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I bought a no-name but decent multimeter, and it came with a thermocouple. Let's say I broke it in some way. Can I just replace it with any other thermocouple, or is each thermocouple calibrated for one specific model of a multimeter?

I have been looking at thermocouples, eg on eBay, e.g. this one. They write some specs but they don't write which multimeter it fits with?

So does any thermocouple fit with any multimeter (or digital thermometer)?

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In general any thermocouple can be used with any meter that handles thermocouples as long as they are compatible. That means that a K type thermocouple must be paired with a meter that is calibrated for K type thermocouples. K type thermocouples are the most common so it is very likely that your meter can handle the one you looked at on Ebay which is a K type. However there are much cheaper K types available than that one which is specified for very high tempertures. As far as calibration goes, there are standard tables of thermocouple voltage versus temperature for each type of thermocouple. This is possible because each type of thermocouple uses the same composition of wires. Thus your multimeter would have been designed to use the table for K type thermocouples. The actual accuracy you can achieve is determined by the specified accuracy of the thermocouple and how well the meter conforms to the standard table values which are nonlinear.

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If the original had a yellow plug it was probably type K (Chromel-Alumel) and you can replace it with any other type K thermocouple. The other common color codes are blue (T) and black (J).

I only say "probably" because there is no way of knowing what some random maker in China might do, but those (ISA) color codes are very widely used despite other standards that existed in Japan and Europe.

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There are a few different types of thermocouples. In general you can replace a thermocouple with the same type, but you can't with different a different type because the calibration constants will be different.

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