Noise is unwanted random signal, but amazingly we can calculate its RMS value or even its peak-to-peak value.
In this blogpost we are going to be looking at non-inverting opamp’s output noise. We’ll see how much each component in the circuit contributes to the noise. In the end there are examples that you can look at.
There are two important sources of noise when designing low noise opamp circuits. The first one is the opamp itself. Opamps have inherent input voltage and input current noise. They are usually amplified with the gain of the opamp and they can be easily modelled with voltage and noise sources as we will see shortly. Other important source of noise is the resistors in the circuit. Larger resistors are noisier. People seem to overlook the noise contribution of the resistors, but the resistors can be the largest contributor of noise sometimes. For example, 1k resistor noise is much larger than most of the low noise opamps’.
In figure.1 noise sources in non-inverting configuration are shown including input voltage and current noise of the opamp and resistor. Resistors’ noise are modeled with voltage sources.
As it’s seen in figure.1 there are a lot of noise sources in non-inverting configuration. Don’t worry about how complex it’s going to be. It’s actually not very difficult to calculate the total output noise. Some of the noise sources are negligible in many situation as we will shortly see.
The superposition principle is going to make it a lot easier than you think. What we are basically going to do is that we will only turn on a single source of noise, turn the others off (set to zero) and calculate its noise contribution to the output one by one. Let’s start with the opamp’s input voltage noise ein.
Input Voltage Noise
Noise is usually measured in RMS and it is characterized by its spectral density in V/√(Hz). In the datasheets spectral noise density is specified and we usually need to calculate the total output noise from that information. Spectral noise density of an opamp is usually in nV’s. It ranges from 10nV/√(Hz) to 0.9nV/√(Hz) for a really quite opamp (We will only talk about white noise here. Flicker noise is ignored because it’s usually is not as relevant as white noise for wide-band applications[>1MHz]). Let us investigate figure.2
The opamp’s input voltage noise is direclty amplified by the gain of the non-inverting stage. Another thing to point out here is that the RMS noise at the output is also proportional to the square root of the bandwidth of the circuit.
Input Current Noise
In many cases input current noise (or noises :D) can be ignored. If you look at any datasheet of an opamp you’ll see that current noise density is at pA/√Hz range. That makes it kind of non-dominant source of noise in many cases. We will talk about it any way.
Inverting input current noise is multiplied by the feedback resistor. If the feedback resistor is high (>1k, yes 1k is high :D) inverting input current noise should be considered, but luckily it’s not amplified by the circuit gain. Non-inverting input current noise is amplified by the circuit as shown in figure.3. It is also multiplied by the source resistance of the input. Considering the current noise is in the pA range you shouldn’t worry about non-inverting current noise when the source resistance is low(<1k). However, it all depends on how sensitive your application is. Generally speaking, current noise is no concern because other sources of noise are dominant.
Resistor Noise
The last thing that we have to cover is resistor noise. They can be quite important when designing low noise circuits. For example, a low noise opamp can have an input voltage noise as low as 0.9nV/√Hz and 50Ω resistor noise at room temperature (300K) is also about 0.9nV/√Hz. Quite often resistor noise is closer to or more than opamp’s inherent noise when designing low noise circuits. Let’s take a look at the following figure.
The contribution of the feedback resistor (Rf) to the output noise is not multiplied by any factor. It just appears at the output directly without multiplication or attenuation. This can be quite useful when high gain is desired because no matter how large the gain is Rf’s noise contribution to the output is the same. R1’s noise ,however, is multiplied by Rf/R1 which is very close to the non-inverting stage gain. R1’s noise “sees” an inverting opamp as shown in figure.4 that’s why it’s multiplied by Rf/R1. When designing low noise circuits with high gain R1 should be chosen in such a way that its noise is lower than the opamp’s input voltage noise and Rf should be used to set the gain. At high gains Rf’s noise contribution is likely to be ignored.
All of the major sources of noise was covered. Let’s take a look at an example. In the example we will ignore the flicker noise and treat all noise sources as white noise and at the and we will compare the calculation with simulation results, which includes flicker noise, and see how they differ or not. :D
Examples
Let’s start with something different. A buffer is a non-inverting opamp in its simplest form. It has no resistors connected to it, so only thing that we have to worry about it that the input voltage noise. We are going to be using LMH6629 from texas instruments it had 0.69nV/√Hz input voltage noise. Ignoring flicker noise and consideting the input voltage noise is 0.69nV/√Hz the RMS output noise shoud be about 0.69nV RMS.
Figure.5 confirms our calculation. Output noise graph shows the spectral output voltage noise density. In this graph we can see that the white noise is about 0.7nV/√Hz and flicker noise is also included in the model, but we ignored it. What total noise graph tells us is that given that we have X Hz bandwidth what is the output noise is going to be in RMS voltage. For example, if you look at the graph for 10kHz bandwidth we have about 50nV RMS noise and for 100kHz bandwidth the output noise is about 200nV RMS. As it’s seen in the figure for 1 MHz bandwidth simualtion results and our calculations are almost exactly the same.
Let’s look at something more challenging. We are going to look at a non-invering opamp circuit with the gain of 2. We can use 2 1k resistors to set the gain to two. If we just use the formula in figure.1 to calculate the resistor noise for 1k resistor (K=300 room temperature) we’ll see that 1k resistor noise is about 4nV/√Hz. We can ignore the flicker noise and current noise (for now) and see how they compare.
As it’s seen in the figure.6 in the first attempt the result wasn't quite close to the simulation that was because input current noise wasn't low enough to be ignored. In our second attempt the calculation is closer, but still it’s not very accurate. That’s because the current noise density is not as constant as the voltage noise. This can be seen in the figure.7. This is taken directly from the datasheet of the opamp.
In figure.7 we can see that the current noise is only white,i.e. constant, after 1MHz. That’s the reason why our calculations are not very close to the simulation results.
In conclusion, one has to be careful when performing noise analysis. One should determine the important sources of noise. Sometimes you shouldn’t worry about a particular resistor’s noise, but sometimes you should. Input current noise is usually not important, but sometimes it is.
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