OpAmp Noise Analysis

The purpose of a noise analysis for an operational amplifier circuit is to determine if the design meets performance criteria. Noise analysis estimates the magnitude various sources of noise produce to better refine the performance of the circuit and ensure a minimum signal-to-noise ratio (SNR). 

Noise is an unwanted signal combined with a wanted signal that influences the wanted signal that distorts the wanted signal in an unwanted way. An electrical circuit may experience noise as a result of interference from outside sources (extrinsic) or from circuit components (intrinsic). It may interfere with or skew signals, resulting in subpar performance or a low SNR. Electromagnetic interference (EMI) from power lines or other sources, static from ground sources, and thermal noise from active components are common sources of noise. Generally speaking, intrinsic noise is usually relatively predictable and can be modeled using a system of equations or other mathematical tools, whereas extrinsic noise is usually unpredictable and very difficult to model accurately.

Intrinsic noise is a kind of noise that a system produces on its own as a result of some innate unpredictability. Thermal noise and flicker noise are a couple of examples of inherent noise. Extrinsic noise is a sort of noise that is brought on by outside elements like crosstalk and electromagnetic interference. Radio frequency interference, static electricity, EMI, and power supply noise are examples of extrinsic noise. 

The magnitude of intrinsic noise can be predicted as it follows a statistical distribution. Even though it’s random, the magnitude can be estimated with some certainty. Extrinsic noise tends to be from sources unknown or hard to predict. Due to its external origins, extrinsic noise is more challenging to predict and measure, making it more challenging to manage. Because of the unpredictability of extrinsic noise, this write-up will focus on intrinsic noise sources.

Generally, there are three different types of noise to consider: White Noise, Flicker Noise, and Burst Noise. 

• White noise (or Broadband Noise) is a type of noise that has a uniform power spectral density. It is usually heard as a hissing or static-like sound.

• Flicker noise (or 1/f Noise) is a type of noise that has a 1/f power spectral density. It is usually heard as a low-level background noise that is abrupt and intermittent.

• Burst noise (or Popcorn Noise) is a type of noise that occurs in short, intense bursts. It is usually heard as a sudden, loud sound that is followed by a period of silence. It’s referred to as Popcorn Noise as when played in a speaker, it sounds like popcorn being popped.

White (Broadband) Noise 

Recall from color theory that the color white is the presence of all colors. That is, all frequencies (at least, in the visible spectrum) combine equally in magnitude to make the color white. White noise gets its name from a similar phenomenon; all frequencies from an infinitely small step past DC out to the cutoff predicted by the Bose-Einsten distribution combine equally to form a random signal (more information in this write-up). 

Generally speaking, white noise is thought to be a combination of all frequencies and has a flat power spectral density. It’s generally considered the dominant noise source in the middle-high frequency range (frequencies greater than 1kHz) and has a Gaussian distribution. Since white noise has no DC component, the peak-to-peak estimate of white noise can be estimated from the ±3σ bounds (99.7% confidence interval), or the ±3.3σ bounds (99.9% confidence interval). 

Flicker noise (1/f or Low-Frequency Noise)

Low-frequency noise, often known as 1/f noise, is distinguished by its power spectrum, which exhibits an inverse power law (1/f) over a wide range of frequencies, mostly less than 1kHz. Numerous systems, including electrical circuits, electronic parts, and even biological systems, exhibit this kind of noise. Due to its spectral power distribution, it is also known as pink noise. 

Low-frequency noise for an OpAmp follows the 1/f slope and is estimated from the part’s datasheet at the lowest frequency published. Low-frequency noise can’t be analyzed at 0Hz (or DC) as the 1/f slope tends to infinity as f approaches DC. This sounds counterintuitive until we recall that zero-frequency corresponds to infinite time. Typically, the value for low-frequency noise is estimated at the lowest frequency published in the datasheet, often 0.1 or 1Hz. 

