Ammeter Design

Shunt resistors used for precision current measurement always have four terminals for the electrical connections, even though normal resistors only have two:

Explain what would be wrong with connecting the voltmeter movement directly to the same two terminals conducting high current through the shunt resistor, like this:

Shunt resistors, being very low in resistance, are usually made from relatively large masses of metal. Their precise resistance is calibrated through a process known as trimming, where a technician takes a metal file and “trims” metal from the shunt conductor until the resistance reaches its correct value. This, of course, only works if the shunt resistor is intentionally manufactured with a resistance that is too low. Like the old carpenter’s joke goes, “I cut the board twice and it’s still too short!”

Being that shunt resistors have such incredibly low resistance values, how do we measure the resistance of a shunt with high accuracy during the “trimming” process? The resistance of a shunt is far too low for an average handheld or even benchtop ohmmeter to measure with precision, and specialized low-resistance ohmmeters such as the Kelvin Double Bridge are quite expensive. If you were given the task of trimming a shunt resistor for use in an ammeter, and you only possessed average pieces of test equipment, how could you do it?

An important step in building any analog voltmeter or ammeter is to accurately determine the coil resistance of the meter movement. In electrical metrology, it is often easier to obtain extremely precise (“standard”) resistance values than it is to obtain equally precise voltage or current measurements. One technique that may be used to determine the coil resistance of a meter movement without need to accurately measure voltage or current is as follows.

First, connect a decade box type of variable resistance in series with a regulated DC power supply, then to the meter movement to be tested. Adjust the decade box’s resistance so that the meter movement moves to some precise point on its scale, preferably the full-scale (100%) mark. Record the decade box’s resistance setting as R₁:

Then, connect a known resistance in parallel with the meter movement’s terminals. This resistance will be known as R_s, the shunt resistance. The meter movement deflection will decrease when you do this. Re-adjust the decade box’s resistance until the meter movement deflection returns to its former place. Record the decade box’s resistance setting as R₂:

The meter movement’s coil resistance (R_coil) may be calculated following this formula:

Your task is to show where this formula comes from, deriving it from Ohm’s Law and whatever other equations you may be familiar with for circuit analysis.

Hint: in both cases (decade box set to R₁ and set to R₂), the voltage across the meter movement’s coil resistance is the same, the current through the meter movement is the same, and the power supply voltage is the same.