Resistor Noise Quiz—the solution

Fun Quiz— What is the total open-circuit noise voltage on a resistor that has a stray parallel capacitance of 0.5pF?

Answer— The total noise voltage of a resistor with parallel capacitance is independent of the resistor value. Although noise density produced by the resistor increases with resistance, the noise bandwidth is reduced.

The -3dB point is at f = 1/2πRC. The effective noise bandwidth of a single real pole is greater than the -3dB frequency because the slow cutoff of a single pole passes significant noise beyond the -3dB point. This increases the effective noise bandwidth to π/2 times -3dB frequency, derived by integrating the noise pass-band of the pole from zero to infinity.

 If you don’t account for the additional noise bandwidth of the single pole (π/2), you arrive at a noise voltage of 72.4uV—not a big error (the square-root of π/2).

The graph below shows a TINA-TI SPICE noise simulation of various resistor values in parallel with 0.5pF. Any point on these lines indicates the total noise of that resistor from 0Hz up to that point in frequency. A voltmeter with infinite bandwidth would be limited only by the single pole and measure the value of these lines at infinity—90.7uV. Without the 0.5pF capacitance the noise would continue to rise indefinitely.

Those familiar with switched-capacitor sampled systems will recognize the result as equal to the noise voltage sampled on a capacitor. A switch (with any series resistance) charging a capacitor has this same noise or uncertainty in the stored voltage when the switch is opened. Aside from the minor contribution of temperature, increased capacitance is the only way to reduce the noise.