Burst (Popcorn) Noise

A type of random noise known as burst or popcorn noise is characterized by brief, isolated energy spikes. It is low-frequency in nature (0.1Hz-1kHz), and frequently appears in highly random signals, such as audio or video transmissions. Burst noise typically has several, short, sharp peaks and valleys and is bimodal or multimodal in nature. As the duration and magnitude of the spikes are frequently unpredictable, burst noise predictability is low. Burst noise can be caused by defects in ICs or from extrinsic sources. Due to the unpredictability of burst noise, it won’t be considered in the noise analysis. 

Synonyms

To add confusion, types of noise are typically referred to with multiple names. They generally refer to the similar things, though there are some nuanced differences between them. For the purposes of this write-up, they can be thought of as synonyms. 

• Broadband Noise is also referred to as White Noise, Johnson Noise, Thermal Noise, and Resistor Noise. 

• Flicker Noise is also referred to as 1/f Noise, Low-Frequency Noise, or Pink Noise. 

• Burst Noise is also referred to as Popcorn Noise, Red Noise, and Random Telegraph Signals (RTS). 

Generally, there are two noise sources to consider when doing an analysis of an OpAmp circuit: Noise sources internal to the OpAmp, and the resistive elements in the input and loop gain networks. The noise sources for an OpAmp are voltage noise and current noise, the magnitude of which can be found in the part's datasheet.

The following image shows two circuits: A non-inverting opamp circuit with a gain of 100 (left), and a noise-equivalent circuit (right). Notice each resistor in the input and feedback networks have an associated noise source, and inside the OpAmp, there is a voltage noise source on the non-inverting input and a current noise source for both the positive and negative input. 

Resistive (or thermal) noise magnitude can be calculated with the equation below. It’s handy to keep a chart on hand for a quick estimate, such as the one below!

For the formula above, kB is Boltzmann’s constant, T is the temperature in Kelvin, R is the resistance of the element, and Δf is the bandwidth of the amplifier circuit. 

In general, OpAmp noise should dominate resistive noise, or the noise introduced by the loop gain. A very low noise amplifier (LNA) may have noise in the range of 1nV/√Hz, which is equivalent to a resistor of about 75Ω at room temperature. LNAs can be expensive, so it doesn’t make much sense to buy an expensive LNA and have resistor noise dominate the noise performance. This statement assumes low noise is the most important consideration in the design, which is a safe assumption if one is looking at an expensive LNA. In reality, the decision will be made by balancing the cost budget, power budget, and noise budget of the product. 

When calculating the total noise of the system, noise sources can’t be added algebraically, but must be summed as vector quantities. That is, the total noise (enT) is the vector sum of each noise component (en). 

A few things to keep in mind when calculating the voltage noise generated by the OpAmp: 

• To estimate the OpAmp Voltage noise, the noise gain must be calculated. 

• The OpAmp voltage noise is always on the non-inverting input. Therefore, the gain seen by the noise voltage source is always on the non-inverting input. 

• The noise gain can be different than the signal gain. 

• Remember to include the noise generated by any resistive elements on the input network to the non-inverting input. 

Take this inverting amplifier as an example: 

Notice the signal source and noise source are connected to a different input. An analysis of the signal gain shows it follows a typical inverting amplifier topology. However, because the noise source is attached to the positive input, analysis shows it follows a non-inverting amplifier topology. This is a case where the signal gain and noise gain are not equal!

Take our inverting amplifier and the OpAmp noise model. We’ll make some approximations to simplify the calculation. 

• Since the inputs are grounded, the OpAmp drives Vo to ground, creating a “virtual” ground. This puts Rf and R1 in parallel. 

• Approximate the noise sources on the positive and negative input as one source between the positive and negative input. This allows the positive input to be grounded, and the voltage noise generated by the OpAmp current noise to be present on the negative input. 

Since Rf and R1 are in parallel,  the voltage at the negative input is simply In*Req. 

The voltage spectral density has units of VRMS. There’s a common misconception that one can integrate the noise voltage spectral density to get total noise, one must integrate the power spectral density. A quick dimensional analysis shows to get the RMS voltage spectral density, the power spectral density must be integrated. 

Integrating the Power Spectral Density and calculating Voltage Noise [with units]: 

Recall from statistics that for a  signal with no DC component, the RMS is equal to the standard deviation. Therefore when estimating the peak-to-peak voltage, the ±3σ or ±3.3σ values can be used. To calculate the 99.7% or 99.9% confidence interval, simply multiply the calculated RMS voltage by 6 or 6.6, respectively.  

