolve 1uV, which equivalent to 0.1ppm resolution in the final result.

**4-pole double-throw switch**It furnishes the substitution.

It can be a scanner, Dataproof type say.

It can be a DIY, like mine:

https://www.eevblog.com/forum/projects/diy-low-thermal-emf-switchscanner-for-comparisons-of-voltage-and-resistor-stand/msg610755/It can be a mechanical, like another one of mine:

The point is: switch quickly and repeatedly.

**Sensitivity analysis**Sensitivity refers to the degree that input affects output.

A sensitivity of 10% means that if one of the input or components change 20ppm, the output will change only 2ppm(10% of 20ppm).

Assume that Rs, Rx, Vref are within +-0.01% nominal, I is within +-0.2% nominal, DMM is also 0.2% in its 100mV range. These conditions are not very difficult to met.

Detailed analysis procedure is omitted(partial differential involved) but here are the results:

short term stability and noise of the 1mA current source: 100%

short term stability and noise of the 10V reference: 100%

Accuracy, noise and stability of the DMM: 0.03% (the change of 100ppm on the DMM reading will only affect 0.03ppm to the final result)

Contact and wire resistance of the switch: negligible if < 1 Ohm

Thermal EMF of the switch: 0.1ppm per 1uV

Leakage current of the switch: 0.1ppm per 100pA

**Other ways to compare two 10k resistor standards** -- Modern DCCs. They are very expensive to amateurs.

-- Old DCC. Like Guildline 9975, I had one, specified as 0.2ppm for comparison on most of the common ranges, but is slow to operate and only compare two 10k in two wire mode.

-- Kelvin Bridge. Like esi 242D, I had one, specified 0.2ppm for 10k comparison, similar drawbacks as 9975.

-- Warshawsky Bridge. Like that employed at NIST, I made one with much better sensitivity but no guard, allowing me to compare 10k resistors within 0.1ppm.

-- Direct measurement of resistors with a 8.5-digit multimeter.

Because the current source is not ideal, often 0.1mA resulting only 1V output, and the linearity etc on 1V range is worse than that of 10V, Some of the DMM and the measurement uncertainty(in parenthesis), assuming 1 year calibrate interval:

3458A(10.5ppm+3ppm, 0.1mA), 8508A(8.0ppm, 0.1mA), 2002(9.8ppm+7.8ppm, 0.096mA), 1281(9.6ppm+5.5ppm, 0.1mA), 6581(8.5ppm+3.1ppm, 1mA), 7081(9.5ppm+?, 1mA), 8081(10.3ppm+5.7ppm?, 1mA)

(Note, the two ppm in parenthesis cannot be simply added together, they should be combined in a Root-Sum-Square summation)

-- Substitute measurement with a 8.5 digits multimeter(transfer).

Transfer uncertainty(some are estimate): 3458A(--0.5ppm), 8508A(0.5ppm), 2002(1.7ppm), 1281(--), 6581(--), 7081(0.3ppm)

the best uncertainty one can achieve is probably 0.5ppm(Fluke 8508A, with 0.1mA current). Solartron 7081 on the other hand do have a 1mA measurement current and 0.3ppm transfer stability but is too old, difficult to find, and very slow to read.

In summary, we need a good standard resistor to compare the unknown standard resistor. We also need a stable(but may not be accurate) current source and a stable 10V voltage reference(but may not be accurate) to do the comparison, apart from a 4-pole double-throw switch and a DMM.

If this properly, the transfer uncertainty can exceed 0.1ppm with ease.