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How Does Noise Affect Circuits? - Understanding Noise Part II

In the first part of the series we made an introduction about the noise and how we can categorize them into device noise, emitted noise, and conducted noise. We first discussed about the most common noise, the Johnson noise coming from a resistor. This noise is also a white noise, due to the fact that it is present at all frequencies. This noise is produced whether or not a resistor is connected in a circuit.

When a resistor is connected to a circuit, say to an amplifier, this resistor will be a source of noise to the circuit. Let's take an inverting operational amplifier as an example that has two resistors connected to it, one at the input and the other one as feedback. The noise due to the resistors can be modeled as noise sources in series with the resistors.

Figure 1 Resistor Noise

In order to appreciate its effect in the circuit, we must be able to understand how these noise affect the behavior and output of the amplifier. Modeling the noise sources as in the figure above will help us easily understand the concept. The ensuing analysis should be straightforward using KVL and good mastery of the operational amplifier. Let's assume at the moment that the op amp is ideal and doesn't contribute noise to the system.  The noise due the resistor R1, eN1, will appear as signal at the input and will be gained up by the factor (RF/R1), because it appears as if an input to the inverting amplifier. But because noise doesn't have polarity, there is no negative sign to it.  Notice that all inputs at are ground when we do the noise analysis, using the superposition technique. If the noise is given as spectral density, Vrms/root Hz, we then proceed to calculating the spectral noise density at the output due to the individual noise sources. The noise due to the feedback resistor, RF, will then just be as it is without any amplification, because the negative input of the op-amp is a virtual short to ground.

The total noise at the output of the amplifiers due to the resistor noise, in  will then be:



If we know the bandwidth  with which we are going to operate our circuit, we can calculate the noise at the output in terms of RMS.

The noise in RMS can be obtained by multiplying the spectral noise density by the square root of the bandwidth. This is a straightforward task because we are dealing with white noise where noise power level is the same at all frequencies. But there is a correction factor that needs to be used because the bandwidth, and the filters as such, aren’t ideal brick wall. The following correction factors:

1.57fC for single pole filter
1.2 fC for the double pole

This means multiplying your bandwidth by 1.57 if the circuit is a single-pole low pass filter in order to obtain the equivalent noise bandwidth. This also means that a single-pole filter allows more noise that would a brick wall filter allows at the same cut-off frequency.

The discussion on noise density, rms noise, and peak-to-peak, are in order in the Part III of this series.


Reference:
MT-048 Tutorial, Op Amp Noise Relationships: 1/f Noise, RMS Noise, and Equivalent Noise Bandwidth 

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