There are two regions to combine when calculating OpAmp voltage and current noise: The low-frequency and broadband regions. For this example, a curve from the TI LM324LV Operational Amplifier is used as the curve shown gives better insight into a typical OpAmp response than the LT1006. Not that the LT1006 is a bad part; in fact, just the opposite. 

A few things to note in the graphs before we get started: 

• f0 is the lowest frequency given on the graph. 

• fL denotes the low-frequency cutoff bandwidth of the amplifier circuit. A typical number is 0.1Hz if no high-pass filter is implemented in the low-end to block DC and very low frequencies. 

• fH denotes the cutoff frequency for the bandwidth of the amplifier circuit. fH is set by the gain bandwidth of the amplifier if no low-pass filter is implemented to limit the bandwidth of the circuit. 

• GBW indicates the limit of the unity-gain bandwidth of the amplifier. If no low-pass filter is implemented, the GBW acts as one.

Low-Frequency Region (1/f)
In the low-frequency noise region of the graph above, notice a few things: 

• The low-frequency noise region extends from just past DC to infinite frequency. A typical low-frequency cutoff is considered at 0.1Hz. 

• The slope of the Voltage Noise Spectral Densiate is 1/root-f. This is because power spectral density is voltage noise spectral density squared, or 1/f, where the low-frequency noise gets its name from. 

• enLF is the noise level given at f0.

• EnLF is the total estimated low-frequency noise and is calculated by the formula given above. 

Broadband Noise Region

• A low-pass filter is added near the end of the broadband region. All circuits will have some kind of low-pass filter, either implemented intentionally in the circuit, or by the gain bandwidth of the amplifier. 

• The broadband noise region extends from just past DC to infinite frequency. 

• The broadband region has a slope of zero for the bandwidth of the circuit, indicating a uniform power spectral density over the bandwidth of the circuit. 

• EnBB is the estimated total broadband noise and is calculated by the formula shown above. 

• eBB is the broadband voltage spectral density noise level (in this example, about 38nV/root-hertz). 

• kN is a correction factor that will be discussed shortly. 

Combining

Notice the noise spectral densities in the graphs above overlap, and therefore must be added once calculated. The noise of the circuit is a combination of low and broadband noise as shown in the figure below. 

• Combining the regions, both are present at low and high frequencies. At low frequencies, the 1/f noise is dominant. At higher frequencies, broadband noise is dominant. 

• EnT denotes the total noise and is the vector sum of the low-frequency noise and broadband noise. 

The broadband noise is limited by the inherent low-pass filter present in every circuit. For a first-order filter, the broadband noise has constant spectral density up to the cutoff frequency set by the low-pass filter, then rolls off at 20dB/decade. This holds for higher-order filters, except the roll-off increases for each order. For a second-order filter, the roll-off is 40dB/decade, the third-order is 60dB/decade, and so on. For each order filter, we can approximate the equivalent spectrum of an infinite-order filter using a correction factor. An infinite order filter has a flat response out to the cutoff frequency, then cuts the amplitude immediately, which resembles a brick wall where the name comes from. While infinite-order filters are impossible to realize, the theory is handy for simplifying calculations!

In the graph below: 

• The light blue areas are equal, and extend to the -3dB frequency. 

• The light red areas are equal. 

• BWn is the bandwidth of the noise spectrum defined by the brick wall filter (BWF). 

Brick Wall Correction Factors
As shown in the table below, the brick wall correction coefficients (kn) tend to approach one for higher-order filters. This is in line with the concept of an infinite-order filter; the higher the order, the steeper the roll-off, and the smaller the correction factor needed. To compute the bandwidth of the brick wall filter, simply multiply the cutoff frequency (f-3dB) with the correction factor (kn). 

Note that this is an approximation. Gain peaking can affect the noise wall. In actual circuits, noise bandwidth may differ somewhat.

Recall the inherent noise sources that are considered when doing noise analysis: The OpAmp voltage noise, the OpAmp current noise, and the resistive thermal noise. The OpAmp voltage noise typically has a low-frequency component (1/f) and broadband component that must be considered. The OpAmp current noise will have a broadband component and may have a low-frequency component. Resistive thermal noise is all broadband noise.  

