# What Determines the Noise Bandwidth of a DC Voltmeter?

Say I have a 1000 Ohm resistor. At room temperature, this has 4 nV/rtHz of Johnson Noise. That means that a measurement with a bandwidth of 100 Hz would see 40 nV of RMS noise.

If I were to connect the resistor to a preamplifier, run the output of the preamplifier through a bandpass filter with 100 Hz bandwidth (say 2950 Hz to 3050 Hz), and then run the output to an oscilloscope sampling at more than 7100 Hz, I would see around 40 nV of RMS noise from the resistor (multiplied by the gain of the preamplifier) plus whatever noise is added by the amplifier.

That case is simple, and easy to understand because a bandpass filter sets the bandwidth. How could I calculate the amount of RMS noise generated by the sample, when I am measuring with, say, the DC Voltmeter setting of a Keithley 2000 Multimeter? In this case, there is no bandpass filter (correct?). I do not have a good understanding of how voltmeters actually work.

Thank you!

• Every circuit limits bandwidth so there will effectively be some sort of bandpass filter. I'd expect some clues in the multimeter's datasheet ... might be useful to link that in the question. Also, what is its resolution on its most sensitive range? Can it even read 40nV? – Brian Drummond Feb 7 '17 at 16:48
• You are correct that it cannot read 40nV. However, if the bandwidth were actually 100 kHz then the noise would be in the uV range and visible. The data sheet is here: research.physics.illinois.edu/bezryadin/labprotocol/…. I know that the AC measurements have a bandwidth of 3/30/300 Hz (lower) - 300 kHz (upper). The only thing they have about DC are the # of power line cycles integrated over. – Jeremy Harrington Feb 7 '17 at 16:58

If you want to know what the bandwidth is, it should be in the datasheet, if its not then it might be hard to find it. One thing you might be able to do is limit the bandwith yourself.

Measuring noise bandwidth is not going to work with a DC volt meter, its measuring near 0Hz (whatever the update rate is). RMS values measure noise across many frequencies and use a bandpass filter.

There are two problems of measuring resistor thermal noise:
1) Other noise sources are going to make it difficult, meters, cables and amplifiers are all noise sources and need to be accounted for when measuring noise
2) The bandwidth of the noise and sampling of the noise needs to be accounted for because we cannot measure the distribution of white noise directly (that would involve measuring across all times to infinity, but we can approximate it).

The noise of a resistor looks like this:

$$\frac{v_n^2}{\Delta f} = 4k_BTR$$

The voltage source is modeled as the noise power and that power changes depending on how much noise sources frequency you sample, like you did in the oscilloscope example.

No matter the instrument, you will need to know the bandpass of the noise or you will not be able to determine the noise power.

It might also be worth noting that you can ground out the inputs and measure the baseline noise (of the pre-amplifier or meter) if your using them and find out how much noise you have from the system. Then connect the resistor that you wish to measure, the RMS noise from the resistor and the system will add by the sum of the squares.

If you want to measure noise with a DC voltmeter you would need a pre amplifier and a way to convert the total noise power to an analog measurement at DC. There are circuits you could use to do this, but most of them would be hard to build yourself, and probably not worth your time. Some manufactures make IC's such as this one(AD737) that can calculate the RMS power, then run that value into the DC Meter.

If the meter has an RMS (AC) button, then just use the AC button after the amplifier and use a preamplifier, the bandwidth of the RMS measurment must be known to calculate the noise power from the resistor.

If your just using a DC meter, your limited on what noise you can measure based on the accuracy of the meter.

Here are some other good resources: Noise sampling Low level measurements Handbook

• This isn't exactly what I am trying to ask. I don't want to measure the noise bandwidth or even the RMS noise. I am just trying to estimate the amount of noise I would expect to see (say, on the display) if I plugged in a 1 kOhm resistor and measured DC voltage. – Jeremy Harrington Feb 7 '17 at 17:11
• The noise bandwidth of a DC voltmeter is actually determined by its integration time. One can model the voltmeter as a moving average filter and from that determine the noise bandwidth (sorry, I don't have the time to write a full answer). – Massimo Ortolano Nov 25 '17 at 10:37

The total capacitance on the node you are measuring will set the total RMS noise, independent of the bandwidth. Use sqrt(K*T/C).

Cap TotalIntegratedNoise_rms

10pF 20uV

100pF 20/3.16

1,000pF 20/10

10,000pF 20/31.6

• The total capacitance on a node.....with only resistive elements contributing to KT noise, sets the TOTAL Integrated Noise and is exactly computable from sqrt(kt/c). – analogsystemsrf Feb 10 '17 at 17:17