Steps used in doing Noise Analysis on the Noninverting Amplifier

October 18, 1999

The particular order that was followed is not critical. However, it is probably easier to "do the bookkeeping" if similar noise sources are analyzed in sequence. The principle of superposition is used to simplify the analysis. Remember, however, that there is an implicit assumption that the circuit being analyzed is linear. This is usually the case since the noise components are sufficiently small that a linear response can be assumed. In this particular case, the noninverting amplifier, it is certainly a valid assumption.

I performed the analysis using the rms-equivalent noise sources, rather than their mean-square sources directly. This was done to make the circuit analysis the same as you have always done with other sources. However, we must determine the total equivalent noise by adding the mean-square noise voltages. This can be done at either the output or by referring them back to the input and doing it there. The normal practice is to refer all the noise seen at the output to an equivalent source at the input. In our case, we have two inputs, but the normal practice is to refer them to the noninverting input. Since we usually know the input signal's characteristics including its signal to noise ratio, having an equivalent input noise source of the instrument system facilitates calculating the output signal to noise ratio.

  1. Draw the circuit schematic for the noise model. In general, this will include thermal noise from resistors and inductors (due to their series resistance), thermal and shot noise from diodes, and voltage and current noise sources for any active components. {If DC voltage sources are shown as part of the operation of the circuit, then the noise inherent to the source should be taken into account. Batteries do have noise associated with them. Electronically derived reference voltages will also have noise. Power supply noise is not usually if it is used to power an operational amplifier. However, if it used elsewhere in the circuit then it would have to be considered.}
  2. Sequentially apply the principle of superposition by selecting one noise source and shorting out all other voltage sources and open circuiting all other current sources.
  3. For each source, determine the output noise voltage.
  4. Refer that voltage back to the input and square it.
  5. Sum the mean-squared voltages and take the square root.

In our example, I used the following sequence of steps:

  1. Analyze the circuit with just the current noise due to the inverting input. All this current must flow through the feedback resistor. Determine the output voltage and divide by the gain, where the gain is that of the noninverting amplifier.
  2. Analyze the circuit with just the current noise of the noninverting input. In this example, all that current must flow through R3. [Note: to reduce the current noise we should let R3 = 0; however, to reduce the offset voltage error due to input bias currents R3 = R1//R2.] Since the voltage created by this noise source is at the noninverting input, there is no need to see what it will be at the output and then refer it back to the noninverting input.
  3. Square these two rms noise voltages, multiply by the bandwidth of the system, and sum them. Strictly speaking it is not necessary to sum them; however, it is very useful in terms of re-designing a circuit to reduce noise if we know which sources dominate the total noise voltage.
  4. The noise voltage of the op amp is also already referred to the noninverting amplifier, so it can be squared and multiplied by the bandwidth.
  5. The thermal noise of R3 is also at the noninverting input, so it can be represented as a mean-square relationship. The actual value can be calculated at this point. Because there is some symbolic simplification that can be done at a later point, many reference books do not do the substitution of values at this point. [However, if you were setting up a spreadsheet or computer program, you would likely not worry about the simplification. This is a holdover from the really olden days of slide rules. It does reduce some keystrokes if you are using a calculator.]
  6. Analyze the circuit with just the voltage noise of R1. This is like any other voltage source at the input of an inverting amplifier. The output voltage will be the noise voltage multiplied by the inverting gain, G'. Referring this back to the noninverting input, we divide by the noinverting gain, G, as done with the output noise voltage due to the inverting input current noise source (step 1). This relationship is squared. The value could be calculated at this point, or left for the symbolic simplification.
  7. Analyze the circuit with just the voltage noise of R2. The output voltage will just be the inverse polarity of the resistor noise voltage. Refer this output voltage to the noninverting input as was done in step 6.
  8. Sum all the mean-square thermal noise voltages that have been referred to the input.
  9. Substitute appropriate values, sum the three noise components (current, op amp voltage noise, and resistor thermal noise). Take the square root. This will be the equivalent input noise voltage.