Let’s explore the noise from the following inverting amplifier that uses the LT1006 from above. This will include the Voltage Noise, Current Noise, and Resistor Noise. 

The two plots below are from the LT1006 datasheet and will be used to calculate the voltage noise. Unless otherwise noted, typical values are: fL=0.1Hz, f0 is the lowest frequency given in the noise graph, fH is the -3dB point from an implemented pole or the pole implemented by the OpAmp. 

The plot below is from the LT1006 datasheet and will be used to calculate the current noise.

The total noise is the vector sum of the OpAmp voltage noise, OpAmp current noise, and resistive thermal noise. Recall up to this point the components of the input noise have been calculated. Once the vector sum is calculated, the noise gain of the circuit must be applied to produce the total output noise. 

From the noise analysis, it’s shown that the OpAmp Voltage Noise is the dominant noise source for the circuit. Further analysis shows the resistive elements in the feedback network could be increased by an order of magnitude to decrease power consumption without significant impact on the noise (ET,OUT would be 285uVRMS vs 251uVRMS). Another option is to limit the bandwidth of the circuit based on the requirements set by the input signal, what the amplifier circuit is driving, and the stability of the circuit. For instance, if a bandwidth of 2kHz was appropriate, the total noise would be smaller (ET,OUT would be 125uVRMS vs 251uVRMS).

Regardless, the purpose of a noise analysis is to highlight the dominant areas of noise. Once those are highlighted, design considerations can be made as to if they’re an issue, or how to handle them.

The noise of the circuit can be simulated to verify calculations. It’s important to do the calculations by hand at least the first time as it builds intuition for where the noise is coming from in the design, which allows for informed design choices. Many spice simulators have the noise simulation feature; since this is an Analog Devices part, the example is shown in LTspice, which has the model built in. 

The image below shows the circuit used for simulation. The noise analysis specifies the output (Vo), the source of noise (V3), that the simulation is a decade sweep with ten points per decade from 100mHz to 1MHz, and the temperature is defined to be around room temperature.

The result of the analysis is in line with our estimate. The simulator calculates the noise over the range to be about 225uVRMS, which is in line with our estimate. Recall that some error is introduced in the hand calculation from estimating curves on the graph, the brick wall correction factor, and simplifying the noise model. 

Introducing a pole to limit the bandwidth to 2kHz, the simulated noise is reduced to about 110uVRMS, in line with the estimate done by hand.  

As shown in the analysis above, the purpose of estimating noise is to choose the correct components for the amplifier circuit. To that end, there are some generalities that can be applied when exploring preliminary parts in the design process to help pick an OpAmp quickly, without having to do the noise analysis many times over. 

General Guidance

• Look at the circuit from a system level. Look at the inputs to the amplifier, and what the amplifier drives. The inputs may have specific requirements, such as a well-defined input impedance. The output may have a drive requirement or force a minimum bandwidth. 

• Define what margin is needed for the circuit (gain and phase). 

• Understand the budgets! The cost of the circuit, allotted power, and permittable noise will all influence the topology of the circuit, and the parts selected. Low-noise designs will cost more, consume more power, or both, low-power designs will cost more, have more noise, or both, low-cost designs will consume more power, have more noise, or both!

• For cost reasons, the amplifier should produce more noise than the resistive elements in the feedback network. Generally, there’s little value to spend money on an expensive LNA only to have the resistive elements produce more noise than a cheaper amplifier. 

• When selecting an amplifier, consider both the voltage and current noise!

Amplifier Topology and Noise

• CMOS amplifiers tend to have lower quiescent current than Bipolar amplifiers. However, bipolar amplifiers tend to have lower noise for an equivalent quiescent current. That is, a bipolar amplifier with the same quiescent current as a CMOS amplifier will tend to have less noise overall. 

• Voltage noise is inversely related to quiescent current. For a given OpAmp, the higher the quiescent current, the lower the voltage noise. 

• Generally speaking, current noise is lower for amplifiers with lower quiescent current, though they are not related.