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Resistor Noise—reviewing basics, plus a Fun Quiz

-December 02, 2012

The noise performance of amplifier circuits is greatly affected by the Johnson noise of resistors—the source resistance and feedback resistors. Most everyone seems to know that resistors have noise but may be a bit foggy on some of the details. Here’s a bite-sized review in preparation for future discussions on amplifier noise:

The Thevenin noise model for a resistor consists of a noiseless resistor in series with a noise voltage, figure 1.

The noise voltage is proportional the root of the resistance, bandwidth and temperature (Kelvin). We often quantify the noise in a 1Hz bandwidth, its spectral density. The theoretical noise of a resistor is “white,” meaning that it is spread uniformly over frequency. It has equal noise voltage in every equal slice of bandwidth.

The noise in each 1Hz band sums randomly according to the root of the sum of the squares. We often refer to the spectral density in volts/root-Hz. The numerical value is the same as for a 1Hz bandwidth. For white noise it’s convenient to multiply by the square-root of a bandwidth to sum the random contribution of each 1Hz band. To measure or quantify the total noise, you need to limit the bandwidth. Without a known cutoff frequency you don’t know how much noise you are integrating.

We instinctively think of spectral plots as having a logarithmic frequency axis—a Bode plot. Note that a Bode plot has more Hertz of bandwidth on the right side than the left side. Considering total noise, the right side of a Bode plot may be much more important than the left side.

Resistor noise is also Gaussian, a description of its amplitude distribution, a probability density function. It’s Gaussian because it’s created by the summation of a gazillion little random events. The Central Limit Theorem explains how this becomes Gaussian. The RMS voltage of AC noise is equal to ±1σ of the amplitude distribution. For 1V RMS noise, there is a 68% (±1σ) probability that the instantaneous voltage will be within a ±1V range. A common misconception is to relate or equate white and Gaussian but they’re unrelated. Filtered resistor noise, for example, is not white but it remains Gaussian. Binary noise is definitely not Gaussian but it can be white. Resistor noise is white and Gaussian.

Purists like to rant that Gaussian noise does not have a defined peak-to-peak value—it’s infinite, they say. True enough because the tails of a Gaussian distribution reach to infinity so any voltage is possible. As a practical matter, the likelihood of noise spikes beyond ±3x the RMS value is pretty small. Many folks use an approximation of 6x the RMS for the peak-to-peak value. You can add a large additional guard-band by using 8x without greatly changing the value.

Some fun points to ponder:  The noise voltages of two resistors in series sum randomly and result is the same noise as for the sum of the resistor values. Similarly, the noise of resistors in parallel results in the noise of the parallel resistance. If it worked out differently, it would be problematic as you think about bisecting a physical resistor and combining them in series or parallel. It all works out. :)

A large value resistor lying on your desk will not arc and spark from unlimited self-generated noise voltage. Stray parallel capacitance will limit the bandwidth and the total voltage. Similarly, the high noise voltage you might imagine on insulators is shunted by parallel capacitance and the resistance of conductors around them.

Fun Quiz—What is the total open-circuit noise voltage on a resistor that has a stray parallel capacitance of 0.5pF?  The solution details will be posted here after someone comments with a correct answer.

Sorry, but I’ve run beyond my self-imposed word limit. If you’ve made it this far, thanks for reading! Comments (and answers) are welcome.

Bruce       email:  thesignal@list.ti.com (Email for direct communications. Comments for all, below.)

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